L-functions for GL_n: Godement–Jacquet and Rankin–Selberg

Draft lesson. Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Public domain (CC0).

For general linear groups, integral representations turn the Euler products of the preceding lesson into meromorphic functions with functional equations. Their poles detect whether two cuspidal representations are contragredient. That analytic test recovers a representation from almost all of its local parameters. It also recovers every constituent of an isobaric sum, including multiplicities and real norm twists.

We state the analytic constructions with their hypotheses and prove the uniqueness deductions. A boundary issue deserves particular attention: convergence for \(\operatorname{Re}s>1\) alone gives a weak local bound. Positivity and the complete self-pairing pole give the strict bound, and a second positivity argument proves nonvanishing on the line one. The prerequisites are Automorphic representations and automorphic L-functions and the adelic theory of Hecke L-functions. Basic free references are the author draft of Getz–Hahn, the author notes of Goldfeld–Jacquet, and the original papers of Jacquet–Shalika and Jacquet–Piatetski-Shapiro–Shalika.

The general local analytic identification in Theorem 2.1 and complete global continuation and pole theorem in Theorem 3.1 are stated below without their full proofs. The arguments in Sections 4–6 are conditional deductions from these precise statements. Section 1 proves general adelic reduction, the entire finite-place matrix zeta ideal and Fourier equation, general archimedean continuation and scalar Fourier identities in actual smooth dual-pair models, the complete canonical determinant-character theory, and general cuspidal admissibility and realization with compatible unitary tensor pairings. It proves the global standard-function argument over every number field from these results and the remaining standard-conductor/factor identification and canonical archimedean inputs. The degree-one and \(GL_2/\mathbb Q\) cases also have earlier proofs.

1. Normalizations and standard L-functions

Throughout, \(F\) is a number field and \(\mathbb A=\mathbb A_F\). Normalize the local absolute value by \(|\varpi_v|_v=q_v^{-1}\) at a finite place. Write \(\nu(g)=|\det g|_{\mathbb A}\), and write \(\widetilde\pi\) for the contragredient. Initially all cuspidal representations are unitary. In particular their central characters are unitary. A real norm twist will be explicitly indicated later.

At an unramified finite place, the normalized Satake parameter of a representation of \(GL_n(F_v)\) has eigenvalues \(\alpha_{1,v},\ldots,\alpha_{n,v}\). The normalization is the one for normalized parabolic induction: an unramified principal series induced from \(\chi_1,\ldots,\chi_n\) has eigenvalues \(\chi_i(\varpi_v)\). Its standard local factor is

At a ramified finite place, the Godement–Jacquet integral defines the corresponding factor. At infinity one obtains products of

We distinguish the finite product \(L_f(s,\pi)\), the product \(L_\infty(s,\pi)\) at infinity, and the complete product

Let \(\mathfrak f(\pi)\) be the finite conductor and put \(A(\pi)=|D_F|^nN\mathfrak f(\pi)\). With this convention the conductor-normalized completion is \(\Lambda(s,\pi)=A(\pi)^{s/2}\mathcal L(s,\pi)\). The two completions are different functions and have the same poles and zeros. The factor \(A^{s/2}\) moves the conductor out of the epsilon factor.

Theorem 1.1 (Godement–Jacquet standard analytic theory). Let \(\pi\) be a unitary cuspidal automorphic representation of \(GL_n(\mathbb A)\). The complete standard L-function continues meromorphically to \(\mathbb C\). If \(n\ge2\), it is entire. For \(n=1\), it is entire unless \(\pi=|\cdot|_{\mathbb A}^{it}\) for some real \(t\); in that exceptional case its only poles are simple poles at \(s=-it\) and \(s=1-it\). There is a constant \(\varepsilon(\pi)\) of absolute value one such that

For the entire cases the completion is bounded in each closed vertical strip; in the exceptional case the same assertion holds after multiplication by \((s+it)(s-1+it)\).

The degree-one proof, including the pure-norm exception, is Hecke L-functions and the Dedekind zeta function, Theorem 10.1. The earlier \(GL_2/\mathbb Q\) proof is Global Whittaker functions and the L-function of a cuspidal representation, Theorems 3.1, 4.2 and 5.1. We now give the matrix-space argument for arbitrary \(n\ge2\) and arbitrary number fields. Its global steps are proved below. Proposition 1.1a proves general adelic reduction; Propositions 1.2–1.3 prove the unramified Fourier theory and the complete finite-place matrix ideal. Theorems 1.11 and 1.13 prove smooth comparison and compatible unitary tensor pairings for actual admissible Hilbert cusp constituents, with reverse comparison in Proposition 1.14 and Lemma 1.15. Theorem 1.17 proves the finite-place scalar Fourier equation in every rank, including ramification. Theorem 1.18 proves the full real/complex determinant-character Schwartz family. Theorem 1.25 proves exact Gaussian ideals, attainment, entire Schwartz division and scalar Fourier equations for all real/complex symmetric powers, their antiholomorphic companions, duals and norm twists. Theorem 1.19 proves realization and admissibility for every abstract cuspidal subquotient, including its unitary norm normalization. Theorems 1.21 and 1.22 prove general archimedean continuation and scalar Fourier identities in the stated smooth dual-pair models. Theorem 1.24 and Corollary 1.24c prove the nonnegative finite-place matrix epsilon exponent, and Corollary 1.24d bounds the reciprocal-factor degree by n. The remaining inputs are standard-factor and standard-conductor identification for general finite-place irreducibles, and the explicit archimedean standard-factor identification in 1.1c; Theorem 1.31 below proves the genuine Gaussian ideal and conjugation package in every prescribed actual dual-pair model; Proposition 1.16 supplies the exact finite-place determinant-character conductor and Tate-product factors. The global proof below is conditional on these precise remaining assertions. The freely accessible treatment being used is Goldfeld–Jacquet, author notes, §§2–5 and 8–9.

The quotient and cusp estimates

Put \(d=[F:\mathbb Q]\), \(G=GL_n\), \(G^1=\ker\nu\), and \([G]^1=G(F)\backslash G^1\). For \(t>0\), let \(a_t\) be the central matrix whose entries at every infinite embedding are the positive real scalar \(t^{1/(nd)}\), and whose finite entries are \(I_n\). Then \(\nu(a_t)=t\), and

The positive center acts on a unitary irreducible representation by \(t^{i\tau}\), for some real \(\tau\). Replacing \(\pi\) by \(\pi_0=\pi\nu^{-i\tau}\) makes this action trivial. We first prove the assertions for \(\pi_0\), and restore the twist at the end.

For \(c>0\), define the positive diagonal cone

Each \(b_i\) is embedded by the same real scalar at every infinite place and by \(1\) at every finite place.

Proposition 1.1a (adelic reduction for every general linear group). Let \(F\) be any number field, \(d=[F:\mathbf Q]\), \(G=GL_n\), and \(G^1=\{g\in G(\mathbb A_F):|\det g|_{\mathbb A}=1\}\). Put \[ K=\prod_{v\text{ real}}O(n)\times \prod_{v\text{ complex}}U(n)\times \prod_{v<\infty}GL_n(\mathcal O_v). \] There are \(c>0\), a compact subset \(\Omega_N\subset N(\mathbb A_F)\), and a compact subset \(\Omega_T\) of the diagonal group whose individual entries have idèle norm one, such that \[ G^1=G(F)\,\Omega_N A(c)\Omega_T K, \] where \(A(c)\) is exactly (1.5b): the entry \(b_i\) is the same positive real scalar at every infinite place and is one at every finite place, \(\prod_i b_i=1\), and \(b_i/b_{i+1}\ge c\). In particular, one such compact-factor set suffices for the covering requested there.

Proof. We use the proofs in The adèle ring of a number field, Theorem 2.2 and equations (10)–(11), giving additive compactness and integral-lattice representatives; Idèles and the idèle class group, Theorem 3.3, giving compact norm-one idèle representatives; and Lattices, Minkowski's theorem and the Minkowski embedding, Proposition 7.2, giving the full integer lattice in \(F_\infty\). We prove the remaining reduction steps below. The covering assertion is the classical one stated in Goldfeld–Jacquet, Section 5, Theorem 5.1; its statement fixes the conventional form of the covering.

Two compact representative sets. Write \(J^1=\{h\in\mathbb A_F^\times:|h|_{\mathbb A}=1\}\). Fix a compact set \(H\subset J^1\) mapping onto \(F^\times\backslash J^1\). This is supplied directly by the norm-one compactness proof: its compact set \(B_\eta\cap J^1\) represents every class. We may adjoin one to \(H\).

For \(\lambda>0\), let \(s_\lambda\) be the idèle equal to \(\lambda^{1/d}\) at every infinite place and to one at every finite place. Then \(|s_\lambda|_{\mathbb A}=\lambda\), since the complex absolute values are squared and \(r_1+2r_2=d\). Every idèle \(t\) can therefore be written \[ t=a\,s_{|t|_{\mathbb A}}h,\qquad a\in F^\times,\quad h\in H. \tag{1.5b.1} \] Also fix a compact additive representative set \(D_0\subset\mathbb A_F\) for \(F\backslash\mathbb A_F\), for example the compact closure of the domain in equation (11) of The adèle ring of a number field.

We shall use the following elementary compactness consequence. Every compact subset of \(\mathbb A_{F,f}^r\) is contained in \(m^{-1}\widehat{\mathcal O}_F^r\) for a positive rational integer \(m\). A finite cover by restricted-product neighborhoods involves only finitely many exceptional finite places; their finitely many coordinates have bounded valuations. A rational integer with sufficiently large powers of the underlying rational primes clears all these bounds. In particular the set \(H H^{-1}\), which is compact in the idèle topology and hence in the additive adèle topology, admits this kind of finite-coordinate bound.

A least adelic projective height. For a row vector \(z\in F_v^r\), use the local quantities \[ \ell_v(z)= \begin{cases} \left(\sum_j|z_j|^2\right)^{1/2},&v\text{ real},\\ \sum_j|z_j|^2,&v\text{ complex},\\ \max_j|z_j|_v,&v<\infty. \end{cases} \] They satisfy \(\ell_v(az)=|a|_v\ell_v(z)\), with the normalized field absolute values. They are invariant under the indicated local compact groups. At a finite place this follows by applying the integral-entry bound first to \(k\), then to \(k^{-1}\); at infinity it is the Euclidean or Hermitian isometry identity.

For \(g\in GL_r(\mathbb A_F)\) and \(0\ne x\in F^r\), define \[ h_g(x)=\prod_v\ell_v(xg_v). \tag{1.5b.2} \] Almost every factor is one, and the product is positive. The product formula makes it unchanged under \(x\mapsto ax\), \(a\in F^\times\), so it is a height on rational lines. For clarity, the product formula here follows from the principal-ideal norm formula proved in Norms of ideals, the ideal class group, and modules over Dedekind domains, Proposition 4.1: the finite norm product is \(|N_{F/\mathbf Q}(a)|^{-1}\), while the real and complex embedding product is \(|N_{F/\mathbf Q}(a)|\).

For fixed \(g\) and \(R>0\), only finitely many rational lines have height at most \(R\). To prove this, choose an idèle \(t\) with \(|t_v|_v=\ell_v(xg_v)\): at infinity use the ordinary Euclidean length as its positive component, and at a finite place choose a uniformizer power with the required valuation. Write \(t=a s_{h_g(x)}h\) as in (1.5b.1), and put \(x'=a^{-1}x\). The finite local lengths of \(x'g\) are bounded by the corresponding components of the fixed compact set \(H\); its ordinary infinite lengths are at most \(C R^{1/d}\), uniformly in \(h\). Multiplying by the fixed \(g^{-1}\) puts the finite coordinates of \(x'\) in one compact subset of \(\mathbb A_{F,f}^r\), and bounds its infinite coordinates.

Thus \(x'\in m^{-1}\mathcal O_F^r\) for one fixed positive integer \(m\), and its Minkowski embedding lies in a fixed bounded set. The integer-lattice proof of Proposition 7.2 shows that this set contains only finitely many such vectors: an inverse lattice-basis matrix bounds their integer coordinates. Every line of height at most \(R\) has one of these representatives. Choosing \(R=h_g(x_0)\) for any nonzero \(x_0\) now proves that the height has a positive attained minimum.

Successively reduce the last row of each block. For any \(GL_r(\mathbb A_F)\) matrix, choose a rational line of least height and extend one of its vectors to the last row of a matrix in \(GL_r(F)\). At each place, right multiplication by the local compact group sends this last row to \((0,\ldots,0,t_r)\). At infinity, extend the normalized row to an orthonormal basis: successively subtract its projections from the standard rows, then normalize the nonzero residuals. Orthogonality makes these rows independent, and the process ends with a full basis. At a finite place, divide the row by an entry of smallest valuation to make it primitive; one coordinate is a unit, and integral column operations carry it to the last coordinate. The upper-left \((r-1)\)-block is invertible.

These local choices form an element of the adelic compact group. Indeed, almost everywhere the matrix after the rational row change already belongs to \(GL_r(\mathcal O_v)\), and we can take its inverse as the local right compact factor, producing the identity matrix there. Now perform the same operation on the upper-left adelic block, and continue down to rank one. Embed every later rational change and right compact change in the preceding block, with identities on the remaining coordinates. This produces \(\gamma\in GL_n(F)\), \(k\in K\), and an upper triangular matrix \[ T=\gamma gk,\qquad T_{ii}=t_i\in\mathbb A_F^\times. \] For its upper-left \(r\)-block \(T^{(r)}\), the last coordinate line has least height: \[ h_{T^{(r)}}(x)\ge h_{T^{(r)}}(e_r)=|t_r|_{\mathbb A} \quad(0\ne x\in F^r). \tag{1.5b.3} \] Here is why subsequent steps preserve this assertion. They multiply the first \(r-1\) rows by a rational invertible matrix, fixing \(e_r\), and multiply the first \(r-1\) columns by a compact matrix, preserving every local length. Rational row multiplication permutes all the rational lines on which the minimum was taken. Thus (1.5b.3), established when the \(r\)-block is first reduced, persists through every smaller-block step.

Write each diagonal idèle as \(t_i=a_i s_{|t_i|}h_i\), and multiply on the left by \(\operatorname{diag}(a_i^{-1})\in GL_n(F)\). This keeps (1.5b.3): on an \(r\)-block it is another rational row change, and the last row is rescaled by \(a_r^{-1}\), whose product of absolute values is one. Rename the resulting triangular matrix \(T\). Its entries are now \[ t_i=b_i h_i,\qquad b_i=|t_i|_{\mathbb A}^{1/d}>0,\quad h_i\in H, \] where \(b_i\) denotes the prescribed equal positive scalar idèle. All compact and rational changes have determinant norm one, so for \(g\in G^1\) we have \(\prod_i b_i=1\).

The adjacent diagonal ratios have a uniform lower bound. Compactness of \(H H^{-1}\) gives a positive rational integer \(D\) and a real \(L\ge1\) such that, for every \(h,h'\in H\), \[ |D h_v/h'_v|_v\le1\quad(v<\infty),\qquad |h_v/h'_v|\le L\quad(v\mid\infty), \tag{1.5b.4} \] where the latter absolute value is the ordinary real or complex modulus.

We also need a uniform simultaneous approximation, and supply its proof. Let \[ U=\{z\in\mathbb A_F: |z_v|<1/4\ (v\mid\infty),\ z_v\in\mathcal O_v\ (v<\infty)\}. \] Its image in the compact additive quotient is open. Finitely many translates of that image cover the quotient; let their number be \(M\ge1\). For any \(u\in\mathbb A_F\), the \(M+1\) classes of \(0,Du,2Du,\ldots,MDu\) have two in the same member of this cover. Subtract them. There are \[ \beta=jD,\quad 1\le j\le M,\qquad \alpha\in F \] with \[ |\beta u_v-\alpha|<1/2\quad(v\mid\infty),\qquad \beta u_v-\alpha\in\mathcal O_v\quad(v<\infty). \tag{1.5b.5} \] This uses only compactness of the additive quotient; \(\alpha\) is not required to be integral or bounded.

Set \(c=(2MDL)^{-1}\). Fix \(i<n\) and put \(r=i+1\). In the \(r\)-block write its \((i,r)\)-entry as \(u t_r\). If \(b_i/b_r<c\), apply (1.5b.5) to this \(u\). The nonzero rational row \(\beta e_i-\alpha e_r\), multiplied by \(T^{(r)}\), has only two possibly nonzero coordinates: \[ \beta t_i,\qquad (\beta u-\alpha)t_r. \] At a finite place, its local length divided by \(|t_r|_v\) is \[ \max\bigl(|\beta t_i/t_r|_v,|\beta u-\alpha|_v\bigr)\le1. \] Indeed \(b_i,b_r\) have finite components one, (1.5b.4) clears the ratio of \(h_i,h_r\), and the rational integer \(j\) is integral everywhere. At each infinite place, \[ |\beta t_i/t_r| \le MDL\,b_i/b_r<1/2,\qquad |\beta u-\alpha|<1/2. \] The Euclidean length ratio is therefore less than \(1/\sqrt2\); at a complex place its normalized local length ratio is the square of that number. Multiplying all these inequalities gives \[ h_{T^{(r)}}(\beta e_i-\alpha e_r)<|t_r|_{\mathbb A}, \] contradicting (1.5b.3). Thus \(b_i/b_{i+1}\ge c\) for every \(i\). The constant depends on the fixed arithmetic compact sets, not on \(g\), its rational flag, or its diagonal entries.

Reduce the unipotent coordinates. Write \(T=v\operatorname{diag}(t_i)\) with \(v\in N(\mathbb A_F)\). Every left \(N(F)\)-coset has a representative whose upper entries lie in \(D_0\). To verify this without invoking unipotent reduction theory, process each row \(i\), with columns \(j=i+1,\ldots,n\) in increasing order. Left multiplication by \(1+aE_{ij}\), \(a\in F\), adds \(a\) times row \(j\) to row \(i\). It changes entry \((i,j)\) by \(a\), does not change that row's earlier entries, and may change only its later upper entries. Choose \(a\) to put the current entry in \(D_0\). Later operations do not disturb the entries already fixed. The set \(\Omega_N\) of upper unitriangular matrices with every upper entry in \(D_0\) is compact, being the continuous coordinate image of the finite product \(D_0^{n(n-1)/2}\).

Consequently some \(q\in N(F)\) gives \[ qT=\omega_N\,\operatorname{diag}(b_i)\,\operatorname{diag}(h_i), \qquad \omega_N\in\Omega_N,\quad \operatorname{diag}(b_i)\in A(c). \] Take \(\Omega_T=\{\operatorname{diag}(h_1,\ldots,h_n):h_i\in H\}\), a compact set with every diagonal entry of norm one. Undoing the rational and right compact changes proves the asserted equality.

For \(n=1\), the same assertion is simply \(J^1=F^\times H\); \(A(c)=\{1\}\) and the unipotent factor is trivial. Nonprincipal ideal classes are retained by the finite components of \(H\), rather than discarded by an assumption of a free integral lattice. Constant Weyl permutations belong to \(K\) at all places. There are no extra unbounded finite or Weyl factors in this proof. Since \(H\) is compact in the idèle restricted product, it has finitely many finite valuation patterns: outside one finite set its entries are units, and inside it each valuation ranges over a bounded set of integers. If desired, splitting \(\Omega_T\) according to those patterns expresses the same covering as finitely many compact-factor sets.

Finally, the positive central splitting in (1.5a) is unchanged: \(a_t\) has the scalar \(t^{1/(nd)}\) at every infinite place, its determinant norm is \(t\), and this proposition is applied to \(a_t^{-1}g\in G^1\). \(\square\)

Here are the consequences of Proposition 1.1a used below. Local Gram–Schmidt decomposition, or triangular reduction of an integral lattice at a finite place, gives \(G=NAK\). Conjugation by \(b\) on the entry \((i,j)\) of \(N\) has additive modulus \(|b_i/b_j|_{\mathbb A}\). Consequently the Iwasawa density is

For the scalars in (1.5b), set \(\rho_i=b_i/b_{i+1}\) and \(\beta(b)=(b_1/b_n)^d=\prod_i\rho_i^d\). In these coordinates \(\delta_B(b)=\prod_i\rho_i^{d\,i(n-i)}\). The integral of this density over \(\rho_i\ge c\) is finite, because each exponent \(d\,i(n-i)\) is positive. The other factors in a Siegel set are compact. Proposition 1.1a therefore proves \(\operatorname{vol}[G]^1<\infty\). Also \(b^{-1}\Omega_Nb\) remains compact: its upper entries are multiplied by \(b_j/b_i\le c^{-(j-i)}\). Thus a Siegel representative has the form \(b\omega\), with \(\omega\) in a fixed compact set. Finally, solving \(\prod b_i=1\) in the \(\rho_i\) shows that every \(b_i^{\pm1}\), and hence every matrix height of \(b\), is bounded by a fixed power of \(\beta(b)\), times a constant.

Lemma 1.1b (rapid decay). Let \(\phi\) be a smooth, finite-level, \(K_\infty\)-finite cusp form in a unitary admissible automorphic realization, of moderate growth, with trivial positive-central action. For every \(M>0\) and every right differential operator \(D\),

on each of the sets above. In particular \(\phi\) is integrable, and its absolute value remains integrable after multiplication by any fixed power of the representative's matrix height.

Proof. We supply the smoothing and Fourier estimates. Fix a finite set of compact types and a finite level containing \(\phi\). Admissibility makes the resulting space \(E\) inside its irreducible representation finite-dimensional. Let \(P_E\) be its compact-type and level projector. A smooth compactly supported approximate identity \(h_\epsilon\) gives \(T_\epsilon=P_E R(h_\epsilon)P_E\to I\) on \(E\). This convergence is in operator norm, since \(E\) is finite-dimensional. For small \(\epsilon\), \(T_\epsilon|_E\) is invertible. Its characteristic polynomial has nonzero constant term, so Cayley–Hamilton expresses \(I|_E\) as a polynomial in \(T_\epsilon\) with zero constant term. The corresponding finite sum of convolution powers is a smooth compactly supported kernel \(f\) on \(G^1\), with

The compact projectors are compactly supported distributions; convolution with \(h_\epsilon\) makes their kernels smooth. Differentiating (1.5e) transfers derivatives to a compactly supported derivative of \(f\). Moderate growth of \(\phi\) therefore gives one polynomial growth exponent for all its right derivatives, with constants depending on the derivative. Right differentiation preserves cuspidality.

Apply Iwasawa decomposition to (1.5e) at \(g=b\omega\), and periodize the \(N\)-coordinate. The kernel becomes

For each subset \(\theta\) of the \(n-1\) simple cuts, let \(V_\theta\) be the upper unipotent radical with entries crossing at least one cut in \(\theta\), and \(N_\theta\) the upper unipotent group within the resulting diagonal blocks. Thus \(N=N_\theta\ltimes V_\theta\). Replace (1.5f) by

The empty-set term is (1.5f). Every other term is left \(V_\theta(\mathbb A)\)-invariant as a function of \(u\): translating \(u\) merely changes the \(v\)-variable. It is also left \(N_\theta(F)\)-invariant. On integrating against \(\phi(ub'k)\), first integrate over \(V_\theta(F)\backslash V_\theta(\mathbb A)\). This is a proper-parabolic constant term and is zero. The compactness needed for these averages is proved in Automorphic representations and automorphic L-functions, Lemma 1.0. It follows directly for these upper unipotent groups by successive upper-diagonal additive quotients and compactness of \(F\backslash\mathbb A\). Thus the replacement does not change the integral.

Write upper unipotent matrices as \(1+X\), with \(X\) strictly upper triangular. If \(\xi\) lies in a block of \(N_\theta\) and \(X\) in the radical, then \((1+\xi)(1+X)=1+\xi+(1+\xi)X\). The change of variables \(X\mapsto(1+\xi)X\) preserves additive measure: it is triangular with diagonal entries one. Apply additive Poisson summation to the block coordinates and integrate over the radical coordinates. Under the trace pairing the dual space consists of strictly lower triangular matrices. Inclusion–exclusion in (1.5g) leaves exactly those dual matrices \(\lambda\) whose nonzero coordinates cross, collectively, every simple cut. Indeed, a frequency supported within the blocks survives for every \(\theta\) disjoint from its crossed cuts; its total coefficient is \((1-1)^{\#\text{uncrossed cuts}}\). Therefore

The asterisk means that every cut is crossed by at least one nonzero coordinate. The Jacobian in (1.5h) is \(\delta_B(b)\), since \(X\mapsto bXb^{-1}\) scales the upper root spaces by that modulus. The trace pairing gives the displayed inverse conjugation on the lower dual space.

Choose compact representatives for \(N(F)\backslash N(\mathbb A)\). Their conjugates \(b^{-1}ub\) stay compact by the root-ratio bound. The support of \(f\) forces \(b^{-1}b'\) to lie in a fixed compact subset of the diagonal group, as follows by applying the diagonal Iwasawa projection to its argument. Consequently the \(H\) in (1.5h), and their Fourier transforms, have uniformly bounded Schwartz seminorms and a common compact finite support. Additive Fourier inversion and Poisson apply to these whole archimedean Schwartz functions, as in Additive characters, self-dual measures and Poisson summation, Theorems 5.4–5.5.

Here is the necessary number-field lattice bound. The finite support forces every coordinate of \(\lambda\) into a fixed fractional ideal. Its image in the full Minkowski space \(F_\infty\) is a lattice. A nonzero lattice vector has Euclidean norm at least \(c_1>0\). This uses the entire infinite block; a lower bound at each individual embedding would be false. An occupied lower coordinate \((j,i)\), \(i<j\), is scaled by \(b_i/b_j\), which is at least \(c^{j-i}\). Uniform Schwartz estimates imply, for arbitrary \(Q\) and sufficiently large \(N\),

To verify this inequality, apply a product Schwartz bound to the coordinate blocks. For each nonzero coordinate, use its lattice minimum \(c_1\) to extract the factor \((b_i/b_j)^{-Q}\); use the fixed lower bound on \(b_i/b_j\) for the remaining summable weight. Zero coordinates need no extracted factor.

The intervals belonging to the occupied coordinates cover all cuts. If a cut occurs \(m_i\ge1\) times, the extracted product is \(\prod_i\rho_i^{-Qm_i}\), bounded by a constant times \((b_1/b_n)^{-Q}\), since \(\rho_i\ge c\) and \(m_i\) has a fixed upper bound. Choose \(Q=dM\). The remaining lattice sum in (1.5i) is finite for sufficiently large \(N\), by lattice-point counting in real dimension \(d\) for each coordinate. We conclude that (1.5h) is \(O(\beta(b)^{-M})\) for every \(M\).

The remaining \(u,k,b' b^{-1}\) integrations range over compact sets. Moderate growth gives \(|\phi(ub'k)|\le C\beta(b)^C\). Absorb this fixed power by choosing \(M\) larger. This proves (1.5d); applying the same argument to right derivatives proves the stated derivative version. The integrability assertions follow from the finite-volume density (1.5c) and the polynomial height bound. \(\square\)

Local data and the good-place calculation

For a Schwartz–Bruhat function \(\Phi_v\) on \(M_n(F_v)\) and a matrix coefficient \(c_v\) of \(\pi_v\), set

Use the trace pairing \((X,Y)\mapsto\psi_v(\operatorname{tr}(XY))\) and its self-dual additive measure to define \(\widehat\Phi_v\), and put \(c_v^\vee(g)=c_v(g^{-1})\). The following is precisely the local theory needed for the global argument.

Local input 1.1c. The local integrals continue meromorphically, and \(H_v(s)=Z_v(s,\Phi_v,c_v)/L_v(s,\pi_v)\) is entire. At a finite place they span the fractional ideal generated by \(L_v=P_v(q_v^{-s})^{-1}\), \(P_v(0)=1\), over \(\mathbb C[q_v^s,q_v^{-s}]\). At an infinite place the \(K_v\)-finite coefficient and Gaussian-polynomial test family generates the corresponding factor over \(\mathbb C[s]\). In particular, at each exceptional place there is a finite identity

where \(p_{v,j}\in\mathbb C[q_v^s,q_v^{-s}]\) or \(\mathbb C[s]\), respectively. The archimedean generator has the gamma-factor normalization used in (1.2). The local functional equation is

For an unramified additive character at a finite place, the epsilon factor is \(w_vq_v^{a_v(1/2-s)}\), with \(a_v\ge0\) the standard conductor exponent. The archimedean epsilon factor is constant in the gamma normalization (1.2). At a good place the spherical integral is the distinguished generator and \(\epsilon_v=1\). The exponents \(a_v\) define \(\mathfrak f(\pi)=\prod_{v<\infty}\mathfrak p_v^{a_v}\) in this normalization. For the standard global trace character and self-dual measures we will consequently obtain

The local normalizations commute with conjugation and duality for unitary representations, and an unramified norm twist shifts \(s\) without changing the finite conductor. Proposition 1.3 proves rational continuation, the principal matrix ideal, entire normalized quotients and generator attainment at every finite place, including ramification. Its generator is denoted \(L_{\mathrm{mat}}\); Theorem 2.3l below identifies it with the rank-one Rankin–Selberg factor for every actual generic irreducible; agreement with the chosen standard parameter remains part of the factor-identification input. Theorem 1.17 proves the scalar Fourier equation for every finite-place smooth admissible irreducible representation. Equations (1.7j)–(1.7m) then give an integral epsilon exponent, character scaling and the compatible-unitary phase. Theorem 1.24 and Corollary 1.24c prove nonnegativity of the matrix epsilon exponent for every finite-place irreducible; Corollary 1.24d proves the degree bound and its exponent-zero consequence. Corollary 2.3ce and Theorems 2.3cf–2.3cg compute the generic standard factor and the exact analytic epsilon factors and conductor exponents from actual cuspidal blocks. Theorems 2.3ci and 2.3cm compute the supplied Weil–Deligne block factors and Artin conductors; Corollary 2.3cn gives the exact reduction to cuspidal parameter compatibility. Theorem 2.3cr identifies the complete matrix factor of every finite-place irreducible from its canonical decreasing data, and Corollary 2.3cs computes its analytic epsilon factor and conductor. Corollary 2.3cu reduces standard parameter comparison to the two single-cuspidal identities (2.38a). Theorem 2.3dg identifies the generic minimal newvector level with the analytic matrix conductor and proves dimension one there. Cuspidal parameter construction and base compatibility and canonical epsilon root compatibility remain required. At infinity, Theorems 1.21–1.22 prove full-Schwartz continuation and a scalar Fourier equation in actual smooth dual-pair models; Theorem 1.31 proves the canonical Gaussian ideal, finite attainment and compatible conjugation normalization in every prescribed actual dual-pair model; explicit standard-factor identification remains unproved. Proposition 1.21b and Corollary 1.22e supply canonical full-Schwartz division and constant epsilon under the precise Gaussian-ideal premise, while Corollary 1.22f proves the unitary scalar critical-line phase. Proposition 1.16 supplies the full finite-place Fourier equation, exact matrix/Tate-product factor and nonnegative conductor exponent for every determinant character, including ramification. Propositions 1.5–1.7 supply polynomial-module closure and the full Gaussian ideal for determinant characters over both real and complex fields; Theorem 1.18 supplies their full-Schwartz continuation, entire division and Fourier equation with exact phases. Theorem 1.25 supplies this entire canonical package for the explicit all-rank symmetric-power families and their duals, including exact Tate-product comparison and conjugation; Theorem 1.31 below extends Gaussian-ideal existence to every prescribed actual irreducible dual-pair model; its explicit standard-factor labels remain open. Proposition 1.2 proves the local functional equation with epsilon factor one at every unramified finite place, in every rank; the Satake inverse/conjugate calculations in Proposition 2.3 give their unramified duality and conjugation normalization. A determinant norm twist shifts the integral variable directly. The exact free source locators for the local integral theory are Goldfeld–Jacquet, §2, Theorem 2.1 and Lemma 2.2; §3, Theorem 3.5, Lemma 3.6 and the subsequent generator construction. The identification of the standard conductor exponent with the epsilon exponent is also part of the required local data; for the rank-one Rankin–Selberg definition of the standard function it is Getz–Hahn, 22 April 2022 draft, Proposition 11.5.5 and Theorem 11.5.6. Agreement of the two standard constructions belongs to this identification input. An unspecified local functional equation would not suffice to deduce (1.4).

The discriminant power in (1.5l) can be verified directly. If \(\psi_{v,a}(x)=\psi_v(ax)\), the self-dual measure on \(M_n(F_v)\) is multiplied by \(|a|_v^{n^2/2}\). Hence \(\widehat\Phi^{\,\psi_{v,a}}(X)=|a|_v^{n^2/2}\widehat\Phi^{\,\psi_v}(aX)\). In the transformed zeta integral change \(g\) to \(a^{-1}g\). The multiplicative matrix measure is unchanged, the coefficient contributes \(\omega_{\pi_v}(a)\), and the determinant weight contributes \(|a|_v^{-n(1-s+(n-1)/2)}\). Comparing the local functional equations gives

At a finite place the trace character has annihilator \(\mathcal O_v^\perp=\mathfrak D_v^{-1}\), as proved in Additive characters, self-dual measures and Poisson summation, Proposition 5.1. Relative to a character with annihilator \(\mathcal O_v\), it is obtained by a parameter \(a\) of valuation \(d_v=\operatorname{ord}_v\mathfrak D_v\). Formula (1.5la) therefore contributes \(q_v^{nd_v(1/2-s)}\), times an \(s\)-independent central-character factor. Since \(\prod_vq_v^{d_v}=|D_F|\), multiplying the finite local conductor factors and the constant infinite factors proves (1.5l), with an \(s\)-independent nonzero constant \(w(\pi)\). Its absolute value will be proved globally below.

We can, however, compute the good-place integral and its convergence explicitly. Let \(v\) be finite, \(K_v=GL_n(\mathcal O_v)\) of volume one, and let the inducing unramified characters be \(\chi_1,\ldots,\chi_n\). In the normalized principal series the spherical function \(\varphi_0\) satisfies \(\varphi_0(uak)=\delta_B(a)^{1/2}\prod_i\chi_i(a_i)\), and the normalized spherical coefficient is \(c_v(g)=\int_{K_v}\varphi_0(kg)\,dk\). This is the coefficient of its spherical constituent, with the Satake identification of the preceding lesson. To verify the assertion even when the principal series is reducible, let \(\eta\) be its character on the one-dimensional compact fixed space. At a representative \(g\) of \(K_vgK_v\), integrating right translates of the fixed vector and evaluating at the identity gives \(\eta(\mathbf1_{K_vgK_v})=\operatorname{vol}(K_vgK_v)\int_{K_v}\varphi_0(kg)\,dk\). The normalized compact fixed vector in the irreducible spherical constituent gives the identical formula for its coefficient, because averaging its translates projects onto that fixed line. The two coefficients are therefore equal on every double coset. Take \(\Phi_v=\mathbf1_{M_n(\mathcal O_v)}\). Its left \(K_v\)-invariance removes the average in \(c_v\). For \(g=uak\), integrating its upper off-diagonal entries gives

Indeed, each entry above the diagonal in column \(i\) is \(u_{ji}a_i\), so there are \(i-1\) independent additive integrals of volume \(|a_i|_v^{-1}\). Multiplying (1.5m), the Haar density \(\delta_B^{-1}\), the normalized induction factor \(\delta_B^{1/2}\), and \(|\det a|^{s+(n-1)/2}\), leaves the exponent \(s\) on every \(a_i\). Thus

The last equality is the geometric series over the valuation of \(a_i\). This proves the value of the spherical integral. Proposition 1.2 below proves the additional generator identification and the full good-place functional equation, using a finite Smith-type basis of Hecke translates.

For a unitary coefficient, \(|c_v(g)|\le C_v\); for normalized unit spherical vectors \(C_v=1\). The same computation with \(c_v=1\) and exponent \(u=s+(n-1)/2\) gives

Formula (1.5o) converges locally for \(\operatorname{Re}s>(n-1)/2\). Thus, for a unitary spherical coefficient, the left side of (1.5n) is holomorphic in this half-plane. The inducing-character computation was initially performed farther to the right, where its individual Tate integrals converge. The identity theorem extends (1.5n) to the unitary half-plane, so no uniform bound on the inducing characters was silently needed.

The product of the good-place absolute integrals converges whenever \(\operatorname{Re}s>(n+1)/2\): every exponent \(u-j\) then has real part greater than one. This uses the already proved convergence of the Dedekind zeta Euler product. At an exceptional place, multiplicative Haar measure on \(GL_n\) is a constant times \(|\det X|_v^{-n}dX\). This follows by applying the Jacobian \(|\det g|_v^n\) of left matrix multiplication. Consequently, if \(\operatorname{Re}s>(n+1)/2\), boundedness of a unitary coefficient reduces absolute convergence to the integrability of a Schwartz function times a positive power of \(|\det X|_v\). At a finite place its support is compact; at infinity \(|\det X|\) grows polynomially. Both integrals are finite, locally uniformly in \(s\); powers of \(\log|\det X|_v\) inserted by differentiation are absorbed by a smaller positive exponent near the singular locus. This proves a common initial half-plane without assuming temperedness.

Why the spherical test is the local generator

Computing one integral does not ordinarily identify the generator of an integral family. In the unramified matrix case we can make that additional step: finite matrix congruences give a triangular basis of Hecke translates of the basic test.

Proposition 1.2 (unramified local standard theory). Let \(k=F_v\) be nonarchimedean, \(K=GL_n(\mathcal O)\), and let \(\tau\) be an irreducible admissible unramified unitary representation. Let \(\psi\) have conductor \(\mathcal O\): it is trivial on \(\mathcal O\), and \(\mathcal O\) is its own annihilator under \(\psi(xy)\). Put \(X=q^{-s}\) and \(R=\mathbb C[X,X^{-1}]\). For every Schwartz–Bruhat \(\Phi\) on \(M_n(k)\) and every smooth matrix coefficient \(c\) of \(\tau\),

The integral family therefore generates exactly \(R\,L(s,\tau)\), with \(L(s,\tau)=\prod_i(1-\alpha_iX)^{-1}\). Its normalized integrals are entire functions of \(s\), and

Thus the good-place standard generator and epsilon factor in 1.1c are proved: the epsilon factor is one and its conductor exponent is zero. No genericity hypothesis is needed for this assertion.

Proof. Write \(u=s+(n-1)/2\). Normalize multiplicative Haar measure by \(\operatorname{vol}K=1\). Let \(v_0\) be a spherical unit vector and \(c_0(g)=\langle\tau(g)v_0,v_0\rangle\). The fixed line, its Hecke character \(\eta\), and its Satake parameter are established in The unramified local correspondence and the Satake isomorphism, Theorem 4.3 and Proposition 5.1. The calculation (1.5n), including its justification for a spherical constituent, gives \(Z(s,\Phi_0,c_0)=L(s,\tau)\), where \(\Phi_0=\mathbf1_{M_n(\mathcal O)}\).

First prove a statement about the test functions, independently of the representation. Let \(\mathcal H=\mathcal H(GL_n(k),K)\) act by \((T*\Phi)(A)=\int_G T(h)\Phi(h^{-1}A)\,dh\). We claim

Here the right side means finite linear combinations of these convolutions. For an integer \(N\ge1\) and a partition \(N\ge\lambda_1\ge\cdots\ge\lambda_n\ge0\), set \(t_\lambda=\operatorname{diag}(\varpi^{\lambda_1},\ldots,\varpi^{\lambda_n})\), \(T_\lambda=\mathbf1_{Kt_\lambda K}\), and \(F_\lambda=T_\lambda*\Phi_0\). The right cosets \(hK\) in this double coset correspond bijectively to the lattices \(L=h\mathcal O^n\subset\mathcal O^n\) satisfying \(\mathcal O^n/L\simeq\bigoplus_i\mathcal O/\varpi^{\lambda_i}\mathcal O\). Indeed, changing the basis of \(L\) is right multiplication by \(K\), and integral row operations give its stated elementary divisors. Every such coset has measure one. Consequently, for \(A\in M_n(k)\),

The number is finite, since every such \(L\) contains \(\varpi^N\mathcal O^n\). The function is zero outside \(M_n(\mathcal O)\) and, inside that set, depends only on \(A\bmod\varpi^N\).

The \(K\times K\)-orbits of matrices modulo \(\varpi^N\) are indexed by the same partitions \(\mu\), allowing exponent \(N\) to represent a zero diagonal entry. To see this directly, choose a nonzero entry of smallest valuation, move it to the first diagonal position, and eliminate its row and column using integral operations. Every other entry is divisible by this pivot. Induction gives a diagonal matrix; permuting the entries orders the exponents. These operations over \(\mathcal O/\varpi^N\) lift to \(K\): a lift of an invertible matrix modulo \(\varpi^N\) has unit determinant. For uniqueness, the quotient \(\mathcal O^n/(A\mathcal O^n+\varpi^N\mathcal O^n)\) has summands \(\mathcal O/\varpi^{\mu_i}\mathcal O\). The dimensions of its successive \(\varpi^{j-1}/\varpi^j\) layers count the \(\mu_i\ge j\), recovering every exponent. This also proves the elementary-divisor assertion for the lattices in (1.5od); equivalently one may use the integral pivot proof in Proposition 1.1 of the preceding lesson.

If \(A\) has truncated type \(\mu\), each lattice counted in (1.5od) gives a quotient of this finite module of type \(\lambda\). Its cardinality is \(q^{|\lambda|}\), where \(|\lambda|=\sum_i\lambda_i\); the original module has cardinality \(q^{|\mu|}\). Hence \(F_\lambda(A)=0\) when \(|\lambda|>|\mu|\). If the cardinalities are equal, a surjection is an isomorphism, so the number is zero unless \(\lambda=\mu\). When they are equal, exactly one lattice occurs: \(L=A\mathcal O^n+\varpi^N\mathcal O^n\). Thus the square matrix of values \(F_\lambda(\mu)\), ordered by total size, is triangular with every diagonal entry one. It follows that the \(F_\lambda\) form a basis of all \(K\times K\)-invariant functions on \(M_n(\mathcal O/\varpi^N)\), extended by zero outside \(M_n(\mathcal O)\).

Every compactly supported locally constant test on \(M_n(\mathcal O)\) factors through one such congruence quotient. Compactness gives a finite cover by neighborhoods of constancy; a common sufficiently small additive subgroup preserves the entire function, including its zero set. More generally, compact support on \(M_n(k)\) lies in \(\varpi^{-r}M_n(\mathcal O)\) for some \(r\ge0\). A central translate reduces it to the integral case. Its characteristic matrix test is also a Hecke translate: \(\mathbf1_{\varpi^{-r}M_n(\mathcal O)}=T_{\varpi^{-r}I}*\Phi_0\). Combining this central translation with the preceding finite expansion proves (1.5oc).

Now apply the integral to these functions. Since \(T_\lambda\) is bi-invariant, the operator \(\tau(T_\lambda)\) has image in the spherical line and vanishes on its orthogonal complement: averaging on both sides gives \(\tau(T_\lambda)=P_K\tau(T_\lambda)P_K=\eta(T_\lambda)P_K\). Consequently \(\int T_\lambda(h)c_0(hx)\,dh=\eta(T_\lambda)c_0(x)\). Changing \(g=hx\) and using that the determinant norm is constant on the double coset gives

The multiplier is a constant times \(X^{|\lambda|}\). A central translate gives an integral power of \(X\), possibly negative. Equation (1.5oc) therefore proves (1.5oa) for all bi-invariant tests with coefficient \(c_0\). For any test with that coefficient, averaging on the left and right over \(K\) preserves its integral; its average is again Schwartz–Bruhat.

It remains to include every coefficient. Irreducibility implies that every smooth vector is a finite linear combination of translates of \(v_0\). Admissibility identifies the smooth contragredient with the conjugate smooth vectors in the invariant Hermitian pairing: a smooth functional fixed by a compact open \(J\) factors through the finite-dimensional space \(\tau^J\), where it is represented by a vector; averaging over \(J\) extends that identity to the entire smooth representation. Thus every smooth coefficient is a finite linear combination of \(c_0(b^{-1}ga)\), \(a,b\in G\). For such a coefficient put \(d_{b,a}=|\det b/\det a|\) and \(\Phi_{b,a}(x)=\Phi(bxa^{-1})\). Unimodularity and substitution give

The determinant norm is an integral power of \(q\), so this adds only a Laurent monomial in \(X\). We have proved rational continuation, the entire normalized quotient, and divisibility by \(L\) for every coefficient. Conversely the basic integral is exactly \(L\); hence the fractional ideal is precisely \(R\,L\). Its prescribed polynomial has constant term one by (1.5n).

Finally prove the functional equation for the whole family, rather than only for the basic test. Self-dual additive measure for conductor-zero \(\psi\) gives \(\widehat\Phi_0=\Phi_0\). For one coordinate this is the compact-character integral: the transform of \(\mathbf1_{\mathcal O}\) is its volume times the indicator of its annihilator, and inversion forces that volume to be one. The finite Cartesian product with the trace pairing merely permutes the matrix coordinates. The inversion and coset-indicator proof is Additive characters, self-dual measures and Poisson summation, Proposition 5.2, after scaling the local character to conductor zero.

Left matrix multiplication by \(h\) has additive Jacobian \(|\det h|^n\). Under the trace pairing,

Write \(u'=(1-s)+(n-1)/2=n-u\). In the dual integral substitute \(x=gh\). Then \(c_0^\vee(xh^{-1})=c_0(hx^{-1})\); integrating the \(h\)-variable uses the same Hecke character \(\eta(T_\lambda)\), rather than an incorrectly inverted determinant multiplier. We obtain

The basic dual integral is \(L(1-s,\widetilde\tau)\), by the already proved good-place calculation applied to the contragredient. This proves (1.5ob) for \(F_\lambda\), and the same substitution proves it for their central translates. All substitutions first take place in a half-plane of absolute convergence of the relevant integral; (1.5og) is an identity of compactly supported tests, and the resulting identities extend as rational functions. No common convergence half-plane for \(s\) and \(1-s\) is needed.

Bi-invariant averaging commutes with Fourier transformation: changing \(A\mapsto k_1Ak_2\) transforms the dual argument to \(k_2^{-1}Yk_1^{-1}\), and the Jacobian is one. Thus (1.5oc) and linearity extend the equation to every \(\Phi\) with coefficient \(c_0\). For the translated coefficient in (1.5of), its dual is \(c_0^\vee(a^{-1}(\,\cdot\,)b)\), and

The dual substitution contributes \(d_{b,a}^{-u'}\), and (1.5oi) contributes \(d_{b,a}^n\). Their product is \(d_{b,a}^{n-u'}=d_{b,a}^u\), exactly the multiplier in (1.5of). This proves the equation for every coefficient and every test. The epsilon factor is one; for the unramified representation the standard epsilon exponent is therefore zero. The additive-character change (1.5la) supplies the equation for any other local character without importing a ramified-representation theorem. \(\square\)

Finite-place matrix zeta ideals

Let \(F\) be any nonarchimedean local field, with integers \(\mathcal O\), uniformizer \(\varpi\), and residue cardinality \(q\). Set \(G=\mathrm{GL}_n(F)\), \(K=\mathrm{GL}_n(\mathcal O)\), and \(\nu(g)=|\det g|\), with \(|\varpi|=q^{-1}\). Normalize \(dg\) by \(\operatorname{vol}(K)=1\). For an irreducible smooth admissible representation \((\pi,V)\), a smooth dual vector \(\ell\), and \(v\in V\), write \[ c(g)=\ell(\pi(g)v),\qquad Z(s,\Phi,c)=\int_G\Phi(g)c(g)\nu(g)^{s+(n-1)/2}\,dg, \qquad X=q^{-s}. \] Here \(\Phi\) is any locally constant compactly supported function on \(M_n(F)\). Admissibility means that \(V^J\) is finite-dimensional for every compact open \(J\). It is an explicit hypothesis throughout.

Proposition 1.3 (finite-place rationality and the fractional ideal). In every rank, all these integrals converge absolutely in a right half-plane and continue to rational functions of \(X\). Their complex linear span is \[ \mathcal I_\pi=\frac{1}{P_\pi(X)}\mathbb C[X,X^{-1}], \qquad P_\pi\in\mathbb C[X],\quad P_\pi(0)=1. \] The polynomial is unique. In particular \(L_{\mathrm{mat}}(s,\pi)=P_\pi(q^{-s})^{-1}\) has no zeros, and its reciprocal is entire in \(s\). Every quotient \(Z(s,\Phi,c)/L_{\mathrm{mat}}(s,\pi)\) is a Laurent polynomial in \(q^{-s}\), hence is entire in \(s\), and a finite sum of actual integrals attains the generator. This assertion includes every ramified admissible irreducible representation; it does not require genericity or a classification of its Jacquet modules.

1. Compact averaging and the dual

For a compact open subgroup \(J\), the average \(e_Jv=\int_J\pi(j)v\,dj\), with probability Haar measure, is a finite sum on each smooth vector and projects onto \(V^J\). Every functional on the finite-dimensional space \(V^J\), composed with \(e_J\), is a smooth dual vector fixed by \(J\). Thus \[ (\widetilde V)^J=(V^J)^* \] under the natural pairing.

The smooth dual \(\widetilde V\) is irreducible. Indeed, a nonzero invariant subspace \(W\subset\widetilde V\) has annihilator zero in \(V\): that annihilator is invariant, and irreducibility of \(V\) excludes any other proper possibility. Consequently the restrictions of \(e_JW\) to \(V^J\) separate \(V^J\). A subspace of its finite-dimensional dual which separates it is the full dual. Hence \((\widetilde V)^J\subset W\) for every \(J\), and smoothness gives \(W=\widetilde V\). The same finite-dimensional pairing identifies the smooth double dual with \(V\).

Every central element acts by a scalar. To see this, choose a nonzero finite-dimensional \(V^J\). Its action preserves this space and has an eigenvector over \(\mathbb C\). The corresponding eigenspace in \(V\) is invariant under \(G\), so is all of \(V\). These scalars form a smooth character \(\omega_\pi:F^\times\to\mathbb C^\times\).

2. A positive diagonal semigroup on a finite fixed space

Fix \(r\geq1\) and put \(J=1+\varpi^rM_n(\mathcal O)\). This subgroup is normal in \(K\). Gaussian elimination using pivots in \(1+\varpi^r\mathcal O\) gives \[ J=J^+J^0J^-, \] where \(J^+\) and \(J^-\) are respectively upper and lower unipotent matrices congruent to the identity modulo \(\varpi^r\), and \(J^0\) is the diagonal subgroup with entries in \(1+\varpi^r\mathcal O\). For completeness, eliminate the last row and last column using the bottom-right unit pivot. Their off-diagonal entries are in \(\varpi^r\mathcal O\); the remaining Schur complement is again the identity modulo \(\varpi^r\). Induction gives the displayed upper-diagonal-lower factorization, with all its factors in \(J\).

For integers \(\lambda_1\geq\cdots\geq\lambda_n\), put \(a_\lambda=\operatorname{diag}(\varpi^{\lambda_1},\ldots, \varpi^{\lambda_n})\). If \(a=a_\lambda\) and \(b=a_\mu\) are both of this form, then \[ aJ^+a^{-1}\subset J^+,\qquad b^{-1}J^-b\subset J^-,\qquad [a,J^0]=[b,J^0]=1. \] The first assertion follows because an upper entry is multiplied by \(\varpi^{\lambda_i-\lambda_j}\) with \(i<j\); the second follows by the corresponding lower-entry calculation. Therefore \[ JaJbJ=JabJ. \tag{1.7a} \] The inclusion from left to right follows by inserting the factorization of the middle \(J\) and absorbing its three factors. The reverse inclusion follows by taking that middle element to be the identity.

The compactly supported probability measure \(\mu_a=e_J*\delta_a*e_J\) is invariant under left and right \(J\). By (1.7a), \(\mu_a*\mu_b\) is a bi-\(J\)-invariant probability measure on \(JabJ\). Such a measure is unique: \(J\times J\) acts transitively on that compact double coset, and averaging a continuous function over \(J\times J\) gives a constant there. Consequently \[ \mu_a*\mu_b=\mu_{ab}. \] On \(E=V^J\), let \(T_a=e_J\pi(a)e_J|_E\). It follows that \[ T_aT_b=T_{ab}. \tag{1.7b} \] In particular the \(n\) commuting operators \[ T_i=T_{\operatorname{diag}(\varpi I_i,I_{n-i})},\quad 1\leq i\leq n, \qquad T_n=\omega_\pi(\varpi)\operatorname{id}_E \tag{1.7c} \] control every dominant diagonal. Negative central powers are allowed, since \(T_n\) is invertible.

3. The Cartan sum and its volume

Elementary row and column operations over the discrete valuation ring give \[ G=\coprod_{\lambda_1\geq\cdots\geq\lambda_n}Ka_\lambda K. \tag{1.7d} \] One proof first multiplies a matrix by a scalar to make its entries integral, moves an entry of smallest valuation into a pivot position, and eliminates its row and column by integral invertible operations. Apply induction to the remaining block and arrange the valuations in decreasing order. Uniqueness follows because the ideals generated by the \(i\)-rowed minors are unchanged by these operations: their valuations recover the successive partial sums of the smallest diagonal valuations. This also proves disjointness in (1.7d).

Put \(d_i=\lambda_i-\lambda_{i+1}\) for \(i<n\). Let \(b_1,\ldots,b_t\) be the sizes of the successive equal-entry blocks in \(\lambda\). The exact volume is \[ \operatorname{vol}(Ka_\lambda K)= q^{\sum_{i=1}^{n-1}i(n-i)d_i}\, \frac{\prod_{j=1}^n(1-q^{-j})} {\prod_{\alpha=1}^t\prod_{j=1}^{b_\alpha}(1-q^{-j})}. \tag{1.7e} \] Here is the counting argument. The volume is the index of \(K\cap a_\lambda Ka_\lambda^{-1}\) in \(K\). The intersection requires its \((i,j)\) entry, for \(i<j\), to be divisible by \(\varpi^{\lambda_i-\lambda_j}\); lower entries have no additional condition. Modulo \(\varpi\) this is the lower block parabolic with the indicated block sizes. Counting ordered bases gives \[ |\mathrm{GL}_m(\mathbb F_q)| =q^{m^2}\prod_{j=1}^m(1-q^{-j}). \] Thus the residue index contributes one power of \(q\) for each cross-block upper entry, together with the ratio of products in (1.7e). Beyond the first congruence, such an entry contributes \(q^{\lambda_i-\lambda_j-1}\). These are independent entry conditions on matrices with invertible residue. Their total exponent is \(\sum_{i<j}(\lambda_i-\lambda_j) =\sum_i i(n-i)d_i\), as claimed.

For an integrable function on \(Ka_\lambda K\), its integral is its volume times its average over \(k_1a_\lambda k_2\), \(k_1,k_2\in K\), with probability measure on \(K\times K\). This follows from the same uniqueness of an invariant probability measure used above. The multiplicative Haar measure is bi-invariant: explicitly it is a positive constant times \(|\det g|^{-n}d g_{\mathrm{add}}\). Left or right multiplication by a matrix \(h\) changes additive matrix measure by \(|\det h|^n\), which proves the assertion.

Choose \(J\) small enough to fix both \(v\) and \(\ell\). Since it is normal in \(K\), every \(k_1,k_2\) gives \(J\)-fixed vectors \(\ell'=\ell\pi(k_1)\), \(v'=\pi(k_2)v\), and there are only finitely many such choices. On a dominant diagonal, \[ c(k_1a_\lambda k_2)=\ell'(T_{a_\lambda}v'). \tag{1.7f} \] If the support of \(\Phi\) lies in \(\varpi^{-B}M_n(\mathcal O)\), then \(\lambda_n\geq-B\) on the Cartan cosets which meet it. Indeed integral invertible left and right multiplication preserves the maximum absolute matrix-entry norm. Write \[ \lambda=(-B,\ldots,-B)+ \sum_{i=1}^n k_i(\underbrace{1,\ldots,1}_{i},0,\ldots,0), \quad k_i\geq0. \] Equations (1.7b), (1.7e), and (1.7f), together with boundedness of \(\Phi\), bound the absolute integral by a constant times \[ \sum_{k_1,\ldots,k_n\geq0} \prod_{i=1}^n \left(q^{i(n-i)-i(\Re s+(n-1)/2)} \max(1,\|T_i\|)\right)^{k_i}. \] For \(i=n\) the term \(i(n-i)\) is zero. Every factor is less than one when \(\Re s\) is sufficiently large. This proves absolute convergence before any averaging of \(\Phi\).

4. A common denominator, independent of the test

We can now average \(\Phi\) on the left and right by \(J\). Since \(c\) is bi-\(J\)-invariant, its zeta integral is unchanged. Choose \(R\geq0\) so that this averaged test is invariant under addition by \(\varpi^RM_n(\mathcal O)\). Such a uniform \(R\) exists for a locally constant compactly supported function: cover its compact support by finitely many constant additive cosets and refine them to one common lattice. Outside their union the function is zero on each coset of that lattice as well.

For each dominant \(\lambda\), let \(h\) be the number of entries \(\lambda_j\geq R\). The remaining entries \[ m_{h+1}\geq\cdots\geq m_n,\qquad -B\leq m_j<R, \] have only finitely many possibilities. On each such region, \(\Phi(k_1a_\lambda k_2)\) is independent of the first \(h\) diagonal entries: replace them by zero. The replacement changes the matrix by an element of \(\varpi^RM_n(\mathcal O)\), even after multiplication by \(k_1,k_2\).

For \(h>0\), take the base \(\lambda^0=(R,\ldots,R,m_{h+1},\ldots,m_n)\). Every member of the region has the unique expression \[ \lambda=\lambda^0+ \sum_{i=1}^h k_i(\underbrace{1,\ldots,1}_{i},0,\ldots,0), \quad k_i\geq0, \] where \(k_i\) for \(i<h\) are successive gaps, and \(k_h=\lambda_h-R\). Split each \(k_i\), \(i<h\), into the cases zero and at least one. On each resulting region the equal-entry block pattern, and hence the product ratio in (1.7e), is constant. If \(h<n\), its boundary gap is already positive because \(R>m_{h+1}\). If \(h=n\), the variable \(k_n\) is central and does not change any gap.

Put \[ z_i(X)=q^{\,i(n-i)-i(n-1)/2}X^i,\qquad D_J(X)=\prod_{i=1}^n\det(1-z_i(X)T_i). \tag{1.7g} \] In every region the Cartan sum is a constant, a Laurent monomial, and a finite-dimensional matrix element of products of series \[ \sum_{k\geq0}(z_iT_i)^k=(1-z_iT_i)^{-1},\qquad \sum_{k\geq1}(z_iT_i)^k=z_iT_i(1-z_iT_i)^{-1}. \] The formulas hold in the right half-plane just proved, and adjugates give their rational continuations. The regions with \(h=0\) contain finitely many diagonals. The \(K\times K\) average is a finite average, since both the coefficient and the averaged test are bi-\(J\)-invariant and \(J\) is normal in \(K\). We have therefore proved \[ Z(s,\Phi,c)\in D_J(X)^{-1}\mathbb C[X,X^{-1}] \quad\hbox{for every }\Phi,\qquad D_J(0)=1. \tag{1.7h} \] \(D_J\) depends on the fixed coefficient space, not on the support or constancy radius of the test.

Choose once and for all \(v_0,\ell_0\) with \(\ell_0(v_0)=1\), and a principal congruence \(J\) fixing them. Irreducibility of \(V\) and \(\widetilde V\) means that every coefficient is a finite linear combination of \(c_0(a^{-1}gb)\), where \(c_0(g)=\ell_0(\pi(g)v_0)\). The substitution \(g=ax b^{-1}\) turns its integral into the integral with coefficient \(c_0\), test \(\Phi(axb^{-1})\), and factor \(\nu(ab^{-1})^{s+(n-1)/2}\). This factor is a nonzero constant times a Laurent monomial in \(X\). Thus (1.7h) supplies one common denominator for every test and every coefficient.

5. From that denominator to the canonical generator

Let \(\mathscr R=\mathbb C[X,X^{-1}]\). Simultaneously replacing \(\Phi(g),c(g)\) by \(\Phi(gh),c(gh)\), with \(\det h=\varpi\), multiplies the integral by \(q^{s+(n-1)/2}\). The inverse translation gives the inverse factor. Therefore the complex span \(\mathcal I_\pi\) is an \(\mathscr R\)-submodule of \(D_J^{-1}\mathscr R\). It contains 1: take \(\Phi=\operatorname{vol}(J)^{-1}1_J\) and \(c=c_0\). The test is a Schwartz function on the matrix space, and on \(J\) the coefficient and determinant norm are both 1.

The Laurent polynomial ring is a principal ideal domain. Indeed clear powers of \(X\) and use the usual Euclidean division in \(\mathbb C[X]\); localizing permits precisely the additional units \(cX^m\). Write \(D_J\mathcal I_\pi=A\mathscr R\). Since \(1\in\mathcal I_\pi\), \(A\) divides \(D_J\). Choose its polynomial representative with nonzero constant term. The quotient \(D_J/A\), normalized to have constant term 1, is \(P_\pi\). This proves Proposition 1.3. If two such normalized polynomials give the same ideal, their ratio is \(cX^m\); their nonzero constant terms force \(m=0\), and normalization forces \(c=1\). The generator is a finite linear combination of actual matrix zeta integrals, because \(\mathcal I_\pi\) was their span. Multiplying an integral by a Laurent monomial can also be absorbed into the simultaneous translations just used. \(\square\)

6. Consequences of a scalar Fourier equation

Take a unitary additive character \(\psi\), the trace-pairing Fourier transform \[ \widehat\Phi(Y)=\int_{M_n(F)}\Phi(X)\psi(\operatorname{tr}(XY))\,dX \] with self-dual additive measure, and \(\check c(g)=c(g^{-1})\). Here inversion can be checked directly, as in the earlier proof in Additive characters, self-dual measures and Poisson summation on the adèles, Propositions 5.1–5.2. The annihilator of \(\mathcal O\) is a fractional ideal \(\varpi^h\mathcal O\). Indeed continuity puts the image of some \(\varpi^r\mathcal O\) in a small arc around 1; a subgroup of the circle contained in such an arc is trivial. The annihilator is therefore open, is an \(\mathcal O\)-submodule, and is proper by nontriviality of \(\psi\), so has the stated form. The annihilator of \(\varpi^r\mathcal O\) is \(\varpi^{h-r}\mathcal O\). Giving \(\mathcal O\) mass \(q^{h/2}\), character orthogonality gives \[ \widehat{1_{a+\varpi^r\mathcal O}}(y) =\operatorname{vol}(\varpi^r\mathcal O)\, \psi(ay)1_{\varpi^{h-r}\mathcal O}(y). \] Applying this twice yields \(1_{a+\varpi^r\mathcal O}(-x)\), since the two subgroup volumes multiply to 1. Every scalar Schwartz function is a finite sum of such indicators. The matrix trace pairing is \(\sum_{i,j}X_{ij}Y_{ji}\), a product of scalar pairings followed by coordinate permutation. Refining a matrix Schwartz function to one common additive matrix lattice gives a finite sum of products of scalar indicators. Thus \(\widehat{\widehat\Phi}(X)=\Phi(-X)\) on the whole matrix space. Theorem 1.17 below proves that one scalar independent of both test and coefficient satisfies \[ Z(1-s,\widehat\Phi,\check c) =\gamma(s,\pi,\psi) Z(s,\Phi,c) \tag{1.7i} \] for all the tests and coefficients. With that existence theorem, the proved fractional ideal gives the following algebraic consequences. Taking a pair whose integral is 1 shows that \(\gamma\) is rational in \(X\). Fourier transformation and coefficient inversion are bijections of the respective families, so (1.7i) identifies their entire fractional ideals. The dual variable \(q^{-(1-s)}=q^{-1}X^{-1}\) generates the same Laurent ring. Hence \[ \gamma(s,\pi,\psi)=\epsilon(s,\pi,\psi) \frac{L_{\mathrm{mat}}(1-s,\widetilde\pi)} {L_{\mathrm{mat}}(s,\pi)},\qquad \epsilon(s,\pi,\psi)=cX^m,\quad c\ne0,\quad m\in\mathbb Z. \tag{1.7j} \] Indeed the ratio after dividing out the two ideal generators must be a unit of \(\mathscr R\).

Fourier inversion gives \(\widehat{\widehat\Phi}(X)=\Phi(-X)\). Applying (1.7i) twice, with the same character \(\psi\), therefore gives \[ \gamma(s,\pi,\psi)\gamma(1-s,\widetilde\pi,\psi) =\epsilon(s,\pi,\psi)\epsilon(1-s,\widetilde\pi,\psi) =\omega_\pi(-1). \tag{1.7k} \] The central sign is essential with this positive trace convention: substituting \(g\mapsto-g\) in \(Z(s,\Phi(-\,\cdot),c)\) multiplies it by \(\omega_\pi(-1)\).

For \(\psi_a(x)=\psi(ax)\), self-dual matrix measure is multiplied by \(|a|^{n^2/2}\). Thus \(\widehat\Phi^{\,\psi_a}(Y)=|a|^{n^2/2}\widehat\Phi^{\,\psi}(aY)\). Substituting \(Y\mapsto a^{-1}Y\) in the dual integral proves \[ \epsilon(s,\pi,\psi_a)= \omega_\pi(a)|a|^{\,n(s-1/2)}\epsilon(s,\pi,\psi). \tag{1.7l} \]

If \(\pi\) has a compatible unitary realization, all its smooth dual vectors are represented by inner products with smooth vectors. To verify this, a \(J\)-fixed dual vector is represented on finite-dimensional \(V^J\) by its Hilbert inner product and extends by the orthogonal average \(e_J\). Consequently \(\widetilde\pi\) is the complex conjugate of \(\pi\), and uniqueness of the normalized ideal generator gives \[ L_{\mathrm{mat}}(s,\widetilde\pi) =\overline{L_{\mathrm{mat}}(\bar s,\pi)}. \] Conjugating Fourier transformation replaces \(\psi\) by \(\bar\psi=\psi_{-1}\). Thus \(\epsilon(s,\widetilde\pi,\bar\psi) =\overline{\epsilon(\bar s,\pi,\psi)}\). At \(s=1/2\), (1.7l) replaces the left side by \(\omega_{\widetilde\pi}(-1)\epsilon(1/2,\widetilde\pi,\psi)\). Together with (1.7k), and \(\omega_\pi(-1)^2=1\), this proves \[ |\epsilon(1/2,\pi,\psi)|=1,\qquad \epsilon(s,\pi,\psi)=wq^{\,m(1/2-s)},\quad |w|=1. \tag{1.7m} \] It proves that \(m\) is an integer, not that \(m\geq0\). Nonnegativity for a character of conductor \(\mathcal O\) still requires a further argument. For a general \(\psi_a\), the integer changes by \(n\,v_F(a)\).

7. A precise compact-mod-center case

Proposition 1.4 (compact-mod-center coefficients). If, in addition to the hypotheses of Proposition 1.3, all coefficients of \(\pi\) have compact support modulo the center and \(V^K=0\), then \(L_{\mathrm{mat}}(s,\pi)=1\).

Fix a coefficient with right \(J\)-fixed vector. Replace its test by \[ \Phi_*(X)=\int_J\Phi(Xj)\,dj-\int_K\Phi(Xk)\,dk. \] The first term has the same integral; the second has zero integral, because its coefficient average replaces \(v\) by \(e_Kv=0\). These substitutions are justified in the convergent half-plane from Proposition 1.3. Both averages have value \(\Phi(0)\) at zero, so \(\Phi_*\) vanishes on a neighborhood of zero. Its support is bounded. If \(\operatorname{supp}(c)\subset F^\times C\) for compact \(C\subset G\), then on the support of \(\Phi_*c\) the absolute value of that central scalar is bounded above and below. This follows by taking the positive minimum and finite maximum of the matrix-entry norm on \(C\). The intersection is therefore contained in a compact subset of \(G\) with finitely many determinant valuations. Integrating on those shells gives a Laurent polynomial in \(X\). All the zeta integrals are Laurent polynomials, and their ideal contains 1 by the earlier test; hence it is \(\mathscr R\). \(\square\)

Corollary 1.4a (the compact-mod-center generator). Using the unramified matrix theorem proved in Proposition 1.2, when \(n>1\) the compact-mod-center hypothesis alone implies \(V^K=0\), and therefore \(L_{\mathrm{mat}}(s,\pi)=1\).

First a real determinant twist makes the central character unitary. Indeed a smooth character has finite image on \(\mathcal O^\times\); choose \(t\in\mathbb R\) with \(q^{-nt}|\omega_\pi(\varpi)|=1\) and replace \(\pi\) by \(\pi\nu^t\). This preserves its compact fixed spaces and the compact-mod-center support of every coefficient.

The twisted representation has a compatible unitary realization. Choose a nonzero smooth dual vector \(\ell\), and put \[ \langle v,w\rangle= \int_{F^\times\backslash G} \ell(\pi(g)v)\overline{\ell(\pi(g)w)}\,d\bar g. \] The integrand is independent of the central representative and has compact support, so this is finite. It is positive definite: the kernel of the coefficient map for this fixed \(\ell\) is invariant in \(V\), and cannot be all of \(V\) since \(\ell\ne0\). Right invariance of quotient Haar measure makes the form invariant. The coefficient embedding also shows its action is continuous before completion, since smooth vectors have open stabilizers. In the Hilbert completion \(H\), compact averaging has range \(H^J=\overline{V^J}=V^J\); the last equality uses admissibility. Hence every smooth vector of \(H\) belongs to \(V\). The completion is irreducible as well: an orthogonal projection onto a closed invariant subspace commutes with \(G\), preserves all \(H^J=V^J\), and on the irreducible \(V\) is zero or the identity. Thus Proposition 1.2 applies if \(V^K\ne0\).

Suppose now that a spherical unit vector \(v_0\) exists and write \(c_0(g)=\langle\pi(g)v_0,v_0\rangle\). Compact support modulo the center gives only finitely many dominant Cartan patterns modulo adding a common integer to every diagonal valuation. This follows also directly because those successive valuation differences are bounded on a compact subset of \(G\). For each pattern choose its representative \(\lambda_n=0\). The cosets in it that meet \(M_n(\mathcal O)\) are then precisely \(a_{\lambda+k(1,\ldots,1)}\), \(k\geq0\). Central covariance, the constant Cartan volume along a central translate, and the determinant weight give \[ Z(s,1_{M_n(\mathcal O)},c_0)= \frac{Q(X)} {1-\omega_\pi(\varpi)q^{-n(n-1)/2}X^n}, \qquad Q\in\mathbb C[X],\quad Q(0)=1. \] The constant term comes from \(K\); every other normalized pattern has positive valuation sum. Proposition 1.2 says that the same integral is \(1/P(X)\), where \(P(X)=\prod_{i=1}^n(1-\alpha_iX)\) and \(\prod_i\alpha_i=\omega_\pi(\varpi)\). All the parameters are nonzero, so \(P\) and the displayed denominator both have degree \(n\). Their equality \(PQ=1-\omega_\pi(\varpi)q^{-n(n-1)/2}X^n\) forces \(Q=1\). Comparing their leading coefficients would give \((-1)^n=-q^{-n(n-1)/2}\), impossible for \(n>1\). This proves \(V^K=0\). Proposition 1.4 applies to the original representation, since the determinant twist does not change its \(K\)-fixed space. \(\square\)

For a general rank greater than two, these propositions do not prove that abstract supercuspidality implies compact support of all coefficients modulo the center. That implication, and the Fourier equation even under the compact-support hypothesis, are separate obligations. In rank two the earlier programme supplies the implication and the full equation.

The freely available Goldfeld–Jacquet author notes, §2, Theorem 2.1 and Lemma 2.2, PDF pp. 6–11, provide the normalization and the traditional reduction route: author-hosted text. The argument above proves the stated ideal assertions directly; the source's statement of the reduction lemma is not used as a substitute for its proof.

The full finite-place theory for determinant characters

We now prove the ramified Fourier equation and its nonnegative conductor exponent for an explicit family in every rank. Let \(k\) be nonarchimedean, let \(\psi\) have conductor \(\mathcal O\), and use the positive trace Fourier transform and self-dual additive matrix measure. For a smooth character \(\chi:k^\times\to\mathbb C^\times\), put \(\tau(g)=\chi(\det g)\). Its conductor \(a=a(\chi)\) is zero when \(\chi\) is trivial on \(\mathcal O^\times\); otherwise it is the least positive integer for which \(\chi(1+\varpi^a\mathcal O)=1\). Define the scalar Tate factor by

The entire scalar theory is proved in Tate's local theory at the finite places, Propositions 7.1 and 7.4 and Theorems 7.2–7.3. The matrix argument below includes the necessary boundary and finite-sum steps.

Proposition 1.16 (determinant characters, all finite places and all ranks). The matrix generator of Proposition 1.3 is

For every matrix Schwartz–Bruhat test, write \(H(s,\Phi,\chi)=Z(s,\Phi,\chi\circ\det)/L_{\mathrm{mat}}(s,\tau)\). Then the whole family satisfies

The epsilon factor is the product of the scalar Tate epsilon factors at the shifts in (1.20b). For \(a=0\), it is one. For \(a>0\), put

Then

In particular, for unitary \(\chi\),

Thus the matrix/Tate-product conductor exponent for this family is exactly \(na\ge0\). This establishes Fourier existence and conductor nonnegativity for determinant characters, including their ramified members; it makes neither assertion for every irreducible representation.

Proof: equivariant distributions and the singular ranks. Write \(V=M_n(k)\), and let \(G\times G\) act by \(X\mapsto AXB^{-1}\). For \(r=0,\ldots,n\), the rank-\(r\) matrices form one orbit \(\mathcal O_r\), with base point \(E_r=\operatorname{diag}(I_r,0)\). Row and column elimination gives transitivity. Its topology is the homogeneous-space topology: on a chart where the leading \(r\)-minor is invertible, a matrix of rank \(r\) has the form

Indeed take \(g_1=\left(\begin{smallmatrix}A&0\\C&I\end{smallmatrix}\right)\) and \(g_2^{-1}=\left(\begin{smallmatrix}I&A^{-1}B\\0&I\end{smallmatrix}\right)\); then \(g_1E_rg_2^{-1}\) is the displayed matrix, giving a continuous local section of the orbit map. Permuting rows and columns supplies all charts.

A distribution here is a linear functional on locally constant compactly supported tests. The space is the union of finite-dimensional spaces with bounded support and fixed additive constancy level; linear functionals and the compact averaging used below are continuous on each such space. For \(t\in\mathbb C\), consider the covariance

If \(\chi\) is ramified, no nonzero distribution supported on the singular matrices has this covariance, for any \(t\). We prove the claim on each orbit first. Choose a unit \(u\) with \(\chi(u)\ne1\). For \(r<n\), the pair

fixes \(E_r\). It lies in \(K\times K\) and normalizes every pair of principal congruence subgroups \(J_b\times J_b\), where \(J_b=I_n+\varpi^bM_n(\mathcal O)\). Choose \(b\) large enough that \(\chi\) is trivial on \(\det J_b\). The compact orbit \(U=(J_b\times J_b)E_r\) is open in \(\mathcal O_r\) by the chart (1.20g); for \(r=0\) it is the one-point orbit. On this compact transitive orbit, averaging any test over \(J_b\times J_b\) gives its mean times \(1_U\). This is a finite averaging operation on each test. Covariance is trivial on these congruence subgroups, so an equivariant distribution on \(U\) is determined by \(D(1_U)\). But \(h\) preserves \(U\), whereas (1.20h) multiplies that value by \(\chi(u)\). Thus \(D(1_U)=0\), and the distribution vanishes on \(U\). Covariance and transitivity give vanishing on every translate of \(U\); a finite compact-open partition of a test's support then proves vanishing on all of \(\mathcal O_r\).

To pass from the orbits to the boundary, let \(F_r\) be the closed set of matrices of rank at most \(r\). A distribution supported on \(F_r\) factors through tests on \(F_r\): an ambient locally constant test zero on \(F_r\) has compact support disjoint from it, so the distribution kills it. Every test on \(F_r\) extends to an ambient test by covering its compact support with finitely many compact-open balls on which its restriction is constant and taking a disjoint refinement. The rank-\(r\) stratum is open in \(F_r\). Its compactly supported tests extend by zero to \(F_r\). Their annihilation puts the remaining distribution on \(F_{r-1}\). Descending through the ranks proves the claim.

On the open orbit \(G\), the space with covariance (1.20h) has dimension one. Indeed put \(w_t(g)=\chi(\det g)|\det g|^{t+n}\). Multiplying tests by \(w_t^{-1}\) turns any such distribution into a left-invariant distribution on \(G\). A left-invariant distribution is a scalar multiple of Haar integration: on a compact open subgroup its compact average is the constant mean, and its value on any translate of a smaller compact open subgroup is determined by the finite index. Refining two subgroups to their intersection makes the scalar independent of the subgroup. Every test is a finite sum of compact-open coset indicators. This proves the assertion without an orbit-distribution uniqueness theorem as an additional input. Since a difference with zero open-orbit restriction is supported on the boundary, the same one-dimensional bound holds on all of \(V\) when \(\chi\) is ramified.

Proof: entireness and the Fourier equation. Up to the positive Haar constant, the matrix integral is the distribution

initially in its convergence half-plane. The singular set has additive measure zero: a point in \(k\) has measure zero since its balls have masses tending to zero; an invertible linear coordinate change then gives measure zero to any proper linear subspace. Successive columns of a matrix almost surely avoid the span of their predecessors, by Fubini on compact matrix boxes. Proposition 1.3 gives rational continuation with one denominator for all tests. If it had a pole, take a highest nonzero Laurent coefficient at that pole; the common denominator bounds its possible order uniformly over tests. Compactly supported tests inside \(G\) have entire integrals, so this coefficient is supported on the singular set. Change of variables gives precisely (1.20h); its highest Laurent coefficient has that covariance at the pole parameter. Boundary vanishing rules it out. Thus every integral is entire when \(\chi\) is ramified. A nonconstant canonical polynomial \(P\), with \(P(0)=1\), would have a nonzero complex root and would give a pole of its attained generator at some \(s\). That is impossible. Hence \(L_{\mathrm{mat}}=1\), proving (1.20b) in the ramified case.

Fourier transformation takes the distribution for \(\chi^{-1}\) and parameter \(t'=-t-n\) to another distribution with covariance (1.20h). In detail, with \(R=|\det A/\det B|\),

Applying the dual covariance contributes \(\chi(\det A/\det B)R^{-t'-n}\); the displayed extra \(R^n\) leaves \(\chi(\det A/\det B)R^{t+n}\). The one-dimensional result therefore proves the existence of one Fourier scalar for every test. A normalized principal-congruence indicator has original zeta integral identically one, so this scalar is a single rational function of \(q^{-s}\), independent of the test. Fourier inversion ensures that it is nonzero. Here the entire assertion and Fourier uniqueness were proved for the whole family, not inferred from one example.

Proof: the scalar and its conductor. Assume \(a>0\), put \(J_a=I_n+\varpi^aM_n(\mathcal O)\), and take

Then \(Z(s,\Phi,\chi\circ\det)=1\). Write \(dg=C|\det X|^{-n}dX\); on \(J_a\) the determinant norm is one, so \(\operatorname{vol}_{dg}(J_a)=Cq^{-an^2}\). The scalar coset-indicator transform proved in Additive characters, self-dual measures and Poisson summation on the adèles, Proposition 5.2, applied to the trace coordinates, gives

In the initial half-plane for the dual integral, substitute \(Y=\varpi^{-a}A\). The Fourier scalar is

Only \(A\in K\) contributes. Here is a cancellation proof for all the other residue cells modulo \(\varpi^a\). For a singular residue cell, choose an integral lift \(A_0\) with Smith form. At least one diagonal Smith entry is divisible by \(\varpi\), or is zero. If \(a\ge2\), choose \(u\in1+\varpi^{a-1}\mathcal O\) with \(\chi(u)\ne1\); minimality of \(a\) supplies it. If \(a=1\), choose any unit with \(\chi(u)\ne1\). Scaling that Smith row by \(u\), and conjugating the row operation back, gives \(E\in K\) with

For every \(A\) in the same cell, \(EA\) is still in that cell and has the same trace modulo \(\varpi^a\). Its determinant norm and additive measure are unchanged. Its multiplicative character is multiplied by \(\chi(u)^{-1}\). Thus the cell integral in (1.20l) is zero. These changes of variables are valid in the convergent dual half-plane, and subsequently by continuation.

On the invertible residue cells all determinant norms are one, so the remaining integral is

Let \(R_a=\mathcal O/\varpi^a\), \(|R_a|=q^a\), and let \(U_n(R_a)\) be its upper unitriangular group. Right multiplication by \(u\in U_n(R_a)\) permutes \(GL_n(R_a)\) and leaves its determinant unchanged. Average the sum over these right multiplications. Since

orthogonality kills every matrix with a nonzero entry below the diagonal. Indeed the annihilator of \(\mathcal O\) is \(\mathcal O\), so for any nonzero residue \(c\) the character \(v\mapsto\psi(\varpi^{-a}cv)\) of \(R_a\) is nontrivial and its sum is zero. The independent upper entries of the surviving triangular matrix are arbitrary, and its diagonal entries are units. It follows that

Combining (1.20l)–(1.20n) gives \(\chi(\varpi)^{an}q^{-ans}\mathcal G_a^n\), which is (1.20e). The scalar Tate formula in the earlier Theorem 7.3 is \(\epsilon_k(z,\chi,\psi)=\chi(\varpi)^a q^{-az}\mathcal G_a\). The shifts \(j-(n+1)/2\) sum to zero, so its product is exactly (1.20e).

For completeness, the size of the primitive scalar sum follows directly from the same positive Fourier inversion. For \(f(x)=\chi(x)^{-1}1_{\mathcal O^\times}(x)\), additive invariance under \(\varpi^a\mathcal O\) puts its transform in \(\varpi^{-a}\mathcal O\). On every shallower shell, multiplying the unit variable by the same \(u\) used in the singular-cell argument leaves its additive phase unchanged and multiplies its character by a nontrivial scalar, so the transform is zero. On the remaining shell \(\widehat f(\varpi^{-a}w)=q^{-a}\chi(w)\mathcal G_a(\chi,\psi)\). Applying Fourier transformation again and evaluating at one gives \(\mathcal G_a(\chi,\psi)\mathcal G_a(\chi^{-1},\psi)=\chi(-1)q^a\). Conjugation of the first sum, followed by \(u\mapsto-u\), gives \(\overline{\mathcal G_a(\chi,\psi)}=\chi(-1)\mathcal G_a(\chi^{-1},\psi)\). The restriction of \(\chi\) to the finite unit quotient has modulus one, so these identities prove \(|\mathcal G_a|=q^{a/2}\). This also proves nonvanishing. Formula (1.20f) and the nonnegative exponent \(na\) follow.

Proof: the unramified characters. First suppose \(\chi\) is unramified and unitary. In the normalized principal series with inducing characters \(\chi|\cdot|^{j-(n+1)/2}\), its spherical function is \(\chi(\det g)\), because these exponents cancel the upper-Borel half-modulus in Iwasawa coordinates. This function spans the determinant-character subrepresentation and its fixed line. The spherical Hecke character therefore has eigenvalues \(\chi(\varpi)q^{(n+1)/2-j}\). Proposition 1.2 gives exactly (1.20b) and the whole Fourier equation with epsilon factor one. Every smooth \(\chi\) is a unitary character times a real determinant norm power; replacing \(\chi\) by \(\chi|\cdot|^b\) replaces \(s\) by \(s+b\), while the dual parameter becomes \(1-s-b\). This supplies the nonunitary unramified case as well, and proves the same norm-twist compatibility in the ramified formula. All scalar multiples of the coefficient are included, since the representation is one dimensional. \(\square\)

For \(n>1\), determinant characters are not the general cuspidal local factors: the upper unipotent subgroup acts trivially, so a nontrivial generic character admits no Whittaker functional on this representation. The proposition supplies a complete family of ramified matrix Fourier and conductor calculations; the general irreducible local assertion in 1.1c remains separate.

The scalar matrix Fourier equation in every finite-place rank

Let \(k\) be any nonarchimedean local field, with integers \(\mathcal O\), uniformizer \(\varpi\) and residue cardinality \(q\). Put \(G=\mathrm{GL}_n(k)\), \(\nu(g)=|\det g|\), and \(u_s=s+(n-1)/2\). Let \((\pi,V)\) be irreducible, smooth and admissible. Its smooth dual is \(\widetilde V\). We use the positive trace convention \[ \widehat\Phi(Y)=\int_{M_n(k)}\Phi(X)\psi(\operatorname{tr}(XY))\,dX, \qquad \check c(g)=c(g^{-1}), \tag{1.21a} \] with a nontrivial unitary additive character and self-dual measure.

Theorem 1.17. There is a nonzero rational function \(\gamma_{\mathrm{mat}}(s,\pi,\psi)\) of \(q^{-s}\), independent of the Schwartz–Bruhat test and the smooth matrix coefficient, such that \[ Z(1-s,\widehat\Phi,\check c) =\gamma_{\mathrm{mat}}(s,\pi,\psi)Z(s,\Phi,c). \tag{1.21b} \] The identity holds for the rational continuations of the entire test/coefficient family in every rank, including all ramified representations. Admissibility is a hypothesis. The theorem does not identify the matrix factor with a parameter or a rank-one Rankin–Selberg factor. Theorem 1.24 and Corollary 1.24c below supply nonnegativity of its matrix conductor exponent.

The already proved Proposition 1.3 supplies rational continuation, one denominator uniform over the entire family, smooth dual admissibility and biduality, and a test/coefficient pair whose integral is identically 1. Its Fourier inversion proof also supplies \(\widehat{\widehat\Phi}(X)=\Phi(-X)\) and preservation of the whole matrix Schwartz space. We use those proved assertions here, without treating the principal ideal as a Fourier equation.

1. A compact-fixed Jacquet lemma

Fix \(0<r<n\), put \(m=n-r\), and let \[ U=\left\{\begin{pmatrix}I_r&B\\0&I_m\end{pmatrix}\right\}, \quad U^-=\left\{\begin{pmatrix}I_r&0\\C&I_m\end{pmatrix}\right\}, \quad M=\mathrm{GL}_r(k)\times\mathrm{GL}_m(k). \] For any smooth representation \(W\), let \(W_U=W/\langle \pi(u)w-w:u\in U,w\in W\rangle\), with its unnormalized \(M\)-action. Write \(p:W\to W_U\) for the quotient. Set \[ J_d=1+\varpi^dM_n(\mathcal O),\quad J_{M,d}=J_d\cap M,\quad J_{U,d}=J_d\cap U,\quad J_{U^-,d}=J_d\cap U^-,\qquad d\geq1. \]

Lemma 1.17a. If \(W\) is smooth and admissible, then \[ p(W^{J_d})=(W_U)^{J_{M,d}}, \tag{1.21c} \] and \(W_U\) is a smooth admissible representation of \(M\).

Proof. Block elimination gives the unique decomposition \[ J_d=J_{U,d}J_{M,d}J_{U^-,d}. \tag{1.21d} \] Indeed for a block matrix \(\left(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\right)\) in \(J_d\), its last block \(D\) is invertible, and the three factors have upper entry \(BD^{-1}\), diagonal blocks \(A-BD^{-1}C,D\), and lower entry \(D^{-1}C\). They have the asserted congruences. The same formulas over \(\mathcal O/\varpi^N\), \(N>d\), give a bijection of the three finite coordinate sets with \(J_d/J_N\). Counting their uniform probabilities, then passing through these finite quotients, proves the probability-average identity \[ e_{J_d}=e_{J_{U,d}}e_{J_{M,d}}e_{J_{U^-,d}}. \tag{1.21e} \] All averages on smooth vectors are finite sums.

Put \(a=\operatorname{diag}(\varpi I_r,I_m)\). It commutes with \(M\), contracts \(U\), and expands \(U^-\). Let \(E=p(W^{J_d})\). This is finite-dimensional and lies in \((W_U)^{J_{M,d}}\). If \(w\in W^{J_d}\), then \(\pi(a)w\) is still fixed by \(J_{M,d}\) and \(J_{U^-,d}\), since \(a^{-1}J_{U^-,d}a\subset J_{U^-,d}\). Equation (1.21e), and the trivial action of \(U\) on \(W_U\), give \[ p(e_{J_d}\pi(a)w)=p(\pi(a)w)=a\,p(w). \] Thus \(aE\subset E\). The action of \(a\) on \(W_U\) is invertible. Its restriction to finite-dimensional \(E\) is therefore bijective, so \(aE=E\).

Now take \(y\in(W_U)^{J_{M,d}}\). Lift it to \(w\in W\) and average over \(J_{M,d}\), obtaining a \(J_{M,d}\)-fixed lift. Smoothness gives an \(N\) for which it is fixed by the lower block subgroup with entries in \(\varpi^NM_{m,r}(\mathcal O)\). For \(j\) sufficiently large, \(\pi(a^j)w\) is fixed by \(J_{U^-,d}\). Indeed, for \(u^-\in J_{U^-,d}\), the lower entries of \(a^{-j}u^-a^j\) are multiplied by \(\varpi^j\), so this conjugate fixes the lift once \(d+j\ge N\). Therefore \(\pi(u^-)\pi(a^j)w=\pi(a^j)w\). It remains \(J_{M,d}\)-fixed. Again (1.21e) yields \[ a^j y=p(e_{J_d}\pi(a^j)w)\in E. \] Since \(a^{-j}E=E\), \(y\in E\). This proves (1.21c). The quotient is smooth under \(M\). Every compact open subgroup of \(M\) contains some \(J_{M,d}\), so its fixed space is contained in the finite-dimensional space in (1.21c). This proves admissibility. \(\square\)

The cases \(r=0\), \(U=\{1\}\), \(M=G\), and \(r=n\) need no Jacquet argument. No cuspidal-support or classification theorem has been used in Lemma 1.17a.

2. The orbit functional and its modular factor

Put \(H=G\times G\), acting on matrix points by \((a,b)\cdot X=aXb^{-1}\). Its test-function action is \[ \rho(a,b)\Phi(X)=\Phi(a^{-1}Xb). \] On \(E=\widetilde V\otimes V\), let \[ \sigma(a,b)(\ell\otimes v)=\widetilde\pi(a)\ell\otimes\pi(b)v, \qquad \chi_s(a,b)=\nu(a)^{u_s}\nu(b)^{-u_s}. \tag{1.21f} \] We will prove that, outside a countable set of \(s\), \[ \dim\operatorname{Hom}_H(\mathcal S(M_n(k))\otimes E,\chi_s) \leq1. \tag{1.21g} \] The Hom space here consists of linear functionals with \(T(\rho(h)f\otimes\sigma(h)e)=\chi_s(h)T(f\otimes e)\); no unproved continuity or irreducible-distribution theorem is implicit in that notation.

For \(0\leq r\leq n\), write \(x_r=\operatorname{diag}(I_r,0)\) and let \(\mathcal O_r\) be its rank-\(r\) orbit. Its stabilizer is \[ S_r= \left\{ \left( \begin{pmatrix}A&B\\0&D\end{pmatrix}, \begin{pmatrix}A&0\\C&D'\end{pmatrix} \right): A\in\mathrm{GL}_r,\ D,D'\in\mathrm{GL}_m \right\},\qquad m=n-r. \tag{1.21h} \] This follows by equating \(a x_r=x_r b\). The unipotent radical is the product of the two additive off-diagonal block spaces. Its conjugation modulus under the displayed Levi is \[ \delta_r(a,b)=|\det A|^m|\det D|^{-r} |\det D'|^r|\det A|^{-m} =|\det D'|^r|\det D|^{-r}. \tag{1.21i} \] The two determinant powers involving \(A\) cancel. In coordinates \(u l\) on \(S_r\), its left Haar measure is \(\delta_r(l)^{-1}du\,dl\), since each general linear Levi factor is unimodular. Thus its right-translation convention is \[ \int_{S_r}f(xp)\,dx=\delta_r(p)\int_{S_r}f(x)\,dx. \tag{1.21j} \] For \(r=0\) or \(n\), the same formulas read \(\delta_r=1\).

Lemma 1.17b (orbit functional). There is a natural identification \[ \operatorname{Hom}_H(\mathcal S(\mathcal O_r)\otimes E,\chi_s) \simeq \{\lambda\in E^*: \lambda(\sigma(p)e)=\chi_s(p)\delta_r(p)\lambda(e) \text{ for }p\in S_r\}. \tag{1.21k} \]

Proof. The quotient \(H/S_r\) identifies with \(\mathcal O_r\) and has local continuous sections. To check this explicitly near \(x_r\), a rank-\(r\) matrix with invertible upper-left block has the form \[ Y=\begin{pmatrix}A&B\\ C&CA^{-1}B\end{pmatrix} =L(Y)x_rR(Y),\quad L(Y)=\begin{pmatrix}A&0\\ C&I_m\end{pmatrix},\quad R(Y)=\begin{pmatrix}I_r&A^{-1}B\\0&I_m\end{pmatrix}. \] The section is \((L(Y),R(Y)^{-1})\). Permuting rows and columns gives sections on the other nonzero-minor charts. Their formulas and inverses are continuous. Above a chart, \(h=s(Y)p\) therefore gives a product homeomorphism with \(S_r\).

Define the surjective \(H\)-map \[ Q:\mathcal S(H)\longrightarrow\mathcal S(H/S_r),\qquad Qf(hS_r)=\int_{S_r}f(hp)\,dp. \] It is well-defined by left invariance of \(dp\). Surjectivity follows using a local section, a compactly supported locally constant function of \(p\) with integral 1, and a finite partition of the compact support into charts. Its kernel is spanned by \[ R(p)f-\delta_r(p)f,\qquad R(p)f(h)=f(hp). \tag{1.21l} \] Here are the details of this kernel assertion. On any chart, a compactly supported locally constant function is a finite sum of products in the quotient and subgroup coordinates: cover its compact support by constant compact-open rectangles and refine that finite cover into a disjoint partition. Refining also in the quotient coordinate makes each vertical function have integral zero if \(Qf=0\). For a locally constant compactly supported function on \(S_r\), choose a compact open subgroup under which it is left invariant. Express it as a finite sum of indicators of left cosets \(Kp\). In the quotient by (1.21l), \([1_{Kp}]=\delta_r(p)^{-1}[1_K]\), exactly their Haar-mass ratio by (1.21j). Refining \(K\) replaces \(1_K\) by its index times \(1_{K'}\), and \(\delta_r\) is 1 on compact subgroups. Consequently the only surviving coordinate is the Haar integral. Its kernel is precisely the span (1.21l). Multiplying by the quotient-coordinate functions proves the assertion on each chart, then on the whole space.

An \(H\)-equivariant functional on \(\mathcal S(H)\otimes E\) is uniquely of the form \[ B_\lambda(f,e)= \int_H f(h)\chi_s(h)\lambda(\sigma(h^{-1})e)\,dh, \qquad \lambda\in E^*. \tag{1.21m} \] This formula needs no growth hypothesis, since \(f\) has compact support and the vector orbit is locally constant. To prove uniqueness rather than assume it, for a compact open \(J\) fixing \(e\) and annihilating \(\chi_s\), set \(\lambda(e)=B(1_J,e)/\operatorname{vol}(J)\). Refining \(J\) and decomposing it into left cosets proves independence of this choice by equivariance. A common such subgroup for finitely many vectors proves linearity. On a compact support the set of vectors \(\sigma(h^{-1})e\) is finite. Refine a right-coset partition of \(f\) to a subgroup fixing all these vectors and lying in the kernel of \(\chi_s\); equivariance on each coset then gives exactly (1.21m). Conversely (1.21m) has the required covariance by substituting \(h\mapsto ah\).

Finally \(H\) is unimodular, so \[ B_\lambda(R(p)f,e) =\chi_s(p)^{-1}B_{\lambda\circ\sigma(p)}(f,e). \] It descends through \(Q\otimes1_E\) exactly when (1.21l) vanishes, namely when \(\lambda\circ\sigma(p)=\chi_s(p)\delta_r(p)\lambda\). This proves (1.21k), including its modular sign. \(\square\)

3. Singular ranks give only a countable obstruction

For \(r<n\), let \(U_r\) be the upper block subgroup in the first component of (1.21h), and put \(W_r=(\widetilde V)_{U_r}\). For \(r=0\), put \(W_0=\widetilde V\) directly. Lemma 1.17a and the proved smooth dual admissibility show that \(W_r\) is admissible for \(M_r=\mathrm{GL}_r\times\mathrm{GL}_{n-r}\).

Suppose a nonzero \(\lambda\) in (1.21k) exists. Its character is trivial on \(U_r\), so it descends in its first tensor factor to \(W_r\). Choose a vector in its second tensor factor for which the resulting functional \(\lambda_0\) on \(W_r\) is nonzero. For \[ p_t=(\operatorname{diag}(I_r,tI_m),I_n)\in S_r, \qquad m=n-r, \] equations (1.21f), (1.21i) and (1.21k) give \[ \lambda_0(t\,w)=|t|^{m(u_s-r)}\lambda_0(w). \tag{1.21n} \] Here \(t\) acts by the central element \(\operatorname{diag}(I_r,tI_m)\) of \(M_r\).

Pick \(w\) with \(\lambda_0(w)\ne0\), fixed by some principal compact subgroup \(J_{M_r,d}\). Its fixed space is finite dimensional, and the central element \(\operatorname{diag}(I_r,\varpi I_m)\) acts on it by an invertible operator \(A_{r,d}\). Restricting (1.21n) to this space shows \[ q^{-m(s+(n-1)/2-r)}\in\operatorname{Spec}(A_{r,d}). \tag{1.21o} \] For a fixed \(r,d\), there are finitely many nonzero eigenvalues. The preimage of any one under the displayed exponential is countable and discrete. Taking the union over \(0\leq r<n\) and integers \(d\geq1\) gives a countable set \(\mathscr E\subset\mathbb C\). Hence every singular-rank Hom space in (1.21k) is zero when \(s\notin\mathscr E\). We need no finite-length theorem and do not assert that this countable set is finite.

On the full-rank orbit, \(S_n\) is diagonal \(G\), \(\delta_n=1\), and \(\chi_s|_{S_n}=1\). Its functionals are invariant pairings on \(\widetilde V\otimes V\). Such a pairing gives an intertwiner \(V\to(\widetilde V)^{\vee}=V\): its value at a vector fixed by \(J\) is a \(J\)-fixed functional on \(\widetilde V\), so lies in the proved smooth bidual. The intertwiner is scalar. Indeed it preserves a nonzero finite-dimensional \(V^J\), where it has an eigenvector; its eigenspace in \(V\) is invariant and therefore all of \(V\). The evaluation pairing is nonzero. Thus the full-rank Hom space has dimension one.

To pass from the strata to the whole matrix space, put \(X_{\geq r}=\{X:\operatorname{rank}X\geq r\}\). These are open and have the finite exact filtration \[ 0\longrightarrow\mathcal S(X_{\geq r+1}) \longrightarrow\mathcal S(X_{\geq r}) \longrightarrow\mathcal S(\mathcal O_r)\longrightarrow0. \tag{1.21p} \] Restriction is onto: a compactly supported locally constant function on the closed subset \(\mathcal O_r\) of \(X_{\geq r}\) extends by a finite compact-open cover in the ambient space, refined into a disjoint partition. Its nonzero level sets are compact, so finitely many such neighborhoods suffice; choose each neighborhood to meet the closed subset only where that level is constant. The kernel is precisely the functions compactly supported in the open complement, extended by zero. Tensoring with \(E\) preserves this exact sequence over \(\mathbb C\).

An equivariant functional which is zero on the full-rank subspace consequently factors successively through the singular-rank quotients in (1.21p). For \(s\notin\mathscr E\) each such quotient has zero Hom space, so the functional is zero. Restriction to the full-rank orbit is injective, and its one-dimensional Hom space proves (1.21g).

4. Both zeta families have the same covariance

For \(e=\ell\otimes v\), let \(c_e(g)=\ell(\pi(g)v)\). In the common initial convergent half-plane, the substitution \(g=ax b^{-1}\) gives \[ Z(s,\rho(a,b)\Phi,c_{\sigma(a,b)e}) =\chi_s(a,b)Z(s,\Phi,c_e). \tag{1.21q} \] Both sides are rational by Proposition 1.3, so the equality continues. That proposition gives a common denominator for all tests and coefficients. Away from its poles we obtain a genuine element \(T_s\) of the Hom space in (1.21g). It is nonzero: the compact subgroup test of Proposition 1.3 has \(T_s(\Phi_0,e_0)=1\) for every \(s\).

Consider the second rational family \[ D_s(\Phi,e)=Z(1-s,\widehat\Phi,\check c_e). \] It is rational by the same proved proposition for \(\widetilde\pi\), with variable \(q^{-(1-s)}=q^{-1}q^s\). Fourier change of variables in the additive matrix space gives \[ \widehat{\rho(a,b)\Phi}(Y) =\nu(a)^n\nu(b)^{-n}\widehat\Phi(b^{-1}Ya). \tag{1.21r} \] Indeed \(X=aZb^{-1}\) has additive Jacobian \(\nu(a)^n\nu(b)^{-n}\), and \(\operatorname{tr}(aZb^{-1}Y)=\operatorname{tr}(Zb^{-1}Ya)\). This is the positive trace Fourier convention, with no extra inverse additive character.

Also \[ \check c_{\sigma(a,b)e}(g)=\check c_e(b^{-1}ga). \] Apply (1.21q) to the dual representation with the two group components exchanged and the parameter \(1-s\). Together with (1.21r), this gives \[ \begin{split} D_s(\rho(a,b)\Phi,\sigma(a,b)e) &=\nu(a/b)^n\,\nu(b/a)^{1-s+(n-1)/2}D_s(\Phi,e)\\ &=\nu(a/b)^{s+(n-1)/2}D_s(\Phi,e) =\chi_s(a,b)D_s(\Phi,e). \end{split} \tag{1.21s} \] The first application may be made in the dual convergent half-plane; rationality then gives the displayed identity everywhere. Thus outside the poles of its own uniform denominator \(D_s\) belongs to the same Hom space.

5. Generic uniqueness gives the entire rational equation

Remove \(\mathscr E\) and the poles of the two uniform denominators. Their union is countable. At every remaining parameter, (1.21g), (1.21q), and (1.21s) give \[ D_s=\gamma(s)T_s,\qquad \gamma(s)=D_s(\Phi_0,e_0), \tag{1.21t} \] because \(T_s(\Phi_0,e_0)=1\). The last expression is a single rational function of \(q^{-s}\), independent of the test/coefficient family. For every fixed \(\Phi,e\), the difference \(D_s(\Phi,e)-\gamma(s)T_s(\Phi,e)\) is rational in \(q^{-s}\) and vanishes outside that countable set. The complement of a countable subset of \(\mathbb C\) is dense in every open disk. On any disk avoiding the finitely many local poles of this particular rational difference it therefore vanishes identically, and rational continuation proves its vanishing everywhere.

Finally \(\gamma\) is not the zero rational function. Fourier transformation is a bijection on the matrix Schwartz space, and smooth biduality makes coefficient inversion a bijection with the dual coefficient family. The latter has a test/coefficient pair whose integral is identically 1. Its inverse image under these bijections gives \(D_s(\Phi,e)=1\) for every \(s\). If \(\gamma\) were zero, (1.21t) at the generic parameters would contradict this pair. Equations (1.21t) are precisely (1.21b). \(\square\)

The rational scalar equation is now proved. The consequences in Proposition 1.3, §6, therefore apply: the matrix epsilon factor is a Laurent unit, the same-character double equation has the central sign \(\omega_\pi(-1)\), and a compatible unitary realization has unit central epsilon phase. Nothing in the rank-stratum argument imposes nonnegativity of that Laurent-unit exponent; Theorem 1.24 and Corollary 1.24c provide the separate positivity proof.

The freely available Goldfeld–Jacquet author notes, §2, Theorem 2.1, PDF pp.6–9, give the precise local analytic package and its normalization. The scalar-existence proof above uses the preceding proved rational family and the explicit compact-average and orbit arguments; a quoted local functional-equation theorem, a general classification or Whittaker uniqueness is not a premise.

Conductor exponents from compact induction

Let \(k\) be any nonarchimedean local field, \(\mathcal O\) its integers, \(\varpi\) a uniformizer, \(q\) the residue cardinality, and \(G=\mathrm{GL}_n(k)\), \(n\geq2\). Write \[ Z=k^\times I_n,\quad K=\mathrm{GL}_n(\mathcal O),\quad J_r=1+\varpi^rM_n(\mathcal O),\quad j(g)=v_k(\det g),\quad X=q^{-s}. \tag{1.24a} \] Multiplicative Haar measure gives \(K\) volume 1. The additive character \(\psi\) has annihilator \(\mathcal O\), and additive matrix measure is self-dual for \[ \widehat\Phi(Y)=\int\Phi(A)\psi(\operatorname{tr}(AY))\,dA . \tag{1.24b} \] Thus \(M_n(\mathcal O)\) has additive volume 1. Put \[ Z_\pi(s,\Phi,c)=\int_G\Phi(g)c(g) |\det g|^{s+(n-1)/2}\,dg,\qquad c^\vee(g)=c(g^{-1}). \tag{1.24c} \]

We use the following preceding results of this lesson: Proposition 1.3 and its §§1–3 prove smooth dual irreducibility and biduality, the normalized rational ideal, the principal-congruence diagonal semigroup, and the elementary Cartan decomposition; Proposition 1.4 and Corollary 1.4a prove that compact-mod-center coefficients imply \(L_{\mathrm{mat}}=1\) in rank \(n>1\); Theorem 1.17 proves the whole-family scalar Fourier equation and Lemma 1.17a proves the two-block probability factorization. The consequences in Proposition 1.3 §6 give \[ \gamma_\pi(s,\psi)=\epsilon_\pi(s,\psi) \frac{L_{\widetilde\pi}(1-s)}{L_\pi(s)},\qquad \epsilon_\pi(s,\psi)=c_\pi X^{a_\pi},\quad c_\pi\ne0,\quad a_\pi\in\mathbb Z . \tag{1.24d} \] Positivity is not assumed in (1.24d).

Zero Jacquet modules imply compact-mod-center coefficients

For \(1\leq i<n\), let \(U_i\) be the upper two-block unipotent subgroup of block sizes \(i,n-i\). Its entries form the additive space \(M_{i,n-i}(k)\). The unnormalized Jacquet quotient is \[ V_{U_i}=V/\langle\pi(u)v-v:u\in U_i,\ v\in V\rangle . \]

Lemma 1.20a. Let \(\pi\) be smooth admissible, and suppose \(V_{U_i}=0\) for all \(i<n\). Every smooth matrix coefficient of \(\pi\) has compact support modulo \(Z\). If \(\pi\) is also irreducible and \(n>1\), then \(L_\pi=L_{\widetilde\pi}=1\).

Proof. Fix \(r\geq1\), and put \(E=V^{J_r}\). For \[ a_i=\operatorname{diag}(\varpi I_i,I_{n-i}),\qquad T_i=e_{J_r}\pi(a_i)e_{J_r}|_E, \] the proved diagonal semigroup identity gives \[ T_i^b=e_{J_r}\pi(a_i^b)e_{J_r}|_E,\qquad b\geq0. \tag{1.24e} \] We show that \(T_i\) is nilpotent. Any \(v\in V\), because its Jacquet class is zero, is a finite sum of differences \(\pi(u_\alpha)w_\alpha-w_\alpha\). Choose a compact additive subgroup \(U_i(\varpi^{-t}\mathcal O)\) containing every \(u_\alpha\). Its probability average kills every difference, by translation invariance of its finite smooth average. Hence it kills \(v\). A basis of the finite-dimensional space \(E\) gives a single \(t\) for which this average kills all of \(E\). Every larger compact subgroup of \(U_i\) also has zero average on \(E\), since averaging again over a larger group preserves the zero obtained by averaging the smaller subgroup.

The two-block factorization, including its probability-average identity, is \[ J_r=(J_r\cap U_i)(J_r\cap M_i)(J_r\cap U_i^-),\qquad e_{J_r}=e_{J_r\cap U_i}e_{J_r\cap M_i}e_{J_r\cap U_i^-}. \tag{1.24f} \] Conjugation by \(a_i^{-b}\) expands the upper entries to \(\varpi^{r-b}\mathcal O\), contracts the lower entries to \(\varpi^{r+b}\mathcal O\), and fixes the block diagonal group. For \(v\in E\), the lower and diagonal averages therefore fix \(v\), so \[ e_{J_r}\pi(a_i^b)v =\pi(a_i^b)e_{a_i^{-b}J_ra_i^b}v =\pi(a_i^b)e_{U_i(\varpi^{r-b}\mathcal O)}v=0 \quad(b\geq r+t). \tag{1.24g} \] This proves \(T_i^{N_i}=0\) for some \(N_i\).

Now choose \(J_r\) fixing the vector and smooth dual vector of a coefficient \(c\). It is normal in \(K\), so their finitely many \(K\)-translates also lie in the same fixed spaces. For \(\lambda_1\geq\cdots\geq\lambda_n\), write \(d_i=\lambda_i-\lambda_{i+1}\), \(i<n\). The diagonal semigroup gives \[ c(k_1a_\lambda k_2) =\ell_{k_1}\!\left( T_n^{\lambda_n}\prod_{i<n}T_i^{d_i}v_{k_2}\right), \qquad T_n=\pi(\varpi I_n)|_E . \tag{1.24h} \] The central operator \(T_n\) is invertible and commutes with every \(T_i\); negative central powers are therefore allowed, even when \(\pi\) is not irreducible. If any \(d_i\geq N_i\), this is zero. Modulo scalar \(\varpi^{\lambda_n}I_n\), only finitely many Cartan double cosets with \(0\leq d_i<N_i\) remain. Each is compact, so their finite union is a compact support bound modulo \(Z\).

In the irreducible case Corollary 1.4a now gives \(L_\pi=1\). Every coefficient of \(\widetilde\pi\) is the inversion of one of \(\pi\), by the proved smooth bidual pairing, so the same support assertion and the same corollary give \(L_{\widetilde\pi}=1\). \(\square\)

Zero proper Jacquet modules therefore give the coefficient-support assertion. A compact-induction realization remains a separate assertion.

Trace integrality in a subgroup compact modulo the center

The next elementary observation supplies the sign of the conductor without an inducing classification or a newvector theorem.

Lemma 1.20b. Let \(H\) be an open subgroup of \(G\) containing \(Z\), with \(H/Z\) compact. Set \(H_j=\{h\in H:j(h)=j\}\). Then \(H_0\) is compact open, every nonempty \(H_j\) is a coset of \(H_0\), and \[ h\in H,\quad j(h)\geq0\quad\Longrightarrow\quad \operatorname{tr}(h)\in\mathcal O . \tag{1.24i} \] Moreover, a single integer \(r_0\) has \[ \bigcup_{j\geq0}H_j\subset\varpi^{-r_0}M_n(\mathcal O). \tag{1.24j} \]

Proof. Use the maximum entry norm \(\|h\|\). The continuous function \[ \rho(h)=\|h\|\,|\det h|^{-1/n} \] is unchanged by multiplying \(h\) by any scalar in \(Z\); it is therefore bounded by a constant \(B\) on \(H\). Likewise \(\rho(h^{-1})\) is bounded there. Thus on \(H_0\) both the matrix entries and the inverse entries are uniformly bounded. The set \(H_0\) is closed: an open subgroup is closed, and \(j\) is continuous to the discrete group \(\mathbb Z\). These entry and inverse-entry bounds put it in a compact subset of \(G\); explicitly, its determinants are units, so the closed bounded matrix set has no singular limit. Hence \(H_0\) is compact. It is open as an intersection of two open subgroups. The coset assertion follows from the homomorphism \(j:H\to\mathbb Z\). The bound \(\|h\|\leq Bq^{-j(h)/n}\leq B\) for \(j(h)\geq0\) proves (1.24j).

Fix such an \(h\). All its nonnegative powers lie in \(H\), and \[ \|h^b\|\leq Bq^{-b j(h)/n}\leq B,\qquad b\geq0. \] Consequently \(\Lambda=\sum_{b\geq0}h^b\mathcal O^n\) is an \(\mathcal O\)-submodule between \(\mathcal O^n\) and \(\varpi^{-R}\mathcal O^n\) for some \(R\). The latter quotient by \(\mathcal O^n\) is finite, so this sum is finitely generated. A submodule of a finite free module over the discrete valuation ring is free: choose a coordinate with an element of least valuation, eliminate that coordinate, and induct on the rank, just as in the preceding Smith reduction. Since \(\Lambda\) contains \(\mathcal O^n\), its rank is \(n\). Also \(h\Lambda\subset\Lambda\). The matrix of \(h\) in a basis of \(\Lambda\) therefore has integral entries. Its trace is integral, and trace is invariant under the change of basis. This proves (1.24i) in every residue characteristic. \(\square\)

Positive conductor for arbitrary compact inducing data

For an open \(H\) as in Lemma 1.20b and a finite-dimensional smooth irreducible representation \(\tau\) of \(H\), compact induction means the space of smooth functions \[ f:G\longrightarrow W_\tau,\qquad f(hg)=\tau(h)f(g), \] with support in finitely many left \(H\)-cosets, acted on by right translation. No classification or irreducibility theorem for such an induction is assumed below: irreducibility and admissibility of this realization are explicit hypotheses.

Theorem 1.20. Suppose \[ \pi\simeq\mathrm{c\!-\!Ind}_H^G\tau \tag{1.24k} \] is irreducible smooth admissible, \(H\supset Z\) is open and compact modulo \(Z\), and \(\tau\) is finite-dimensional smooth irreducible. For conductor-zero \(\psi\), the matrix exponent satisfies \[ a_\pi>0. \tag{1.24l} \] More precisely, choose \(r\geq1\) such that \(J_r\subset H\), \(\tau(J_r)=1\), and (1.24j) holds with \(r\) in place of \(r_0\). Then \[ 1\leq a_\pi\leq nr,\qquad a_\pi\in j(H)\subset\mathbb Z. \tag{1.24m} \] This includes all positive-depth data satisfying (1.24k).

Proof. First \(\tau^{H_0}=0\). Otherwise this nonzero space is \(H\)-stable, because \(H_0\) is normal, and irreducibility would make \(\tau\) trivial on \(H_0\). The group \(H/H_0=j(H)=d\mathbb Z\) is cyclic, with \(d\mid n\), since \(\varpi I_n\in H\). A finite-dimensional irreducible complex representation of a cyclic group is one-dimensional: an eigenvector of its generator gives an invariant line. Thus \(\tau(h)=b^{j(h)/d}\). Choose \(a\in\mathbb C^\times\) with \(a^d=b\), and put \(\chi(g)=a^{j(g)}\). It extends \(\tau\). There is then the explicit nonzero equivariant map \[ f\longmapsto\sum_{Hg\in H\backslash G}\chi(g)^{-1}f(g) \tag{1.24n} \] from the compact induction to the one-dimensional \(\chi\). The finite sum is independent of the left-coset representatives; substitution \(g\mapsto gx\) multiplies it by \(\chi(x)\). Irreducibility would identify \(\pi\) with \(\chi\). This is impossible: \(H\backslash G\) is infinite, so its compact induction has infinite dimension. Indeed otherwise \(G/Z\) would be a finite union of translates of the compact \(H/Z\). But the continuous scalar-invariant function \(\|g\|\|g^{-1}\|\) takes the unbounded values \(q^b\) on \(\operatorname{diag}(\varpi^b,1,\ldots,1)\), \(b\geq0\); thus \(G/Z\) is not compact. This proves the assertion.

Every coefficient of \(\pi\) has compact support modulo \(Z\). Here is the complete dual-support justification. For \(\lambda\in W_\tau^*\), evaluation \(\ell_\lambda(f)=\lambda(f(1))\) is a smooth dual vector: an open subgroup of \(H\) fixing \(\lambda\) fixes this functional. The span of all its \(G\)-translates is a nonzero invariant subspace of the irreducible smooth dual, hence the whole smooth dual by Proposition 1.3 §1. A coefficient using evaluation is \(g\mapsto\lambda(f(g))\), supported in finitely many \(H\)-cosets. Translated evaluation and a finite linear combination give the same conclusion with finitely many translated \(H\)-cosets. These sets have compact image modulo \(Z\). Corollary 1.4a therefore gives \[ L_\pi=L_{\widetilde\pi}=1,\qquad \gamma_\pi=\epsilon_\pi. \tag{1.24o} \]

Choose \(v_0\in W_\tau\) and \(\lambda\in W_\tau^*\) with \(\lambda(v_0)=1\). Let \(v\) be supported on \(H\), with \(v(h)=\tau(h)v_0\), and use \(\ell_\lambda\). The coefficient is exactly \[ c(g)= \begin{cases} \lambda(\tau(g)v_0),&g\in H,\\ 0,&g\notin H. \end{cases} \tag{1.24p} \] Our choice of \(r\) makes both vectors \(J_r\)-fixed. The test \(\Phi=\operatorname{vol}_G(J_r)^{-1}1_{J_r}\) has \(Z_\pi(s,\Phi,c)=1\). Since \(J_r=I_n+\varpi^rM_n(\mathcal O)\) as an additive matrix coset, character orthogonality with the positive trace pairing gives \[ \widehat\Phi(Y)=A_r\psi(\operatorname{tr}Y) 1_{\varpi^{-r}M_n(\mathcal O)}(Y),\qquad A_r=\frac{q^{-rn^2}}{\operatorname{vol}_G(J_r)}. \tag{1.24q} \] In fact \(A_r=\prod_{i=1}^n(1-q^{-i})\), independently of \(r\): \([K:J_r]=|\mathrm{GL}_n(\mathcal O/\varpi^r)|= q^{rn^2}\prod_i(1-q^{-i})\), by reduction modulo \(\varpi\), its \(q^{n^2}\)-element successive kernels, and ordered-basis counting over the residue field.

Consider the dual integral with this test. On a shell \(H_j\), \(j\geq0\), its test is the constant \(A_r\): the lattice condition holds by our choice of \(r\), and the trace character is 1 by Lemma 1.20b. This shell contributes zero, because \[ \int_{H_j}\lambda(\tau(g^{-1})v_0)\,dg=0. \tag{1.24r} \] To verify (1.24r), write \(H_j=h_jH_0\) and use the probability average \(e_{H_0}\) of \(\tau\). It is the projection onto \(\tau^{H_0}=0\). Inversion preserves Haar measure on the compact group \(H_0\), so the integral is \(\operatorname{vol}_G(H_0)\lambda(e_{H_0}\tau(h_j^{-1})v_0)=0\). Every individual shell is compact; these cancellations are valid first in the absolutely convergent dual half-plane.

If \(g\in\varpi^{-r}M_n(\mathcal O)\), expansion of its determinant gives \(j(g)\geq-nr\). Thus only the finitely many negative shells can remain, and the Fourier equation with the integral-1 test is the polynomial identity \[ \epsilon_\pi(s,\psi)= \sum_{j=-nr}^{-1} A_r q^{-j(n+1)/2} X^{-j} \int_{H_j\cap\varpi^{-r}M_n(\mathcal O)} \psi(\operatorname{tr}g)\lambda(\tau(g^{-1})v_0)\,dg . \tag{1.24s} \] Empty shells are zero. Every exponent on the right lies between 1 and \(nr\). The left side is a nonzero Laurent monomial by the proved scalar Fourier equation and ideal normalization (1.24d). Hence exactly one shell coefficient is nonzero; its exponent is \(a_\pi=-j\). Since \(j\in j(H)\), also \(-j\in j(H)\). This proves (1.24l)–(1.24m). No unitarity hypothesis, restriction on the residue characteristic, or generic newvector was used. \(\square\)

Equation (1.24s) is also an exact inducing-data interpretation: \(-a_\pi\) is the unique determinant shell carrying the nonzero matrix Gauss average displayed there. This assertion is independent of the chosen sufficiently large \(r\), \(v_0\) and \(\lambda\) with \(\lambda(v_0)=1\), since all give the same scalar \(\epsilon_\pi\). It identifies that shell, not an Artin conductor or the minimal newvector level.

Corollary 1.20c (a depth-zero realization). Let \(\sigma\) be an irreducible representation of \(\mathrm{GL}_n(\mathbb F_q)\) with zero invariants under the unipotent radical of every proper parabolic. Inflate it to \(K\). Extend its scalar central character on \(\mathcal O^\times\) to a smooth character \(\omega:k^\times\to\mathbb C^\times\); one may choose \(\omega(\varpi)\) arbitrarily. Then \[ \tau(zk)=\omega(z)\sigma(\bar k),\qquad \pi=\mathrm{c\!-\!Ind}_{ZK}^G\tau \tag{1.24sa} \] is irreducible smooth admissible, and \(a_\pi=n\).

Proof. The displayed \(\tau\) is well-defined: two decompositions differ by a scalar unit, on which its two characters agree. It is finite-dimensional smooth irreducible. We prove the hypotheses on \(\pi\), rather than use a compact-induction irreducibility criterion.

For a \(J_r\)-fixed function \(f\), if \(f(g)\ne0\), every \(u\in K\cap gJ_rg^{-1}\) fixes \(f(g)\) under \(\tau\), because \(ug=gj\) implies \(\tau(u)f(g)=f(ug)=f(gj)=f(g)\). Write \(g=k_1a_\lambda k_2\) in Cartan form. Since \(J_r\) is normal in \(K\), \(k_2\) does not change the intersection condition; \(k_1\) just conjugates its subgroup in \(K\). If some gap \(\lambda_i-\lambda_{i+1}\geq r\), the opposite two-block unipotent group with all its lower off-block entries in \(\mathcal O\) lies in \(K\cap a_\lambda J_ra_\lambda^{-1}\). Indeed conjugation by \(a_\lambda^{-1}\) multiplies its \((b,a)\) entry, \(b>i\geq a\), by \(\varpi^{\lambda_a-\lambda_b}\), which belongs to \(\varpi^r\mathcal O\). Its reduction is the full opposite unipotent radical of that proper parabolic. The assumption on \(\sigma\), also for conjugate parabolics, forces \(f(g)=0\).

Thus every gap on the support of a \(J_r\)-fixed function is smaller than \(r\). After removing the central \(\varpi^{\lambda_n}I_n\), only finitely many Cartan forms remain. Each gives finitely many left \(ZK\)- and right \(J_r\)-cosets, since \(K/J_r\) is finite. Values in the finite-dimensional \(W_\tau\) therefore prove \(\dim\pi^{J_r}<\infty\). Every compact open subgroup contains a \(J_r\), proving admissibility.

For \(r=1\), the support condition says that all gaps vanish, so \(\pi^{J_1}\) consists exactly of the functions supported on \(ZK\). Evaluation at 1 identifies this space with \(\sigma\) as a \(K\)-module. If a nonzero invariant subspace of \(\pi\) contains \(f\), translate a nonzero value to 1 and average over \(J_1\). The averaged value at 1 remains that value, since \(\tau(J_1)=1\). Thus the subspace meets \(\pi^{J_1}\) nontrivially, contains it by irreducibility of \(\sigma\), and contains all of \(\pi\): its translates span the functions on every finite collection of \(ZK\)-cosets. This proves irreducibility.

Apply Theorem 1.20 with \(H=ZK\) and \(r=1\). Its hypotheses hold, including (1.24j), since \(H_j=\varpi^{j/n}K\) for \(j\in n\mathbb Z\). Consequently \(1\leq a_\pi\leq n\) and \(a_\pi\in n\mathbb Z\), so \(a_\pi=n\).

For completeness its exact positive-trace epsilon constant is \[ \epsilon_\pi(s,\psi)= C_n\,\omega(\varpi)\,q^{n(n+1)/2}\,\mathcal G_\sigma X^n, \quad C_n=\prod_{i=1}^n(1-q^{-i}), \tag{1.24sb} \] where the finite-group operator \[ \frac1{|\mathrm{GL}_n(\mathbb F_q)|} \sum_{h\in\mathrm{GL}_n(\mathbb F_q)} \psi_{\mathrm{res}}(\operatorname{tr}h)\sigma(h^{-1}) =\mathcal G_\sigma I \tag{1.24sc} \] is scalar by conjugation invariance and the finite-dimensional Schur argument. Here \(\psi_{\mathrm{res}}(\bar x)=\psi(\varpi^{-1}x)\) is well-defined on the residue field. Formula (1.24sb) is (1.24s) on its sole negative shell \(\varpi^{-1}K\). The nonzero scalar Fourier theorem proves \(\mathcal G_\sigma\ne0\), so no unproved nonvanishing theorem for finite Gauss sums has been used. \(\square\)

The polynomial correction in a parabolic reduction

The algebraic step in a general reduction can be proved without assuming that epsilon exponents are already nonnegative.

Lemma 1.20d (conditional parabolic transfer). Let \(\sigma\) and \(\pi_1,\ldots,\pi_t\) be irreducible smooth admissible representations with the matrix ideal and scalar Fourier theorem. Suppose the following additional identities have been proved for this data, with the same conductor-zero additive character: \[ \gamma_\sigma(s,\psi)=\prod_i\gamma_{\pi_i}(s,\psi),\qquad L_\sigma(s)=R(X)\prod_iL_{\pi_i}(s),\qquad L_{\widetilde\sigma}(s)=\widetilde R(X) \prod_iL_{\widetilde\pi_i}(s), \tag{1.24t} \] where \(R,\widetilde R\in\mathbb C[X]\) and \(R(0)=\widetilde R(0)=1\). Then \[ \deg R=\deg\widetilde R=d,\qquad a_\sigma=\sum_i a_{\pi_i}+d . \tag{1.24u} \] In particular nonnegative exponents for the inducing data give a nonnegative exponent for \(\sigma\).

Proof. Substituting (1.24d) into the first identity gives \[ \frac{\epsilon_\sigma(s)}{\prod_i\epsilon_{\pi_i}(s)} =\frac{R(X)}{\widetilde R(q^{-1}X^{-1})}. \tag{1.24v} \] The left side is \(bX^e\), with \(e=a_\sigma-\sum_i a_{\pi_i}\). The order at \(X=0\) of the right side is \(\deg\widetilde R\): \(R\) has nonzero constant term, while the reversed polynomial in the denominator has order \(-\deg\widetilde R\). Thus \(e=\deg\widetilde R=d\). The identity \[ R(X)=bX^d\widetilde R(q^{-1}X^{-1}) \tag{1.24w} \] has degree exactly \(d\), because \(\widetilde R(0)=1\). Hence \(\deg R=d\) as well, and (1.24u) follows. \(\square\)

It is essential to include the factor \(X^d\) in (1.24w). The unqualified identity \(\widetilde R(X)=R(q^{-1}X^{-1})\) cannot hold for two nonconstant polynomials with constant term 1. We do not use such a reversed-polynomial assertion as an inducing theorem. The identities (1.24t) themselves are not proved by this algebra, by the scalar Fourier theorem, or by the existence of the ideal.

Theorem 1.20 includes positive-depth inducing data, whenever the stated compact induction is irreducible and admissible. It does not assert that every cuspidal representation has such a realization. Lemma 1.20a proves the coefficient-support direction from zero proper Jacquet modules; it does not construct an inducing subgroup. Theorem 1.23 and Corollary 1.23c below prove the parabolic identities (1.24t), and Theorem 1.23e proves cuspidal-support embedding. Lemmas 1.23f–1.23g construct compact inducing data under their precise orthogonality or compact-intertwining hypotheses. Universal existence of such data is not asserted. The direct proof in Theorem 1.24 and the parabolic transfer in Corollary 1.24c prove general nonnegativity without requiring those inducing data.

The exponent in (1.24s) has a precise matrix-Gauss-shell meaning. Identification with an Artin conductor requires a parameter construction and epsilon compatibility; identification with a minimal newvector level requires a generic newvector theorem. The latter cannot be extended to every irreducible representation. For the usual subgroup \[ K_1(\varpi^a)=\{g\in K:g_{nj}\in\varpi^a\mathcal O\ (j<n),\quad g_{nn}\equiv1\pmod{\varpi^a}\},\qquad a\geq1, \] every \(\operatorname{diag}(u,1,\ldots,1)\), \(u\in\mathcal O^\times\), belongs to \(K_1(\varpi^a)\). A ramified determinant character in rank at least two therefore has no fixed vector at any such level or at level zero \(K\), although Proposition 1.16 proves its finite epsilon exponent.

Further reading on the normalization and the general cuspidal-support reduction is Goldfeld–Jacquet, author notes, §2.1 and Theorem 2.1/Lemma 2.2, PDF pages 5–9.

Cuspidal support and parabolic matrix factors

Fix a nonarchimedean local field \(k\), integers \(\mathcal O\), uniformizer \(\varpi\), and residue cardinality \(q\). All representations are complex, smooth and admissible, and irreducible when so stated. Let \[ G=\mathrm{GL}_n(k),\quad K=\mathrm{GL}_n(\mathcal O),\quad \nu(g)=|\det g|,\quad X=q^{-s}. \tag{1.27a} \] Multiplicative Haar measure gives each standard maximal compact volume 1. Additive matrix measures are self-dual for the positive trace pairing with the same conductor-zero character \(\psi\). Thus every integral matrix lattice has additive volume 1.

We use Proposition 1.3’s complete matrix ideal and Theorem 1.17’s scalar Fourier equation. Every \(L\)-factor in this section denotes the normalized matrix-ideal generator; identification with parameter factors is a separate question. The tensor factorization used below is the fully general local-unit algebra theorem, Theorem 4.1 of Restricted tensor products and the tensor product theorem. Its proof concerns arbitrary algebras and finite-dimensional corners; its use here does not import the rank-two restriction of that lesson's subsequent adelic application.

Iwasawa integration and normalized induction in every rank

Let \(n=r+t\), \(r,t>0\), and put \[ P=MU,\quad M=\mathrm{GL}_r(k)\times\mathrm{GL}_t(k),\quad U=\left\{u(B)=\begin{pmatrix}I_r&B\\0&I_t\end{pmatrix}\right\}. \] For \(m=\operatorname{diag}(A,D)\), the conjugation modulus is \[ \delta_P(m)=|\det A|^t|\det D|^{-r}. \tag{1.27b} \] This follows from \(B\mapsto ABD^{-1}\) on its \(rt\) additive coordinates.

Here \(G=PK\). To see it directly, let \(W\subset k^n\) be the row span of the last \(t\) rows of \(g\). The lattice \(\Lambda=W\cap\mathcal O^n\) is saturated: if \(a x\in W\), \(a\ne0\), then \(x\in W\). A basis of \(\Lambda\) extends to an integral basis of \(\mathcal O^n\), by the following explicit argument. If \(W\ne0\), scale a nonzero vector of \(W\) so its coordinates are integral and at least one is a unit. Move that unit coordinate to the last position and use integral elementary coordinate changes to send the vector to \(e_n\). These changes belong to \(\mathrm{GL}_n(\mathcal O)\). Then \[ W=k e_n\oplus(W\cap k^{n-1}),\qquad W\cap\mathcal O^n=\mathcal O e_n \oplus(W\cap k^{n-1}\cap\mathcal O^{n-1}). \] Induction on \(\dim W\) constructs the desired basis and an integral complement. This also proves that the rank is \(t\) and that the quotient lattice is free. Let these basis vectors be the rows of \(k_0\in K\), with the last \(t\) spanning \(\Lambda\). Expressing the rows of \(g\) in this basis gives \(g=p k_0\), \(p\in P\): the last \(t\) rows have zero first \(r\) coordinates, and both diagonal blocks are invertible because \(g\) is invertible.

With \(dB\) giving \(M_{r,t}(\mathcal O)\) volume 1, left Haar measure on \(P\), in the order \(u(B)m\), is \[ dp=\delta_P(m)^{-1}\,dB\,dm . \tag{1.27c} \] Left translation by \(U\) is additive translation; left translation by \(m_0\) scales \(dB\) by \(\delta_P(m_0)\), which cancels the extra inverse-modulus factor. This proves left invariance, and \(P\cap K\) has volume 1.

For every integrable function on \(G\) the exact Iwasawa formula is \[ \int_G F(g)\,dg =\int_K\int_M\int_{M_{r,t}(k)} F(u(B)m k)\delta_P(m)^{-1}\,dB\,dm\,dk . \tag{1.27d} \] Indeed first average \(F\) on the right over \(K\). A right \(K\)-invariant function integrates as a sum on the discrete quotient \(G/K\), each coset of volume 1. The map \(P/(P\cap K)\to G/K\) is a bijection by \(G=PK\). Its cosets have volume 1 for the left Haar measure (1.27c). This proves (1.27d), first for nonnegative functions by counting cosets, then for integrable functions by linearity.

For a smooth admissible \(M\)-representation \(\rho\), normalized induction \(I_P(\rho)\) consists of smooth functions \(f:G\to V_\rho\) with a common right open stabilizer and \[ f(u m g)=\delta_P(m)^{1/2}\rho(m)f(g). \tag{1.27e} \] Right translation is the action. The quotient \(P\backslash G\) is compact, as the image of \(K\).

Lemma 1.23a. Normalized induction is admissible, and its smooth dual is \(I_P(\widetilde\rho)\), with invariant pairing \[ \langle\widetilde f,f\rangle =\int_K\langle\widetilde f(k),f(k)\rangle\,dk . \tag{1.27f} \] It preserves injections and is exact.

Proof. Fix a compact open \(J\subset K\). The double quotient \(P\backslash G/J\) is finite: a finite right \(J\)-cover of the compact \(K\) supplies representatives \(k_\alpha\in K\). A \(J\)-fixed section is determined by its values at these representatives. Such a value is fixed by \(\rho\) under the \(M\)-projection of \(P\cap k_\alpha Jk_\alpha^{-1}\); the modulus on this compact group is 1. Its projection is compact and contains a compact open subgroup of \(M\), since the intersection contains \(M\cap k_\alpha Jk_\alpha^{-1}\). Consequently its fixed space is finite-dimensional. There are finitely many representatives, proving admissibility.

We verify the invariant pairing without assuming an invariant measure on \(P\backslash G\). A smooth scalar function \(F\) with \(F(pg)=\delta_P(p)F(g)\) is the finite sum, at its right level \(J\), of the functions \[ Q\eta(g)=\int_P\eta(pg)\,dp,\qquad \eta\in C_c^\infty(G). \tag{1.27g} \] For \(\eta=1_{k_\alpha J}\), this function is supported on \(P k_\alpha J\) and has value \(\operatorname{vol}_P(P\cap k_\alpha Jk_\alpha^{-1})>0\) at \(k_\alpha\). Left Haar measure satisfies \(\int_P a(pp_0)\,dp=\delta_P(p_0)\int_P a(p)\,dp\) by (1.27c), so \(Q\eta\) has precisely the required left transformation. The finite representative values therefore prove the spanning assertion. Formula (1.27d) gives \[ \int_K Q\eta(k)\,dk=\int_G\eta(g)\,dg . \] Right translation commutes with \(Q\), and the right side is right invariant, since the matrix Haar measure on \(G\) is bi-invariant. Thus \(\int_K F(k)\,dk\) is a right \(G\)-invariant functional on these scalar densities. The product in (1.27f) has exactly this transformation; hence (1.27f) is invariant.

It is nondegenerate. A nonzero section has a nonzero value \(v\) at some \(k_\alpha\). Choose a dual vector detecting that value and average it over the compact projected stabilizer. Because \(v\) is fixed, its detected value is unchanged. The dual section supported on \(P k_\alpha J\) with that value exists by (1.27e). Its pairing with \(f\) is a nonzero constant times the positive measure of the corresponding subset of \(K\). This gives an injection of the dual induction into the smooth dual. At every \(J\) the two inducing fixed-value spaces are finite-dimensional dual spaces, by compact averaging. Thus the fixed-space dimensions agree, and the injection is onto every fixed space, hence onto the entire smooth dual.

Injections are preserved pointwise. For a surjection of smooth \(M\)-modules, a \(J\)-fixed target section has only finitely many representative values. Lift each value, then average over its compact projected stabilizer; this supplies a fixed lift of that value. The sections defined from these values by (1.27e), zero on the other double cosets, give a preimage. For kernels, the section takes values in the kernel exactly when its image section is zero. This proves exactness. \(\square\)

The complete two-block matrix descent

Let \(\rho=\pi_1\otimes\pi_2\), with irreducible smooth admissible \(\pi_1,\pi_2\) on \(\mathrm{GL}_r(k),\mathrm{GL}_t(k)\). Set \(I=I_P(\rho)\). For \(f\in I\) and \(\widetilde f\in I_P(\widetilde\rho)\), its coefficient is \[ c_I(g)=\int_K\langle\widetilde f(k),f(kg)\rangle\,dk . \tag{1.27h} \] For \(\Phi\in\mathcal S(M_n(k))\), define \[ \Phi^P_{k,k'}(A,D)= \int_{M_{r,t}(k)} \Phi\left(k^{-1}\begin{pmatrix}A&C\\0&D\end{pmatrix}k'\right)dC, \qquad c_{k,k'}(A,D)= \langle\widetilde f(k),\rho(A,D)f(k')\rangle . \tag{1.27i} \] Each \(\Phi^P_{k,k'}\) is a Schwartz–Bruhat function on the product of the two matrix spaces. These functions and the two section values have only finitely many possibilities as \((k,k')\) ranges over \(K\times K\). A common sufficiently small principal congruence subgroup fixes \(\Phi\) on both sides and fixes both sections on the right. This makes the dependence constant on finitely many compact cosets.

For a Levi test \(\Psi\) and coefficient \(b\), put \[ Z_\rho(s,\Psi,b)= \int_{\mathrm{GL}_r(k)\times\mathrm{GL}_t(k)} \Psi(A,D)b(A,D) |\det A|^{s+(r-1)/2}|\det D|^{s+(t-1)/2}\,dA\,dD . \tag{1.27j} \] Every such integral is a finite sum of products of the two factor zeta integrals. A product-space Schwartz function is a finite sum of rectangular product tests, by partition into cosets of a common additive product lattice. A coefficient has the same finite-product property: the vectors are finite sums of tensors, and a smooth dual vector fixed by a product compact subgroup is in the dual of its finite-dimensional fixed space, the tensor product of the factor fixed spaces. Compact averaging extends its finite tensor expression to the factor smooth duals.

Lemma 1.23b (matrix descent). In an absolute-convergence right half-plane, and consequently for the rational continuations, \[ Z_I(s,\Phi,c_I) =\int_{K\times K}Z_\rho(s,\Phi^P_{k,k'},c_{k,k'})\,dk\,dk' . \tag{1.27k} \] Every induced coefficient integral is rational and its entire ideal is \[ \mathcal I_I=L_{\pi_1}(s)L_{\pi_2}(s)\, \mathbb C[X,X^{-1}] . \tag{1.27l} \] Here the ideal is the span of all induced tests and coefficients; no irreducibility of \(I\) is required.

Proof. Insert (1.27h) and substitute \(h=kg\), then apply (1.27d) to \(h=u(B)m k'\). The section law supplies \(\delta_P(m)^{1/2}\rho(m)\), leaving \(\delta_P(m)^{-1/2}\nu(m)^{s+(n-1)/2}\). In \[ u(B)m=\begin{pmatrix}A&BD\\0&D\end{pmatrix} \] put \(C=BD\); then \(dB=|\det D|^{-r}dC\). The resulting two determinant powers are exactly \[ s+(n-1)/2-t/2=s+(r-1)/2,\qquad s+(n-1)/2+r/2-r=s+(t-1)/2 . \tag{1.27m} \] This proves (1.27k).

It also justifies all these rearrangements initially: with absolute values inserted, the \(C\)-integral is again a bounded compactly supported locally constant Levi test. There are finitely many section values and tests on \(K^2\). Factor absolute convergence therefore supplies a common right half-plane. The finite factor expansion gives rational continuation and the inclusion \(\mathcal I_I\subset L_{\pi_1}L_{\pi_2}\mathbb C[X,X^{-1}]\).

We prove the reverse inclusion by constructing actual sections and tests; it is not an assertion of generator attainment. Given factor vectors and dual vectors, form \(v=v_1\otimes v_2\), \(\ell=\ell_1\otimes\ell_2\). Given factor tests \(\Phi_1,\Phi_2\), take a compact matrix test \[ \Phi\begin{pmatrix}A&C\\W&D\end{pmatrix} =\Phi_1(A)\Phi_2(D)\eta(C)\zeta(W), \quad \int\eta(C)\,dC=1,\quad \zeta(0)=1 . \tag{1.27n} \] For example normalized lattice indicators supply \(\eta,\zeta\). Then \(\Phi^P_{1,1}=\Phi_1\Phi_2\).

Choose \(J=J_a\) small enough that \(\Phi\) is bi-\(J\)-invariant and that \(J\cap M\) fixes \(v,\ell\). Such matrix-test invariance is explicit: if the support is in \(\varpi^{-B}M_n(\mathcal O)\) and the additive constancy lattice is \(\varpi^R M_n(\mathcal O)\), take \(a-B\geq R\). Left and right multiplication by \(J_a\) changes a matrix in this ball by that constancy lattice and preserves the ball, since both a matrix in \(J_a\) and its inverse are integral. Smoothness gives the remaining vector condition.

Let \(f,\widetilde f\) be the sections supported on \(PJ\) with respective values \(v,\ell\) at 1, extended by (1.27e). They are well-defined: the \(M\)-projection of \(P\cap J\) fixes those values, its modulus is 1, and \(U\) acts trivially. Only \(k,k'\in (P\cap K)J\) contribute to (1.27k). Write \(k=pj\), \(k'=p'j'\), \(p,p'\in P\cap K\). Bi-\(J\)-invariance removes \(j,j'\) from the test. Write \(m_p,m_{p'}\) for the diagonal blocks of \(p,p'\). Both have unit determinants. The upper off-diagonal integration is changed by an affine bijection preserving its additive measure, so \[ \Phi^P_{p,p'}(m)=\Phi^P_{1,1}(m_p^{-1}m m_{p'}),\qquad c_{p,p'}(m)=\langle\ell,\rho(m_p^{-1}m m_{p'})v\rangle . \tag{1.27o} \] The Levi change of variable \(m\mapsto m_p^{-1}m m_{p'}\) preserves Haar measure and both determinant weights. Every contributing \(Z_\rho\) therefore has the same value \[ Z_{\pi_1}(s,\Phi_1,c_1)Z_{\pi_2}(s,\Phi_2,c_2). \] If \(b=\operatorname{vol}_K((P\cap K)J)>0\), (1.27k) is exactly \(b^2\) times this product. Thus every product of actual factor integrals is an actual induced integral up to a nonzero constant. The product of their entire ideals is generated by these products. Simultaneous determinant translations also make \(\mathcal I_I\) a Laurent-ring module, by the same change of variable as in Proposition 1.3. This proves (1.27l). \(\square\)

Fourier descent and the gamma factor

Let \(\mathcal F_M\) denote the product of the factor matrix Fourier transforms with the same positive trace convention. The exact elementary identity is \[ (\widehat\Phi)^P_{k,k'} =\mathcal F_M\bigl(\Phi^P_{k',k}\bigr). \tag{1.27p} \] To prove it first put \(k=k'=1\). In the trace pairing, the upper-right block of the Fourier variable pairs with the lower-left block of the original variable. Integrating over that Fourier block therefore sets the original lower-left block to zero. What remains is the factor Fourier transform of the upper-right integral in (1.27i). This statement uses ordinary finite Fourier orthogonality, not a distributional delta assertion: partition a Schwartz–Bruhat test into additive matrix-lattice cosets; the scalar coset Fourier formula of Proposition 1.3 §6 proves the identity for each rectangular coset indicator. Refining to a common lattice and adding proves it for every test.

More explicitly, for a rectangular product \(\Phi(A,C,W,D)=\phi_1(A)\eta(C)\zeta(W)\phi_2(D)\), its transform is \[ \widehat\phi_1(A')\widehat\phi_2(D') \widehat\eta(W')\widehat\zeta(C'). \] The rectangular Fourier transforms use the dual trace pairing between \(M_{r,t}\) and \(M_{t,r}\). Putting \(W'=0\) and integrating \(C'\) gives \(\widehat\phi_1(A')\widehat\phi_2(D') (\int\eta)\zeta(0)\), exactly the Levi transform of \(\Phi^P_{1,1}\). The identity \(\int\widehat\zeta=\zeta(0)\) follows from the same lattice-coset orthogonality formula and the chosen self-dual measures; no extra block-volume constant occurs.

For \(k,k'\in K\), the matrix substitution \(A=k^{-1}Bk'\) has additive Jacobian 1, and the trace is cyclic. The Fourier covariance swaps \(k,k'\): \[ \widehat{\Phi(k^{-1}(\,\cdot\,)k')}(Y) =\widehat\Phi(k'^{-1}Yk). \] Combining this with the identity at 1 proves (1.27p), including the index order.

Theorem 1.23. For every induced test and coefficient, \[ Z_{\widetilde I}(1-s,\widehat\Phi,c_I^\vee) =\gamma_{\pi_1}(s,\psi)\gamma_{\pi_2}(s,\psi) Z_I(s,\Phi,c_I). \tag{1.27q} \] Thus the induced gamma factor is the product, in every residue characteristic and for the full ramified family.

Proof. The invariant pairing (1.27f) identifies \(c_I^\vee\) with the coefficient on \(\widetilde I\) whose vector is \(\widetilde f\) and whose dual vector is \(f\). Its descended coefficient at indices \(k,k'\) is \[ \langle\widetilde\rho(m)\widetilde f(k'),f(k)\rangle =c_{k',k}(m^{-1}). \tag{1.27r} \] Apply (1.27k) to the dual integral, use (1.27p), and apply the proved whole-family factor Fourier equations to each of the finitely many factor test/coefficient terms. They multiply every term by \(\gamma_{\pi_1}(s,\psi)\gamma_{\pi_2}(s,\psi)\). Finally exchange \(k,k'\) in the compact integral; (1.27k) gives (1.27q). Both descent formulas are rational identities, so their initial convergence half-planes need not intersect. The resulting identity holds for their rational continuations. \(\square\)

Corollary 1.23c (every irreducible constituent). If \(\sigma\) is any irreducible subquotient of \(I_P(\pi_1\otimes\pi_2)\), then \[ \gamma_\sigma=\gamma_{\pi_1}\gamma_{\pi_2},\qquad L_\sigma(s)=R(X)L_{\pi_1}(s)L_{\pi_2}(s),\quad R(0)=1, \quad R\in\mathbb C[X]. \tag{1.27s} \] The dual identity has an analogous \(\widetilde R\in\mathbb C[X]\), \(\widetilde R(0)=1\). No finite-length assumption is needed.

Proof. For a subrepresentation, every smooth dual functional on it extends smoothly to the ambient module: choose a compact open \(J\) fixing the functional, extend its linear restriction from the subspace of \(J\)-fixed vectors to the ambient \(J\)-fixed space, and compose with \(e_J\). The restriction remains the original functional. For a quotient, lift the vector and compose the dual functional with the quotient map. Thus every coefficient of any subquotient is a coefficient of the original induction, through a submodule and a quotient as needed. Its integral and inverse-coefficient Fourier equation are identical to the ambient ones. The nonzero integral-1 test fixes its scalar, giving the first identity in (1.27s).

The same lifting gives \(\mathcal I_\sigma\subset\mathcal I_I\). Since their normalized generators are \(L_\sigma=1/P_\sigma\) and \(L_I=1/(P_{\pi_1}P_{\pi_2})\), (1.27l) implies \[ R(X)=\frac{P_{\pi_1}(X)P_{\pi_2}(X)}{P_\sigma(X)} \in\mathbb C[X,X^{-1}]. \] It is regular at 0 with value 1, so has no negative powers. This proves the polynomial assertion.

Smooth duality is exact for admissible smooth modules: the same extension by compact averaging proves surjectivity of restriction for an injection, and composition proves the other direction. Every submodule and quotient is admissible, since compact averages lift its fixed vectors in the quotient. Lemma 1.23a identifies the dual induction. Consequently \(\widetilde\sigma\) is a subquotient of \(I_P(\widetilde\pi_1\otimes\widetilde\pi_2)\), and the same ideal argument proves its polynomial identity. \(\square\)

With the scalar Laurent-unit epsilon already proved, these identities supply the precise correction without any remaining parabolic premise: \[ \deg R=\deg\widetilde R=d,\qquad a_\sigma=a_{\pi_1}+a_{\pi_2}+d,\qquad R(X)=bX^d\widetilde R(q^{-1}X^{-1}),\quad b\ne0. \tag{1.27t} \] Indeed divide (1.27s)'s gamma identity by the two \(L\)-factor ratios. The epsilon ratio is a Laurent monomial. Its order at \(X=0\) is exactly \(\deg\widetilde R\), since \(R(0)=1\). The last identity then gives \(\deg R=d\).

General cuspidal-support embedding without classification

For a two-block \(P=MU\), write \(r_U(V)=\delta_P^{-1/2}V_U\). The unnormalized quotient \(V_U\) is the quotient by all \(\pi(u)v-v\).

Lemma 1.23d (reciprocity and a noncuspidal embedding). For smooth modules there is the exact normalized adjunction \[ \operatorname{Hom}_G(V,I_P(\rho)) =\operatorname{Hom}_M(r_U(V),\rho). \tag{1.27u} \] If \(V\) is irreducible admissible and \(V_U\ne0\), there are irreducible admissible \(\pi_1,\pi_2\) on the two blocks and an embedding \(V\hookrightarrow I_P(\pi_1\otimes\pi_2)\).

Proof. Evaluate an intertwining section at 1. Its value map \(T:V\to V_\rho\) has \[ T(\pi(u)v)=T(v),\qquad T(\pi(m)v)=\delta_P(m)^{1/2}\rho(m)T(v), \] so factors through the indicated normalized quotient. Conversely a map on that quotient gives \((Av)(g)=T(\pi(g)v)\). The displayed identities prove the section law (1.27e), and an open subgroup fixing \(v\) fixes this section on the right. It is equivariant under right translation. Evaluation and this construction are inverse, proving (1.27u).

The quotient \(r_U(V)\) is finitely generated under \(M\). Choose a nonzero cyclic vector \(v\) of \(V\). Its finite-dimensional compact \(K\)-orbit span has a finite basis \(v_\alpha\), because an open stabilizer has finite index in \(K\). By \(G=PK\), their Jacquet classes generate \(V_U\) under \(M\); \(U\) has become trivial. Twisting by \(\delta_P^{-1/2}\) preserves finite generation. The actual compact-fixed Jacquet lemma, Lemma 1.17a, proves that this quotient is admissible.

It has a simple nonzero quotient \(S\). In Zorn's argument the union of a chain of proper submodules cannot contain all its finite set of generators unless one member already does; hence a maximal proper submodule exists. The quotient \(S\) is smooth and admissible by compact averaging. The actual arbitrary-algebra tensor theorem factors it as \(S=\pi_1\otimes\pi_2\) with irreducible admissible block representations. For completeness its hypotheses apply here: the two group Hecke algebras have local units \(e_J\); finite product test functions identify the product group Hecke algebra with their algebraic tensor product, and \((e_{J_1}\otimes e_{J_2})S=S^{J_1\times J_2}\) is finite. A Hecke-invariant subspace is group-invariant, since on a \(J\)-fixed vector translation by \(g\) is the action of \(\operatorname{vol}(J)^{-1}1_{gJ}\). Thus simplicity and admissibility are exactly the algebra theorem's hypotheses. Its proved Theorem 4.1 supplies the factors.

Here the abstract factor modules really give smooth group representations. For a factor vector \(x=e_Jx\) define \[ \pi(g)x=h_{g,J}x,\qquad h_{g,J}=\operatorname{vol}(J)^{-1}1_{gJ}. \] If \(L\subset J\), convolution gives \(h_{g,L}e_J=h_{g,J}\); a common subgroup of two choices therefore proves independence of \(J\). Also \(e_{gJg^{-1}}h_{g,J}=h_{g,J}\), so the image is fixed at that conjugate level. With \(L\subset g_2Jg_2^{-1}\) the convolution identity \(h_{g_1,L}h_{g_2,J}=h_{g_1g_2,J}\) proves the group law. This constructs a smooth action. Partitioning a compact test into finitely many right \(J\)-cosets shows that its original algebra action is the integral of this group action. Hence group-fixed spaces are exactly the local-unit corners and invariant subspaces are the same; the factors are indeed irreducible admissible group representations.

Applying (1.27u) to the quotient map gives a nonzero map \(V\to I_P(S)\). Its kernel is zero by irreducibility. \(\square\)

Normalized induction is transitive with its exact square-root moduli. If \(Q\) is a standard parabolic of \(M\), and \(R=Q U\subset P\), then \[ I_P^G(I_Q^M(\tau))\simeq I_R^G(\tau). \tag{1.27v} \] Here \(\delta_R(q)=\delta_P(q)\delta_Q(q)\), since conjugation on the two successive unipotent radicals multiplies their coordinate determinants. An explicit isomorphism sends \[ f_R\longmapsto \bigl[g\mapsto(m\mapsto\delta_P(m)^{-1/2}f_R(mg))\bigr], \qquad F\longmapsto[g\mapsto F(g)(1)]. \tag{1.27w} \] Substituting \(q m\) verifies the inner \(Q\)-section law; substituting \(u m_0 g\) verifies the outer \(P\)-section law and its factor \(\delta_P(m_0)^{1/2}\). The maps are inverse and commute with right translation. If \(J\) fixes \(f_R\) on the right, it also fixes the outer section on the right. For each fixed \(g\), its inner section is fixed by the compact open subgroup \(M\cap gJg^{-1}\), after shrinking \(J\) to a compact open subgroup. The modulus is 1 on that subgroup. This proves both smoothness requirements. For the inverse a right stabilizer of \(F\) fixes its evaluated section. Inducing simultaneously in both blocks also commutes with their tensor product: sections on the two compact flag quotients have finite rectangular locally constant partitions and finite tensor values, so are finite sums of products. This proves the required vector-space and action identification, not a statement about arbitrary smooth functions on noncompact products.

Theorem 1.23e (cuspidal support). Every irreducible smooth admissible representation of \(\mathrm{GL}_n(k)\) embeds in normalized induction from an outer tensor product of irreducible admissible block representations having zero Jacquet modules for every proper two-block parabolic.

Proof. Induct on the rank. In rank 1 there is no proper Jacquet condition, and an irreducible smooth admissible representation is a character. Indeed a nonzero compact fixed space is invariant under the abelian group \(k^\times\), so irreducibility makes it the whole finite-dimensional space; commuting complex matrices have a common eigenline, which must be the whole space. In higher rank, if all such modules are zero, take the representation itself as the sole block. Otherwise apply Lemma 1.23d. Both block ranks are smaller, so the induction hypothesis embeds each factor into an induction of the indicated cuspidal factors. Induction preserves injections by Lemma 1.23a, tensoring injections over a field preserves injections, and (1.27v)–(1.27w) combine the successive inductions. This proves the embedding in finitely many steps. \(\square\)

The condition on the terminal blocks also means zero Jacquet module for every proper parabolic. For a proper standard multiblock parabolic choose a cut between two of its blocks. Its unipotent radical contains the radical of the resulting two-block parabolic, so its coinvariant quotient is a further quotient of that already zero two-block coinvariant module. Every parabolic of \(\mathrm{GL}_n(k)\) stabilizes a flag; choosing a basis adapted to that flag conjugates it to a standard parabolic. Coinvariant vanishing is unchanged by conjugation. Thus no extra nonstandard-parabolic condition has been left implicit.

Combining this embedding with the two-block descent and gamma theorem proves, for every such irreducible \(\sigma\), with any resulting cuspidal block factors \(\rho_i\), \[ \gamma_\sigma=\prod_i\gamma_{\rho_i},\qquad L_\sigma=R(X)\prod_iL_{\rho_i},\quad L_{\widetilde\sigma}=\widetilde R(X)\prod_iL_{\widetilde\rho_i}, \quad R(0)=\widetilde R(0)=1, \tag{1.27x} \] and \[ a_\sigma=\sum_i a_{\rho_i}+\deg R . \tag{1.27y} \] For a general multiblock induction one applies the same two-block formulas iteratively; when an intermediate induction is reducible, its descent formula is applied to its finite products of cuspidal coefficient families, without requiring it to be irreducible. The kernel proof uses only its explicitly known family formula, ideal and dual pairing, so the iteration remains valid.

Parabolic reduction and the direct positivity proof

The parabolic formulas hold for the entire smooth admissible matrix family and every irreducible constituent. Theorem 1.23e provides cuspidal-support embedding in arbitrary rank and characteristic. Lemma 1.20a and Proposition 1.4 give factor 1 for each higher-rank terminal block, while the rank-one scalar Tate proof gives its exact character factor and nonnegative exponent. Theorem 1.24 below proves strict positivity directly for every higher-rank terminal block. Formula (1.27y) therefore proves the universal nonnegative matrix exponent in Corollary 1.24c. This route uses no universal compact-induction construction. Standard-factor, generic minimal-\(K_1\)-level and Artin conductor identifications remain separate.

Further reading is Goldfeld–Jacquet’s author notes, §2, Lemma 2.2, PDF page 9. The full matrix-kernel and embedding arguments are proved above, with the positive-trace convention fixed in (1.27p)–(1.27r).

Compact intertwining and finite inducing data

Let \(G=\mathrm{GL}_n(k)\), \(n\geq2\), \(Z=k^\times I_n\), and let \(\pi\) be irreducible smooth admissible with compact-mod-center coefficients. A real determinant twist makes its central character unitary; the preceding compact-coefficient proof, Corollary 1.4a, then constructs a compatible unitary realization. A determinant twist shifts \(s\) and leaves the epsilon exponent unchanged by direct substitution in the matrix integral. We may therefore work in that unitary realization.

Lemma 1.23f (orthogonal finite inducing subspace). Suppose there is an open \(H\supset Z\), compact modulo \(Z\), and a nonzero finite-dimensional \(H\)-invariant subspace \(W\subset V\) which is irreducible as an \(H\)-module, such that \[ \langle\pi(g)w,w'\rangle=0 \quad(g\notin H,\ w,w'\in W). \tag{1.28a} \] Then \[ V\simeq\mathrm{c\!-\!Ind}_H^G(W). \tag{1.28b} \] Consequently the earlier compact-induction theorem proves \(a_\pi>0\).

Proof. For a function in the compact induction supported on the left coset \(Hg\), let its value at \(g\) be \(w\). Send this function to \(\pi(g^{-1})w\). The map does not depend on the representative: replacing \(g\) by \(hg\) replaces the value by \(\pi(h)w\), and \(\pi((hg)^{-1})\pi(h)w=\pi(g^{-1})w\). A compactly induced vector has finitely many such cosets, so addition defines a linear map. Under right translation by \(x\) the support coset is \(Hg x^{-1}\), with the same value \(w\); its image is \(\pi(x)\pi(g^{-1})w\). Thus it is equivariant.

Different coset subspaces have orthogonal images. Indeed for \(Hg_i\ne Hg_j\), \[ \langle\pi(g_i^{-1})w_i,\pi(g_j^{-1})w_j\rangle =\langle\pi(g_jg_i^{-1})w_i,w_j\rangle=0 \] by (1.28a), since \(g_jg_i^{-1}\notin H\). Each individual image preserves the norm of its value. The norm square of a finite sum is therefore the sum of the norm squares of its values. The map is injective. Its image is a nonzero invariant algebraic subspace of the irreducible smooth module \(V\), hence equals \(V\). This proves (1.28b). The untwisted realization follows by twisting the finite-dimensional inducing representation; the subgroup is unchanged. \(\square\)

From compact intertwining to that realization

The extension step in a simple-type route can itself be proved here, with a weaker hypothesis than an already constructed inducing representation.

Lemma 1.23g (compact intertwining criterion). In the unitary realization above suppose that a compact open \(J\subset G\) has an irreducible smooth representation \(\tau\) occurring in \(V|_J\). Define \[ I_G(\tau)= \left\{g\in G: \operatorname{Hom}_{J\cap g^{-1}Jg} \bigl(\tau(h),\tau(ghg^{-1})\bigr)\ne0\right\}. \tag{1.28c} \] Suppose \(J\subset H\), \(Z\subset H\), where \(H\) is an open subgroup compact modulo \(Z\), and \(I_G(\tau)\subset H\). Then \(V\) is compactly induced from a finite-dimensional irreducible representation of \(H\).

Proof. Let \(E\) be the full \(\tau\)-isotypic subspace of \(V|_J\). It is finite-dimensional. Indeed, a smooth irreducible representation of a compact group factors through a finite quotient: the orbit of a vector is finite and spans its whole irreducible module; intersecting the stabilizers of a finite basis gives an open normal kernel \(J_0\). Thus \(E\subset V^{J_0}\), which is finite-dimensional by admissibility. Compact-group representations on that space split into irreducibles by invariant orthogonal complements. This also shows that \(E\) is a sum of finitely many copies of \(\tau\). In the Hilbert completion \(E\) is closed; write \(P_E\) for its orthogonal projection. The projection commutes with \(J\), since the subspace and its orthogonal complement are \(J\)-invariant.

For \(g\in G\), put \(A_g=P_E\pi(g)|_E\). If \(h\in J\cap g^{-1}Jg\), then \[ A_g\pi(h)|_E =\pi(ghg^{-1})|_E A_g. \tag{1.28d} \] Decompose both domain and range into their copies of \(\tau\). If \(A_g\ne0\), a nonzero coordinate of this operator is an intertwiner in (1.28c). Hence \(A_g=0\) for \(g\notin H\), and all coefficients between vectors of \(E\) vanish there.

The span \(W_0=\operatorname{span}\pi(H)E\) is finite-dimensional and \(H\)-invariant. To see finiteness, \(ZJ\) is an open subgroup of \(H\). The quotient \(H/(ZJ)\) is discrete and is the image of the compact space \(H/Z\), so has finitely many cosets \(h_i ZJ\). The center acts by its scalar character and \(J\) preserves \(E\), giving \(W_0=\sum_i\pi(h_i)E\).

For \(g\notin H\), \(h_i^{-1}gh_j\notin H\). Unitarity and the preceding vanishing therefore give \[ \langle\pi(g)\pi(h_j)e_j,\pi(h_i)e_i\rangle=0 \quad(e_i,e_j\in E). \tag{1.28e} \] By linearity every coefficient between vectors of \(W_0\) vanishes outside \(H\). Choose an irreducible \(H\)-subspace \(W\subset W_0\): a nonzero invariant subspace of minimal positive dimension exists, and is irreducible. It satisfies all hypotheses of Lemma 1.23f. That lemma proves the claimed compact induction, including injectivity and surjectivity, by the explicit equivariant map and orthogonality proof. \(\square\)

Thus a sufficient remaining existence theorem is precise: every abstract Jacquet-cuspidal \(\pi\) must contain some compact-open type \((J,\tau)\) whose full intertwining set is contained in an open subgroup compact modulo the center. Lemma 1.23g supplies the ensuing finite inducing data and its actual compact-induction realization. The existence of that type is still unproved here.

Compact coefficient support does imply that the set of \(g\) with \(P_E\pi(g)|_E\ne0\) is compact modulo \(Z\): use a finite basis of \(E\) and its finitely many coefficients. It does not imply that the full abstract intertwining set \(I_G(\tau)\) is compact, because the reverse implication in (1.28d) has not been proved and is not assumed. Even compactness of the actual nonzero-compression set does not place it in a compact subgroup, as the following explicit example explains.

What compact support actually gives

For any chosen finite-dimensional \(W\subset V\), compact support of its finitely many basis coefficients gives a compact set \(C_W\subset G/Z\) outside which all \(\langle\pi(g)w,w'\rangle\) vanish. This is a finite union of support bounds. It does not show that \(C_W\) is contained in a compact subgroup of \(G/Z\), or that one can choose \(W,H\) satisfying (1.28a).

The distinction is mathematical. Even a compact subset of \(\mathrm{PGL}_2(k)\) can generate a noncompact subgroup. For example, let \(K=\mathrm{GL}_2(\mathcal O)\) and \(a=\operatorname{diag}(\varpi,1)\). The images of \(K\) and \(aKa^{-1}\) in \(G/Z\) are compact. Their generated group is noncompact modulo \(Z\). To verify this, it contains the two elementary matrices \[ u(1)=\begin{pmatrix}1&1\\0&1\end{pmatrix},\qquad l(\varpi^{-1})=\begin{pmatrix}1&0\\\varpi^{-1}&1\end{pmatrix}. \] Put \(t=1+\varpi^{-1}\), of valuation \(-1\). The identity \[ u(1)l(\varpi^{-1})= l(\varpi^{-1}/t)\operatorname{diag}(t,t^{-1})u(1/t) \] has its other two factors in \(K\). Thus the generated group contains every power of \(\operatorname{diag}(t,t^{-1})\). The scalar-invariant entry-norm product on those powers is \(q^{2b}\), \(b\geq0\), which is unbounded. Replacing a compact support bound by its generated subgroup is therefore not a valid completion of the realization proof.

The criteria just proved do not establish universal existence of compact inducing data. Theorem 1.24 proves the matrix exponent positive directly for every Jacquet-cuspidal irreducible of rank greater than one, making that existence theorem unnecessary for nonnegativity. A freely accessible account of the type-construction route is Sécherre–Stevens, author arXiv preprint math/0607298, Theorems 4.1 and 5.21 and Corollary 5.22(i). Its simple-character and intertwining constructions are not proved here. The generic minimal-\(K_1\)-level and Artin conductor identifications also remain separate.

The normalized measure modulo the scalar centre

For a nonarchimedean local field \(k\) of either characteristic, put \(G=GL_n(k)\), \(Z=k^\times I_n\), \(K=GL_n(\mathcal O)\), and \(Q=Z\backslash G\). Matrix Haar measure has \(\operatorname{vol}_G(K)=1\); central multiplicative Haar measure has \(\operatorname{vol}_Z(\mathcal O^\times)=1\). We first construct the compatible measure on the quotient.

Lemma 1.23h. There is a positive bi-invariant Haar measure on \(Q\), uniquely normalized by \(\operatorname{vol}_Q(\bar K)=1\). It satisfies the exact fiber formula

Proof. The quotient map is open: the inverse image of the image of an open set \(A\subset G\) is \(ZA\), a union of open translates. The scalar centre is closed, since its matrix entries satisfy the closed conditions that every off-diagonal entry is zero and all diagonal entries agree. If \(xZ\ne yZ\), continuity of multiplication gives an identity neighbourhood \(W\) with \(W^{-1}x^{-1}yW\cap Z=\varnothing\). The images of \(xW\) and \(yW\) in the quotient are disjoint open neighbourhoods: an intersection would give \(w_1^{-1}x^{-1}yw_2\in Z\). Thus \(Q\) is Hausdorff. On the open locus with a specified matrix entry nonzero, dividing by that entry supplies a continuous local section with that entry equal to \(1\). These charts identify open sets with open subsets of affine matrix hyperplanes, so \(Q\) is locally compact and has a compact-open basis. It is a group because \(Z\) is central. The images \(\bar J_r\) of \(J_r=1+\varpi^rM_n(\mathcal O)\) form a basis of compact open identity subgroups and are contained in \(\bar K\).

Define the mass of every left coset \(x\bar J_r\) to be

The finite index is well defined because \(\bar J_r\) is open in the compact group \(\bar K\). Refinement to a smaller subgroup partitions each coset into exactly its subgroup index many equal pieces; multiplication of finite indices proves consistency of these masses. Every compactly supported locally constant function on \(Q\) is a finite linear combination of indicators of cosets of one sufficiently small \(\bar J_r\): take a common right-invariance subgroup by compactness, then partition its compact support into finitely many such open cosets. Thus (1.39b) defines a positive left-invariant integral with \(\bar K\)-mass one. It is the unique such integral, since left invariance and the same coset partitions determine every compact-open test.

For \(g\in G\), the intersection of \(\bar K\) and \(\bar g\bar K\bar g^{-1}\) is exactly the image of \(K\cap gKg^{-1}\). Indeed if \(k=zghg^{-1}\), \(k,h\in K\), then determinant valuation gives \(n\,v(z)=0\), so \(v(z)=0\) and \(z\in\mathcal O^\times I_n\). Such a scalar belongs to both compact groups and can be absorbed into \(h\). Also \(Z\cap K=\mathcal O^\times I_n\) is contained in that intersection. Hence the two quotient indices equal the corresponding ordinary indices:

Bi-invariance of \(G\)'s matrix measure makes the two ordinary compact groups have the same volume \(1\), so these indices agree. Therefore conjugation preserves the normalized quotient measure: the ratio of the masses of \(\bar K\) and its conjugate is the ratio of these two indices. Conjugation followed by left translation gives right translation, proving bi-invariance. The valuation equation here is an equation of integers; it is valid also when the field characteristic divides \(n\).

For \(f\in C_c^\infty(G)\), its central-fiber integral \(qf(\bar g)=\int_Z f(zg)\,dz\) is well defined, compactly supported, and locally constant. Right invariance by a common compact open subgroup of \(f\) gives this last assertion on its quotient image. The functional \(F(f)=\int_Qqf\,d\bar g\) is positive and left \(G\)-invariant. Left invariance fixes its value on each translate of a compact-open subgroup. Partitioning \(K\) into cosets of a sufficiently small common subgroup and then partitioning arbitrary test supports proves that any such functional is a scalar multiple of matrix Haar integration. Here \(q1_K=1_{\bar K}\): the relevant central fiber is a translate of \(\mathcal O^\times I_n\), whose Haar mass is exactly \(1\). Thus \(F(1_K)=1=\operatorname{vol}_G(K)\). The scalar is \(1\), proving (1.39a). \(\square\)

Use this quotient measure in the coefficient and convolution integrals below. In Lemma 1.23r, \(c_\rho>0\) is the constant defined by the Schur-orthogonality integral and \(d_\rho=c_\rho^{-1}\). The identities \(f_{vw}*f_{v'w'}=H(v',w)f_{vw'}\), \(\rho(f_{vw})u=H(u,w)v\), and \(e=f_{ww}\) for \(H(w,w)=1\) consequently contain every normalization constant. Rescaling the quotient Haar measure rescales \(c_\rho\) and \(d_\rho\) inversely, leaving these identities unchanged.

Write \(q\) for the residue cardinality, \(\varpi\) for a uniformizer, and \(\nu(g)=|\det g|\). Let \(P\) be the upper parabolic of block sizes \(n_1,\ldots,n_t\), and let \(I_P\) denote its normalized induction with covariance \(f(umg)=\delta_P(m)^{1/2}\rho(m)f(g)\) and right translation. A representation with unitary scalar centre is coefficient-tempered when every smooth coefficient lies in \(L^{2+\epsilon}\) modulo that centre for every \(\epsilon>0\). Its real norm twist is essentially coefficient-tempered. The exact modulus is \[ \delta_P(\operatorname{diag}(m_1,\ldots,m_t)) =\prod_{i<j}|\det m_i|^{n_j}|\det m_j|^{-n_i}. \tag{1.39d} \]

Compact support, duality and actual cuspidal quotients

Lemma 1.23i. If a smooth admissible representation \(V\) has every smooth coefficient compactly supported modulo \(Z_n\), then \(V_{U_i}=0\) for every standard proper two-block upper radical \(U_i\). Consequently the condition of zero proper Jacquet modules is preserved by ordinary smooth contragredience of irreducible admissible representations.

Proof. Fix \(i<n\), \(r\geq1\), \(J=J_r\), \(E=V^J\), and

Choose bases of \(E\) and its dual. Extending a dual basis functional by composition with \(e_J\) gives an actual \(J\)-fixed smooth dual vector. All finitely many resulting coefficients vanish at \(a^b\) for every sufficiently large \(b\): its two elementary-divisor levels have gap \(b\), bounded on any compact subset of \(Z_n\backslash G_n\). A single \(N\) works for all these coefficients. Thus

the second equality being the proved probability semigroup identity.

Let \(p:V\to V_{U_i}\) be the ordinary quotient. Block elimination in each finite congruence quotient gives the probability-average identity

Conjugating by \(a^{-b}\) expands upper entries to \(\varpi^{r-b}\mathcal O\), fixes the block diagonal group, and contracts lower entries to \(\varpi^{r+b}\mathcal O\). For \(v\in E\), the diagonal and contracted lower averages fix \(v\); the expanded upper average has the same Jacquet class as \(v\). Therefore

where \(a\) on the right is its invertible ordinary Levi action on the quotient. Equations (1.39f)–(1.39h) imply \(p(v)=0\). Every smooth vector is fixed by some \(J_r\), so \(p(V)=0\).

Now let \(V\) be irreducible admissible and Jacquet-zero. Lemma 1.20a gives compact-mod-center coefficient support. Smooth biduality identifies every coefficient of \(V^\vee\) with a coefficient of \(V\) composed with inversion. Inversion preserves compactness modulo the center. The assertion just proved gives zero proper two-block Jacquet modules for \(V^\vee\); further coinvariants and flag conjugation give every proper parabolic. In rank one the condition is vacuous. This proof uses neither Jacquet duality, second adjointness nor transpose-inverse identification. \(\square\)

Exact ordinary smooth duality

Lemma 1.23j. Ordinary smooth duality is exact on admissible smooth representations; it preserves admissibility and irreducibility, and evaluation is smooth biduality.

Proof. Restriction identifies \((V^\vee)^J\) with \((V^J)^*\), with inverse \(a\mapsto a\circ e_J\); these spaces are finite-dimensional. Given \(W\hookrightarrow V\), extend a \(J\)-fixed functional from \(W^J\) to \(V^J\), then compose with \(e_J\). Its restriction to \(W\) is the original functional. Restriction of smooth duals is therefore onto. The remaining exactness assertions follow from composition with a quotient and annihilators. Compact averaging lifts fixed vectors through a surjection, so submodules and quotients of admissible modules are admissible.

Evaluation \(V\to(V^\vee)^\vee\) is an isomorphism on every fixed space by finite-dimensional biduality, hence on their union. If \(V\) is irreducible and \(0\ne W\subset V^\vee\) is invariant, its annihilator in \(V\) is proper and invariant, hence zero. Detection of a \(J\)-fixed vector by \(W\) remains detection by its \(J\)-average. Thus \(W^J\) has zero annihilator in \(V^J\), so finite dimension gives \(W^J=(V^\vee)^J\). Taking the union proves \(W=V^\vee\). \(\square\)

Theorem 1.23k (actual unordered cuspidal quotient). Every irreducible smooth admissible \(\pi\) of \(G_n\) is an actual quotient

where each \(\rho_j\) is an irreducible admissible representation of \(G_{n_j}\) having zero every proper Jacquet module.

Proof. Apply Theorem 1.23e to the irreducible admissible \(\pi^\vee\), obtaining an embedding into \(I_P(\sigma_1\boxtimes\cdots\boxtimes\sigma_t)\) with Jacquet-zero factors. Lemma 1.23j and Lemma 1.23a give an onto map

The factors are irreducible admissible by Lemma 1.23j and Jacquet-zero by Lemma 1.23i. This is (1.39i). The same upper \(P\) and same block order are used in normalized induction duality. No reflected permutation or inverse intertwiner is inserted. Transitivity was proved in (1.27v)–(1.27w). Finite length is unnecessary. \(\square\)

Real normalization and compatible unitary cuspidal blocks

Lemma 1.23l. An irreducible admissible Jacquet-zero \(\rho\) of \(G_d\) has a unique real number

such that \(\tau=\rho\nu^{-e(\rho)}\) has unitary central character. The representation \(\tau\) has a compatible irreducible unitary Hilbert realization; every smooth coefficient is compactly supported modulo the center and lies in every \(L^p(Z_d\backslash G_d)\), \(0<p<\infty\). In particular it is square-integrable and coefficient-tempered.

Proof. A commuting endomorphism is scalar: restrict to a nonzero finite-dimensional fixed space, choose an eigenvalue, and use the invariant nonzero kernel of its difference from that scalar. Central elements therefore act by a smooth character \(\omega_\rho\). On \(\mathcal O^\times\) this character has finite image because an open central subgroup fixes a nonzero vector and is in its kernel. That image has modulus one. Since \(\nu(\varpi I_d)=q^{-d}\), the modulus of the central character of \(\rho\nu^{-e}\) at \(\varpi\) is \(|\omega_\rho(\varpi)|q^{de}\). It is one exactly for (1.39j). Units and the powers of \(\varpi\) exhaust the center. This proves existence and uniqueness, including the negative sign and division by \(d\). The real determinant twist preserves Jacquet vanishing and coefficient supports.

For \(d>1\), Lemma 1.20a gives compact coefficient support modulo the center. For \(d=1\), the quotient by the center is one point. With unitary central character, the absolute value of a coefficient is well defined on that quotient and bounded on its compact support. It lies in every stated \(L^p\).

Fix \(0\ne\ell\in\tau^\vee\) and define

The integrand is representative-independent and compactly supported. The quotient measure is right invariant: the matrix Haar density \(|\det g|^{-d}d^{d^2}g\) is bi-invariant, and right translation commutes with integration first over the central fibers. Thus this form is invariant. Its kernel is the invariant kernel of the coefficient map for \(\ell\); it is proper since \(\ell\ne0\), and hence zero by irreducibility.

The Hilbert completion \(H\) carries a strongly continuous unitary action: the dense smooth subspace has open stabilizers, and the unitary norm extends continuity to \(H\). Compact probability averaging is an orthogonal projection, and

by admissibility. Every smooth vector of \(H\) therefore belongs to \(\tau\). The orthogonal projection onto a closed invariant subspace would commute with the action and preserve these fixed spaces. On the irreducible smooth \(\tau\) it is scalar and idempotent, hence zero or identity; density gives the same on \(H\). The completion is irreducible. A \(J\)-fixed smooth dual vector is represented by a vector in the finite-dimensional Hermitian space \(\tau^J\), and composing with the orthogonal projection \(e_J\) represents it on the entire \(\tau\). This proves compatibility. \(\square\)

Theorem 1.23k and Lemma 1.23l give, for every irreducible admissible \(\pi\), an actual quotient

with actual unitary square-integrable Jacquet-zero \(\tau_j\). The exponents have no proved ordering. Each supplied block's exponent is fixed by (1.39j); the entire list for \(\pi\) has no proved uniqueness here.

A compact section generates the full induction

Lemma 1.23m. If \(\rho=\rho_1\boxtimes\cdots\boxtimes\rho_t\) is the outer tensor product of irreducible admissible blocks, \(I_P(\rho)\) is finitely generated. One compact big-cell section generates it.

Proof. The tensor theorem used in Lemma 1.23d makes \(\rho\) irreducible and smooth under \(M\), so a nonzero \(v\) generates it. The block centers act by scalars.

Let \(N^-\) be the opposite lower block radical. Block elimination at the invertible successive trailing principal block pivots gives a unique factorization and continuous inverse for \(P\times N^-\to PN^-\). Its open image in \(P\backslash G_n\) is a chart identified with \(N^-\). Compactly supported locally constant functions \(N^-\to V_\rho\) give induced sections by the covariance law and extension by zero. Their common right stabilizer follows from a finite cover of the compact chart support on which that law and the open stabilizers of the finitely many vector values are fixed. The support is compact and open in the flag space, hence closed there too, so the zero extension is smooth.

Choose a compact open subgroup \(H\subset N^-\), with all lower block entries in a sufficiently small common power of \(\varpi\). Let \(F_v\) restrict to \(1_Hv\). Put

Conjugation by \(a\) contracts its \((i,j)\) lower block, \(i>j\), by \(\varpi^{i-j}\). Thus \(a^hHa^{-h}\), \(h\geq0\), is cofinal among identity neighborhoods of \(N^-\). With \(\rho(a)=\lambda\,\mathrm{id}\), \(\lambda\ne0\), (1.39d) gives

The section law at \(u a^h=a^h(a^{-h}ua^h)\) gives

Right translation by \(m\in M\) multiplies the value by \(\delta_P(m)^{1/2}\rho(m)\) and conjugates the support by \(m\). As \(a\) commutes with \(m\), the span of \(F_v\) contains, after multiplication by a nonzero scalar, the function with value \(\rho(m)v\) and support \(m a^hHa^{-h}m^{-1}\), for every \(m,h\).

For any compact open subgroup \(L\subset N^-\), choose \(h\) with this support subgroup \(H'\subset L\). A finite right-coset partition of \(L\) by \(H'\), followed by right translation inside \(N^-\) and summation, gives \(1_L\rho(m)v\). Finite sums of \(\rho(m)v\) span \(V_\rho\), so the span contains \(1_Lw\) for every \(w\). Further right translations give \(1_{Lx}w\). Every compactly supported locally constant vector-valued function is a finite sum of these: it has finitely many values and one small common left-invariance subgroup on its compact support. We have generated every section supported compactly in this chart.

Translated charts cover the flag space, which is compact by the proved Iwasawa decomposition. Images of sufficiently small compact-open matrix neighborhoods give a compact-open basis of that space. Refine a finite chart cover to a finite disjoint compact-open partition, each piece lying compactly inside a chart. Any section is the sum of its restrictions to these pieces, extended by zero. Each is a translate of a section already generated, so \(F_v\) generates the entire induction. For \(t=1\), induction is \(\rho\), and its cyclicity proves the claim. \(\square\)

The compact picture alone would not prove finite generation: smooth sections have arbitrarily deep right levels. The contraction and coset partition explicitly generate those levels.

Finite heads and precise compact-corner tests

Lemma 1.23n. A nonzero finitely generated smooth admissible representation \(V\) has a finite semisimple maximal quotient. Choose a compact open \(J\) fixing a finite set of generators and \(E=V^J\). Then

where \(Q_i\) are distinct irreducible admissible quotients. Every map to a finite semisimple representation factors through this quotient. At least one \(Q_i\) exists. No finite-length assertion about \(V\) is made.

Proof. Set \(\mathcal H=C_c^\infty(G)\), \(e=e_J\) and \(A=e\mathcal He\). A group translate of a \(J\)-fixed vector is its Hecke translate by the normalized characteristic function of \(gJ\). Hence \(V=\mathcal HE\), and every quotient is generated by the image of \(E\). Averaging lifts fixed vectors, so any nonzero quotient has nonzero \(e\)-corner. Zorn's argument using finite generators produces a maximal proper submodule. Its irreducible quotient is admissible.

For an irreducible quotient \(Q\), \(eQ\) is a nonzero simple \(A\)-module: a nonzero \(A\)-submodule \(W\) generates \(Q\) under \(\mathcal H\), and applying \(e\) gives \(eQ=e\mathcal HeW=W\). This simple corner is a quotient of the finite-dimensional \(A\)-module \(E\). Decreasing dimension constructs a finite composition series of \(E\). Under a map from \(E\) onto any simple corner, the first term of this series with nonzero image gives an isomorphism from its simple successive quotient to that corner. Thus only finitely many simple quotient types occur.

A simple \(\mathcal H\)-module with nonzero \(e\)-corner is determined by that corner. For a simple \(A\)-module \(M\), form \(\mathcal He\otimes_A M\). Its corner is \(M\), which generates the module. Every proper submodule has zero corner, since a nonzero corner would equal \(M\) and generate the entire module. The sum of all proper submodules still has zero corner and is proper. It is the unique maximal submodule, and every simple module with that corner is a quotient by it. Thus only finitely many simple quotient types \(Q_1,\ldots,Q_s\) of \(V\) occur.

Restriction injects

as \(E\) generates \(V\). The target is finite-dimensional. Choose a basis of each space on the left and take the product of all these finitely many maps. Its image \(D\) is a submodule of the finite direct sum \(\bigoplus_iQ_i^{m_i}\).

A finite direct sum of simple modules and every submodule thereof are semisimple. One proof takes a simple submodule \(S\) of a nonzero submodule, obtainable by the finite-length descending argument inside that finite sum, and projects \(S\) to an ambient simple summand where it is nonzero. This gives a retraction onto \(S\); splitting it and iterating proves the statement. Thus \(D\) is finite semisimple. The product's kernel is the intersection of the kernels of every map to a simple quotient, by the chosen bases and finite list of types. Every map to a finite semisimple representation factors through \(D\).

Each \(Q_i\) has scalar endomorphisms by the compact-fixed eigenvalue argument. Consequently its multiplicity in \(D\) is \(\dim\operatorname{Hom}_G(D,Q_i)=\dim\operatorname{Hom}_G(V,Q_i)\). Averaging \(V\twoheadrightarrow D\) makes \(eD\) a quotient of \(E\). Its dimension gives (1.39q). This identifies \(D=\operatorname{hd}(V)\). \(\square\)

Lemmas 1.23a and Lemma 1.23m let Lemma 1.23n apply to every induction in (1.39m): it has a nonzero finite semisimple head and \(\pi\) is one of its actual head constituents.

Corollary 1.23o (unique-head test). If \(V=\mathcal HeV\) is admissible and \(eV\) is simple over \(e\mathcal He\), then \(V\) has a unique maximal proper submodule and one irreducible quotient, with multiplicity one in the head.

Proof. A proper submodule \(W\) has \(eW=0\): otherwise its corner is all of \(eV\), forcing \(W=V\). The sum of all proper submodules has zero corner and is proper; it is the unique maximal submodule. Its quotient is simple. \(\square\)

Corollary 1.23p (finite-length test). If an admissible \(V\) has a compact open \(J\) such that every nonzero subquotient has nonzero \(J\)-fixed vectors, then \(V\) has finite length at most \(\dim V^J\).

Proof. Fixed vectors are exact by averaging. For every strict inclusion \(W\subsetneq W'\), the nonzero \((W'/W)^J\) gives \(\dim W'^J>\dim W^J\). An ascending or descending chain can therefore have at most \(\dim V^J\) strict steps. This ensures existence of a maximal proper submodule and termination of recursive maximal-submodule descent in a composition series of the claimed length. \(\square\)

Finite generation detects every nonzero quotient at the generating corner. The common-level test additionally requires that corner to detect every subquotient. Theorem 1.23v below establishes finite length by a finite family of Jacquet functors.

Rank-one data and the obstruction to arbitrary exchange

An irreducible smooth admissible \(G_1\)-representation is a character: a nonzero compact-fixed space is invariant under the abelian group, hence the entire finite-dimensional representation; commuting complex matrices have an eigenline, which is the whole irreducible space. Formula (1.39j) uniquely writes it as \(\chi_0|\cdot|^e\), with \(\chi_0\) unitary. Induction is then the representation itself. The classification, unique quotient and parameter uniqueness are completely proved in rank one.

The same one-block assertion holds for any Jacquet-zero irreducible \(\pi\). In fact it cannot be a quotient of a proper induction \(I_P(\rho)\) from an admissible module: such a quotient would dualize by Lemma 1.23j and Lemma 1.23a to a nonzero embedding \(\pi^\vee\hookrightarrow I_P(\rho^\vee)\). Normalized reciprocity would then give a nonzero map from the Jacquet module of \(\pi^\vee\), which is zero by Lemma 1.23i. This is impossible. Hence any ordered quotient datum for \(\pi\) must have the sole block \(\pi\), whose unique real normalization is Lemma 1.23l. This proves the full one-block classification assertion without an unproved exclusion of cuspidal subquotients from every proper induction.

Arbitrary exchange of inducing blocks already fails to give isomorphic modules in rank two. Put

Here \(\delta_B^{1/2}(\operatorname{diag}(a,d))=|a/d|^{1/2}\). Thus \(I_-\) has trivial left covariance and contains the constant section \(\mathbf1\). The actually proved normalized reciprocity gives

The characters are different because \(q>1\), so \(I_-\not\simeq I_+\). Smooth duality gives an actual \(I_+\twoheadrightarrow\mathbf1\), whereas \(\operatorname{Hom}_{G_2}(I_-,\mathbf1)=0\) by dualizing (1.39t). This uses no rank-two constituent classification. Thus an unproved blanket invariance of induction under adjacent exchanges cannot order (1.39m).

Coefficient idempotents and finite-length induction

Use the normalized quotient measure of Lemma 1.23h and the compact-coefficient unitary construction of Lemma 1.23l. The two-parabolic orbit calculation in Lemma 2.3p, equations (2.9cc)–(2.9ch), supplies its section filtration, additive-fibre kernel and exact density without a finite-length premise.

Lemma 1.23q (relative central-character convolution). Fix a smooth unitary character \(\omega:Z_n\to\mathbb C^\times\). Let \(\mathcal H_\omega(G_n)\) be the locally constant functions \(f:G_n\to\mathbb C\) compactly supported modulo \(Z_n\) with \(f(zg)=\omega(z)^{-1}f(g)\). They have convolution

On any smooth module \(V\) whose center acts by \(\omega\), they act by

These are well-defined associative operations, with actual finite vector-valued integrals.

Proof. Replacing \(x\) by \(zx\) in (1.39u) multiplies the first factor by \(\omega(z)^{-1}\) and the second by \(\omega(z)\), so their product is independent of the representative. The same cancellation holds in (1.39v). Local sections of the quotient exist explicitly: on the open set where the \((i,j)\)-entry is nonzero, choose the representative whose entry is \(1\), by dividing every matrix entry by that entry. These sections identify open subsets of \(Z_n\backslash G_n\) with open subsets of an affine matrix hyperplane. Thus the quotient is an \(l\)-space, and the displayed integrands are locally constant there. Their compact supports have finite compact-open partitions. The vector integrand in (1.39v) consequently takes finitely many vector values and its integral is a finite linear combination of those values.

The quotient group is unimodular: left and right translation commute with integration over central fibers of the bi-invariant matrix Haar measure. Products of compact quotient supports are compact. Finite partitions of such supports prove Fubini and the substitution rules directly, so the usual convolution calculation proves associativity and \(\pi_V(f*h)=\pi_V(f)\pi_V(h)\). Smaller common compact-open stabilizers of the functions and vectors justify every local-constancy refinement used in these calculations. No integration over the noncompact center itself is present. \(\square\)

Lemma 1.23r (Schur orthogonality with its proof). Let \(\rho\) be an irreducible admissible Jacquet-zero representation with unitary central character \(\omega\), and let \(H\) be its invariant Hermitian form from Lemma 1.23l, linear in the first variable. For a positive real constant \(c_\rho\), all smooth \(v,w,u,z\) satisfy

Proof. Each coefficient is compactly supported modulo the center, and the integrand has central weight \(\omega^{-1}\omega=1\), so the integral is finite. For fixed \(w,u\), call the resulting sesquilinear form in \(v,z\) \(B_{w,u}(v,z)\). Left Haar invariance makes it invariant under simultaneous translation of \(v,z\).

This invariant form is represented by an actual smooth commuting endomorphism, not an assumed bounded Hilbert operator. If \(v\) is \(J\)-fixed, invariance gives \(B_{w,u}(v,z)=B_{w,u}(v,e_Jz)\). The finite-dimensional Hermitian space \(\rho^J\) has a unique vector \(Tv\) representing this functional. Its value on every smooth \(z\) then represents the original form, and different choices of \(J\) give the same vector by nondegeneracy. The resulting linear map \(T\) commutes with \(G_n\). The admissible smooth Schur argument in Lemma 1.23l shows that it is scalar, say \(B_{w,u}(v,z)=C(w,u)H(v,z)\).

Right translation in the integral makes \(C(w,u)\) invariant under simultaneous translation of \(w,u\). It is linear in \(u\) and conjugate-linear in \(w\). The same compact-fixed representation argument gives \(C(w,u)=c_\rho H(u,w)\). Taking \(v=w=u=z\ne0\) makes the integral the squared absolute value of a coefficient, which is positive on a neighborhood of the identity and finite. Hence \(c_\rho>0\). This proves (1.39w) entirely within the actual smooth representation. \(\square\)

Put \(d_\rho=c_\rho^{-1}\), and define the relative coefficient kernels

They belong to \(\mathcal H_\omega(G_n)\); the inverse central weight follows from conjugate-linearity of \(H\)'s second variable. Equations (1.39u) and (1.39w) give exactly

For the first equality, use \(H(v',\rho(x^{-1}g)w')=H(\rho(x)v',\rho(g)w')\) inside the integral; (1.39w) then gives the scalar \(H(v',w)\). For the second, pair the vector integral with every smooth \(z\) and use (1.39w). All constants and conjugations are fixed by these substitutions.

The span of the \(f_{vw}\) is a two-sided coefficient ideal. For \(h\in\mathcal H_\omega(G_n)\),

These identities follow from (1.39u) by left translation and the substitution \(x=gy^{-1}\), respectively; the latter conjugation is necessary because \(w\mapsto f_{vw}\) is conjugate-linear. The unitary central character ensures \(h^*\in\mathcal H_\omega(G_n)\).

Lemma 1.23s (actual projectivity and embeddings). In the category of smooth modules with central character \(\omega\), the functor \(\operatorname{Hom}_{G_n}(\rho,-)\) is exact. Every irreducible subquotient isomorphic to this Jacquet-zero \(\rho\) of a module \(V\) gives an actual embedding \(\rho\hookrightarrow V\).

Proof. Choose \(w\) with \(H(w,w)=1\), and set \(e=f_{ww}\). Equation (1.39y) makes \(e\) an idempotent, whose action on \(\rho\) is the rank-one projection \(u\mapsto H(u,w)w\). The functor \(V\mapsto eV\) is exact: for an onto map, lift a vector and apply \(e\); for kernels, intersect with its image.

For \(u\in eV\), define

Left translation gives \(L_a f_{vw}=f_{\rho(a)v,w}\), and (1.39v) gives \(\pi_V(a)\pi_V(f)=\pi_V(L_af)\). Thus \(T_u\) is a \(G_n\)-map. Moreover \(T_u(w)=eu=u\). Conversely, if \(T:\rho\to V\) is equivariant, \(Tw\in eV\) and \(T(v)=T(\rho(f_{vw})w)=\pi_V(f_{vw})Tw\). Consequently evaluation at \(w\) identifies

naturally, proving exactness. If \(\rho\) is a subquotient of \(V\), exactness of \(e(-)\) makes \(eV\ne0\). A nonzero \(u\in eV\) gives the nonzero map \(T_u\), whose kernel is zero by irreducibility. This is the required embedding. \(\square\)

Corollary 1.23t. A proper parabolic induction of irreducible admissible Jacquet-zero blocks has no irreducible admissible Jacquet-zero subquotient.

Proof. Its center is the single smooth scalar character \(\omega=\prod_j\omega_{\rho_j}\). Twist the entire induction by \(\nu^s\), where \(s=\log|\omega(\varpi)|/(n\log q)\), so its central character becomes unitary. This simultaneously twists every inducing block by \(\nu^s\), preserving Jacquet vanishing and all subquotient relations. Any Jacquet-zero irreducible subquotient \(S\) now has this unitary center. Lemma 1.23l and Lemma 1.23s give an embedding of \(S\) into the full induction.

Normalized reciprocity for any block parabolic follows from exactly the written value-at-\(1\) proof: a \(G_n\)-map to \(I_P(\rho)\) evaluates to a map \(\delta_P^{-1/2}S_{U_P}\to\rho\), and conversely such a map \(T\) gives the section \(g\mapsto T(\pi(g)v)\). Its common right stabilizer is that of \(v\). Since \(P\) is proper and \(S\) is Jacquet-zero, this Hom space is zero. This contradicts the embedding. Untwisting proves the claim for the original induction. \(\square\)

Lemma 1.23u (the actual finite Jacquet filtration). Let \(I=\rho_1\times\cdots\times\rho_t\) with each \(\rho_j\) irreducible admissible and Jacquet-zero, of rank \(n_j\). For \(1\le l<n\), the normalized \(P_{l,n-l}\)-Jacquet module has a finite filtration indexed by subsets \(A\subset\{1,\ldots,t\}\) with \(\sum_{j\in A}n_j=l\). The corresponding quotient is

with the original relative order retained in both products.

Proof. The proof of Lemma 2.3p applies to arbitrary smooth inducing modules: the orbit decomposition, locally constant section restriction, exact additive unipotent coinvariants, fiber integration kernel, and density calculation use no irreducibility of an inducing module. Thus it applies when the second inducing module is itself an induction. Refine the partition \((n_1,\ldots,n_t)\) successively. At each step its finite orbit filtration assigns ranks \(i_j\) to the first Jacquet block. A rank \(0<i_j<n_j\) makes the corresponding Jacquet module of \(\rho_j\) zero. Hence a nonzero quotient assigns each whole block to one side, giving precisely the subsets in the statement.

Normalized induction is exact, as proved by the compact representative-value lifts, so it refines these successive filtrations. Induction in stages is the actual isomorphism (1.27v)–(1.27w), with its tensor section construction. The two negative half-moduli of the inducing Jacquet modules and the positive half-moduli of each new induction agree exactly with the exponents in (2.9ch); when a whole block is assigned, its own Jacquet half-modulus is one. No twist remains in (1.39ac). This proves the finite filtration and its precise ordered quotient modules, without a finite-length assertion about \(I\). \(\square\)

Theorem 1.23v (finite length, all ranks and both characteristics). Every induction of finitely many irreducible admissible Jacquet-zero blocks has finite length. If \(t\ge2\), its length is bounded by

The equality is the finite filtration length formula; the lower-rank lengths on its right are defined inductively. Parabolic induction preserves finite length for admissible data. Every Jacquet module of a finite-length admissible representation has finite length.

Proof. Induct on total rank \(n\). In rank one the only such induction is its irreducible block and has length one; the same holds in any rank when \(t=1\). Assume \(t\ge2\) and the theorem on cuspidal induction in all smaller ranks. Lemma 1.23u gives finite filtrations of each \(r_{l,n-l}I\) with quotients (1.39ac). Both factors have strictly smaller rank and therefore finite length. Tensoring their composition filtrations is exact over \(\mathbb C\); each successive outer tensor product of irreducible admissible factors is irreducible admissible by the tensor theorem used in Lemma 1.23d. Thus (1.39ac) has length \(\ell(I_A)\ell(I_{A^c})\), and the finite filtration gives the equality in (1.39ad). In particular all these Jacquet modules have finite length.

Every nonzero subquotient \(V\) of \(I\) has an irreducible subquotient: choose a nonzero vector, take its cyclic submodule, and apply the finite-generator Zorn argument to that cyclic module. Its simple quotient is admissible by compact averaging. Corollary 1.23t says it is not Jacquet-zero. Some proper standard two-block Jacquet module of that quotient is therefore nonzero. Exactness of that Jacquet functor implies \(r_{l,n-l}V\ne0\) for at least one \(l\).

For a submodule \(W\subset I\), let

The terms are finite because they are submodules of the already finite-length \(r_{l,n-l}I\). If \(W\subsetneq W'\), the nonzero subquotient \(W'/W\) is detected by at least one of those Jacquet functors. Exactness and additivity of length give \(b(W')>b(W)\). Since \(0\le b(W)\le b(I)<\infty\), every strict chain of submodules has at most \(b(I)\) steps. This provides maximal proper submodules and termination of their descending recursion, hence a composition series of \(I\) and the inequality in (1.39ad). The rank induction is complete. This argument detects every subquotient by a finite collection of exact functors; it does not assert an unproved universal compact level.

For arbitrary irreducible admissible inducing blocks, Theorem 1.23e embeds each in an induction of Jacquet-zero irreducibles. Exactness and transitivity of induction and exactness of tensor products give an embedding of their whole induction into one of the cuspidal inductions just proved finite-length. A submodule of a finite-length module is finite-length, so induction from arbitrary irreducible admissible blocks is finite-length. For finite-length admissible inducing data, induce their finite composition filtrations and use exactness, reducing to the preceding assertion.

Finally, an irreducible admissible \(\pi\) embeds in a cuspidal induction \(I\). For a proper standard two-block parabolic, its Jacquet module embeds by exactness in the finite-length module furnished by Lemma 1.23u. Thus all such Jacquet modules of \(\pi\) are finite-length. Their simple factors are admissible representations of a product Levi and factor as actual outer tensor products by the tensor theorem. Iterating on the smaller ranks gives the same assertion for every standard multiblock Jacquet module. Coinvariants in stages follow directly by successively quotienting by the normal unipotent subgroups, with the product of their normalizing moduli. Flag conjugation gives every parabolic. An arbitrary finite-length admissible representation then follows by exactness along its finite composition series. \(\square\)

Multiblock Jacquet filtrations and intrinsic cuspidal support

Lemma 1.23w (multiblock whole-factor filtration). Let \(P\) have block ranks \(d_1,\ldots,d_s\), and let \(I=\sigma_1\times\cdots\times\sigma_t\), where each \(\sigma_j\) is Jacquet-cuspidal of rank \(m_j\). Its normalized \(P\)-Jacquet module has a finite filtration indexed by ordered partitions of the positions

The corresponding quotient is

where each product retains the original relative order of its positions.

Proof. Cut the first \(d_1\) coordinates from the remaining coordinates. Lemma 1.23u gives the finite subset filtration, with the assigned factors and remaining factors. Apply the two-block Jacquet functor inside the second Levi block, then continue refining it. These functors are exact by compact averaging; they commute with tensor factors on which they act trivially, because the quotient relations are precisely the tensor product of the original relation subspace with that other factor. Thus they refine the finite filtration at each stage.

Coinvariants in stages are the actual successive quotient: the full refined upper radical is the product of the original normal radical and the new radical in its Levi, so quotienting successively imposes exactly all its relations. The normalizing moduli multiply, since their root-coordinate determinants multiply. This identifies the resulting normalized functor with \(r_P\), rather than with a twisted version. A nonzero term assigns each whole \(\sigma_j\) to one final block; a partial assignment would have a zero proper Jacquet module. The successive choices are exactly (1.39af), and transitive induction identifies their quotients with (1.39ag). As in Lemma 1.23u, the normalized density calculation cancels all intermediate half-moduli. This proves the stated finite filtration. \(\square\)

Theorem 1.23x (intrinsic cuspidal support). For every irreducible smooth admissible \(\pi\) of \(G_n\), there is a unique unordered multiset of actual Jacquet-cuspidal irreducible admissible representations such that \(\pi\) is a subquotient of their normalized induction. Multiplicities, ranks, representations and all norm twists are retained.

Proof. Theorem 1.23e gives an actual embedding

with Jacquet-cuspidal \(\rho_\alpha\) of ranks \(d_\alpha\). Normalized reciprocity sends this nonzero map to a nonzero map

The target is an actual irreducible admissible Levi representation by the tensor theorem, so this map is onto.

Suppose also that \(\pi\) is any actual subquotient of \(I=\sigma_1\times\cdots\times\sigma_t\) with Jacquet-cuspidal \(\sigma_j\). Exactness makes \(r_P\pi\) a subquotient of \(r_PI\). Consequently the tensor in (1.39ai) is an irreducible subquotient of that latter module.

An irreducible subquotient of a module with a finite filtration occurs as an actual subquotient of one of its successive quotients. Indeed, if \(C/D\) is the simple subquotient, intersect \(C\) with the filtration and take the first term having nonzero image in \(C/D\). That image is the whole simple module and the preceding term maps to zero. The intersection term modulo its preceding term embeds in the ambient successive quotient and maps onto \(C/D\). This proves the assertion without a virtual Grothendieck-group substitution.

Apply this fact to Lemma 1.23w. Each \(I_{A_\alpha}\) has finite length by Theorem 1.23v, so tensoring their actual composition series gives a finite filtration of (1.39ag) by simple outer tensor products. The same subquotient argument shows that the tensor of the \(\rho_\alpha\) is isomorphic to \(S_1\boxtimes\cdots\boxtimes S_s\), where each \(S_\alpha\) is an irreducible constituent of \(I_{A_\alpha}\). Uniqueness of actual tensor factors gives \(S_\alpha\simeq\rho_\alpha\).

If \(|A_\alpha|\ge2\), that \(I_{A_\alpha}\) is a proper parabolic induction of Jacquet-cuspidal blocks, so Corollary 1.23t excludes its Jacquet-cuspidal constituent \(\rho_\alpha\). Every \(A_\alpha\) is nonempty because its assigned rank \(d_\alpha\) is positive. Hence each \(A_\alpha\) has exactly one position \(j\), and \(\rho_\alpha\simeq\sigma_j\) with the same rank. The disjoint partition covers all positions, proving \(s=t\) and equality of the unordered multisets including multiplicities and actual twists. This proves uniqueness and existence. \(\square\)

Corollary 1.23y (the full Jacquet supercuspidality criterion). For an irreducible smooth admissible representation, zero all proper Jacquet modules is equivalent to not occurring as a subquotient of any proper parabolic induction from irreducible admissible Levi data.

Proof. If a proper two-block Jacquet module is nonzero, the actual Lemma 1.23d constructs an embedding into a proper induction, hence a subquotient occurrence. Conversely, suppose a Jacquet-zero irreducible \(S\) occurred in a proper induction from arbitrary irreducible admissible blocks. Embed each inducing block in its actual cuspidal induction from Theorem 1.23e. Exactness, tensor injections and induction in stages embed the whole original induction in an induction from Jacquet-cuspidal blocks. Because the original parabolic was proper, the final list has at least two blocks. Transitivity of subquotient occurrence would put \(S\) in that proper cuspidal induction, contradicting Corollary 1.23t. Nonstandard parabolics are conjugated to standard ones by a flag basis, which transports induction and coinvariants by actual inner conjugation. \(\square\)

The arbitrary-algebra tensor results used here have complete earlier proofs in Restricted tensor products and the tensor product theorem, Lemmas 3.1–3.2 and Theorem 4.1. Their finite interpolation, simple-corner reconstruction and admissible tensor factorization apply to every product Levi in these arguments.

Direct positivity of the cuspidal matrix epsilon exponent

This proof treats every nonarchimedean local field and every rank \(n\geq2\). It uses the full matrix Fourier family and zero proper Jacquet modules. It does not construct an inducing type or assume a compact-induction realization.

Let \(k\) have integer ring \(\mathcal O\), uniformizer \(\varpi\), residue cardinality \(q\), and valuation \(v(\varpi)=1\). Put \[ G=\mathrm{GL}_n(k),\quad Z=k^\times I_n,\quad K=\mathrm{GL}_n(\mathcal O),\quad j(g)=v(\det g),\quad X=q^{-s}. \tag{1.29a} \] The character \(\psi\) has annihilator \(\mathcal O\), and the matrix Fourier pairing is positive: \[ \widehat\Phi(Y)=\int_{M_n(k)}\Phi(A)\psi(\operatorname{tr}(AY))\,dA. \tag{1.29b} \] The additive measures give every integral coordinate lattice volume 1; they are self-dual. Multiplicative Haar measure gives \(K\) volume 1. Write \[ \alpha_n=\prod_{i=1}^n(1-q^{-i}),\qquad dg=\alpha_n^{-1}|\det g|^{-n}\,d_{\rm add}g. \tag{1.29c} \] Indeed additive matrix measure scaled by \(|\det g|^{-n}\) is left and right invariant: multiplication by \(a\) has additive Jacobian \(|\det a|^n\). The additive volume of \(K\) is the fraction of invertible matrices over the residue field, namely \(q^{-n^2}\prod_{i=0}^{n-1}(q^n-q^i)=\alpha_n\). This proves (1.29c) with the exact constant.

Proposition 1.3, Proposition 1.4, Corollary 1.4a, Theorem 1.17 and Lemma 1.20a are the preceding matrix proofs used here. They supply the entire rational matrix ideal, the scalar Fourier equation, the Laurent-unit epsilon factor, and compact-mod-center coefficient support from zero two-block Jacquet modules. In rank greater than one that support also gives \[ V^K=0,\qquad L_\pi=L_{\widetilde\pi}=1. \tag{1.29d} \] We also use the complete proofs of the unique extended absolute value and Newton factorization in Extensions of complete valued fields, Theorems 1.2 and 5.1. Those actual proofs explicitly include inseparable extensions and polynomials in positive characteristic.

Fix an irreducible smooth admissible \(\pi\) with \(V_{U_r}=0\) for every upper two-block radical \(U_r\), \(1\leq r<n\). Let \(c(g)=\ell(\pi(g)v)\). Both \(c\) and \(c^\vee(g)=c(g^{-1})\) have compact support modulo \(Z\).

Constant terms and compact determinant shells

If a locally constant function \(f\) has support in \(ZC\) for compact \(C\subset G\), its support on every shell \(j(g)=j\) is compact. In fact \(j(C)\) is a finite set, and \(j(zc)=nv(z)+j(c)\) confines \(v(z)\) to finitely many integers. The corresponding subsets \(\varpi^b\mathcal O^\times\) are compact, so their products with \(C\) are compact. In particular all integrals below on a fixed shell are absolutely convergent.

Lemma 1.24a (vanishing along the full unipotent fiber). For \(m=\operatorname{diag}(A,D)\), \(k_0\in K\), and \[ u(B)=\begin{pmatrix}I_r&B\\0&I_t\end{pmatrix}, \quad B\in M_{r,t}(k),\quad r+t=n, \] both \(f=c\) and \(f=c^\vee\) satisfy \[ \int_{M_{r,t}(k)} f(k_0u(B)m k_0^{-1})\,dB=0. \tag{1.29e} \] Moreover \[ \int_{j(g)=j}f(g)\,dg=0\qquad(j\in\mathbb Z). \tag{1.29f} \]

Proof. If a vector \(w\) has zero Jacquet class, write it as a finite sum of \((\pi(u_i)-1)w_i\). Any sufficiently large compact additive subgroup of \(U_r\) contains all \(u_i\); its probability average kills every summand by translation invariance. Thus its average kills \(w\).

The integrand in (1.29e) has compact support in \(B\). To check this from compact support modulo \(Z\), conjugation by \(k_0\) preserves compactness, so consider \(u(B)m=z h\) with \(h\) in a fixed compact subset of \(G\). The fixed invertible diagonal blocks \(A,D\) bound \(|z|\) below, using the bound on the entries of \(h\); the fixed diagonal blocks of the inverse bound \(|z|\) above, using the bound on the entries of \(h^{-1}\). The upper-right block \(BD\) is then bounded, hence so is \(B\). Closedness of the support and local compactness prove the assertion.

For \(c\), averaging the vector \(\pi(mk_0^{-1})v\) over a compact subgroup containing this support gives zero, proving (1.29e). For the inverse coefficient the integrand is \[ c^\vee(k_0u(B)m k_0^{-1}) =\ell\bigl(\pi(k_0m^{-1})\pi(u(-B))\pi(k_0^{-1})v\bigr). \] Averaging the vector \(\pi(k_0^{-1})v\) proves the same result. This does not require an unproved dual-Jacquet identity.

Finally use (1.29d). The shell is invariant under right \(K\) translation, and its integral for \(c\) equals the integral of the right \(K\)-average, which replaces \(v\) by \(e_Kv=0\). For \(c^\vee\), use left \(K\)-translation instead; it likewise replaces \(v\) by \(e_Kv=0\). Compact shell support and the compact \(K\)-orbit of each smooth vector justify Fubini. This proves (1.29f). \(\square\)

Splitting matrices according to root absolute values

Extend the absolute value uniquely to an algebraic closure of \(k\). For \(g\in G\), call its characteristic polynomial equal-valued if all its roots, counted with algebraic multiplicity, have the same absolute value. In that case their common valuation is \(j(g)/n\). Consequently \[ j(g)\geq0,\quad g\text{ equal-valued} \quad\Longrightarrow\quad \operatorname{tr}(g)\in\mathcal O. \tag{1.29g} \] Indeed the trace is the sum of those roots, all of absolute value at most 1, and belongs to \(k\).

If the roots are not equal-valued, let \(R\) be their largest absolute value, let \(r\) be the number of roots of that value, and put \(t=n-r>0\). Newton factorization, including polynomial multiplicities, gives coprime monic polynomials in \(k[T]\): \[ p_g=p_+p_-,\quad \deg p_+=r,\quad\deg p_-=t, \quad |\text{roots of }p_+|=R> |\text{each root of }p_-|. \tag{1.29h} \] There is a unique canonical \(r\)-dimensional subspace \[ W_g=\ker p_+(g). \tag{1.29i} \] For completeness, Bézout for \(p_+,p_-\), together with Cayley–Hamilton, splits \(k^n\) as \(\ker p_+(g)\oplus\ker p_-(g)\). The polynomial \(p_+\) kills the first summand and \(p_-\) the second. Over a splitting field an operator is upper triangularizable: choose an eigenvector, pass to its quotient, and induct on the dimension. Applying the annihilating polynomial to such a triangular form shows that its diagonal roots belong to the roots of that polynomial. The two summand characteristic polynomials therefore have disjoint root sets and multiply to \(p_g\); all the multiplicities of \(p_+\) must be in the first and all those of \(p_-\) in the second. They are \(p_+,p_-\) respectively. Their dimensions are therefore \(r,t\). The same reasoning shows that any invariant subspace carrying all \(r\) of the largest-value roots is killed by \(p_+(g)\), by Cayley–Hamilton on that subspace, and equals (1.29i). This proves uniqueness.

The Cayley–Hamilton identity used here also follows directly from \((TI-g)\operatorname{adj}(TI-g)=p_g(T)I= \operatorname{adj}(TI-g)(TI-g)\). The two identities show that each coefficient matrix of the adjugate commutes with \(g\); evaluating this polynomial matrix identity at \(T=g\) gives \(p_g(g)=0\). For the direct-sum assertion, a Bézout identity \(a p_++b p_-=1\) writes each vector as the sum of its \(b(g)p_-(g)\) image in \(\ker p_+(g)\) and its \(a(g)p_+(g)\) image in \(\ker p_-(g)\); their intersection is zero by the same identity.

The root-valuation multiset is locally constant as a function of an invertible matrix. Here is the needed topological detail. At a polynomial with nonzero leading and constant coefficients, the finitely many coefficient points defining its Newton polygon have fixed finite heights. A sufficiently small coefficient change keeps each nonzero coefficient's valuation fixed and puts every formerly zero coefficient strictly above that polygon. The polygon is unchanged. The preceding Newton theorem identifies its slopes and lengths with the root valuations and multiplicities. Characteristic coefficients are polynomial functions of matrix entries, so the assertion follows.

Write \(\Omega_r\) for matrices whose largest-value group has dimension \(r<n\). The \(\Omega_r\) are disjoint open sets covering the matrices that are not equal-valued. Define the open block domain \[ \mathcal D_r= \{(A,D)\in\mathrm{GL}_r(k)\times\mathrm{GL}_t(k): \text{all roots of }A\text{ have one absolute value, larger than every root absolute value of }D\}. \tag{1.29j} \]

We give finite integral charts for \(W_g\). Any \(r\)-subspace \(W\subset k^n\) has a saturated lattice \(\Lambda=W\cap\mathcal O^n\), and a basis of \(\Lambda\) extends to an integral basis of \(\mathcal O^n\). An explicit proof is to scale a nonzero vector in \(W\) until one coordinate is a unit, use integral elementary coordinate changes to send it to \(e_n\), and split \[ W=k e_n\oplus(W\cap k^{n-1}),\quad \Lambda=\mathcal O e_n\oplus(W\cap k^{n-1}\cap\mathcal O^{n-1}). \] Induction supplies the asserted integral basis.

Thus \(\Lambda/\varpi\Lambda\) determines an \(r\)-plane \(\overline W\) in the residue-field vector space. For each of the finitely many such planes choose \(k_{\overline W}\in K\) carrying the first coordinate \(r\)-plane to it on reduction. Every \(W\) with that reduction has a unique form \[ W=k_{\overline W} \{(x,Bx):x\in k^r\},\qquad B\in\varpi M_{t,r}(\mathcal O). \tag{1.29k} \] Indeed after \(k_{\overline W}^{-1}\), the projection of an integral basis of \(\Lambda\) to its first \(r\) coordinates is invertible modulo \(\varpi\), hence belongs to \(\mathrm{GL}_r(\mathcal O)\). Normalizing that projection to the identity makes its lower block the unique \(B\), which is zero modulo \(\varpi\).

Put \(l(B)=\left(\begin{smallmatrix}I_r&0\\B&I_t\end{smallmatrix}\right)\) and \(k_B=k_{\overline W}l(B)\in K\). The map \[ F_{\overline W}(B,A,C,D) =k_B\begin{pmatrix}A&C\\0&D\end{pmatrix}k_B^{-1}, \quad B\in\varpi M_{t,r}(\mathcal O),\quad (A,D)\in\mathcal D_r,\quad C\in M_{r,t}(k), \tag{1.29l} \] is a bijection onto the part of \(\Omega_r\) with \(\overline{W_g}=\overline W\). Its canonical subspace is exactly \(k_B k^r\); uniqueness in (1.29i) determines \(B\), and reading the conjugated blocks determines \(A,C,D\). Conversely any \(g\) in that part preserves its canonical subspace, so this block reading gives (1.29l) and (1.29j). The charts are disjoint; no Weyl multiplicity or finite representative factor is omitted.

The measure calculation, including its analytic justification

Define the Sylvester map on the lower rectangular block: \[ S_{A,D}:M_{t,r}(k)\longrightarrow M_{t,r}(k), \qquad H\longmapsto HA-DH. \tag{1.29m} \] It is invertible on \(\mathcal D_r\). If \(HA=DH\), then \(Hp_A(A)=p_A(D)H=0\), where \(p_A\) is the characteristic polynomial of \(A\). Its coprimality with the characteristic polynomial of \(D\) makes \(p_A(D)\) invertible by Bézout. Hence \(H=0\), which proves invertibility in finite dimension.

The exact chart measure is \[ dg=\alpha_n^{-1}|\det A\,\det D|^{-n} |\det S_{A,D}|\,dB\,d_{\rm add}A\,dC\,d_{\rm add}D. \tag{1.29n} \] Most importantly its density is independent of \(B,C\). To calculate it, discard the fixed integral conjugation \(k_{\overline W}\), whose additive Jacobian is 1. Conjugation by \(l(B)\) also has additive Jacobian 1. Since \(l(B)^{-1}dl(B)= \left(\begin{smallmatrix}0&0\\dB&0\end{smallmatrix}\right)\), the differential of (1.29l), after that conjugation, is \[ \begin{pmatrix} dA-C\,dB&dC\\ dB\,A-D\,dB&dD+dB\,C \end{pmatrix}. \tag{1.29o} \] As a linear map in \(dA,dC,dD,dB\), its determinant has absolute value \(|\det S_{A,D}|\): the lower output is \(S_{A,D}(dB)\), and the other three outputs have identity coefficients on \(dA,dC,dD\). Finally \(\det F_{\overline W}=\det A\,\det D\), and (1.29c) gives (1.29n).

Here the change of variables is valid in every characteristic. We include the finite-dimensional analytic fact being used, instead of importing it from a manual. If a polynomial map \(F:k^d\to k^d\) has invertible derivative \(L\) at \(x_0\), write \[ L^{-1}(F(x_0+h)-F(x_0))=h+R(h). \] The polynomial \(R\) has no constant or linear terms. On a sufficiently small ball \(\varpi^N\mathcal O^d\), its finitely many coefficients and the telescoping formula for a difference of monomials give \[ \|R(h)-R(h')\|\leq q^{-1}\|h-h'\|, \qquad R(\varpi^N\mathcal O^d)\subset \varpi^{N+1}\mathcal O^d. \tag{1.29p} \] For \(y\) in this ball, \(h\mapsto y-R(h)\) is a contraction of that complete ball. Successive iterates are Cauchy, with errors bounded by the geometric powers \(q^{-i}\); their limit is its unique fixed point. This proves a bijection of the ball under \(h\mapsto h+R(h)\), and proves continuity of its inverse. The same argument on every contained coset ball of radius \(q^{-a}\), \(a\geq N\), sends it onto the coset ball of the same radius around its image. It therefore preserves additive Haar measure: these coset balls form a basis, and locally constant compact tests are finite linear combinations of their indicators. Translation and the linear map \(L\) then supply the measure factor \(|\det L|\). The latter factor follows from integral row and column elimination and diagonal scalings.

The derivative determinant remains of that same absolute value on the sufficiently small ball. Thus this is the Jacobian change of variables there. A nonarchimedean local field has a countable basis of such coordinate balls: each quotient \(k/\varpi^a\mathcal O\) is countable. Partitioning an open domain into countably many sufficiently small disjoint balls proves the formula globally for nonnegative measurable functions, then for integrable functions. On (1.29l) its derivative is invertible everywhere, and the previously proved bijection removes all multiplicities. This proves (1.29n) and justifies its use for all the integrals below, without a separability or residual-characteristic restriction.

The nonnegative determinant Gauss integrals vanish

Lemma 1.24b. For \(f=c\) or \(f=c^\vee\), \[ \int_{j(g)=j}\psi(\operatorname{tr}g)f(g)\,dg=0 \qquad(j\geq0). \tag{1.29q} \]

Proof. By (1.29f) it suffices to integrate \[ h(g)f(g),\qquad h(g)=\psi(\operatorname{tr}g)-1. \] On the equal-valued matrices in this shell, (1.29g) and the conductor of \(\psi\) give \(h=0\). On each remaining \(\Omega_r\), use the finitely many disjoint charts (1.29l). The determinant and the trace are \(\det A\,\det D\) and \(\operatorname{tr}A+\operatorname{tr}D\). The shell condition, the root condition (1.29j), \(h\), and the measure density (1.29n) are consequently all independent of \(C\). The inner integral is \[ \int_C f\!\left(k_B \begin{pmatrix}A&C\\0&D\end{pmatrix} k_B^{-1}\right)\,dC =|\det D|^r\int_Q f(k_Bu(Q)m k_B^{-1})\,dQ=0, \tag{1.29r} \] by \(C=QD\) and Lemma 1.24a. This proves the desired vanishing on each chart. Compact shell support, boundedness of \(h\), and (1.29n) justify the absolute Fubini rearrangement. Summing over the finite charts and \(1\leq r<n\) proves (1.29q). \(\square\)

The subtraction of 1 is essential: its shell integral is zero by the compact \(K\)-average, while the difference vanishes exactly where the trace is known to be integral. The argument does not identify a coefficient with a character distribution and does not infer scalar operator vanishing from a trace limit.

The Fourier equation now forces strict positivity

Theorem 1.24 (every Jacquet-cuspidal block). If \(n\geq2\) and \(\pi\) is irreducible smooth admissible with all proper two-block Jacquet modules zero, then its conductor-zero, positive-trace matrix epsilon factor has \[ \epsilon_\pi(s,\psi)=c_\pi X^{a_\pi},\qquad c_\pi\ne0,\qquad a_\pi>0. \tag{1.29s} \] This holds over every nonarchimedean local field.

Proof. The scalar Fourier equation and the Laurent-unit argument have already proved \(a_\pi\in\mathbb Z\). Choose \(v,\ell\) with \(\ell(v)=1\), and let \(J_N= 1+\varpi^N M_n(\mathcal O)\), \(N\geq1\), fix \(v,\ell\). Every deeper \(J_N\) does too. The actual test \[ \Phi_N=\operatorname{vol}_G(J_N)^{-1}1_{J_N} \] has \(Z_\pi(s,\Phi_N,c)=1\). Additive lattice orthogonality and (1.29c) give exactly \[ \widehat\Phi_N(Y)= \alpha_n\psi(\operatorname{tr}Y)\, 1_{\varpi^{-N}M_n(\mathcal O)}(Y). \tag{1.29t} \] There is no \(N\)-dependent scalar: the additive and multiplicative volumes of \(J_N\) have ratio \(\alpha_n\).

By (1.29d) gamma equals epsilon. Apply the whole-family Fourier equation to this test. Its exact Laurent expansion is \[ c_\pi X^{a_\pi} =\alpha_n\sum_{j\in\mathbb Z} q^{-j(n+1)/2}X^{-j} \int_{\substack{j(g)=j\\ g\in\varpi^{-N}M_n(\mathcal O)}} \psi(\operatorname{tr}g)c^\vee(g)\,dg. \tag{1.29u} \] For clarity this sum is finite. Its lower bound is \(j\geq-nN\), by the determinant expansion. For large positive \(j\), compact-mod-center support of \(c^\vee\) places its entire shell support in \(M_n(\mathcal O)\): write \(g=z h\), \(h\) in a fixed compact subset of \(G\); the finite values of \(j(h)\) force \(v(z)\to+\infty\) uniformly as \(j\to+\infty\). Its entries therefore tend uniformly to zero. The ball indicator is then 1 and \(\psi(\operatorname{tr}g)=1\); (1.29f) makes that shell coefficient zero. Thus (1.29u) is a genuine finite Laurent identity, obtained first in a convergence half-plane and then as the proved rational continuation.

Suppose \(a_\pi\leq0\), and put \(j_0=-a_\pi\geq0\). The support of \(c^\vee\) on that one shell is compact. Choose \(N\) sufficiently large that its support lies in \(\varpi^{-N}M_n(\mathcal O)\), retaining the fixed-vector condition. The coefficient of \(X^{a_\pi}=X^{-j_0}\) on the right of (1.29u) is then \[ \alpha_n q^{-j_0(n+1)/2} \int_{j(g)=j_0}\psi(\operatorname{tr}g)c^\vee(g)\,dg=0 \] by Lemma 1.24b. On the left it is the nonzero number \(c_\pi\). This contradiction proves \(a_\pi>0\). \(\square\)

Only one shell needs eventual containment. No uniform support bound over every vector, coefficient, test or determinant shell is assumed in this final extraction.

Every irreducible representation: nonnegative exponent and factor degree

Corollary 1.24c (universal nonnegative matrix exponent). Every irreducible smooth admissible \(\sigma\) embeds in normalized induction from terminal Jacquet-cuspidal blocks \(\rho_i\), by Theorem 1.23e. Its exact polynomial correction (1.27x)–(1.27y) gives \[ a_\sigma=\sum_i a_{\rho_i}+\deg R,\qquad R(0)=1, \quad R\in\mathbb C[X]. \tag{1.29v} \] Each block of rank greater than one has positive exponent by Theorem 1.24. Each rank-one block is a character and has nonnegative scalar Tate exponent by Proposition 1.16 in rank one. Since \(\deg R\geq0\), this proves \[ a_\sigma\geq0 \] for every smooth irreducible admissible \(\mathrm{GL}_n(k)\), in all ranks and all nonarchimedean characteristics. This proves the corollary. \(\square\)

This closes nonnegativity of the matrix epsilon exponent. It does not identify it with a generic minimal \(K_1\)-level or an Artin conductor, and does not assert existence of compact inducing data for every cuspidal representation. Those separate statements are unnecessary for the proof here.

Corollary 1.24d (degree bound and exponent zero). Write \(L_\sigma(s)=P_\sigma(q^{-s})^{-1}\), \(P_\sigma(0)=1\). For a cuspidal-support embedding as in Theorem 1.23e, let \(\chi_1,\ldots,\chi_h\) be its unramified rank-one blocks, listed with multiplicity. Then \[ P_\sigma(X)R(X)=\prod_{i=1}^{h}(1-\chi_i(\varpi)X), \qquad \deg P_\sigma\leq h\leq n. \tag{1.29w} \] In particular \(L_\sigma=1\) if there are no such blocks. If \(a_\sigma=0\), every block has rank one and is unramified, \(h=n\), \(R=1\), and \[ P_\sigma(X)=\prod_{i=1}^{n}(1-\chi_i(\varpi)X). \tag{1.29x} \]

Proof. Lemma 1.20a and Proposition 1.4 give factor 1 for each cuspidal block of higher rank. Proposition 1.16 in rank one gives factor 1 for a ramified character and \((1-\chi(\varpi)X)^{-1}\) for an unramified character. Substitution in (1.27x), followed by multiplication by \(P_\sigma\prod_i(1-\chi_i(\varpi)X)\), proves (1.29w). Degrees add in a product of nonzero polynomials, so its degree is \(h\), proving the bound. If \(h=0\), both polynomials have constant term 1 and their product is 1; hence both equal 1.

When \(a_\sigma=0\), (1.29v) is a sum of nonnegative integers. Every summand is therefore zero. Theorem 1.24 excludes a higher-rank block. The exact rank-one conductor in Proposition 1.16 makes every remaining character unramified. Thus \(h=n\). Also \(\deg R=0\) and \(R(0)=1\), so \(R=1\), proving (1.29x). \(\square\)

This is a conclusion about the cuspidal-support embedding and the matrix factor. A proof that \(\sigma\) itself has a \(K\)-fixed vector would require an additional argument. The algebraic compact-induction criteria in Lemmas 1.23f–1.23g retain \(n\geq2\) for strict positivity: an unramified rank-one character has exponent zero.

Further reading is Getz–Hahn, 22 April 2022 author draft, §8.7, Proposition 8.7.1, printed pages 207–208. The finite Newton charts, Sylvester Jacobian and full-fiber cancellation above provide the proof, including matrices that are not semisimple and fields of positive characteristic.

Matrix tests at a real or complex place

For \(F=\mathbb R\) use \(|x|_F=|x|\); for \(F=\mathbb C\) use \(|z|_F=z\bar z\). Put \(G=\mathrm{GL}_n(F)\), \(\nu(g)=|\det g|_F\), and \(u_s=s+(n-1)/2\). On the additive matrix space use ordinary real Lebesgue measure, or the product of \(2\,d\Re z\,d\Im z\) at a complex place. Multiplicative Haar measure is \(dg=C_F\nu(g)^{-n}dX\), where \(C_F>0\) records the chosen normalization. This formula follows from the left and right additive Jacobians \(|\det h|_F^n\).

These arguments use smooth matrix coefficients of a specified smooth realization. Wherever differentiation of a zeta integral is used, the coefficients and the indicated Lie derivatives are assumed to have polynomial growth. No algebraic-to-smooth or Hilbert-to-smooth comparison theorem is inferred here. Polynomial growth here means a bound by \(C(1+\|g\|+\|g^{-1}\|)^N\), with a possibly different \(C,N\) for each of the finitely many derivatives used in an argument.

1. Polynomial multiplication and tests supported on the group

Proposition 1.5 (polynomial-module closure). Suppose a matrix coefficient \(c\) and its Lie derivatives have polynomial growth. In the initial domain of convergence, the complex span of its matrix zeta integrals, allowing all coefficient vectors and Schwartz tests, is stable under multiplication by \(s\). The same is true if the tests are Gaussian times polynomials and the coefficient vectors are bi-\(K\)-finite, where \(K=O(n)\) or \(U(n)\).

Choose a real Lie algebra element \(A\) with \[ \left.\frac{d}{dt}\log\nu(e^{tA})\right|_{t=0}=1: \quad A=\operatorname{diag}(1,0,\ldots,0)\ \ (F=\mathbb R), \quad A=\operatorname{diag}(1/2,0,\ldots,0)\ \ (F=\mathbb C). \] For functions on \(G\) or on the matrix space let \(R_Af(g)=\left.\frac d{dt}f(ge^{tA})\right|_{0}\). Right invariance of Haar measure and the substitution \(g\mapsto ge^{-tA}\) give \[ Z(s,R_A\Phi,c)+Z(s,\Phi,R_Ac) =-\left(s+\frac{n-1}{2}\right)Z(s,\Phi,c). \tag{1.8a} \] Polynomial growth of the coefficients and their derivatives, together with the Schwartz bounds, justifies differentiation in a sufficiently far right half-plane. Near singular matrices, the same determinant weight and polynomial bound in \(g^{-1}\) give a common integrable majorant after increasing \(\Re s\); for example Cartan polar coordinates, or the column Gram–Schmidt coordinates below, bound powers of \(g^{-1}\) by a power of \(\nu(g)^{-1}\) times a polynomial in the matrix entries. More explicitly, the adjugate identity gives \(\|g^{-1}\|\leq C\|g\|^{n-1}|\det g|_{\mathrm{ordinary}}^{-1}\); the Gram–Schmidt integral of a sufficiently large positive determinant power is finite. Both estimates are uniform when \(t\) ranges over a compact interval.

Equation (1.8a) expresses \(sZ\) as a sum of three actual integrals. Iterating proves the assertion for any polynomial in \(s\). Differentiating a polynomial times the basic Gaussian again gives a polynomial times that Gaussian. A Lie derivative of a \(K\)-finite vector remains \(K\)-finite: if its original \(K\)-span is \(W\), the span after differentiation lies in the image of the finite-dimensional space \(\operatorname{span}\{\operatorname{Ad}(k)A:k\in K\}\otimes W\). Apply the same observation to the dual vector. \(\square\)

Proposition 1.6 (an entire test nonzero at a prescribed parameter). For every \(s_0\in\mathbb C\) and a coefficient with \(c(1)=1\), there is a Schwartz test supported inside \(G\) for which \(Z(s_0,\Phi,c)\ne0\). Integrals with such compact support are entire in \(s\).

Take a nonzero nonnegative smooth function \(\eta\) supported in a sufficiently small compact neighborhood of 1 in \(G\), and set \[ \Phi(g)=\eta(g)\overline{c(g)}\,\nu(g)^{-s_0-(n-1)/2}. \] Extend it by zero to the matrix space. Its support stays away from the singular locus, so the extension is smooth and Schwartz. Then \[ Z(s_0,\Phi,c)=\int_G\eta(g)|c(g)|^2\,dg>0. \] On a compact subset of \(G\), the real function \(\log\nu(g)\) is bounded. This bounds every parameter derivative uniformly on compact sets of \(s\), proving entireness. \(\square\)

In particular, if a common meromorphic generator \(L\) has already been shown to divide every zeta integral with entire quotient, Proposition 1.6 proves that it has no zeros. The entire-quotient assertion is indispensable: nonvanishing of a compact-support integral does not prove that division theorem.

2. The Gaussian ideal for determinant characters in every rank

Let \(\pi(g)=\chi(\det g)\), with \[ \chi(x)=\operatorname{sgn}(x)^e|x|^a,\quad e\in\{0,1\}, \quad F=\mathbb R; \qquad \chi(z)=(z/|z|)^k|z|_F^a,\quad k\in\mathbb Z, \quad F=\mathbb C. \] All matrix coefficients are scalar multiples of this character. These formulas cover every smooth character of \(F^\times\): on positive radii the character, written in the logarithmic coordinate, satisfies \(h'(t)=h'(0)h(t)\) and is an exponential. Over \(\mathbb R\) its value at \(-1\) has square one. Over \(\mathbb C\), the same differential equation on the angular coordinate and period \(2\pi\) force the exponent to be \(ik\) with \(k\in\mathbb Z\); the radial exponent is written as \(2a\) because the normalized complex norm is the squared radius. Let \(\mathcal S_0\) be the polynomials times \(e^{-\pi\operatorname{tr}(XX^t)}\) over \(\mathbb R\), or the polynomials in \(X,\overline X\) times \(e^{-2\pi\operatorname{tr}(XX^*)}\) over \(\mathbb C\).

Proposition 1.7 (the Gaussian ideal for determinant characters). The span of the matrix zeta integrals with tests in \(\mathcal S_0\) is exactly \(L_\chi(s)\mathbb C[s]\), where a constant normalization independent of \(s\) is permitted and \[ L_\chi(s)= \begin{cases} \displaystyle\prod_{j=1}^n \Gamma_{\mathbb R}\!\left(s+a+e+j-\frac{n+1}{2}\right), &F=\mathbb R,\\[4pt] \displaystyle\prod_{j=1}^n \Gamma_{\mathbb C}\!\left(s+a+\frac{|k|}{2} +j-\frac{n+1}{2}\right), &F=\mathbb C. \end{cases} \tag{1.8b} \] Here \(\Gamma_{\mathbb R}(w)=\pi^{-w/2}\Gamma(w/2)\) and \(\Gamma_{\mathbb C}(w)=2(2\pi)^{-w}\Gamma(w)\). This is an assertion about every Gaussian-polynomial test, not just a calculation of one test.

Proof: averaging the polynomial. Replace a test by \[ \Phi^\chi(X)=\int_{K\times K}\chi(\det k_1)\chi(\det k_2) \Phi(k_1Xk_2)\,dk_1\,dk_2. \tag{1.8c} \] The measures here are probability measures. Change of variables in the absolutely convergent integral shows that its zeta integral is unchanged. The basic Gaussian is bi-\(K\)-invariant; averaging its polynomial factor gives another polynomial of the same degree bound. Write this factor as \(P_\chi\). It has the transformation law \[ P_\chi(h_1Xh_2)= \chi(\det h_1\det h_2)^{-1}P_\chi(X) \quad(h_1,h_2\in K). \tag{1.8d} \]

Over \(\mathbb R\), restrict to diagonal matrices with entries \(x_1,\ldots,x_n\). Changing the sign of an individual entry shows that its polynomial degree in that entry has parity \(e\). Conjugating by permutation matrices shows that \[ P_\chi(\operatorname{diag}(x_j)) =(x_1\cdots x_n)^e Q(x_1^2,\ldots,x_n^2) \] with \(Q\) symmetric. Over \(\mathbb C\), restrict to diagonal entries \(z_j\). An independent unitary phase in entry \(j\) shows that every surviving monomial has \(\deg z_j-\deg\bar z_j=-k\). Thus its restriction is \(\prod_j\bar z_j^k Q(|z_1|^2,\ldots,|z_n|^2)\) if \(k\geq0\), and \(\prod_jz_j^{-k}Q(|z_1|^2,\ldots,|z_n|^2)\) if \(k<0\), again with \(Q\) symmetric.

Every symmetric polynomial in \(n\) variables is a polynomial in their elementary symmetric functions. An elementary proof subtracts the product of elementary symmetric functions with the same highest lexicographic monomial, and repeats; each subtraction decreases that monomial and terminates at bounded degree. Apply this to \(Q\). Its value on the squared singular values is therefore a polynomial in the characteristic coefficients of \(X^*X\), or \(X^tX\). These coefficients are themselves polynomials in the matrix entries.

Every matrix admits a real or complex singular-value decomposition. To obtain it, diagonalize the positive symmetric or Hermitian matrix \(X^*X\): the maximum of its quadratic form on the unit sphere gives an eigenvector, its orthogonal complement is invariant, and induction gives an orthonormal eigenbasis. The positive-eigenvalue columns of \(X\), divided by their singular values, are orthonormal; complete them to an orthonormal basis. This gives the desired two compact factors, also for singular matrices. The transformation law (1.8d) therefore identifies the full polynomial as \[ P_\chi(X)= \begin{cases} (\det X)^e Q_{\mathrm{Gram}}(X^tX),&F=\mathbb R,\\ (\overline{\det X})^k Q_{\mathrm{Gram}}(X^*X),&F=\mathbb C,\ k\geq0,\\ (\det X)^{-k}Q_{\mathrm{Gram}}(X^*X),&F=\mathbb C,\ k<0. \end{cases} \tag{1.8e} \] The notation \(Q_{\mathrm{Gram}}\) means a polynomial in those characteristic coefficients. None of the displayed powers has a negative exponent.

Proof: all the Gaussian moments. The real Gaussian matrix measure and the complex Gaussian matrix measure just specified both have total mass one: the one-entry integrals are respectively \(\int_{\mathbb R}e^{-\pi x^2}dx=1\) and \(\int_{\mathbb C}e^{-2\pi|z|^2}2\,d\Re z\,d\Im z=1\). Perform Gram–Schmidt on the independent Gaussian columns and write \(X=UR\), with \(R\) upper triangular and positive real diagonal entries \(r_1,\ldots,r_n\). The singular set has measure zero: successively, a Gaussian column lies in the span of preceding columns with probability zero, since that is a proper linear subspace.

Conditional on the preceding columns, the next Gaussian column in any orthonormal frame has the same product Gaussian law. Its coordinates along those columns and its orthogonal component are independent, and their distributions do not depend on the preceding columns. Thus the upper entries of \(R\) are independent one-entry Gaussians, and \(r_j\) is the length of a Gaussian vector in dimension \(d=n-j+1\). The variables so obtained are independent. Radial integration shows that \(r_j^2\) has gamma density with shape \(d/2\) and rate \(\pi\) in the real case, or shape \(d\) and rate \(2\pi\) in the complex case. The normalizing constants follow by dividing the radial integral by its value at exponent zero. This uses only the one-dimensional gamma integral.

Consequently, for the test in (1.8e) with \(Q_{\mathrm{Gram}}=1\), put \[ \kappa=s+a+e-\frac{n+1}{2}\quad(F=\mathbb R),\qquad \beta=s+a+\frac{|k|}{2}-\frac{n+1}{2}\quad(F=\mathbb C). \] The sign or angular character cancels the determinant polynomial. Including the factor \(\nu(g)^{-n}\) in Haar measure, its integral is \[ \begin{split} Z_{\mathrm{base}}(s) &=C_{\mathbb R}\pi^{-n\kappa/2} \prod_{j=1}^n\frac{\Gamma((\kappa+j)/2)}{\Gamma(j/2)} &&(F=\mathbb R),\\ Z_{\mathrm{base}}(s) &=C_{\mathbb C}(2\pi)^{-n\beta} \prod_{j=1}^n\frac{\Gamma(\beta+j)}{\Gamma(j)} &&(F=\mathbb C). \end{split} \tag{1.8f} \] Initially these formulas hold respectively for \(\Re\kappa>-1\) and \(\Re\beta>-1\). Their ratios to (1.8b) are the positive constants \[ C_{\mathbb R} \frac{\pi^{\,n(n+1)/4}}{\prod_{j=1}^n\Gamma(j/2)}, \qquad C_{\mathbb C} \frac{(2\pi)^{\,n(n+1)/2}}{2^n\prod_{j=1}^n\Gamma(j)}. \tag{1.8g} \]

For a general \(Q_{\mathrm{Gram}}\), substitute \(X^*X=R^*R\) and integrate the polynomial in the strictly upper entries of \(R\). Their Gaussian moments are finite, so the result is a polynomial in the \(r_j\). It is a polynomial in the squares \(r_j^2\): in the real case, changing the sign of row \(j\) of \(R\) preserves \(R^tR\) and the Gaussian law of all its upper entries; in the complex case the analogous row phase preserves \(R^*R\) and forces equal holomorphic and antiholomorphic degrees in its diagonal variable. It is thus a finite sum of monomials \(\prod_jr_j^{2m_j}\), with \(m_j\geq0\).

The ratio of each corresponding real radial moment to the base moment is \(\pi^{-m_j}((\kappa+d)/2)_{m_j}\); the complex ratio is \((2\pi)^{-m_j}(\beta+d)_{m_j}\). The rising factorial \((w)_m=w(w+1)\cdots(w+m-1)\) follows by integrating the gamma integral by parts \(m\) times. These ratios are polynomials in \(s\). Hence every Gaussian-polynomial integral is a polynomial multiple of (1.8f), and continues meromorphically by that formula. The base test attains a nonzero constant times (1.8b). Proposition 1.5 then supplies every polynomial multiple. This proves the full asserted \(\mathbb C[s]\)-ideal. \(\square\)

The one-dimensional gamma integral, its recurrence and its absence of zeros are proved in the earlier Tate's local theory at the infinite places, the section “Mellin continuation without a functional equation” and Proposition 8.2. In particular (1.8b) is zero-free, its reciprocal is entire, and it has finitely many poles in any bounded vertical strip. Indeed the gamma poles form decreasing real arithmetic progressions after a fixed complex shift. The polynomial Gaussian calculation (1.8f) will now be extended to every Schwartz test in Theorem 1.18.

3. The full Schwartz family for determinant characters

Let \(F=\mathbb R\) or \(\mathbb C\), \(G=GL_n(F)\), and \(\nu(g)=|\det g|_F\). The complex absolute value is squared: \(|z|_F=z\bar z\). Use additive measure \(dX\) equal to real Lebesgue measure, or the product of \(2\,d\Re z\,d\Im z\) on complex entries, and write \[ dg=C_F\nu(g)^{-n}dX,\qquad C_F>0. \] Let \[ \chi(x)=\operatorname{sgn}(x)^e|x|^a \quad(F=\mathbb R,\ e=0,1),\qquad \chi(z)=(z/|z|)^k|z|_F^a \quad(F=\mathbb C,\ k\in\mathbb Z), \] where \(a\in\mathbb C\), and put \(\pi=\chi\circ\det\). Set \[ Z_\chi(s,\Phi)=\int_G\Phi(g)\chi(\det g)\nu(g)^{s+(n-1)/2}\,dg. \] The factor already obtained from all polynomial Gaussian tests is \[ L_\chi(s)= \begin{cases} \displaystyle\prod_{j=1}^n \Gamma_{\mathbb R}\!\left(s+a+e+j-\frac{n+1}{2}\right),&F=\mathbb R,\\[3pt] \displaystyle\prod_{j=1}^n \Gamma_{\mathbb C}\!\left(s+a+\frac{|k|}{2}+j-\frac{n+1}{2}\right),&F=\mathbb C. \end{cases} \tag{1.22a} \] Here \(\Gamma_{\mathbb R}(w)=\pi^{-w/2}\Gamma(w/2)\) and \(\Gamma_{\mathbb C}(w)=2(2\pi)^{-w}\Gamma(w)\).

Theorem 1.18 (the full Schwartz family for determinant characters). In every rank, \(Z_\chi(s,\Phi)\) converges absolutely for every \(\Phi\in\mathcal S(M_n(F))\) when \(\Re(s+a)>(n-1)/2\). It continues meromorphically, and \[ \mathcal D_\chi(s)(\Phi)=Z_\chi(s,\Phi)/L_\chi(s) \tag{1.22b} \] is an entire family of tempered distributions. More precisely, for every compact set \(B\subset\mathbb C\) there is a Schwartz seminorm \(p_B\) such that \[ \sup_{s\in B}|\mathcal D_\chi(s)(\Phi)|\le p_B(\Phi); \tag{1.22c} \] the same assertion holds for every parameter derivative.

Use the programme characters \[ \psi_{\mathbb R}(x)=e^{-2\pi ix},\qquad \psi_{\mathbb C}(z)=e^{-2\pi i(z+\bar z)} \] and the positive trace argument in the Fourier kernel: \[ \widehat\Phi(Y)=\int\Phi(X)\psi_F(\operatorname{tr}(XY))\,dX. \] Then, for every Schwartz test, \[ \frac{Z_{\chi^{-1}}(1-s,\widehat\Phi)}{L_{\chi^{-1}}(1-s)} =\epsilon_\chi\, \frac{Z_\chi(s,\Phi)}{L_\chi(s)},\qquad \epsilon_\chi= \begin{cases} (-i)^{ne},&F=\mathbb R,\\ (-i)^{n|k|},&F=\mathbb C. \end{cases} \tag{1.22d} \] All matrix coefficients of this one-dimensional representation are scalar multiples of the coefficient used here, so the assertion includes the whole coefficient family.

The polynomial Gaussian family still has exactly the ideal \(L_\chi(s)\mathbb C[s]\), with an attaining test; that is the earlier Proposition 1.7, whose proof applies to this same normalization. The present theorem supplies division and the Fourier equation on the full Schwartz space.

1. Normalized Mellin distributions with seminorm bounds

For an even Schwartz function \(f\) on \(\mathbb R\), initially define \[ \mathcal M_w(f)=\frac1{\Gamma(w/2)} \int_0^\infty f(r)r^{w-1}\,dr,\qquad\Re w>0. \] This is an entire family of continuous functionals on the even Schwartz space, locally bounded by a finite number of its seminorms.

Here is the complete continuation argument. Choose an even compactly supported smooth cutoff \(\rho\), equal to one near zero. For an integer \(N\ge1\), subtract \[ \rho(r)\sum_{j=0}^{N-1}\frac{f^{(2j)}(0)}{(2j)!}r^{2j}. \] The remainder is \(O(r^{2N})\) at zero, uniformly with the requisite derivative bounds, and remains rapidly decreasing at infinity. Its Mellin integral is holomorphic for \(\Re w>-2N\). Each subtracted term has Mellin integral \[ \frac1{w+2j}+\text{an entire function of }w, \] if the region on which \(\rho=1\) is chosen as \([0,1]\); another such region changes only an entire term and a nonzero exponential factor in the pole term. The reciprocal \(\Gamma(w/2)^{-1}\) has a simple zero at each \(w=-2j\), so all these poles cancel.

The Taylor remainder on \([0,1]\) is bounded by a finite derivative seminorm. On \([1,\infty)\), a sufficiently high Schwartz decay seminorm bounds the integral uniformly on any fixed compact parameter set. The same estimates absorb each power of \(\log r\) introduced by parameter differentiation. The canceled pole terms are bounded on compact sets as entire functions. Increasing \(N\) proves the assertion everywhere, with the required seminorm bounds. The gamma recurrence, its poles, and the entire reciprocal are actually proved in Tate's local theory at the infinite places, the section “Mellin continuation without a functional equation”; no gamma nonvanishing assertion is being inferred from a functional equation.

For \(h\in\mathcal S(\mathbb R^n)\) even in every variable, this construction in each variable gives \[ \mathcal M_{\boldsymbol w}(h)= \frac1{\prod_j\Gamma(w_j/2)} \int_{(0,\infty)^n}h(\boldsymbol r)\prod_j r_j^{w_j-1}\,d\boldsymbol r. \tag{1.22e} \] It is jointly entire and satisfies the analogous seminorm bounds. To justify this without a tensor-product assertion, apply the preceding cutoff and finite Taylor subtraction in each coordinate. The resulting finite expansion consists of products of the canceled pole terms and integrals of remainders on coordinate rectangles. Each remainder has the independently prescribed order of vanishing at each small coordinate and rapid decay in every large coordinate. On a compact subset of \(\mathbb C^n\), choose all subtraction orders sufficiently large and a common sufficiently large decay exponent. The product bounds give absolute majorants for these integrals and all parameter derivatives. The finite expression proves both joint holomorphy and continuity on the entire even Schwartz space.

2. QR coordinates on the whole matrix space

Put \[ \delta=\dim_{\mathbb R}F,\quad b_F=\begin{cases}\pi,&F=\mathbb R,\\[2pt] 2\pi,&F=\mathbb C,\end{cases} \quad d_j=n-j+1,\quad m=\begin{cases}e,&F=\mathbb R,\\|k|,&F=\mathbb C.\end{cases} \] Every invertible matrix has a unique decomposition \(X=UR\), where \(U\in K=O(n)\) or \(U(n)\), and \(R\) is upper triangular with positive real diagonal \(r_j\). The strictly upper entries form a vector \(Y\) over \(F\). Successive column Gram–Schmidt proves this decomposition.

Its additive integration formula is \[ dX=A_F\,dU\,dY\prod_{j=1}^n r_j^{\delta d_j-1}\,dr_j,\qquad A_F=\frac{2^n b_F^{\delta n(n+1)/4}} {\prod_{j=1}^n\Gamma(\delta j/2)}, \tag{1.22f} \] where \(dU\) is probability Haar measure and each complex upper entry uses measure \(2\,d\Re y\,d\Im y\).

For completeness, obtain the exponents by taking polar coordinates in the first column, then decomposing each subsequent column into its coordinates along the preceding orthonormal columns and its orthogonal component. That component has real dimension \(\delta d_j\), so radial measure contributes \(r_j^{\delta d_j-1}dr_j\). Changes to an orthonormal frame have real Jacobian one. The successive angular probability measures give a probability measure on complete orthonormal frames invariant under left \(K\); averaging over \(K\) shows it is Haar probability. These steps prove (1.22f) up to a positive constant. To find it, integrate the basic Gaussian \(e^{-b_F\|X\|^2}\), whose mass is one. Every strictly upper Gaussian integral has mass one, and the \(j\)-th radial integral is \[ \frac12 b_F^{-\delta d_j/2}\Gamma(\delta d_j/2). \] Their product gives the displayed constant. The excluded singular set has additive measure zero: in successive column integration, a column lying in the span of its predecessors belongs to a proper real linear subspace.

Write \(\xi(x)=\operatorname{sgn}(x)^e\) or \(\xi(z)=(z/|z|)^k\). For arbitrary diagonal entries \(z_j\in F\), define \[ H_\Phi(\boldsymbol z)= \int_K\int_{F^{n(n-1)/2}}\xi(\det U)\, \Phi\bigl(U R(\boldsymbol z,Y)\bigr)\,dY\,dU. \tag{1.22g} \] This is a continuous linear map from \(\mathcal S(M_n(F))\) to \(\mathcal S(F^n)\). Indeed \(\|UR\|=\|R\|\); differentiation in the diagonal entries gives a finite combination of derivatives of \(\Phi\) with coefficients bounded uniformly in \(U\). A Schwartz bound of arbitrarily large order in \((\boldsymbol z,Y)\), integrated over \(Y\) and compact \(K\), bounds every weighted derivative in \(\boldsymbol z\). This proves the assertion, with each target seminorm bounded by finitely many source seminorms.

Changing row \(j\) of \(R\) by a real sign gives \[ H_\Phi(\ldots,-r_j,\ldots)=(-1)^e H_\Phi(\ldots,r_j,\ldots). \] Over \(\mathbb C\), changing that row by \(e^{i\theta}\), rotating its upper entries in the \(Y\)-integral, and replacing \(U\) by \(U\operatorname{diag}(1,\ldots,e^{i\theta},\ldots,1)\), gives \[ H_\Phi(\ldots,e^{i\theta}z_j,\ldots) =e^{-ik\theta}H_\Phi(\ldots,z_j,\ldots). \tag{1.22h} \] In the Taylor expansion at \(z_j=0\), this retains only monomials \(z_j^p\bar z_j^q\) with \(p-q=-k\). All derivatives of total order less than \(|k|\) vanish, and restriction to the real axis has parity \((-1)^{|k|}\).

It follows that on the real diagonal \[ H_\Phi(\boldsymbol r)=\prod_j r_j^m\,h_\Phi(\boldsymbol r), \qquad h_\Phi\in\mathcal S(\mathbb R^n), \tag{1.22i} \] with \(h_\Phi\) even in each variable; this division is continuous on the indicated subspace. Explicitly, near \(r_j=0\), division by \(r_j^m\) is given by the integral Taylor remainder \[ \frac{f(r_j)}{r_j^m} =\frac1{(m-1)!}\int_0^1(1-\theta)^{m-1} f^{(m)}(\theta r_j)\,d\theta\quad(m>0). \] On \(|r_j|\ge1\), differentiate the ordinary quotient. Both formulas bound all Schwartz seminorms, using ordinary derivatives on the bounded coordinate and rapid decay in the other coordinates. Repeating in the coordinates gives (1.22i). For \(m=0\) no division is required.

3. Entire division for every Schwartz test

Put \[ t=s+a-\frac{n+1}{2},\qquad w_j=\delta(t+d_j)+m. \] For \(\Re t>-1\), QR integration is absolutely convergent before the compact angular average: the smallest radial dimension is \(\delta\), and \(\delta(t+1)>0\) is precisely the worst radial integrability condition. Infinity is controlled by Schwartz decay. Equations (1.22f)–(1.22i) give \[ Z_\chi(s,\Phi)=C_FA_F \int_{(0,\infty)^n}h_\Phi(\boldsymbol r) \prod_j r_j^{w_j-1}\,d\boldsymbol r. \tag{1.22j} \] Let \(\iota_F=0\) over \(\mathbb R\) and \(1\) over \(\mathbb C\). Reindexing \(d_j\) in (1.22a) gives exactly \[ L_\chi(s)= 2^{n\iota_F} b_F^{-\sum_jw_j/2} \prod_j\Gamma(w_j/2). \] Consequently \[ \mathcal D_\chi(s)(\Phi)= C_FA_F\,2^{-n\iota_F} b_F^{\sum_jw_j/2}\mathcal M_{\boldsymbol w}(h_\Phi). \tag{1.22k} \] The Mellin construction and continuity of \(\Phi\mapsto h_\Phi\) prove (1.22b)–(1.22c), including their derivative versions. Multiplication by \(L_\chi\) then gives the meromorphic continuation of the original integral. Angular cancellation may enlarge the convergence domain of (1.22j); it has not been used to assert absolute convergence of the original integral in that larger domain.

4. A uniqueness lemma for determinant-equivariant distributions

Fix \(t\in\mathbb C\) with \(\Re t>0\), and let a distribution \(T\) on \(M_n(F)\) satisfy \[ T(\Phi(hX))= \xi(\det h)^{-1}|\det h|_F^{-(t+n)}T(\Phi), \qquad T(\Phi(Xh))= \xi(\det h)^{-1}|\det h|_F^{-(t+n)}T(\Phi) \tag{1.22l} \] for every \(h\in G\). Such distributions form a one-dimensional space, spanned by the regular distribution \(\Phi\mapsto\int\Phi(X)\xi(\det X)|\det X|_F^t\,dX\).

First consider restriction to \(G\). Multiplying the distribution by the reciprocal of the smooth character \[ W(X)=\xi(\det X)|\det X|_F^{t+n} \] makes it left invariant. A left-invariant distribution is a constant times Haar measure. Here is a distribution-level proof of that assertion. Choose compactly supported smooth approximate identities \(\rho_j\) of Haar integral one. Left invariance gives \[ S(\phi*\rho_j)=S(\rho_j)\int_G\phi\,dg. \] Convolution with \(\rho_j\) tends to the identity in the compactly supported smooth topology: in a group coordinate chart this follows by Taylor estimates uniformly on a common compact support. Fixing one \(\phi_0\) of integral one shows \(S(\rho_j)\to S(\phi_0)\), and the displayed equality then proves the assertion for every \(\phi\). Since \(dg\) is a constant times \(\nu^{-n}dX\), the restriction of \(T\) has the claimed density.

It remains to prove that no nonzero distribution satisfying (1.22l) can be supported on singular matrices. Induct downwards on the largest rank \(r<n\) in its support. Near a rank-\(r\) matrix, choose an invertible \(r\)-by-\(r\) minor, after fixed row and column permutations. Put \(m_0=n-r\) and use the chart \[ X= \begin{pmatrix} A&Ab\\ cA&cAb+W \end{pmatrix}, \quad A\in GL_r(F),\quad b\in M_{r,m_0}(F),\quad c\in M_{m_0,r}(F). \tag{1.22m} \] Here \(\operatorname{rank}X=r+\operatorname{rank}W\). Left lower-block unipotents translate \(c\), and right upper-block unipotents translate \(b\), with determinant one. Thus the chart distribution factors as Lebesgue measure in \(b,c\), times a distribution \(U\) in \(A,W\).

For clarity, translation invariance gives this factorization without a disintegration theorem. In one variable, a compactly supported smooth function of integral zero is the derivative of its compactly supported primitive. In several variables choose fixed integral-one bumps and subtract successive coordinate averages. The difference from the product bump times the total integral is a sum of coordinate derivatives, with compact support and smooth dependence on all other parameters. A translation-invariant distribution annihilates these derivatives. This proves the asserted factorization; constants in the complex coordinate measures do not change it.

The support of \(U\) lies in \(W=0\). On every compact \(A\)-subchart it has finite normal order: \[ U(f)=\sum_{|\alpha|\le N}U_\alpha\bigl(\partial_W^\alpha f(A,0)\bigr), \tag{1.22n} \] for distributions \(U_\alpha\) in \(A\). To prove this assertion, take a finite-order bound for \(U\) on a compact coordinate set. If all normal derivatives through that order vanish at \(W=0\), multiply \(f\) by a cutoff supported in \(\|W\|\le\varepsilon\), equal to one near zero. Taylor's estimate makes its bounded derivatives tend to zero as \(\varepsilon\to0\), while its value under \(U\) is unchanged. Thus \(U(f)=0\). Subtracting the finite Taylor polynomial in \(W\), multiplied by a fixed cutoff equal to one near zero, proves (1.22n), including continuity of its coefficients.

Now apply left multiplication by \(\operatorname{diag}(I_r,h)\), \(h\in GL_{m_0}(F)\). In (1.22m) this sends \((c,W)\) to \((hc,hW)\). The \(c\)-integration contributes \(|\det h|_F^{-r}\), so (1.22l) implies \[ U(f(A,hW))= \xi(\det h)^{-1}|\det h|_F^{-(t+m_0)}U(f). \tag{1.22o} \] Take \(h=zI_{m_0}\) with \(z>1\) real. The angular character is one. A normal jet of degree \(j\) on the left of (1.22o) is multiplied by \(z^j\), while the right multiplier is \[ z^{-\delta m_0(t+m_0)}. \] Its absolute value is less than one, whereas \(z^j\ge1\). Tests with one prescribed normal Taylor monomial isolate each coefficient in (1.22n), so every \(U_\alpha\) is zero. The distribution therefore vanishes near the rank-\(r\) stratum. Descending the rank, including \(r=0\), proves the uniqueness lemma. This argument uses only positive real scalar elements of the stabilizer and works over both fields with all angular characters.

5. Fourier equation on the whole Schwartz space

The Fourier transform preserves the Schwartz space continuously, has square \(\Phi(X)\mapsto\Phi(-X)\), and fixes the basic Gaussian. These are the actual earlier whole-space proofs in Additive characters, self-dual measures and Poisson summation on the adèles, Proposition 5.2 and Lemma 5.4A, applied in real dimension \(n^2\) or \(2n^2\). The trace pairing permutes entry coordinates; over \(\mathbb C\) its real coefficient matrix has eigenvalues \(2,-2\) on each scalar pair, matching the factor two in the self-dual measure.

Consider the two entire tempered-distribution families \[ \mathcal D_\chi(s)(\Phi),\qquad \mathcal B_\chi(s)(\Phi)=\mathcal D_{\chi^{-1}}(1-s)(\widehat\Phi). \] The first satisfies (1.22l) with \(t=s+a-(n+1)/2\), first by substitution in its integral and then everywhere by continuation. The second satisfies the same laws. Indeed \[ \widehat{\Phi(h\,\cdot)}(Y)=|\det h|_F^{-n} \widehat\Phi(Yh^{-1}), \] and the dual determinant parameter is \(t'=-t-n\). Its right-equivariance factor therefore combines with the Fourier Jacobian to give \(\xi(\det h)^{-1}|\det h|_F^{-(t+n)}\). The other side is identical with the two matrix multiplication sides exchanged. Notice that the trace pairing requires \(Yh^{-1}\), not a conjugate transpose.

When \(\Re t>0\), the uniqueness lemma makes these two distributions proportional. Its conclusion on compactly supported smooth tests also determines a tempered distribution on the entire Schwartz space. Indeed, multiply a Schwartz test by a smooth cutoff \(\rho(X/R)\), equal to one on \(\|X\|\le R\). Each seminorm of the discarded tail is bounded by \(O(R^{-1})\) times finitely many seminorms with one additional decay power; the derivatives of the cutoff satisfy the same bound. Thus the cutoff tests converge in Schwartz topology, and continuity extends the distribution identity. Determine the scalar using the attaining test \[ \Phi_\chi(X)= \begin{cases} (\det X)^e e^{-\pi\|X\|^2},&F=\mathbb R,\\ (\overline{\det X})^k e^{-2\pi\|X\|^2},&F=\mathbb C,\ k\ge0,\\ (\det X)^{-k}e^{-2\pi\|X\|^2},&F=\mathbb C,\ k<0. \end{cases} \] QR integration, or the earlier Gaussian calculation, gives \[ \mathcal D_\chi(s)(\Phi_\chi)=K_F,\qquad K_{\mathbb R}=C_{\mathbb R} \frac{\pi^{n(n+1)/4}}{\prod_j\Gamma(j/2)},\quad K_{\mathbb C}=C_{\mathbb C} \frac{(2\pi)^{n(n+1)/2}}{2^n\prod_j\Gamma(j)}. \tag{1.22p} \] Both constants are nonzero and independent of \(s,a,e,k\).

With the negative-exponent programme character and positive trace argument, \[ \widehat{\Phi_\chi}=(-i)^{ne}\Phi_{\chi^{-1}} \quad(F=\mathbb R),\qquad \widehat{\Phi_\chi}=(-i)^{n|k|}\Phi_{\chi^{-1}} \quad(F=\mathbb C). \tag{1.22q} \] Here \(\Phi_{\chi^{-1}}\) denotes the opposite angular attaining polynomial, and is independent of the radial exponent. Over \(\mathbb R\), multiplication by a coordinate transforms to \(-(2\pi i)^{-1}\) times the appropriate derivative of the Fourier Gaussian. Every determinant monomial uses distinct coordinates, giving exactly \((-i)^n\det Y\), with transpose harmless. Over \(\mathbb C\), the scalar calculation gives \(\widehat{\bar z\,e^{-2\pi|z|^2}}=-iz\,e^{-2\pi|z|^2}\) and its conjugate counterpart. Successive Wirtinger differentiation of a holomorphic or antiholomorphic polynomial introduces no lower contractions: the resulting polynomial depends on the coordinate independent of that derivative. Hence a homogeneous such polynomial of degree \(N\) has phase \((-i)^N\); determinant powers have degree \(n|k|\). This proves (1.22q) in all ranks.

Equations (1.22p)–(1.22q) determine the proportionality scalar as the value in (1.22d). The identity initially obtained for \(\Re t>0\) extends to every \(s\): for each Schwartz test both sides are entire by (1.22k) and Fourier continuity, so the ordinary identity theorem applies. No overlapping convergence half-plane for the original and transformed integrals is required. This proves the full theorem.

For the positive-exponent characters \(e^{2\pi ix}\) and \(e^{2\pi i(z+\bar z)}\), the same proof replaces \(-i\) by \(i\). In either convention the product of the two epsilon phases is \(\chi((-1)^n)\), exactly the central sign from applying Fourier twice.

For any nonzero scalar \(b\in F\), let \(\psi_b(x)=\psi_F(bx)\). Self-dual matrix measure is multiplied by \(|b|_F^{n^2/2}\), and \[ \widehat\Phi^{\,\psi_b}(Y)=|b|_F^{n^2/2}\widehat\Phi^{\,\psi_F}(bY). \] Substitution \(Y=b^{-1}X\) in the dual integral gives \[ \epsilon_\chi(s,\psi_b)= \chi(b^n)|b|_F^{n(s-1/2)}\epsilon_\chi. \tag{1.22r} \] This includes every nontrivial archimedean additive character by the actual local self-duality proof in Proposition 5.1 of the preceding additive-character lesson.

Replacing \(\chi(x)\) by \(\chi(x)|x|_F^b\) replaces \(s\) by \(s+b\) in the original integral and in (1.22a), while the dual parameter becomes \(1-s-b\). If \(\chi\) is unitary, then \(\Re a=0\); the inverse has radial parameter \(-a=\bar a\) and the same \(e\) or \(|k|\). The gamma definition, initially by its real integral and then by continuation, therefore gives \[ L_{\chi^{-1}}(s)=\overline{L_\chi(\bar s)}. \] The canonical phases in (1.22d) have absolute value one, and (1.22r) retains that property at \(s=1/2\) for every additive character when \(\chi\) is unitary.

6. Preceding proofs and further reading

• Tate's local theory at the infinite places, “Mellin continuation without a functional equation,” equations (5)–(11): actual gamma reciprocal, pole and scalar angular Mellin proofs. Section 1 above supplies the needed stronger multivariable seminorm argument.
• Additive characters, self-dual measures and Poisson summation on the adèles, equations (1)–(2), Proposition 5.2 and Lemma 5.4A: actual character conventions, Gaussian transforms, Schwartz continuity and inversion.
• Proposition 1.7 of this lesson: the complete polynomial Gaussian ideal, including its Gram–Schmidt moment proof. The theorem above adds every Schwartz test and its full Fourier equation.
• B. Rubin, Zeta integrals and integral geometry in the space of rectangular matrices, arXiv:math/0406289, Lemma 2.7, printed p. 13; Lemma 4.2, printed p. 18; Theorem 4.3, printed pp. 19–21. These give the QR/Mellin method and Fourier theorem for the unsigned real determinant case when the matrix is square. Sections 1–5 above include the topology, sign and complex angular arguments rather than importing that source's result for them.
• D. Goldfeld and H. Jacquet, Automorphic representations and L-functions for GL(n), author-hosted notes, Theorem 3.5, printed pp. 15–16, and Lemma 3.6, printed p. 17, formulate the general irreducible archimedean target. Their Fourier definition on printed p. 15 uses \(\psi(-\operatorname{tr}(XY))\); phases from that convention require reversing the character to use (1.22d).

The general irreducible target is unchanged: prove its Gaussian gamma ideal and attaining family, full-Schwartz division and Fourier equation, with the necessary realization, induced-representation and factor-identification arguments. The scalar equivariance in (1.22l) is specific to determinant characters; this proof does not supply those representation-valued arguments.

Canonical matrix gamma ideals for symmetric powers in every archimedean rank

Let \(F=\mathbb R\) or \(\mathbb C\), \(G=GL_n(F)\), and \(\nu(g)=|\det g|_F\). Write \[ \rho_j=\frac{n+1}{2}-j,\qquad B_F(s)=\prod_{j=1}^n\Gamma_F(s+\rho_j), \qquad \Gamma_{\mathbb R}(z)=\pi^{-z/2}\Gamma(z/2),\quad \Gamma_{\mathbb C}(z)=2(2\pi)^{-z}\Gamma(z). \tag{1.30a} \] We use the actual programme characters \(\psi_{\mathbb R}(x)=e^{-2\pi ix}\) and \(\psi_{\mathbb C}(z)=e^{-2\pi i(z+\bar z)}\), self-dual additive matrix measure, and the positive trace argument \[ \widehat\Phi(Y)=\int_{M_n(F)}\Phi(X)\psi_F(\operatorname{tr}(XY))\,dX . \tag{1.30b} \] Put \(dg=C_F\nu(g)^{-n}dX\), where \(C_F>0\) is fixed. The matrix integral is \[ Z_\pi(s,\Phi,c)= \int_G\Phi(g)c(g)\nu(g)^{s+(n-1)/2}\,dg. \tag{1.30c} \] Our Gaussian is \(G_{\mathbb R}(X)=e^{-\pi\|X\|^2}\) or \(G_{\mathbb C}(X)=e^{-2\pi\|X\|^2}\). A polynomial Gaussian means a polynomial in all real coordinates times this Gaussian; over \(\mathbb C\) these are exactly the polynomials in \(X,\bar X\) times \(G_{\mathbb C}\).

For \(m\geq0\) and \(t\in\mathbb C\), consider the actual finite-dimensional smooth representation \[ \pi_{m,t}(g)=\nu(g)^t\operatorname{Sym}^m(g). \tag{1.30d} \] Over \(\mathbb C\) we also allow the antiholomorphic representation \(\nu(g)^t\operatorname{Sym}^m(\bar g)\). The theorem below includes their contragredients. It is a theorem about these explicit families in every rank, not a classification of all admissible representations.

Theorem 1.25. Put \(m=2l+e\), \(e\in\{0,1\}\), in the real case. The canonical Gaussian-polynomial matrix ideals for (1.30d) have generators \[ \begin{aligned} L_{\pi_{m,t}}(s) &=\Gamma_{\mathbb R}(s+t+\rho_1+m+e) \prod_{j=2}^n\Gamma_{\mathbb R}(s+t+\rho_j), &&F=\mathbb R,\\ L_{\pi_{m,t}}(s) &=\Gamma_{\mathbb C}(s+t+\rho_1+m) \prod_{j=2}^n\Gamma_{\mathbb C}(s+t+\rho_j), &&F=\mathbb C . \end{aligned} \tag{1.30e} \] The antiholomorphic complex family has the same formula. For the contragredients, \[ \begin{aligned} L_{\pi_{m,t}^{\vee}}(s) &=\Gamma_{\mathbb R}(s-t-\rho_1-m+e) \prod_{j=2}^n\Gamma_{\mathbb R}(s-t-\rho_j), &&F=\mathbb R,\\ L_{\pi_{m,t}^{\vee}}(s) &=B_{\mathbb C}(s-t), &&F=\mathbb C . \end{aligned} \tag{1.30f} \] The complex antiholomorphic dual again has the same formula. More precisely, the \(\mathbb C[s]\)-span of all polynomial-Gaussian tests and matrix coefficients is exactly \(L_\pi(s)\mathbb C[s]\). One test and one coefficient attain a nonzero constant multiple of the displayed generator. For every Schwartz test and every coefficient, \(Z_\pi/L_\pi\) is entire; on each compact parameter set it has a bound by a fixed finite sum of Schwartz seminorms and a norm on the finite coefficient space.

For \(c^\vee(g)=c(g^{-1})\), the complete scalar functional equation is \[ Z_{\pi^\vee}(1-s,\widehat\Phi,c^\vee) =\epsilon_\pi\frac{L_{\pi^\vee}(1-s)}{L_\pi(s)} Z_\pi(s,\Phi,c),\qquad \epsilon_\pi= \begin{cases}(-i)^e,&F=\mathbb R,\\(-i)^m,&F=\mathbb C.\end{cases} \tag{1.30g} \] The same phases hold for the contragredient and antiholomorphic families. These statements hold for every \(n\geq1\), every \(m\geq0\), and every complex norm exponent \(t\).

Exactly which preceding results enter

The determinant-character theorem of this lesson, Theorem 1.18, supplies the following proved entire distribution family: \[ T_F(z)(\Psi)= \frac{C_F}{B_F(z)} \int_{M_n(F)}\Psi(X)|\det X|_F^{\,z-(n+1)/2}\,dX . \tag{1.30h} \] It is initially the integral in its convergence region, extends to every \(z\), and has compact-parameter Schwartz seminorm bounds. Every polynomial Gaussian has polynomial \(T_F(z)\)-value, and \[ T_F(z)(G_F)=K_F\ne0 \tag{1.30i} \] independently of \(z\). The proof of Theorem 1.18 gives the exact constant \(K_F\), but its value will cancel throughout this argument. It also proves the full determinant functional equation \[ T_F(1-z)(\widehat\Psi)=T_F(z)(\Psi). \tag{1.30j} \] The Gram–Schmidt measure and its first-column gamma moments are the proved calculation of Proposition 1.7, “Proof: all the Gaussian moments,” equations (1.8f)–(1.8g) and the ensuing radial-moment argument.

The scalar gamma recurrence, its pole set and absence of zeros, and the real parity and complex angular Tate normalizations are the actual proofs in Tate’s local theory at the infinite places, “Mellin continuation without a functional equation” and Propositions 8.2–8.4. Full-Schwartz Fourier continuity, inversion, and the basic Gaussian transform are proved in Additive characters, self-dual measures and Poisson summation on the adèles, Proposition 5.2 and Lemma 5.4A. No general induction, classification, globalization or asymptotic theorem is used below.

Representations and coefficient spans

Identify \(\operatorname{Sym}^m(\mathbb C^n)\) with homogeneous polynomials of degree \(m\) in formal variables. It is irreducible under \(\mathfrak{gl}_n(\mathbb C)\): the commuting diagonal operators have distinct monomial joint weights. From any nonzero invariant subspace, polynomial spectral projections in those diagonal operators extract a nonzero monomial. The operators \(x_i\partial_{x_j}\) move one unit of exponent from \(j\) to \(i\), with nonzero coefficient whenever that exponent is positive. They connect every degree-\(m\) monomial to every other. Thus the invariant subspace is the entire space. The differentiated real \(GL_n(\mathbb R)\) action has this complex enveloping-algebra action, and the differentiated holomorphic \(GL_n(\mathbb C)\) action does too. Norm twisting changes no invariant subspaces. Antiholomorphic conjugation and duality preserve irreducibility.

These are actual complete smooth moderate-growth realizations with their finite-dimensional topology and the usual continuous dual pairing. Indeed \(\|\operatorname{Sym}^m(g)\|\leq C\|g\|^m\), and \(|\nu(g)^t|=\nu(g)^{\operatorname{Re}t}\) is bounded by a power of \(1+\|g\|+\|g^{-1}\|\). Every differentiated action is a fixed matrix. The same estimates with \(g^{-1}\) give the dual. No abstract globalization assertion is needed.

Pure powers \(v^{\otimes m}\), \(v\in F^n\), span the symmetric power. If a linear functional killed every pure power, its homogeneous polynomial in the coordinates of \(v\) would vanish identically. A polynomial vanishing on \(\mathbb R^n\), or on \(\mathbb C^n\), has every coefficient zero, by the one-variable polynomial identity successively in each coordinate. The same argument applies to the dual. Consequently all coefficients, with the norm character removed, are spanned by \[ c_{u,v}(X)=(u^tXv)^m,\qquad u,v\in F^n. \tag{1.30k} \] The dual coefficients are spanned by \((u^tX^{-1}v)^m\). Over \(\mathbb C\) complex conjugation gives the corresponding antiholomorphic spans.

In particular, the coefficient \((X_{11})^m\) has left and right \(G\)-translates spanning the entire coefficient space. This last fact uses the full group, not the restriction to \(O(n)\); that restriction of a real symmetric power can be reducible.

Removing exactly the finite polynomial from the base generator

Put \(p=z-(n+1)/2\). In the real case let \[ a=(p+n)/2=(z+\rho_1)/2; \] in the complex case let \[ a=p+n=z+\rho_1. \] At negative integral values of \(a\), (1.30j) makes (1.30h) particularly simple.

Lemma 1.25a (the point distributions). At \(a=-k\), \(k=0,1,\ldots\), \(T_{\mathbb R}(z)\) is a nonzero scalar multiple of \((\det\partial)^{2k}\delta_0\). At \(a=-k\), \(T_{\mathbb C}(z)\) is a nonzero scalar multiple of \((\det\partial_X)^k(\det\partial_{\bar X})^k\delta_0\).

Proof. For the real parameter \(z=-2k-\rho_1\), the dual exponent in (1.30h), at \(1-z\), is \(2k\). Its kernel is the polynomial \(\det(Y)^{2k}\), divided by the nonzero number \(B_{\mathbb R}(1-z)\). All its gamma arguments are positive. In the complex case \(z=-k-\rho_1\) gives dual exponent \(k\), and the dual kernel is \(\det(Y)^k\overline{\det(Y)}^k/B_{\mathbb C}(1-z)\), again with a nonzero denominator. The Fourier transform of a coordinate polynomial is the corresponding derivative of \(\delta_0\), with a nonzero constant. This follows by differentiating (1.30b) and applying its proved Schwartz inversion; entry transposition in the trace pairing leaves the determinant differential operator unchanged. Equation (1.30j) proves the claimed identities as identities on the whole Schwartz space. \(\square\)

Multiplication by \(X_{11}^m\) kills the real point distribution when \(m>2k\), since every term of \((\det\partial)^{2k}\) has total derivative degree \(2k\) in its first row. It kills the complex point distribution when \(m>k\), since the holomorphic first-row derivative degree is \(k\). These assertions hold after applying the derivative to \(X_{11}^m\Psi\): every resulting evaluation at zero vanishes.

The distribution (1.30h) is relatively equivariant under \(X\mapsto AXB\). A direct change of variables proves this in its initial half-plane, and the identity theorem proves it everywhere. Thus if a coefficient polynomial \(c\) annihilates \(T_F(z)\), then every left and right \(G\)-translate of \(c\) also annihilates it. For clarity, the relative scalar is \[ T_F(z)(\Psi(AXB)) =|\det A\det B|_F^{-(z+\rho_1)}T_F(z)(\Psi). \tag{1.30l} \] It is nonzero for every invertible \(A,B\). The coefficient-span result of Section 2 therefore proves the same annihilation for every coefficient of the symmetric power, including arbitrary linear combinations.

For (1.30d), use \(z=s+t\) in this discussion. By (1.30h), every \(Z_{\pi_{m,t}}(s,\Phi,c)/B_F(s+t)\), with the norm factor removed from \(c\), is a polynomial for a polynomial-Gaussian \(\Phi\). It is divisible by \[ \begin{cases} (a)_{l+e},&F=\mathbb R,\quad a=(s+t+\rho_1)/2,\\ (a)_m,&F=\mathbb C,\quad a=s+t+\rho_1, \end{cases} \qquad (a)_r=\prod_{h=0}^{r-1}(a+h). \tag{1.30m} \] The roots here are distinct and were just proved to be zero values for every coefficient and every Schwartz test. The gamma recurrence gives exactly \[ \frac{L_{\pi_{m,t}}(s)}{B_F(s+t)}= \begin{cases} \pi^{-(l+e)}(a)_{l+e},&F=\mathbb R,\\ (2\pi)^{-m}(a)_m,&F=\mathbb C. \end{cases} \tag{1.30n} \] So every polynomial-Gaussian integral is a polynomial multiple of the proposed canonical generator, rather than merely an entire multiple of a larger gamma product.

One coefficient and one Gaussian attain the bound

In the QR calculation \(X=UR\), the first column is \(r_1u\), where \(u\) is a compact uniform unit vector. With determinant weight, the first radial gamma argument is the \(a\) of Section 3. Thus its even moment ratio is \(\pi^{-h}(a)_h\) over \(\mathbb R\), or \((2\pi)^{-h}(a)_h\) over \(\mathbb C\). This is the explicit first-column specialization of Proposition 1.7's Gram–Schmidt measure proof.

For real \(m=2l\), take \[ c(X)=\left(\sum_iX_{i1}^2\right)^l,\qquad \Phi=G_{\mathbb R}. \tag{1.30o} \] This is a coefficient: apply the degree-\(m\) polynomial \((\sum_i x_i^2)^l\) as the dual functional to \(\operatorname{Sym}^m(X)e_1^{\otimes m}\). Its QR value is \(r_1^{2l}\), so \[ Z_{\pi_{m,t}}(s,G_{\mathbb R},\nu^t c) =K_{\mathbb R}L_{\pi_{m,t}}(s). \tag{1.30p} \] For real \(m=2l+1\), take \[ c(X)=X_{11}\left(\sum_iX_{i1}^2\right)^l,\qquad \Phi=X_{11}G_{\mathbb R}. \tag{1.30q} \] Now the angular average is \(\int u_1^2\,du=1/n\): orthogonal invariance makes all coordinate second moments equal and their sum is one. The radial moment is \(r_1^{2l+2}\). Hence \[ Z_{\pi_{m,t}}(s,X_{11}G_{\mathbb R},\nu^t c) =\frac{K_{\mathbb R}}{n}L_{\pi_{m,t}}(s). \tag{1.30r} \]

For the holomorphic complex family, take \[ c(X)=X_{11}^m,\qquad \Phi=\bar X_{11}^mG_{\mathbb C}. \tag{1.30s} \] The angular moment is \(m!/(n)_m\). Here is a proof of its constant. The product complex Gaussian gives the moment \(m!/(2\pi)^m\) in one coordinate. Polar coordinates for the whole \(n\)-vector give radius moment \((n)_m/(2\pi)^m\); the radial gamma integral follows by the same substitution as the one-coordinate integral. Dividing the two equal descriptions of the Gaussian integral yields the angular moment. Therefore \[ Z_{\pi_{m,t}}(s,\bar X_{11}^mG_{\mathbb C},\nu^t c) =\frac{K_{\mathbb C}m!}{(n)_m}L_{\pi_{m,t}}(s). \tag{1.30t} \] For the antiholomorphic family interchange \(X\) and \(\bar X\). All constants in (1.30p), (1.30r), (1.30t) are nonzero and independent of \(s,t\).

To finish the ideal assertion, the family is closed under multiplication by \(s\). Let \(E\) be real scalar dilation on matrix space and \(d=[F:\mathbb R]\). Differentiating a change of variable \(X\mapsto rX\) in an absolutely convergent integral gives \[ Z_{\pi_{m,t}}(s,E\Phi,c) =-\{dn(s+t+\rho_1)+m\}Z_{\pi_{m,t}}(s,\Phi,c), \tag{1.30u} \] where \(c\) on both sides denotes the coefficient including the norm factor. The meromorphic identity continues, and \(E\Phi\) is again a polynomial Gaussian. Thus all polynomials times the attained generator belong to the span. Combined with (1.30n), this proves the exact \(\mathbb C[s]\)-ideal and finite attainment for the original family.

Every dual Gaussian and every dual Schwartz test

The matrix determinant lemma, proved by multilinearity of columns, says \[ \det(X+hvu^t)=\det X\,(1+h\,u^tX^{-1}v). \tag{1.30v} \] Let \(D_{v u^t}\) be differentiation along this rank-one matrix. Over \(\mathbb C\) use its holomorphic directional derivative; for the antiholomorphic family use the conjugate derivative. On invertible matrices, \[ D_{v u^t}^{\,m}|\det X|_F^p =(p)^{\downarrow}_m(u^tX^{-1}v)^m|\det X|_F^p, \qquad (p)^{\downarrow}_m=\prod_{h=0}^{m-1}(p-h). \tag{1.30w} \] In the complex case \(|\det X|_{\mathbb C}^p\) has the local form \(\det(X)^p\overline{\det(X)}^p\), and the holomorphic derivative differentiates only the first factor. In the real case, on either determinant component, differentiating \(|\det X|^p\) gives the same formula.

The identity holds as a distribution for sufficiently large \(\operatorname{Re}p\), by integration by parts. To justify this without a boundary assumption, take \(\operatorname{Re}p>m+1\) on each compact set. Each derivative through order \(m\) is a sum of a polynomial times a power of \(|\det X|\) whose real exponent remains positive. The derivatives extend by zero continuously across \(\det X=0\). Restricting to coordinate lines and using the fundamental theorem of calculus proves they are the distributional derivatives: generic lines have finitely many determinant zeros, and general lines follow by continuous approximation. Schwartz cutoffs remove the boundary at infinity. The identity theorem then proves (1.30w) meromorphically on the whole Schwartz space, using (1.30h).

For a dual coefficient use \(z=s-t\), \(p=z-(n+1)/2\). Equation (1.30w) and the coefficient span prove that every dual polynomial-Gaussian integral has the form \[ B_F(s-t)\frac{Q(p)}{(p)^{\downarrow}_m},\qquad Q\in\mathbb C[p]. \tag{1.30x} \] Indeed a constant-coefficient directional derivative preserves polynomial Gaussians, so the numerator is a polynomial by (1.30h).

At \(p=k\geq0\), the complex normalized determinant kernel is the polynomial \(\det X^k\overline{\det X}^k/B_{\mathbb C}(k+(n+1)/2)\). Equation (1.30v) makes its holomorphic rank-one degree \(k\). Thus its \(m\)-th rank-one derivative is zero when \(k<m\); every root of the denominator in (1.30x) cancels. This proves the dual bound \(B_{\mathbb C}(s-t)\), including all coefficients and all polynomial Gaussian tests.

At real \(p=2h\geq0\), the kernel is the polynomial \(\det X^{2h}/B_{\mathbb R}(2h+(n+1)/2)\). Its rank-one degree is \(2h\), so the derivative vanishes if \(2h<m\). The even roots of the denominator cancel. The remaining factor is \[ R_m(p)=\prod_{h=1}^l(p-(2h-1)). \tag{1.30y} \] With \(b=s-t-\rho_1=p+1\), gamma recurrence gives \[ L_{\pi_{m,t}^{\vee}}(s) =B_{\mathbb R}(s-t)\frac{(2\pi)^l}{R_m(p)}. \tag{1.30z} \] Consequently every dual Gaussian integral is a polynomial multiple of (1.30f). It is not necessary, or correct, to cancel the odd real roots by regarding \(|\det X|^{2h+1}\) as a polynomial.

These arguments also prove entire division for every Schwartz test. For the original family, \(T_F(s+t)(c\Phi)\) is entire and continuous, because multiplication by the fixed coefficient polynomial is a continuous Schwartz operator. Its zeros (1.30m) hold for every Schwartz test by Lemma 1.25a, so division by the finite polynomial is entire. For the dual family, directional derivatives of \(T_F(s-t)\) are entire continuous families. The canceled roots in (1.30x) are zeros of these distributions on every Schwartz test. Dividing by their finite polynomial leaves an entire family; the remaining real odd-root denominator is exactly the one absorbed by (1.30z).

Here is the continuity detail. If a continuous entire distribution family \(A(s)\) vanishes at a root \(s_0\), then each scalar \(A(s)(\Phi)/(s-s_0)\) has a removable singularity. On a small circle around \(s_0\), Cauchy's formula bounds the quotient on an inner disk by a fixed constant times the supremum of \(A(s)(\Phi)\) on that circle. The latter has the compact-parameter Schwartz seminorm bound of (1.30h), also after a polynomial multiplier or directional derivative. Remove the finitely many roots successively and cover a compact parameter set by finitely many disks and their complement. This gives the asserted locally uniform seminorm bounds. The coefficient space is finite dimensional, so a basis makes these bounds jointly continuous in the coefficient. This proves full-Schwartz entire division directly, without assuming a general density reduction.

The scalar Fourier equation and dual attainment

For the rank-one matrix \(M=vu^t\), differentiation of the Fourier kernel gives \[ D_M^m\widehat\Phi=(-2\pi i)^m \widehat{(u^tXv)^m\Phi}. \tag{1.30aa} \] In the complex holomorphic or antiholomorphic case the corresponding Wirtinger derivative gives the same formula. This is the reason for retaining the positive trace argument in (1.30b); entry transposition in the trace makes \(D_M\) pair with \(u^tXv\), not with a Hermitian coefficient.

Put \(w=s+t+\rho_1\), and let \(\gamma_0(s+t)=B_F(1-s-t)/B_F(s+t)\). At the dual variable \(1-s-t\), the determinant exponent is \(-w\). Apply (1.30w), integration by parts, (1.30aa), and the whole-family determinant Fourier equation (1.30j). This gives \[ (-w)^{\downarrow}_m Z_{\pi^\vee}(1-s,\widehat\Phi,\nu^{-t}(u^tX^{-1}v)^m) =(2\pi i)^m\gamma_0(s+t) Z_\pi(s,\Phi,\nu^t(u^tXv)^m). \tag{1.30ab} \] All operations are identities of meromorphic distributions. There is no need for overlapping original convergence half-planes. By the coefficient span, the same scalar equation holds for every coefficient. Since \((-w)^{\downarrow}_m=(-1)^m(w)_m\), its scalar is \[ \frac{(-2\pi i)^m}{(w)_m}\gamma_0(s+t). \tag{1.30ac} \]

For \(F=\mathbb C\), (1.30n) and (1.30f) immediately turn this into (1.30g). For \(F=\mathbb R\), split the factors of \((w)_m\) into their even and odd offsets. With \(h=l+e\), \[ \begin{aligned} \frac{L_\pi(s)}{B_{\mathbb R}(s+t)} &=(2\pi)^{-h}\prod_{j=0}^{h-1}(w+2j),\\ \frac{L_{\pi^\vee}(1-s)}{B_{\mathbb R}(1-s-t)} &=(-1)^l(2\pi)^l \left\{\prod_{j=1}^{l}(w+2j-1)\right\}^{-1}. \end{aligned} \tag{1.30ad} \] Their ratio, times \((-i)^e\), is the multiplier \((-2\pi i)^m/(w)_m\), since \((-i)^e(-1)^l=(-i)^m\). This proves the real phase in (1.30g).

The original attaining test has Fourier transform equal to the following dual attaining test times exactly \(\epsilon_\pi\): \[ \begin{array}{c|c|c} F,m&\Phi&\Phi^\vee\\ \hline \mathbb R,\ m\ {\rm even}&G_{\mathbb R}&G_{\mathbb R}\\ \mathbb R,\ m\ {\rm odd}&X_{11}G_{\mathbb R}&X_{11}G_{\mathbb R}\\ \mathbb C,\ {\rm holomorphic}&\bar X_{11}^mG_{\mathbb C}&X_{11}^mG_{\mathbb C}\\ \mathbb C,\ {\rm antiholomorphic}&X_{11}^mG_{\mathbb C}&\bar X_{11}^mG_{\mathbb C}. \end{array} \tag{1.30ae} \] The real transform is the differentiated Gaussian transform. In the complex case successive Wirtinger differentiation of the Gaussian introduces no contractions, giving degree-\(m\) phase \((-i)^m\). Use \(c^\vee(X)=c(X^{-1})\) for the attaining coefficient of Section 4. Equation (1.30g) and (1.30p), (1.30r) or (1.30t) now show that \((\Phi^\vee,c^\vee)\) attains the dual generator with the same nonzero constant. The dual analogue of (1.30u), with coefficient degree \(-m\) and norm exponent \(-t\), proves polynomial-module stability. Together with Section 5 this finishes the exact dual Gaussian ideal.

Fourier inversion gives the equation for the dual representation too. Its phase must be \(\omega_\pi(-I)/\epsilon_\pi\). Here \(\omega_\pi(-I)=(-1)^m\), so this is \((-i)^e\) over \(\mathbb R\) and \((-i)^m\) over \(\mathbb C\). Thus the two phases multiply to the required central sign.

Standard character data and canonical dual/conjugation

These gamma factors agree with the Tate factors of an explicit normalized Borel embedding, without using an all-rank classification. Let \(B\) be upper triangular and \(\delta_B^{1/2}(b)=\prod_j|b_{jj}|_F^{\rho_j}\). The functional taking the \(e_n^{\otimes m}\)-coordinate obeys \(\ell(\operatorname{Sym}^m(b)v)=b_{nn}^m\ell(v)\). The map \[ v\longmapsto f_v(g)=\ell(\pi_{m,t}(g)v) \tag{1.30af} \] therefore embeds (1.30d) into smooth normalized Borel induction with characters \[ \chi_n(x)=x^m|x|_F^{\,t-\rho_n},\qquad \chi_j(x)=|x|_F^{\,t-\rho_j}\quad(j<n). \tag{1.30ag} \] It is injective because the representation is irreducible and \(\ell\ne0\). The scalar action and differentiated action in (1.30af) are actual smooth group actions. On the compact group the embedding is continuous, and finite dimensionality gives a closed image.

The unordered multiset (1.30ag) consists of \[ x^m|x|_F^{\,t+\rho_1}, \quad |x|_F^{\,t+\rho_2},\ldots,|x|_F^{\,t+\rho_n}. \tag{1.30ah} \] The first real character is \(\operatorname{sgn}(x)^e|x|^{t+\rho_1+m}\). Its Tate factor is the first real factor of (1.30e). The first complex character has angular weight \(m\) and radial exponent \(t+\rho_1+m/2\); its Tate factor is the first complex factor of (1.30e). Antiholomorphic conjugation changes angular weight to \(-m\), whose absolute angular degree is still \(m\). Thus the direct matrix ideal calculation has proved exactly these Tate-product factors, and not merely a gamma product with unspecified shifts.

For the dual representation, the functional taking the first symmetric coordinate transforms under \(B\) by \(b_{11}^{-m}\). The same embedding construction gives \[ \chi'_1(x)=x^{-m}|x|_F^{-t-\rho_1},\qquad \chi'_j(x)=|x|_F^{-t-\rho_j}\quad(j>1). \tag{1.30ai} \] This multiset is the inverse of (1.30ag). The real special factor has parity \(e\) and radial exponent \(-t-\rho_1-m\), giving (1.30f). The complex special factor has angular weight \(-m\) and radial exponent \(-t-\rho_1-m/2\); its absolute angular shift cancels the \(-m/2\), giving \(B_{\mathbb C}(s-t)\). Replacing the absolute angular degree by its signed value would produce a wrong dual factor. The product of the actual scalar Tate phases is precisely (1.30g).

These particular embeddings and the direct integral calculations identify the matrix factors with the stated Tate products for the explicit family. They do not assert that an arbitrary irreducible subquotient of a Borel induction has that induction's entire product as its matrix generator.

Gamma conjugation is exact, since the Euler integral and continuation prove \(\overline{\Gamma(\bar z)}=\Gamma(z)\). Over \(\mathbb R\), conjugation sends \(t\) to \(\bar t\); over \(\mathbb C\), it additionally interchanges the holomorphic and antiholomorphic families. Equations (1.30e)–(1.30f) thus give \[ L_{\bar\pi}(s)=\overline{L_\pi(\bar s)},\qquad L_{\overline{\pi^\vee}}(s)=\overline{L_{\pi^\vee}(\bar s)} \tag{1.30aj} \] with no undetermined exponential or scalar. The phases satisfy \(\epsilon_{\bar\pi}=\omega_\pi(-I)\overline{\epsilon_\pi}\), as also follows by conjugating the Fourier kernel.

If \(n>1\) and \(m>0\), (1.30d) admits no invariant positive Hermitian form: \(\operatorname{diag}(e^r,e^{-r},1,\ldots,1)\), of determinant one, has symmetric-power eigenvalues \(e^{mr}\) and \(e^{-mr}\). A unitary operator has eigenvalues of modulus one, so \(r\ne0\) contradicts unitarity. For \(n=1\), unitarity is exactly \(\operatorname{Re}t=-m\) over \(\mathbb R\) and \(\operatorname{Re}t=-m/2\) over \(\mathbb C\). In those cases (1.30e)–(1.30f) explicitly yield \[ L_{\pi^\vee}(s)=\overline{L_\pi(\bar s)},\qquad |\epsilon_\pi|=1 . \tag{1.30ak} \] For \(m=0\), the unitary condition is \(\operatorname{Re}t=0\) in every rank, and the same identity follows by the symmetry of the \(\rho_j\)'s. This treats every unitary member of the explicit family.

Finally, for \(\psi_b(x)=\psi_F(bx)\), self-dual matrix measure changes by \(|b|_F^{n^2/2}\), and \(\widehat\Phi^{\,\psi_b}(Y)=|b|_F^{n^2/2}\widehat\Phi^{\,\psi_F}(bY)\). Substitute \(X=bY\) in the dual integral. Since \(\omega_\pi(bI)=b^m|b|_F^{nt}\) for the holomorphic or real family, this gives \[ \epsilon_\pi(s,\psi_b)= \omega_\pi(bI)|b|_F^{n(s-1/2)}\epsilon_\pi. \tag{1.30al} \] Use \(\bar b^m|b|_F^{nt}\) for the antiholomorphic family. At \(s=1/2\) this retains absolute value one for every unitary member. An arbitrary determinant norm twist shifts \(s\) as displayed in (1.30e)–(1.30f).

The proof has established a complete canonical Gaussian and whole-Schwartz matrix package for symmetric powers, their antiholomorphic companions, their duals and all norm twists in every real and complex rank. The general infinite-dimensional irreducible generator and the general higher-rank standard-module factor identification remain separate.

Determinant differential operators and the full archimedean matrix family

Let \(F=\mathbb R\) or \(\mathbb C\), let \(d=[F:\mathbb R]\), and put \(G=GL_n(F)\), \(K=O(n)\) or \(U(n)\). Write \(\nu(g)=|\det g|_F=|\det g|^d\). Fix an additive Haar measure \(dX\) on \(M_n(F)\) and \(dg=c_F\nu(g)^{-n}dX\), with \(c_F>0\).

Here is the precise realization hypothesis. The representations \(V,W\) are complete Fréchet smooth representations of \(G\), with continuous moderate-growth action and a continuous invariant bilinear pairing \[ B:V\times W\longrightarrow\mathbb C. \tag{1.25a} \] Their \(K\)-finite cores \(V_0,W_0\) are respectively an irreducible admissible \((\mathfrak g,K)\)-module and its algebraic admissible contragredient, and \(B\) restricts to that duality. Invariance means \(B(\pi(g)v,\widetilde\pi(g)w)=B(v,w)\). For every continuous seminorm \(p\) on \(V\), moderate growth means that \[ p(\pi(g)v)\le C\bigl(1+\|g\|+\|g^{-1}\|\bigr)^N p'(v) \tag{1.25b} \] for some continuous seminorm \(p'\), integer \(N\), and constant \(C\); the analogous condition holds on \(W\). These are conditions on actual supplied realizations. Their existence and comparison for an abstract module are not inferred here.

Put \(c_{v,w}(g)=B(\pi(g)v,w)\), and \[ Z_\pi(s,\Phi;v,w)= \int_G\Phi(g)c_{v,w}(g)\nu(g)^{s+(n-1)/2}\,dg. \tag{1.25c} \] The integral always uses the whole Schwartz space, not just its Gaussian-polynomial subspace.

Dense cores and scalar central operators

The cores \(V_0,W_0\) are dense. To see this, average a vector over a continuous probability kernel on \(K\) supported in a sufficiently small identity neighbourhood. Continuity of its compact orbit makes the average tend to the vector in every prescribed finite list of seminorms. Uniformly approximate the kernel by a finite sum of compact matrix coefficients. This is the actual proved Peter–Weyl theorem in Matrix coefficients and the Peter–Weyl theorem, Theorem 4.1. The error in a seminorm \(p\) is at most the uniform kernel error times \(\sup_{k\in K}p(\pi(k)v)\). A finite coefficient kernel has image in a finite sum of \(K\)-types, hence in \(V_0\). Successively treating the first \(j\) seminorms gives a sequence converging to \(v\). The proof on \(W\) is identical.

Every endomorphism of \(V_0\) commuting with \((\mathfrak g,K)\) is a scalar. Indeed, it preserves a nonzero finite-dimensional \(K\)-isotypic component. An eigenvector there supplies a nonzero kernel of \(T-\lambda\). That kernel is a submodule, so irreducibility makes it all of \(V_0\). A central complexified enveloping element is fixed by the adjoint action of \(K\) in these matrix groups: the complex group \(GL_n(\mathbb C)\) is generated by elementary unipotents and diagonal matrices, all complex exponentials, and a central element is fixed by each exponential. The real compact group is contained in this complex group; for the complex group regarded as real apply this observation to both complexified summands. Thus the central operator does commute with the full \(K\), including its real disconnected component. In particular the centre \(Z(U(\mathfrak g_\mathbb C))\) acts by a character. Continuity of differentiation and density extend each scalar central identity to \(V\). The same statements hold for \(W\).

The admissible contragredient is simple: the annihilator of a proper submodule of \(W_0\) is a nonzero submodule of \(V_0\). Nonzeroness follows on a \(K\)-type on which the submodule is proper, since the pairing of corresponding finite-dimensional isotypic components is perfect. Invariance makes the annihilator a submodule. Thus it is all of \(V_0\), and the original submodule is zero.

The central group elements also act scalarly: they preserve the core and commute with \((\mathfrak g,K)\). Density extends the result to the realization. Denote the resulting group character by \(\omega_\pi\).

A central determinant operator

For real matrices put \[ D=\det\left(\frac{\partial}{\partial x_{ij}}\right),\qquad C=(\det X)D. \tag{1.25d} \] For complex matrices use Wirtinger derivatives and put \[ D_+=\det(\partial/\partial z_{ij}),\quad D_-=\det(\partial/\partial\bar z_{ij}),\quad C_+=(\det Z)D_+,\quad C_-=(\overline{\det Z})D_-. \tag{1.25e} \] On \(G\), \(C\), respectively \(C_+,C_-\), are central invariant differential operators.

Here is a proof that does not assume a Capelli formula. The chain rule gives \[ D\bigl(f(aXb)\bigr)=(\det a)(\det b)(Df)(aXb). \tag{1.25f} \] Multiplication by \(\det X\) cancels these two determinant factors, so \(C\) commutes with both translations. The holomorphic and antiholomorphic chain rules give the same assertions for \(C_+,C_-\). Every left-invariant differential operator is an element of the enveloping algebra of right differentiations: choose an ordered Lie basis, subtract the ordered products with its principal symbol at the identity, and repeat at lower orders. Its coefficients are determined everywhere by left invariance. A bi-invariant operator then commutes with every right first derivative, so its enveloping element is central. This also proves the assertion for the two complexified summands in \(\mathfrak{gl}_n(\mathbb C)_{\mathbb R}\otimes\mathbb C \simeq\mathfrak{gl}_n(\mathbb C)\oplus\mathfrak{gl}_n(\mathbb C)\).

Twist the differentiated representation by \(\nu^t\). In the real case this replaces \(d\pi(A)\) by \(d\pi(A)+t\operatorname{tr}(A)\); in the complex case it makes that replacement on each complexified summand. The central operators therefore act by polynomials \[ b_\pi(t),\qquad b_{\pi,+}(t),\quad b_{\pi,-}(t). \tag{1.25g} \] Each polynomial is monic of degree \(n\). Polynomiality follows by substitution in an enveloping element of order \(n\). Its leading coefficient is the principal symbol on the trace functional: at \(X=I\), \(\partial_{ij}\log\det X=\delta_{ij}\), and the determinant of that diagonal derivative matrix is one.

In particular, on the invertible locus, \[ D\bigl(c(X)|\det X|^t\bigr) =b_\pi(t)(\det X)^{-1}c(X)|\det X|^t . \tag{1.25h} \] The sign character is locally constant, so a second application gives \[ D^2\bigl(c(X)|\det X|^{t+2}\bigr) =b_\pi(t+2)b_\pi(t+1)c(X)|\det X|^t . \tag{1.25i} \] For complex matrices, \(D_+\) does not differentiate an antiholomorphic determinant. Consequently \[ D_+D_-\bigl(c(Z)|\det Z|^{2(t+1)}\bigr) =b_{\pi,+}(t+1)b_{\pi,-}(t+1)c(Z)|\det Z|^{2t}. \tag{1.25j} \] Neither identity assumes that the representation is a principal series, tempered, spherical, finite dimensional or unitary.

Bounds and distributional integration by parts

Lemma 1.21a. In a sufficiently far right half-plane, (1.25c) converges absolutely, is holomorphic, and satisfies bounds \[ \sup_{s\in\Omega}|Z_\pi(s,\Phi;v,w)| \le C_\Omega\,p_\Omega(\Phi)\,p_V(v)\,p_W(w) \tag{1.25k} \] on each compact parameter set \(\Omega\) there. The same assertion holds with any fixed number of parameter derivatives.

Proof. Continuity of \(B\) bounds it by a product of seminorms. Applying (1.25b) yields a uniform polynomial bound for \(c_{v,w}\). Adjugates give \[ 1+\|X\|+\|X^{-1}\| \le C_n(1+\|X\|)^n(1+|\det X|^{-1}). \tag{1.25l} \] Since the additive density of (1.25c) is \(c_Fc_{v,w}(X)|\det X|^{d(s-(n+1)/2)}\), a sufficiently large real part makes its possible negative determinant power disappear. The remaining bound is a polynomial in \(\|X\|\), absorbed by a Schwartz seminorm. This proves absolute convergence and (1.25k). Parameter differentiation adds powers of \(d\log|\det X|\). Near zero, \(x^\epsilon|\log x|^j\) is bounded for every \(\epsilon>0\); at infinity an extra polynomial weight absorbs the logarithm. Dominated differentiation proves holomorphy and the derivative assertions. \(\square\)

For clarity, integration by parts across the singular matrices needs justification. Every coordinate derivative of \(c(X)\) is a finite sum of coefficients of differentiated vectors, with polynomial expressions in the entries of \(X^{-1}\). This follows by writing an additive coordinate tangent at \(X\) as a right tangent \(XA\), where \(A=X^{-1}\dot X\), and iterating the chain rule. For each fixed derivative order, (1.25l), moderate growth and the derivative seminorms therefore give another bound by a polynomial in \(\|X\|\) times a finite negative determinant power. The same is true after differentiating \(|\det X|^{dt}\).

Take \(\operatorname{Re}t\) sufficiently large for all derivatives up to order \(2n\). On every compact set, these derivatives tend to zero at the singular locus. Extending the function by zero there gives a \(C^{2n}\) function. One elementary verification is to restrict to coordinate lines on which the determinant polynomial is not identically zero. Such lines have finitely many singular points; the fundamental theorem of calculus, and the vanishing one-sided derivatives, prove the derivative identity across them. Lines of this kind are dense among parallel lines, because the determinant polynomial is not identically zero on any open set. Continuity then proves the derivative identity on every line, successively up to order \(2n\).

These extended derivatives have polynomial growth at infinity. Integrate first with a compact cutoff and then let its radius tend to infinity; all cutoff terms tend to zero against a Schwartz test. Thus (1.25i) and (1.25j) hold as distributional identities in a far right half-plane. The transposed differential operator has the same sign: its total order is \(2n\).

Put \(c_n=(n+1)/2\), and define \[ (a,A,B_\pi(s))= \begin{cases} (2,D^2,b_\pi(s-c_n+2)b_\pi(s-c_n+1)),&F=\mathbb R,\\ (1,D_+D_-,b_{\pi,+}(s-c_n+1)b_{\pi,-}(s-c_n+1)), &F=\mathbb C. \end{cases} \tag{1.25m} \] Then \[ B_\pi(s)Z_\pi(s,\Phi;v,w) =Z_\pi(s+a,A\Phi;v,w). \tag{1.25n} \] This is initially a proved integral identity, not a presupposed functional equation.

A common gamma majorant for every smooth coefficient

Theorem 1.21. Factor the monic degree-\(2n\) polynomial \(B_\pi(s)=\prod_{\ell=1}^{2n}(s-\beta_\ell)\), including multiplicities, and put \[ H_\pi(s)=\prod_{\ell=1}^{2n} \Gamma\left(\frac{s-\beta_\ell}{a}\right). \tag{1.25o} \] For all \(\Phi\in\mathcal S(M_n(F)),v\in V,w\in W\), \[ Z_\pi(s,\Phi;v,w)=H_\pi(s)Q_\pi(s,\Phi;v,w), \tag{1.25p} \] where \(Q_\pi\) is entire in \(s\), trilinear, and obeys a bound of the form (1.25k) on every compact subset of \(\mathbb C\). Thus (1.25c) has meromorphic continuation as a continuous tempered family with a fixed gamma pole majorant in every rank. On any closed vertical strip, after removing its finitely many possible poles by a fixed polynomial, the resulting family decreases faster than every inverse power of \(1+|\operatorname{Im}s|\), with joint seminorm bounds.

Proof. The gamma recurrence gives exactly \[ H_\pi(s+a)=a^{-2n}B_\pi(s)H_\pi(s). \tag{1.25q} \] The gamma poles, entire reciprocal and absence of zeros are the actual scalar proofs in Tate’s local theory at the infinite places, “Mellin continuation without a functional equation”. No general representation-theoretic local factor is imported from that scalar result.

Choose a right half-plane beyond all roots of \(B_\pi\) and all thresholds used above. There \(Z/H_\pi\) is holomorphic, and (1.25n) becomes \[ Q_\pi(s,\Phi)=a^{-2n}Q_\pi(s+a,A\Phi). \tag{1.25r} \] For a compact parameter set choose \(N\) so that its translate by \(Na\) lies in this half-plane, and define \[ Q_\pi(s,\Phi;v,w) =a^{-2nN} \frac{Z_\pi(s+Na,A^N\Phi;v,w)}{H_\pi(s+Na)}. \tag{1.25s} \] Increasing \(N\) does not change this expression, by (1.25r) in the original half-plane. These formulas therefore patch to an entire family. The differential operator \(A^N\) is continuous on the Schwartz space. Lemma 1.21a bounds the numerator uniformly on a compact set, and \(1/H_\pi\) is bounded there. This proves the required joint seminorm bounds. Meromorphic continuation and uniqueness follow from agreement with the original integral in its convergence domain.

For the strip assertion, a bounded real-part interval meets only finitely many poles of the finitely many gamma factors in (1.25o). Choose a polynomial \(P_{\mathrm{strip}}\) canceling these poles with their full possible multiplicities. Iteration of the already proved meromorphic identity (1.25n) gives \[ Z_\pi(s,\Phi;v,w)= \frac{Z_\pi(s+Na,A^N\Phi;v,w)} {\prod_{k=0}^{N-1}B_\pi(s+ka)}. \tag{1.25sa} \] Choose \(N\) large enough to put the entire shifted strip in the absolute-convergence domain. The modulus of \(\nu(g)^{i\operatorname{Im}s}\) is one, so the numerator is bounded uniformly in the imaginary part by a fixed Schwartz seminorm and the same coefficient seminorms used in Lemma 1.21a. For large imaginary part the denominator is bounded below by a positive constant times \((1+|\operatorname{Im}s|)^{2nN}\). Increasing \(N\) makes this dominate the degree of \(P_{\mathrm{strip}}\) and any prescribed inverse power. On the remaining compact part of the strip, \(P_{\mathrm{strip}}H_\pi Q_\pi\) is holomorphic and has the previously proved local seminorm bound. This proves the strip assertion, without assuming a functional equation. \(\square\)

The factor \(H_\pi\) is a majorant, not the canonical standard gamma generator. Confusing these would create a false proof. For instance, in rank one with trivial real character, \[ B(s)=s(s+1),\quad H(s)=\Gamma(s/2)\Gamma((s+1)/2),\quad Z(s,e^{-\pi x^2})=\pi^{-s/2}\Gamma(s/2). \tag{1.25t} \] Its quotient by \(H\) is \(\pi^{-s/2}/\Gamma((s+1)/2)\). It has infinitely many zeros and is not a polynomial. Taking a polynomial gcd of these majorant quotients would not be legitimate.

The exact Gaussian-to-Schwartz division lemma

Let \(\mathcal S_0\) be the span of the polynomial multiples of \(e^{-\pi\operatorname{tr}(XX^t)}\) over \(\mathbb R\), or of \(e^{-2\pi\operatorname{tr}(XX^*)}\) over \(\mathbb C\). It is dense in \(\mathcal S(M_n(F))\). The actual proof is Hermite functions, tempered distributions and the Schwartz kernel theorem, Lemma 3.1 and Theorem 3.2. After a real scalar dilation, its finite Hermite sums are exactly Gaussian-polynomial tests in the real coordinates; complex polynomials in \(Z,\bar Z\) are the same polynomial algebra. That theorem proves convergence in every Schwartz seminorm, not merely \(L^2\) density.

Proposition 1.21b (conditional canonical division, with the missing hypothesis isolated). Suppose a nonzero meromorphic function \(L_\pi(s)\) has the following Gaussian ideal property: \[ \operatorname{span}_{\mathbb C} \{Z_\pi(s,\Phi;v,w): \Phi\in\mathcal S_0,\ v\in V_0,\ w\in W_0\} =L_\pi(s)\mathbb C[s]. \tag{1.25u} \] Then every full-Schwartz, full-smooth-vector quotient \(Z_\pi/L_\pi\) is entire, with local joint seminorm bounds. Moreover \(L_\pi\) has no zeros.

Proof. Equality (1.25u) supplies a finite attaining family: \(L_\pi=\sum_{i=1}^r Z_\pi(\Phi_i;v_i,w_i)\). Theorem 1.21 therefore shows that \[ h_\pi(s)=L_\pi(s)/H_\pi(s) \tag{1.25v} \] is an entire nonzero function. For every Gaussian-core test, \(Q_\pi=h_\pi P\) with \(P\in\mathbb C[s]\).

Approximate \(\Phi,v,w\) in their respective topologies by Gaussian-polynomial and core vectors. Theorem 1.21's joint compact-parameter bounds make the corresponding \(Q_\pi\)'s converge locally uniformly in \(s\). If \(h_\pi\) has a zero of order \(m\) at \(s_0\), all approximating functions and their first \(m-1\) derivatives vanish there. Cauchy's formula makes the derivatives converge too. The limit is divisible by \((s-s_0)^m\). Consequently \(Q_\pi/h_\pi\), and hence \(Z_\pi/L_\pi\), is entire.

This also proves continuity, rather than just pointwise division. On a sufficiently small circle about \(s_0\), \(h_\pi\) has no zeros. For \(s\) inside the circle the entire quotient equals the Cauchy integral of \(Q_\pi(\zeta)/h_\pi(\zeta)\). Theorem 1.21 bounds that integral by fixed Schwartz and vector seminorms. Away from the isolated zeros, divide directly by the nonvanishing \(h_\pi\). A finite cover treats every compact parameter set.

Finally, at any prescribed \(s_0\), choose a coefficient nonzero at some \(g_0\). Take a nonnegative bump \(\eta\) supported in a small compact subset of \(G\), where that coefficient is nonzero, and set \[ \Phi(g)=\eta(g)\overline{c_{v,w}(g)} \nu(g)^{-i\operatorname{Im}s_0}. \tag{1.25w} \] It extends by zero to a Schwartz function on the matrix space. The entire compact-support integral at \(s_0\) is strictly positive, up to the fixed positive Haar constant. If \(L_\pi\) had a zero there, the entire division just proved would force every such integral to vanish. This contradiction proves the absence of zeros. \(\square\)

The hypothesis (1.25u) has not been proved for a general irreducible representation here. Theorem 1.21 does not imply it: its normalized functions lie in \(\mathcal O(\mathbb C)\), not necessarily in \(\mathbb C[s]\). Proposition 1.21b supplies the full continuity and multiplicity-sensitive density argument once a genuine Gaussian generator and finite attaining family have been established.

A scalar Fourier equation for all archimedean matrix coefficients

Use the actual smooth dual-pair realizations, measures and integrals (1.25a)–(1.25c) of Determinant differential operators and the full archimedean matrix family. The argument below supplies a meromorphic scalar Fourier equation in all ranks over \(\mathbb R,\mathbb C\). It does not identify the canonical Gaussian generator.

Fix a nontrivial unitary additive character \(\psi\), its self-dual additive measure, and the positive-trace convention \[ \widehat\Phi(Y)=\int_{M_n(F)}\Phi(X)\psi(\operatorname{tr}(XY))\,dX. \tag{1.26a} \] The actual whole-space Schwartz continuity and inversion proof is Additive characters, self-dual measures and Poisson summation on the adèles, Proposition 5.2 and Lemma 5.4A, applied in real dimension \(dn^2\). Trace permutes the matrix coordinates, so \(\widehat{\widehat\Phi}(X)=\Phi(-X)\).

Finite central algebra on unipotent coinvariants

Lemma 1.22a. Let \(\mathfrak g=\mathfrak{gl}_n(\mathbb C)\), and let \(\mathfrak p=\mathfrak l+\mathfrak u\) be the two-block upper parabolic of sizes \(r,m=n-r\). If the centre of \(U(\mathfrak g)\) acts on a module \(M\) by a character \(\alpha\), there is a monic polynomial \(q_{\alpha,r}\) such that \[ q_{\alpha,r}(z_m)(M/\mathfrak uM)=0,\qquad z_m=\operatorname{diag}(0_r,I_m). \tag{1.26b} \] For \(\mathfrak{gl}_n(\mathbb C)\) regarded as a real Lie algebra the same assertion holds for the sum of the two complexified block-centre elements.

Proof. Ordered monomials in a Lie basis span its enveloping algebra by replacing inverted pairs \(YX\) by \(XY+[Y,X]\). They are independent for the Lie algebras here: realize them as invariant differential operators on the corresponding complex matrix group. Their highest symbols at the identity are the independent commutative tangent monomials. Descending through the orders proves independence. This proves the required PBW decomposition, in particular \[ U(\mathfrak g) =U(\mathfrak u)\,U(\mathfrak l)\,U(\mathfrak u^-) \quad\text{as an ordered vector-space decomposition.} \tag{1.26c} \] Its centres have associated graded algebras \[ \operatorname{gr}Z(U(\mathfrak{gl}_a)) =S(\mathfrak{gl}_a)^{GL_a} =\mathbb C[e_1,\ldots,e_a]. \tag{1.26d} \] Indeed, a central highest symbol is invariant. Conversely symmetrization intertwines the adjoint Lie action, so an invariant polynomial symmetrizes to a central element with that symbol. For the last equality restrict to diagonal matrices: the dense open set of distinct-eigenvalue matrices is diagonalizable, so restriction is injective; permutation invariance makes the restriction symmetric. Successively subtracting products of elementary symmetric functions with the same lexicographic leading monomial expresses every such polynomial in the \(e_i\)'s. These extend as characteristic- polynomial coefficients.

Project a central element onto its \(U(\mathfrak l)\) term in (1.26c), and call the projection \(\Gamma\). Every other term lies in \(\mathfrak uU(\mathfrak g)\). To see this, use a block-centre element whose adjoint weights are positive on \(\mathfrak u\) and negative on \(\mathfrak u^-\). A weight-zero ordered monomial containing a negative part must contain a positive part. Thus \[ z-\Gamma(z)\in\mathfrak uU(\mathfrak g). \tag{1.26e} \] Projection is \(\mathfrak l\)-equivariant, so \(\Gamma(z)\) is central in \(U(\mathfrak l)\). It is an algebra homomorphism on the centre: the right ideal \(\mathfrak uU(\mathfrak g)\) is preserved by left multiplication by \(U(\mathfrak l)\); use (1.26e) on two factors and project their product. Its leading-symbol map is restriction to block diagonal matrices.

The algebra \(Z(U(\mathfrak l))\) is a finite module over \(\Gamma Z(U(\mathfrak g))\). Here are the details. The block invariant polynomial ring is \(\mathbb C[x_1,\ldots,x_n]^{S_r\times S_m}\). Each \(x_i\) satisfies the degree-\(n\) monic polynomial with the full symmetric coefficients. Reducing every power \(x_i^n\) shows that the monomials with each exponent \(<n\) span the entire polynomial ring over the full symmetric ring. Average this finite set over \(S_r\times S_m\). Its averages span the block invariant ring, because the coefficients in the full symmetric ring are already invariant. Lift these homogeneous averages to central elements of \(U(\mathfrak l)\) by (1.26d), and lift the full elementary symmetric generators to \(Z(U(\mathfrak g))\). Subtract a lifted expression for the highest symbol and repeat at lower filtered degree. Induction proves finite generation of the central enveloping algebra.

By (1.26e), \(\Gamma(z)\) acts on \(M/\mathfrak uM\) as \(\alpha(z)\). The quotient central algebra by these scalar relations is finite dimensional. Multiplication by \(z_m\) obeys its monic characteristic polynomial; that polynomial acts as zero on every module over the quotient. This proves (1.26b). If the quotient is zero, use the polynomial \(1\). For the complex group regarded as real, apply the argument to both complexified summands. Their tensor product central quotient is finite dimensional, and the sum of the block-centre elements again obeys its monic characteristic polynomial. \(\square\)

This statement applies to the entire supplied smooth realization as a Lie module: central scalar identities extend from its dense core by continuity. No smooth/core coinvariant comparison or Jacquet admissibility is assumed.

Relative descent and normal jets

Lemma 1.22b (relative descent). Let \(H\) be a unimodular Lie group, \(P\) a closed subgroup, \(E\) a complete locally convex smooth \(H\)-module, and \(J\) a finite-dimensional \(P\)-module with action \(\tau\). Assume the orbit projection has local smooth product sections. In the application below these are the explicit minor-chart sections (1.26l). Jointly continuous relative functionals on compactly supported smooth sections of \(H\times_PJ\), together with \(E\), are in bijection with continuous \(\lambda\) on \(E\otimes J\) satisfying \[ \lambda(\sigma(p)e\otimes\tau(p)j) =\chi(p)\delta_P(p)\lambda(e\otimes j),\qquad \delta_P(p)=|\det(\operatorname{Ad}(p);\operatorname{Lie}P)|. \tag{1.26f} \]

Proof. For left Haar measure on \(P\), \(\int f(tp)\,dt=\delta_P(p)\int f(t)\,dt\). In exponential coordinates the conjugation Jacobian is the displayed determinant; right translation is the composition of left translation and conjugation. This fixes the modular sign.

A lifted test \(\Psi\in C_c^\infty(H,J)\) maps to the section whose value in the frame \(h\) is \[ Q\Psi(h)=\int_P\tau(p)\Psi(hp)\,dp. \tag{1.26g} \] It obeys \(Q\Psi(hp_0)=\tau(p_0)^{-1}Q\Psi(h)\). Use the supplied local smooth product sections. A finite partition of unity on each compact support, and a compact fibre bump of integral one, show that \(Q\) is onto. The partitions needed here can be constructed in the Euclidean orbit charts: choose finitely many smooth compact bumps whose positive sets cover the compact support, and divide them by their positive sum.

Its kernel is the closed span of \[ R_p\Psi-\delta_P(p)\tau(p)^{-1}\Psi. \tag{1.26h} \] Here is the local kernel calculation. In a real coordinate rectangle, a compact scalar test with zero integral is a sum of compactly supported coordinate derivatives: choose unit-mass one-variable bumps, successively subtract each marginal times its bump, and take the compact primitive of each zero-marginal remainder. Insert the Haar density to obtain Haar divergences. Right-invariant Lie fields span the tangent space; expressing a vector field in that frame makes its Haar divergence a sum of right Lie derivatives with their constant modular correction. Each corrected derivative is the limit of differences in (1.26h). A finite partition of unity handles compact support. The integrals of the pieces are moved to one fixed bump through overlapping charts in a component; right translation also moves between components, with precisely the factor \(\delta_P\). For the \(J\)-valued version apply this argument componentwise after multiplication by \(\tau(p)\). In local product charts this is exactly the condition of zero fibre integral in (1.26g). Finite partitions on the base prove the stated kernel.

The lifted relative functional has the regular-group form \[ D(\Psi,e)=\int_H \chi(h)\lambda(\sigma(h)^{-1}e\otimes\Psi(h))\,dh. \tag{1.26i} \] To derive it, permit compact smooth \(E\otimes J\)-valued tests and conjugate the diagonal regular action by \((JF)(h)=\chi(h)^{-1}\sigma(h)F(h)\). The relative functional composed with \(J\) is left invariant. A scalar left-invariant distribution is a multiple of Haar integration: convolve it with a compact smooth approximate identity. Invariance makes each convolution a constant smooth function. Testing one fixed unit-mass test makes those constants converge; convergence of the convolutions to the distribution proves the claim. Apply it to every fixed vector, obtaining a continuous \(\lambda\), and undo \(J\). This gives (1.26i).

The vector-valued extension just used can be checked without assuming a nuclearity theorem. In a compact coordinate cube, cut off and periodize a smooth vector-valued test. Its Fourier coefficients in any continuous seminorm decrease faster than every power: integrate by parts sufficiently many times. Taking that order larger than the dimension plus the distribution order gives an absolutely convergent sum of finite vector-valued Fourier tests for the relevant seminorm and derivatives. A joint bound therefore extends the functional uniquely. For the coefficient variables one uses the completed projective tensor, whose seminorm on a finite tensor is the infimum of \(\sum_i p(v_i)q(w_i)\); the same estimate proves the extension there.

Finally, (1.26i) kills (1.26h) exactly when \[ \chi(p)^{-1}\lambda(\sigma(p)e\otimes j) =\delta_P(p)\lambda(e\otimes\tau(p)^{-1}j). \] Replace \(j\) by \(\tau(p)j\) to get (1.26f). Conversely that equation makes (1.26i) descend by the proved kernel statement. All estimates are continuous on compact supports. \(\square\)

Lemma 1.22c (normal jets). A distribution supported on a smooth closed submanifold and locally of order at most \(N\) depends only on its first \(N\) normal jets. Its highest nonzero normal part is a distribution on the submanifold with the dual of \(\operatorname{Sym}^j(N^*)\), for some \(j\le N\). The assertion includes jointly continuous coefficient variables.

Proof. Write the submanifold locally as \(y=0\) in coordinates \((x,y)\). A test whose normal derivatives through order \(N\) vanish is, by Taylor's integral remainder, a sum of \(y^\alpha f_\alpha\), \(|\alpha|=N+1\). Multiply by a cutoff supported in \(|y|<2\epsilon\) and equal to one in \(|y|<\epsilon\). Support makes the distribution value unchanged. Every derivative through order \(N\) of the cutoff remainder is \(O(\epsilon)\), so the order bound makes the value tend to zero. Arbitrary finite jets are attained by \(\chi(y)\sum y^\alpha f_\alpha(x)/\alpha!\). The highest normal degree transforms under coordinate change by the induced normal-bundle action. This proves the assertion. Its estimates carry the unchanged coefficient seminorms, which proves the vector-valued assertion too. \(\square\)

Rank strata and generic uniqueness

Let \(H=G\times G\) act on matrices by \(aXb^{-1}\), on tests by \(\rho(a,b)\Phi(X)=\Phi(a^{-1}Xb)\), and on \(E=W\widehat\otimes V\) by \[ \sigma(a,b)=\widetilde\pi(a)\otimes\pi(b). \quad u_s=s+\frac{n-1}{2},\qquad \chi_s(a,b)=(\nu(a)/\nu(b))^{u_s}. \tag{1.26j} \] The tensor completion encodes the joint coefficient seminorm bounds, rather than any representation- theoretic tensor classification.

The rank-\(r\) orbit has base point \(x_r=\operatorname{diag}(I_r,0_m)\), with stabilizer \[ a=\begin{pmatrix}A&B\\0&D\end{pmatrix},\qquad b=\begin{pmatrix}A&0\\C&D'\end{pmatrix}. \tag{1.26k} \] It is a smooth orbit, with explicit sections on invertible-minor charts: \[ \begin{pmatrix}U&V\\W&WU^{-1}V\end{pmatrix} =\begin{pmatrix}U&0\\W&I_m\end{pmatrix} x_r \begin{pmatrix}I_r&U^{-1}V\\0&I_m\end{pmatrix}. \tag{1.26l} \] Permuting rows and columns covers the orbit. The projective tensor action is smooth: on finite tensors its derivatives are the finite Leibniz sums of the continuous derivative operators on \(V,W\). On a compact group-coordinate set their seminorm bounds extend to the tensor completion. Approximate by finite tensors and use the fundamental theorem of calculus in each coordinate to pass these derivative identities to the limit, successively at every order. Its modular factor is \[ \delta_r(a,b)=|\det D'/\det D|^{dr}. \tag{1.26m} \] Indeed \(B_0\mapsto AB_0D^{-1}\) has determinant \(|\det A|^{dm}|\det D|^{-dr}\); \(C_0\mapsto D'C_0A^{-1}\) has determinant \(|\det D'|^{dr}|\det A|^{-dm}\). Conjugation on each reductive block has determinant one, proving (1.26m). The normal space is the bottom-right matrix block, with action \(Y\mapsto DY(D')^{-1}\). The two stabilizer unipotent blocks act trivially on that normal space and its symmetric jet powers.

Proposition 1.22d (generic uniqueness). Outside a countable subset of \(s\), the space of jointly continuous functionals satisfying \[ T(\rho(a,b)\Phi,\sigma(a,b)e) =\chi_s(a,b)T(\Phi,e) \tag{1.26n} \] has dimension at most one. This concerns the full Schwartz and smooth coefficient spaces in (1.25a).

Proof. First consider a boundary-supported relative functional. Take its highest support rank \(r<n\), on the open set of rank at least \(r\). Its rank-\(r\) orbit is a closed smooth submanifold there. Joint continuity bounds test order by a finite \(N\) near one base point, uniformly in the coefficient variables. Covariance transports this bound to every point of the orbit. Lemma 1.22c gives a highest nonzero normal degree \(j\le N\). Lemma 1.22b then gives a nonzero \(\lambda\) satisfying (1.26f) for \(J=\operatorname{Sym}^j(N^*)\).

Vary only the first-factor upper unipotent block in (1.26k), with second component the identity. Its character, modular factor and action on normal jets are trivial. Differentiating shows that \(\lambda\) factors through \(W/\mathfrak uW\) in its first variable. Lemma 1.22a supplies a monic polynomial \(q_{\widetilde\pi,r}\) annihilating the radial block-centre operator on that quotient.

Take \(a_t=\operatorname{diag}(I_r,tI_m)\), \(b=I_n\), with real \(t>0\). A normal degree-\(j\) jet has weight \(t^{-j}\), while \(\chi_s\delta_r\) has weight \(t^{dm(u_s-r)}\). Equation (1.26f) says \[ \lambda(\widetilde\pi(a_t)w\otimes v\otimes\xi) =t^{dm(u_s-r)+j}\lambda(w\otimes v\otimes\xi). \tag{1.26o} \] Differentiate at one and apply the annihilating polynomial. A nonzero \(\lambda\) requires \[ q_{\widetilde\pi,r}\bigl(dm(u_s-r)+j\bigr)=0. \tag{1.26p} \] For each \(r,j\) its nonconstant affine argument allows only finitely many parameters. The union over \(0\le r<n,\ j\ge0\) is countable. Off that union the highest normal part vanishes. Descending in the normal degree and then in the finitely many ranks proves that every boundary- supported relative distribution is zero.

For the full-rank orbit the stabilizer is diagonal \(G\), with trivial character and modular factor. Lemma 1.22b identifies its relative functionals with continuous invariant pairings on \(W\times V\), a one-dimensional space. Indeed a core pairing gives a \((\mathfrak g,K)\)-map \(V_0\to(W_0)^\sim=V_0\): \(K\)-invariance makes each functional \(K\)-finite, and finite-dimensional isotypic duality proves this bidual equality. The scalar-endomorphism proof under “Dense cores and scalar central operators” applies. Density of the cores extends the scalar identity.

Two relative distributions with proportional full-rank restrictions thus have zero difference off the countable set. This proves the result for compactly supported tests. These tests are dense in Schwartz space: cut off outside a radius-\(R\) ball; rapid decay makes every Schwartz seminorm of the error tend to zero, including the cutoff derivatives. A continuous Schwartz functional is determined by its compact-test restriction, proving the stated result. \(\square\)

This proof uses neither finite length nor a principal-series embedding or asymptotic expansion. It does not assert that the exceptional set is finite.

The scalar Fourier equation

Theorem 1.22. A nonzero meromorphic scalar \(\gamma_{\rm mat}(s,\pi,\psi)\) exists, independent of all tests and coefficient vectors, such that \[ Z_{\widetilde\pi}(1-s,\widehat\Phi;w,v) =\gamma_{\rm mat}(s,\pi,\psi) Z_\pi(s,\Phi;v,w). \tag{1.26q} \] It holds for every full Schwartz test and all smooth vectors in the actual dual pair (1.25a).

Proof. Theorem 1.21 supplies meromorphic continuous families for both sides. Substitution in the original convergence domain, followed by continuation, gives \[ Z_{\pi,s}(\rho(a,b)\Phi,\sigma(a,b)e) =\chi_s(a,b)Z_{\pi,s}(\Phi,e). \tag{1.26r} \] The positive Fourier chain rule is \[ \widehat{\rho(a,b)\Phi} =(\nu(a)/\nu(b))^n\rho(b,a)\widehat\Phi. \tag{1.26s} \] Change \(X=aYb^{-1}\); its additive Jacobian is \((\nu(a)/\nu(b))^n\), and trace is cyclic. Swap the coefficient variables and use \[ n-\left(1-s+\frac{n-1}{2}\right) =s+\frac{n-1}{2}. \tag{1.26t} \] The dual Fourier family therefore has the same character \(\chi_s\).

Off Proposition 1.22d's countable set and the discrete majorant pole sets, the two families are proportional. Choose a fixed compact test supported inside \(G\) and coefficient pair whose integral is nonzero at a real parameter in the convergence domain. Its integral \(A(s)\) is entire and not identically zero. Evaluate the Fourier family on this same triple, obtaining meromorphic \(D(s)\), and define \(\gamma_{\rm mat}=D/A\). This is meromorphic on \(\mathbb C\). On the dense complement of the exceptional set, poles and zeros of \(A\), it is the proportionality scalar. For each other triple, the difference in (1.26q) is a meromorphic function vanishing on that dense set. The identity theorem proves the equation everywhere.

The scalar cannot vanish identically. Otherwise the dual Fourier family vanishes on every triple. Fourier transformation and swapping the coefficient spaces are bijections. The entire dual original family would then vanish, contradicting an ordinary nonzero compact-support integral in its convergence domain. \(\square\)

The exact reflection and character-change identities are \[ \gamma_{\rm mat}(s,\pi,\psi) \gamma_{\rm mat}(1-s,\widetilde\pi,\psi) =\omega_\pi(-I_n), \tag{1.26u} \] \[ \gamma_{\rm mat}(s,\pi,\psi_a) =\omega_\pi(aI_n)|a|_F^{n(s-1/2)} \gamma_{\rm mat}(s,\pi,\psi), \qquad \psi_a(x)=\psi(ax). \tag{1.26v} \] Applying (1.26q) twice reflects the test; substituting \(g\mapsto-g\) contributes \(\omega_\pi(-I_n)\), proving (1.26u). For (1.26v), the self-dual measure scales by \(|a|_F^{n^2/2}\), and \(\widehat\Phi_{\psi_a}(Y)=|a|_F^{n^2/2} \widehat\Phi_\psi(aY)\). In the dual integral substitute \(Y=aX\). The inverse coefficient contributes \(\omega_\pi(aI_n)\), and its determinant power contributes \(|a|_F^{-n(1-s+(n-1)/2)}\). The product is the exponent in (1.26v). These are meromorphic identities, initially verified by substitution in the convergent dual integral and then continued.

Consequences of a genuine Gaussian generator

Corollary 1.22e (conditional canonical epsilon). For \(\psi_{\mathbb R}(x)=e^{-2\pi ix}\) and \(\psi_{\mathbb C}(z)=e^{-4\pi i\operatorname{Re}z}\), suppose the Gaussian ideal (1.25u) holds for both \(\pi\) and \(\widetilde\pi\). Then Proposition 1.21b gives canonical full- Schwartz division and absence of zeros, and \[ \gamma_{\rm mat}(s,\pi,\psi) =\epsilon_\pi\,L_{\widetilde\pi}(1-s)/L_\pi(s), \qquad \epsilon_\pi\in\mathbb C^\times \text{ constant}. \tag{1.26w} \]

Proof. Differentiating the basic Gaussian Fourier identity shows that Fourier transformation preserves polynomial Gaussian tests for these standard characters. Trace transpose and the complex pairing's real sign just change polynomial coordinates. Swap the core coefficient spaces too. Taking the Gaussian family spans in (1.26q) therefore gives \[ \gamma_{\rm mat}(s,\pi,\psi)L_\pi(s)\mathbb C[s] =L_{\widetilde\pi}(1-s)\mathbb C[s]. \tag{1.26x} \] The ratio in (1.26w) multiplies \(\mathbb C[s]\) onto itself. Applying this to \(1\) in both directions makes the ratio and its inverse polynomials. Polynomial units are nonzero constants, proving the assertion. General character dependence follows from (1.26v). \(\square\)

Theorem 1.31 below supplies the general Gaussian ideal and its finite attaining family in every prescribed actual complete smooth moderate dual-pair model. The common gamma majorant, scalar Fourier equation and polynomial-module closure do not establish it. These arguments do not identify the canonical factors with the gamma factors of a local Langlands parameter.

Unitary conjugation and the critical line

Corollary 1.22f. Suppose the dual pair is the smooth pair furnished by an actual admissible irreducible unitary Hilbert representation, with the compatible Hilbert pairing. For every nontrivial unitary additive character, the scalar in Theorem 1.22 has neither a zero nor a pole on \(\operatorname{Re}s=1/2\), and \[ |\gamma_{\rm mat}(1/2+it,\pi,\psi)|=1\qquad(t\in\mathbb R). \tag{1.26y} \] If both Gaussian ideals exist and their chosen generators additionally obey \(L_{\widetilde\pi}(s)=\overline{L_\pi(\bar s)}\), the constant in Corollary 1.22e has absolute value one.

Proof. Hilbert conjugation identifies the conjugate smooth representation with its compatible contragredient. This follows directly from invariance of the Hilbert form: a conjugate vector defines its paired functional, differentiation commutes with this map, and on each finite-dimensional compact type it is the ordinary dual identification. Thus conjugating a matrix coefficient of \(\pi\) gives a coefficient of \(\widetilde\pi\), with the swapped dual pair. Conjugation of the positive Fourier integral gives \(\overline{\widehat\Phi_\psi}=\widehat{\bar\Phi}_{\psi^{-1}}\), since the additive measure is real. Conjugating (1.26q) in its meromorphic continuation and using uniqueness of its scalar therefore proves \[ \overline{\gamma_{\rm mat}(s,\pi,\psi)} =\gamma_{\rm mat}(\bar s,\widetilde\pi,\psi^{-1}) =\omega_\pi(-I_n)\gamma_{\rm mat}(\bar s,\widetilde\pi,\psi). \tag{1.26z} \] The second equality is (1.26v) with \(a=-1\); its norm factor is one and the inverse central sign equals the same sign. On the critical line \(\bar s=1-s\). At every regular point (1.26u) now gives the square modulus \(\omega_\pi(-I_n)^2=1\), because \((-I_n)^2=I_n\). A Laurent expansion at a putative zero or pole on that line would force the modulus along the line to tend to zero or infinity. The regular values have modulus one, so neither can occur. Finally, under the stated generator conjugation, \(L_{\widetilde\pi}(1-s)/L_\pi(s)\) has modulus one at a regular critical-line point. Substitution in (1.26w) gives \(|\epsilon_\pi|=1\). This last assertion uses the specified generator normalization; the scalar Fourier equation alone does not identify it. \(\square\)

The all-rank archimedean inducing Gaussian kernel

Let \(F=\mathbb R\) or \(\mathbb C\), \(G=GL_n(F)\), \(K=O(n)\) or \(U(n)\), \(B\) the upper triangular subgroup, and \(M=B\cap K\). Put \(d=[F:\mathbb R]\), \(\nu(g)=|\det g|_F\), and \(\rho_j=(n+1)/2-j\). Every smooth character has the form \[ \chi_j(x)=\operatorname{sgn}(x)^{e_j}|x|^{t_j} \quad(F=\mathbb R),\qquad \chi_j(z)=(z/|z|)^{\ell_j}|z|_{\mathbb C}^{t_j} \quad(F=\mathbb C), \tag{1.31a} \] with \(e_j\in\{0,1\}\), \(\ell_j\in\mathbb Z\), \(t_j\in\mathbb C\). Indeed a smooth homomorphism on the positive radial group, transported by logarithm, solves \(f'(u)=cf(u)\). On the real sign group it has value \(1\) or \(-1\). On the complex unit circle the same differential equation and period \(2\pi\) force an integer angular exponent.

Use the programme's negative-exponent characters \(\psi_{\mathbb R}(x)=e^{-2\pi ix}\), \(\psi_{\mathbb C}(z)=e^{-2\pi i(z+\bar z)}\), their self-dual measures, and the positive trace Fourier argument. With \(dg=C_F\nu(g)^{-n}dX\), \[ Z(s,\Phi,c)=\int_G\Phi(g)c(g)\nu(g)^{s+(n-1)/2}\,dg. \tag{1.31b} \] The fixed matrix Gaussians throughout are \[ G_{\mathbb R}(X)=e^{-\pi\operatorname{tr}(XX^t)},\qquad G_{\mathbb C}(X)=e^{-2\pi\operatorname{tr}(XX^*)}. \tag{1.31ba} \] The self-dual additive measures are ordinary Lebesgue measure in each real entry and \(2\,du\,dv\) in each complex entry. For each scalar diagonal coordinate use exactly \(d^\times x=dx/|x|\) over \(\mathbb R\) and \(d^\times z=(2\,du\,dv)/(\pi|z|_{\mathbb C})\) over \(\mathbb C\). These are the actual NT-ADL-08 normalizations; the positive constant \(\kappa_F\) in the matrix decomposition absorbs only the remaining matrix Haar normalization and compact angular redundancy. Write \[ \begin{aligned} \mathcal I(\boldsymbol\chi) &=\{f\in C^\infty(G): f(bg)=\delta_B(b)^{1/2}\boldsymbol\chi(b)f(g)\},\\ \delta_B(b)^{1/2} &=\prod_j|b_{jj}|_F^{\rho_j},\qquad \boldsymbol\chi(b)=\prod_j\chi_j(b_{jj}). \end{aligned} \tag{1.31c} \] The topology is the usual \(C^\infty\) topology on the compact model \[ \{f\in C^\infty(K):f(mk)=\boldsymbol\chi(m)f(k)\}. \tag{1.31d} \] The extension between (1.31c) and (1.31d) is defined by row Gram--Schmidt, with positive diagonal, and is unique. This is an actual complete smooth moderate-growth realization, not a formal induced module.

Put \[ L_{\boldsymbol\chi}(s)= \begin{cases} \prod_j\Gamma_{\mathbb R}(s+t_j+e_j),&F=\mathbb R,\\ \prod_j\Gamma_{\mathbb C}(s+t_j+|\ell_j|/2),&F=\mathbb C, \end{cases} \qquad \epsilon_{\boldsymbol\chi}= \begin{cases} (-i)^{\sum e_j},&F=\mathbb R,\\ (-i)^{\sum |\ell_j|},&F=\mathbb C. \end{cases} \tag{1.31e} \]

Theorem 1.26. For the paired actual models \(\mathcal I(\boldsymbol\chi)\) and \(\mathcal I(\boldsymbol\chi^{-1})\), the matrix zeta family has:

1. the full polynomial-Gaussian ideal \(L_{\boldsymbol\chi}(s)\mathbb C[s]\), over all compact-finite coefficient vectors;
2. one polynomial Gaussian and one coefficient attaining \(\kappa_F L_{\boldsymbol\chi}(s)\), where \(\kappa_F>0\) depends only on the measures;
3. entire normalized division for every Schwartz test and every smooth paired coefficient, with compact-parameter joint seminorm bounds;
4. the full scalar positive-trace Fourier equation \[ Z_{\boldsymbol\chi^{-1}}(1-s,\widehat\Phi,c^\vee)= \epsilon_{\boldsymbol\chi} \frac{L_{\boldsymbol\chi^{-1}}(1-s)} {L_{\boldsymbol\chi}(s)} Z_{\boldsymbol\chi}(s,\Phi,c). \tag{1.31f} \]

This holds in every rank, for all the character parameters in (1.31a); the induced representation need not be irreducible. It therefore proves the intended exact Gaussian generator for every irreducible full Borel induction in this actual model, without a classification of when the induction is irreducible.

Step 1: The actual compact model and its pairing

Row Gram--Schmidt writes \(g=T k\), with \(T\) upper triangular and positive real diagonal. The coefficients and derivatives of the Gram--Schmidt maps are bounded by powers of \[ H(g)=1+\|g\|+\|g^{-1}\|. \tag{1.31g} \] Here is the estimate. Each orthogonal component of a row has length at most \(\|g\|\) and at least \(\|g^{-1}\|^{-1}\): subtracting later rows amounts to forming a row combination with one coefficient equal to one, whose norm is bounded below by the smallest singular value of \(g\). Every fixed derivative of the recursion has a polynomial numerator in the entries and previously computed quantities and a denominator consisting of finitely many powers of these positive lengths. Character differentiation gives only further powers of the same lengths. On the compact \(k\)-variable, this proves \[ p_N(\mathcal I(g)f)\le C_N H(g)^{A_N}p_N(f) \tag{1.31h} \] for increasing compact smooth seminorms \(p_N\), with fixed \(A_N\). Differentiating the action in \(g\) gives continuous maps with finitely more compact derivatives. The closed space (1.31d) is complete, and these formulas prove a smooth moderate-growth action.

The compact pairing is \[ \langle f,\widetilde f\rangle=\int_K f(k)\widetilde f(k)\,dk, \qquad \widetilde f\in\mathcal I(\boldsymbol\chi^{-1}), \tag{1.31i} \] with Haar probability. It is nondegenerate, including on paired finite compact types: \(\boldsymbol\chi|_M\) is unitary, so the conjugate of a nonzero \(f\) has the inverse \(M\)-covariance and its pairing with \(f\) is positive.

For completeness, the pairing is \(G\)-invariant with exactly this normalization. Identify \(B\backslash G=M\backslash K\). If \(kg=b(k,g)k_g\), the Jacobian of \(Bk\mapsto Bkg\) in compact Haar quotient density is \(\delta_B(b(k,g))\). To compute it, use right translation by \(k\) and \(k_g\) to identify the two tangent spaces with \(\mathfrak g/\mathfrak b\). The differential is \(\operatorname{Ad}(b^{-1})\) on that quotient. For diagonal \(b\), its lower-entry weights are \(b_{jj}/b_{ii}\), \(i>j\), whose product of normalized absolute values is \(\delta_B(b)\). For an upper unipotent generator \(\exp(xE_{ij})\), its adjoint action on the quotient is the exponential of a nilpotent operator and has determinant one; these generators exhaust the unipotent subgroup. Thus the Jacobian formula holds for every \(b\). The tangent identifications use the right-\(K\)-invariant quotient density, so there is no additional point-dependent scalar. Changes of the factorization by \(M\) have unit Jacobian.

The two multiplier factors in (1.31c) multiply to \(\delta_B(b)\). The computed Jacobian then proves (1.31i) invariant by change of variable. The inverse coefficient is accordingly \[ c^\vee(g)=c(g^{-1}) =\langle \mathcal I(\boldsymbol\chi^{-1})(g)\widetilde f,f\rangle \quad\text{if }c(g)=\langle\mathcal I(\boldsymbol\chi)(g)f,\widetilde f\rangle . \tag{1.31j} \]

The compact-finite cores are admissible and dense by the elementary compact-polynomial argument written in Step 1 of the subquotient argument below. Its evaluation-at-identity argument bounds the multiplicity of a compact type in the scalar function space by the type's dimension; the line-covariance subspace (1.31d) can only reduce it. Its explicit central polynomial approximate identities converge in all compact smooth seminorms and have compact-finite range. Thus no general Peter--Weyl or globalization statement is needed as a separate unproved input.

Step 2: The all-rank upper-triangular measure

For \(a=(a_1,\ldots,a_n)\in(F^\times)^n\) and strict upper entries \(x=(x_{ij})_{i<j}\), let \(T(a,x)\) be upper triangular with that diagonal. The Haar integration formula is \[ dg=\kappa_F\prod_{i=1}^n|a_i|_F^{\,i-n}\, \prod_i d^\times a_i\prod_{i<j}dx_{ij}\,dk . \tag{1.31k} \] This formula includes the redundant unit-diagonal factors in \(a_i\); \(\kappa_F>0\) accounts for their fixed multiplicative angular masses.

Here is a derivation in every rank. Orthogonalize rows in the order \(n,n-1,\ldots,1\). At step \(i\), the row components along the already chosen later rows are \(n-i\) additive coordinates over \(F\). Its remaining orthogonal component lies in an \(i\)-dimensional \(F\)-space. Polar coordinates there contribute \(r_i^{di-1}dr_i\) and its unit-vector measure. Successively these unit vectors form a complete orthonormal frame. Their product measure is invariant under right-\(K\) rotation of the ambient row space, hence a fixed positive multiple of Haar measure. The orthogonal coordinate changes have real Jacobian one. Consequently additive matrix measure is a positive constant times \(\prod_i r_i^{di}\,dr_i/r_i\prod dx_{ij}\,dk\). Division by \(\nu(g)^n=\prod_i r_i^{dn}\) gives the powers in (1.31k). Finally replace positive \(r_i\) by arbitrary \(a_i\) using the diagonal compact phases/signs: multiply column \(j\) of \(T\) by its unit \(m_j\) and replace \(k\) by \(m^{-1}k\). The upper-entry additive measures and Haar probability are unchanged. The multiplicative angular masses are constants, giving precisely (1.31k). This derives the density without importing an Iwasawa integration theorem.

Step 3: The matrix kernel

Define the Schwartz diagonal marginal \[ \phi_{\Phi;h,k}(a)= \int_{F^{n(n-1)/2}} \Phi(h^{-1}T(a,x)k)\,\prod_{i<j}dx_{ij} \tag{1.31l} \] on the whole additive space \(F^n\), including zero coordinates, and the kernel \[ \mathcal K_{\Phi,\boldsymbol\chi}(h,k;s)= \int_{(F^\times)^n}\phi_{\Phi;h,k}(a) \prod_i\chi_i(a_i)|a_i|_F^s\,d^\times a_i. \tag{1.31m} \] Restriction to a linear coordinate subspace and integration in other coordinates map the Schwartz space continuously to itself. For example, after differentiating in \(a\), choose an upper-variable decay power greater than its real dimension plus the desired seminorm degree, integrate that bound, and retain the required decay in \(a\). Linear substitutions by compact \(h,k\) have uniform seminorm bounds. This proves continuity and compact uniformity of (1.31l).

The matrix coefficient integral is exactly \[ Z(s,\Phi,c_{f,\widetilde f}) =\kappa_F\int_{K\times K} \mathcal K_{\Phi,\boldsymbol\chi}(h,k;s) \widetilde f(h)f(k)\,dh\,dk . \tag{1.31n} \] In fact expand the pairing coefficient by (1.31i), substitute \(g\mapsto h^{-1}g\), and apply (1.31k). The power of each \(|a_i|_F\) is \[ \rho_i+\frac{n-1}{2}+(i-n)+s=s. \tag{1.31o} \] Thus (1.31m) is the resulting diagonal integral with no extra shift. The same calculation with absolute values proves every integral absolutely convergent in a common far right half-plane; compact \(h,k\) and the bounds of (1.31l) make all uses of Fubini valid.

Step 4: Full-Schwartz division and polynomial Gaussian quotients

The scalar Tate normalized distributions \[ A_{\chi}(s)(f)=Z_F(s,f,\chi)/L_F(s,\chi) \tag{1.31p} \] are entire continuous Schwartz families with compact-parameter seminorm bounds. The actual proof of Tate’s local theory at the infinite places supplies their Mellin continuation, pole cancellation and entire reciprocal gamma. Its Taylor proof also gives the seminorm bound needed here: expand the parity or angular projection to order \(N\) at zero; its finitely many coefficient functionals are continuous derivatives at zero, and the remainder is bounded by a fixed \(C^N\) seminorm times the radial power \(N\). Choose \(N\) for a fixed compact \(s\)-set. The integral outside a unit ball is bounded by a fixed Schwartz seminorm. Multiplication by the entire gamma reciprocal removes the finite polar terms, with bounded coefficients on the compact set. This proves the assertion without a family-level continuity assumption.

The same estimate holds with Schwartz variables as parameters: apply it to every derivative in the other variables after multiplying by their desired polynomial weights. It defines a continuous entire operator from \(\mathcal S(F^r)\) to \(\mathcal S(F^{r-1})\). This is entireness in its Schwartz topology: the Taylor-subtracted integrals can be differentiated in \(s\) with arbitrary fixed Schwartz seminorms under domination, and their Cauchy formulas converge in each seminorm on smaller parameter disks. The finitely many removed polar terms obey the same formulas after multiplication by the entire gamma reciprocal. Iterate this in the \(n\) diagonal coordinates. Hence \[ \mathcal K_{\Phi,\boldsymbol\chi}(h,k;s)/ L_{\boldsymbol\chi}(s) \tag{1.31q} \] is entire, continuous in \(\Phi\), and uniformly bounded by finitely many Schwartz seminorms for compact \(h,k,s\). Equation (1.31n) proves entire normalized division for all smooth paired coefficient vectors, with a bound by such a test seminorm times \(\|f\|_\infty\|\widetilde f\|_\infty\).

If \(\Phi=P G_F\) is a polynomial Gaussian, compact multiplication preserves the Gaussian. Integrating its upper-entry polynomial gives a polynomial in \(a,\bar a\) times the diagonal Gaussian; its degree is bounded independently of \(h,k\). Every real monomial of degree \(r\) has zero scalar Tate integral unless \(r=e+2q\), in which case its quotient by \(\Gamma_{\mathbb R}(s+t+e)\) is \(\pi^{-q}((s+t+e)/2)_q\). Every complex monomial \(a^u\bar a^v\) has zero integral unless \(v-u=\ell_j\), in which case its quotient is \((2\pi)^{-\min(u,v)}(s+t+|\ell_j|/2)_{\min(u,v)}\). These are proved gamma recurrence calculations with nonnegative integers. Therefore (1.31q), and then the normalized matrix integral in (1.31n), is a polynomial in \(s\). Its coefficients are smooth functions of \(h,k\), and the finite degree bound permits the compact integrations coefficient by coefficient.

Step 5: A Gaussian attaining the whole product

Take the scalar fundamental angular Gaussian polynomials \[ p_j(z)= \begin{cases} z^{e_j},&F=\mathbb R,\\ \bar z^{\ell_j},&F=\mathbb C,\quad \ell_j\ge0,\\ z^{-\ell_j},&F=\mathbb C,\quad \ell_j<0, \end{cases} \tag{1.31r} \] In the following formula evaluate these polynomials on \(X_{jj}\): \[ \Phi_0(X)=\prod_jp_j(X_{jj})\,G_F(X). \tag{1.31s} \] At \(h=k=e\), its upper marginal is the tensor of the fundamental scalar Tate tests: each upper-coordinate Gaussian has mass one in its self-dual measure. Thus \[ \mathcal K_{\Phi_0,\boldsymbol\chi}(e,e;s) =L_{\boldsymbol\chi}(s). \tag{1.31t} \]

It remains to realize this evaluation by actual finite coefficient vectors, not by a delta distribution on \(K\). The kernel (1.31m) has the compact covariances \[ \mathcal K(mh,k;s)=\boldsymbol\chi(m)\mathcal K(h,k;s),\qquad \mathcal K(h,mk;s)=\boldsymbol\chi(m)^{-1}\mathcal K(h,k;s). \tag{1.31u} \] For the first, replace \(T\) by \(m^{-1}T\); for the second replace it by \(Tm\). The diagonal changes supply the indicated characters, and all compact unit Jacobians are one.

For (1.31s), the normalized kernel is a polynomial in \(s\) and in the entries and conjugate entries of \(h,k\). It lies in a fixed finite-dimensional tensor product \(E_h\otimes E_k\), independent of \(s\), with \[ E_h\subset\mathcal I(\boldsymbol\chi)|_K,\qquad E_k\subset\mathcal I(\boldsymbol\chi^{-1})|_K. \tag{1.31v} \] To see this exactly, start with the finite-dimensional spaces of coordinate polynomials of the bounded degrees in \(h,k\). Project their left \(M\)-actions onto the two characters in (1.31u). These compact averages preserve degree and commute with right \(K\). Applying both projections to the kernel fixes it, so their images are the required finite compact spaces.

Evaluation at \(e\) on \(E_h\) is represented by pairing with a single \(\widetilde f\in\overline{E_h} \subset\mathcal I(\boldsymbol\chi^{-1})|_K\); evaluation on \(E_k\) is represented by pairing with a single \(f\in\overline{E_k}\subset\mathcal I(\boldsymbol\chi)|_K\). This is just finite-dimensional nondegenerate Gram duality: choose a basis, invert its positive Haar Gram matrix, and form the vector representing the evaluation functional. These choices do not depend on \(s\). The tensor of the two evaluations therefore gives \[ \int_{K\times K}\mathcal K_{\Phi_0,\boldsymbol\chi}(h,k;s) \widetilde f(h)f(k)\,dh\,dk =\mathcal K_{\Phi_0,\boldsymbol\chi}(e,e;s). \tag{1.31w} \] Equations (1.31n) and (1.31t) prove the claimed one-coefficient attainment.

Polynomial-module closure follows directly from scalar dilation. For the real dilation Euler operator \(E\), \[ Z(s,E\Phi,c) =-\{dn(s+(n-1)/2)+d\sum_jt_j\}Z(s,\Phi,c). \tag{1.31x} \] The coefficient's positive-central homogeneity is \(r^{d\sum t_j}\). Prove (1.31x) by differentiating the change of variable \(X\mapsto rX\) in a convergence half-plane and continue meromorphically. \(E\) preserves polynomial Gaussians, so multiplication by \(s\) preserves the span. The polynomial bound and the attained product now prove the exact ideal of Theorem 1.26. In particular polynomial coefficients in an attaining identity can be absorbed into new polynomial Gaussian tests: the operator \[ \mathscr S=-\frac{E}{dn}-\frac{n-1}{2} -\frac{\sum_jt_j}{n} \tag{1.31xa} \] satisfies \(Z(s,\mathscr S\Phi,c)=sZ(s,\Phi,c)\). It preserves the fixed polynomial-Gaussian space. Thus any finite polynomial combination of matrix integrals is a finite unweighted sum of integrals with explicitly changed tests.

Step 6: The all-rank diagonal Fourier marginal

At \(h=k=e\), the identity of Schwartz functions is \[ \phi_{\widehat\Phi;e,e} =\widehat{\phi_{\Phi;e,e}}, \tag{1.31y} \] where the right transform is the \(n\)-coordinate diagonal transform. The trace pairing pairs an upper Fourier entry \(Y_{ij}\), \(i<j\), with the lower original entry \(X_{ji}\). Integrating in all upper Fourier entries sets all those lower original entries to zero by partial Schwartz Fourier inversion. The upper original entries are then integrated out, and the diagonal entries retain exactly their ordinary bilinear Fourier transform. One can perform this as successive continuous partial Fourier transforms, restriction and integration on Schwartz spaces, so no nonabsolutely convergent iterated integral is being asserted.

For compact \(h,k\), \[ \widehat{\Phi(h^{-1}Xk)}(Y)=\widehat\Phi(k^{-1}Yh). \tag{1.31z} \] Apply (1.31y) to \(\Phi(k^{-1}Xh)\). The \(n\) scalar Tate equations then yield \[ \mathcal K_{\widehat\Phi,\boldsymbol\chi^{-1}}(h,k;1-s) =\left(\prod_j\gamma_F(s,\chi_j,\psi_F)\right) \mathcal K_{\Phi,\boldsymbol\chi}(k,h;s). \tag{1.31aa} \] The scalar equations are the actual whole-Schwartz proof of NT-ADL-08; iterating them in parameter variables is justified by the operators of Step 4 of this argument. Compact integration and interchanging \(h,k\) in (1.31n) prove (1.31f) for the whole smooth coefficient family. Formula (1.31e) is the product of those exact scalar normalizations. This finishes Theorem 1.26.

There are two useful sign checks. Fourier inversion gives \(\widehat{\widehat\Phi}(X)=\Phi(-X)\). Applying (1.31f) twice must therefore give the central sign of the coefficient. In fact \[ \epsilon_{\boldsymbol\chi}\epsilon_{\boldsymbol\chi^{-1}} =(-1)^{\sum e_j}\quad(\mathbb R),\qquad =(-1)^{\sum|\ell_j|}=\prod_j\chi_j(-1)\quad(\mathbb C), \tag{1.31ab} \] as required. If \(\psi_F\) is replaced by \(x\mapsto\psi_F(ax)\) with \(a\in F^\times\), the self-dual matrix measure is multiplied by \(|a|_F^{n^2/2}\), and \(\widehat\Phi_a(Y)=|a|_F^{n^2/2}\widehat\Phi(aY)\). Changing \(g\) to \(a^{-1}g\) in the dual matrix integral gives \[ \epsilon_{\boldsymbol\chi}(s,\psi_{F,a}) =\omega_{\boldsymbol\chi}(a)|a|_F^{n(s-1/2)} \epsilon_{\boldsymbol\chi}, \qquad \omega_{\boldsymbol\chi}(a)=\prod_j\chi_j(a). \tag{1.31ac} \] This uses matrix Haar invariance and the exact exponent \(n^2/2-n((n+1)/2-s)=n(s-1/2)\); it does not change the Gaussian generator for the original fixed Gaussian convention.

Scope of this proof

Theorem 1.26 proves the all-rank inducing kernel and its actual smooth paired realization, Gaussian polynomial quotient, attainment, full Schwartz continuity and Fourier equation. It does not assert that every irreducible admissible module embeds in this principal series, or that an arbitrary constituent has the full product generator. The subquotient argument below proves the exact finite polynomial correction and its attaining identity for every algebraic constituent that is actually supplied. The identification of those corrections with the intended standard-module gamma data for every irreducible remains a further obligation.

The earlier programme inputs used here are the actual scalar NT-ADL-08 theory and the whole-Schwartz Fourier proof of Additive characters, self-dual measures and Poisson summation on the adèles. Goldfeld--Jacquet's freely accessible author notes, §3, Theorem 3.5 and Lemma 3.6 at Goldfeld–Jacquet, freely accessible author notes, locate the general target; their inducing reduction is not being used as an unproved premise.

Actual closed principal-series subquotients and their Gaussian ideals

Use the actual compact models \(I=\mathcal I(\boldsymbol\chi)\) and \(\widetilde I=\mathcal I(\boldsymbol\chi^{-1})\), compact pairing, measures and gamma factors of the preceding Theorem 1.26. Write \(I_0,\widetilde I_0\) for their compact-finite cores.

Theorem 1.26a. For every pair of algebraic \((\mathfrak g,K)\)-submodules \(W_0\subset V_0\subset I_0\), their compact-smooth closures \(W\subset V\subset I\) are closed \(G\)-subrepresentations. The quotient \[ Q=V/W \tag{1.32a} \] is an actual complete smooth moderate-growth realization with core \(Q_0=V_0/W_0\). The inclusion and quotient are actual continuous \(G\)-maps. If \(Q_0\) is irreducible, \(Q\) is topologically irreducible. There is a paired actual smooth realization \[ Q^\vee=\operatorname{Ann}(W)/\operatorname{Ann}(V) \quad\text{inside }\widetilde I, \tag{1.32b} \] whose core is the admissible contragredient of \(Q_0\). Every smooth paired coefficient lifts to an induced coefficient; every compact-finite coefficient has compact-finite lifts.

Theorem 1.26b. If \(Q_0\ne0\), its exact Gaussian ideal is \[ L_Q(s)\mathbb C[s],\qquad L_Q(s)=P_Q(s)L_{\boldsymbol\chi}(s), \tag{1.32c} \] where \(P_Q\) is a nonzero monic polynomial defined by the actual Gaussian integrals. A finite sum of polynomial multiples of Gaussian coefficient integrals equals \(L_Q\). For every Schwartz test and paired smooth coefficient, \(Z_Q/L_Q\) is entire, with compact-parameter joint seminorm bounds. \(L_Q\) has no zeros. With \(b=\deg P_Q\), \[ P_{Q^\vee}(1-s)=(-1)^bP_Q(s),\qquad \epsilon_Q=(-1)^b\epsilon_{\boldsymbol\chi}, \tag{1.32d} \] and \[ Z_{Q^\vee}(1-s,\widehat\Phi,c^\vee) =\epsilon_Q\frac{L_{Q^\vee}(1-s)}{L_Q(s)} Z_Q(s,\Phi,c). \tag{1.32e} \] Thus \(|\epsilon_Q|=1\) for the programme's basic characters. Conjugation gives the construction in \(I(\overline{\boldsymbol\chi})\), and exactly \[ L_{\bar Q}(s)=\overline{L_Q(\bar s)}. \tag{1.32f} \] If \(Q_0\) has an invariant positive Hermitian form, also \[ L_{Q^\vee}(s)=\overline{L_Q(\bar s)} \tag{1.32g} \] in this monic polynomial normalization.

These statements cover every supplied algebraic principal-core subquotient, with its actual models and maps constructed below. They do not assume an abstract globalization functor. They do not assert that every irreducible admissible module occurs in a principal core, or identify \(P_Q\) with the intended standard-module factors.

Step 1: Compact polynomials, finite packets and density

On \(K=O(n)\) or \(U(n)\) put \[ q(k)=\frac{1+\operatorname{Re}\operatorname{tr}(k)/n}{2} =1-\frac{\|k-I\|_{\mathrm{HS}}^2}{4n},\qquad A_Nf(k)=\frac{\int_Kq(h)^Nf(kh)\,dh}{\int_Kq(h)^N\,dh}. \tag{1.32h} \] Here \(0\le q\le1\), and \(q=1\) only at the identity. These are approximate identities. On the complement of a fixed identity neighborhood let \(q\le c<1\); a smaller neighborhood has \(q\ge c'>c\) and positive Haar mass. The relative mass on that complement is at most \((c/c')^N/\operatorname{mass}(U')\), tending to zero. Uniform continuity gives uniform convergence \(A_Nf\to f\).

The class function \(q^N\) makes \(A_N\) commute with right \(K\) and its differentiated actions. Applying uniform convergence to each compact derivative proves convergence in every compact smooth seminorm. The operator preserves the left \(M\)-character of \(I\). After changing variable \(y=kh\), its kernel is \(q(k^{-1}y)^N\), a polynomial in entries and conjugate entries of \(k\). Hence \(A_Nf\) is a coordinate polynomial restriction, of bounded degree, and has finite-dimensional right-\(K\) orbit. This proves core density. It works for disconnected \(O(n)\), including \(O(1)\).

Conversely every compact-finite function is a coordinate polynomial. Its finite orbit splits into irreducibles by averaging a Hermitian form. On each type, central convolution \(A_N\) is scalar by the finite-dimensional Schur lemma. Its scalar tends to one, hence is nonzero for large \(N\). Invert the finitely many scalars to write the original function as \(A_Nf'\), a coordinate polynomial. The elementary facts used here can be checked directly. Averaging any positive Hermitian form over compact \(K\) preserves positivity and makes the form invariant; invariant orthogonal complements give the finite irreducible decomposition by induction on dimension. An endomorphism of a nonzero complex irreducible representation has an eigenvalue; its eigenspace is nonzero and invariant, hence the whole space. It is therefore scalar. The integral defining \(A_N\) preserves each such irreducible copy, and its centrality makes it an intertwiner there, so the scalar argument applies to every copy.

Every compact type has finite multiplicity in \(C^\infty(K)\): an equivariant map \(T:E_\tau\to C^\infty(K)\) is determined by \(\ell(v)=(Tv)(e)\), since \((Tv)(k)=\ell(\tau(k)v)\). Its multiplicity is at most \(\dim E_\tau\). The compact-type projector \[ P_\tau f(k)=\dim E_\tau\int_K \overline{\operatorname{tr}\tau(h)}f(kh)\,dh \tag{1.32i} \] is continuous and has finite-dimensional range. Its projection property follows by averaging a linear map between finite-dimensional irreducibles: the average is zero or scalar, and trace fixes the scalar. Changing variable \(y=kh\) in (1.32i) exhibits its finite matrix-coefficient range directly. The same statements hold in the line-covariance space \(I\). This supplies all the compact finiteness used below without a general globalization theorem. More explicitly, for irreducibles \(\sigma,\tau\), the map \(\int_K\tau(h)A\sigma(h)^{-1}\,dh\) intertwines them. It is zero when they are inequivalent, since a nonzero intertwiner has invariant kernel and image and is an isomorphism. For \(\sigma=\tau\), the scalar is \(\operatorname{tr}(A)/\dim E_\tau\). Taking \(A\) to be the matrix units gives the matrix-entry orthogonality and exactly the character projector (1.32i).

Step 2: Uniform analytic exponentials on the principal core

For fixed \(X\in\mathfrak g_{\mathbb R}\) and any \(f\in I_0\), \[ t\longmapsto \mathcal I(\exp(tX))f \tag{1.32j} \] has a Taylor series converging in the compact smooth topology for \(|t|<r_X\), with \(r_X>0\) independent of \(f\).

Indeed \(f\) is a coordinate polynomial on \(K\). Factor \(k\exp(tX)=T(t,k)k'(t,k)\) by row Gram--Schmidt with positive diagonal. Squared pivots are ratios of consecutive trailing row-Gram determinants. The Gram matrix on real \(t\) is \[ k\exp(tX)\exp(tX^*)k^*. \tag{1.32k} \] Replace \(t\) by a complex variable in this formula, keeping \(X^*\) fixed. At zero all those determinants are one, uniformly in \(k\). On a small disk they stay near one and avoid zero; choose their logarithm and square-root branches starting at one. All remaining Gram--Schmidt entries are algebraic expressions divided by these nonzero pivots. The conjugate trajectory, with the conjugated fixed coefficients, similarly gives a holomorphic extension of the actual conjugate entries of \(k'(t,k)\). Thus every polynomial in \(k',\bar k'\) has a holomorphic extension in \(t\). The inducing multiplier is \(\prod_j r_j(t,k)^{d(t_j+\rho_j)}\), using the same logarithms; the diagonal is positive on real \(t\), so its angular characters are one.

Every fixed number of compact derivatives is bounded on a smaller closed complex disk. Its radius is determined by the Gram determinants, not by the degree of \(f\). Cauchy's coefficient estimate proves Taylor convergence in every compact smooth seminorm, on a common disk. The coefficients are \((d\mathcal I(X))^af/a!\), by differentiation of the actual action.

The same argument near any \(g_0\in G\), replacing \(k\) by \(kg_0\), proves real analyticity of its core matrix coefficients. Pivots at \(g_0\) are uniformly positive by the singular-value estimates in Theorem 1.26. The construction extends in all real group coordinates in a small complex polydisk, and continuous compact pairing and uniform compact bounds preserve analyticity after integration.

Step 3: Algebraic closures are genuinely group stable

Let \(V=\overline{V_0}\). For \(v\in V_0\), every term of the Taylor series in (1.32j) lies in \(V_0\), by Lie stability. Its sum therefore lies in \(V\) for \(|t|<r_X\). The radius is independent of \(v\). Approximation by \(V_0\) and continuity show the same inclusion for every \(v\in V\). Use negative \(t\) and finite products of small intervals to obtain equality for every real \(t\). The closure is also \(K\)-stable.

These elements generate the whole group. Polar decomposition gives \(g=k\exp H\): diagonalize the positive matrix \((g^*g)^{1/2}\) by the spectral theorem and take the logarithms of its positive eigenvalues. The resulting \(H\) is real symmetric or complex Hermitian and belongs to the real Lie algebra. Any negative real determinant is carried by \(k\in O(n)\), so both real components are included. Thus \(V\), and likewise \(W\), is an actual closed \(G\)-subrepresentation. Only elementary finite-dimensional spectral theory is needed: a Hermitian matrix has an eigenvector by maximizing its real Rayleigh quotient on the unit sphere and differentiating there. The orthogonal complement of that eigenvector is invariant, so induction gives an orthonormal eigenbasis. For \(g^*g\), each eigenvalue is strictly positive. Define its positive square root and logarithm in this basis; \(k=g(g^*g)^{-1/2}\) is orthogonal or unitary, and \(H=\log((g^*g)^{1/2})\) gives the claimed factorization.

Its core is exactly \(V_0\). The projector \(P_\tau\) maps \(V_0\) into its finite-dimensional type space \(V_0(\tau)\), closed in the ambient finite packet. The projection of any limit in \(V\) therefore remains in this space. A compact-finite limit is a finite sum of such projections, and lies in \(V_0\). The reverse inclusion is immediate. The same argument gives \(W_0\).

Step 4: Complete quotients and exact continuous core maps

The Fréchet quotient \(Q=V/W\) is complete. For an explicit argument, use increasing seminorms \(p_j\). A Cauchy quotient sequence has a subsequence whose successive differences have quotient \(p_j\)-seminorm less than \(2^{-j}\). Choose lifts of those differences with \(p_j<2^{-j+1}\). Their series converges in every fixed seminorm of the complete \(V\); its sum lifts the quotient limit, and the original Cauchy sequence has the same limit.

All smooth orbit maps and differentiated actions pass to the quotient. Its seminorms \[ \bar p_N(v+W)=\inf_{w\in W}p_N(v+w) \tag{1.32l} \] inherit the moderate bound (1.31h), since \(\mathcal I(g)W=W\). For a compact-finite quotient vector, apply the finite sum of its compact-type projectors to any lift. This produces a lift in \(V_0\), with kernel \(W_0\). Thus the quotient core is exactly \(V_0/W_0\), and the inclusion and quotient are the actual continuous maps asserted in Theorem 1.26a. The operators \(A_N\) give core density.

If that core is irreducible, a nonzero closed group-stable subspace of \(Q\) has a nonzero core vector by these approximate identities. Difference quotients give Lie stability of its core. Irreducibility makes it contain all \(Q_0\), and density makes it equal to \(Q\). This proves topological irreducibility.

Step 5: Actual duals, annihilators and coefficient lifts

Under the continuous invariant ambient pairing let \[ \operatorname{Ann}(V)= \{\widetilde f\in\widetilde I: \langle v,\widetilde f\rangle=0\text{ for all }v\in V\}. \tag{1.32m} \] It and \(\operatorname{Ann}(W)\) are closed group-stable spaces. Their cores are the algebraic annihilators of \(V_0,W_0\). They are the closures of those cores: \(A_N\) preserves the annihilator, since its compact adjoint preserves the original \(K\)-stable space. Its outputs are core vectors and converge in the smooth topology. Pairing with the dense original core is equivalent to pairing with its closure.

Finite-dimensional packet duality gives \[ W_0^\perp/V_0^\perp\simeq(V_0/W_0)^\vee. \tag{1.32n} \] To check surjectivity, pull a finite-packet functional back to \(V_0\), extend it to the ambient finite packet by linear algebra, and represent it by the compact pairing. Extend by zero to all other packets. It annihilates \(W_0\); its restriction is the given functional, and the kernel of restriction is \(V_0^\perp\). Invariance of the pairing makes this identification Lie- and compact-equivariant, so no exact dual functor is being assumed.

There is also exact double-annihilator recovery in these models. If every finite compact projection of \(v\in I\) lies in \(V_0\), then each finite-range \(A_Nv\) lies in \(V_0\), and their smooth convergence puts \(v\) in \(V\). If a projection lies outside \(V_0\), finite-packet duality provides a core vector in \(\operatorname{Ann}(V)\) pairing nontrivially with it; compact adjunction gives such a vector pairing nontrivially with \(v\). Consequently \(\operatorname{Ann}(\operatorname{Ann}(V))=V\), and likewise for \(W\). Thus the constructed double dual is literally paired with \(Q\); coefficient inversion loses no core coefficients in Step 8 of this argument.

The ambient pairing descends jointly continuously to \(Q\times Q^\vee\). Minimize its seminorm bound over both lifts to obtain quotient seminorm bounds. It is nondegenerate on the whole models: a nonzero vector has a nonzero finite compact projection, by convergence of \(A_N\), and the finite packet pairing detects it. Compact adjunction then detects the original vector.

In particular every smooth paired coefficient is literally \[ \langle Q(g)(v+W),\widetilde f+\operatorname{Ann}(V)\rangle =\langle I(g)v,\widetilde f\rangle,\quad v\in V,\ \widetilde f\in\operatorname{Ann}(W). \tag{1.32o} \] Both core vectors have core lifts by Step 4 of this argument. Changing either lift changes the function by zero for all \(g\), by group stability. All normalized bounds from Theorem 1.26 descend by infima over lifts, and its complete Fourier equation descends too. This completes Theorem 1.26a, including its genuine continuous maps.

Step 6: The polynomial correction and its finite attainment

Let \(J_Q\) be the span in \(\mathbb C[s]\) of \[ Z_Q(s,\Phi,c)/L_{\boldsymbol\chi}(s),\quad \Phi\text{ polynomial Gaussian},\quad c\text{ paired core coefficient}. \tag{1.32p} \] These values are polynomials by Theorem 1.26 and (1.32o). The radial identity (1.31x) makes their span an ideal. It is nonzero: choose core vectors with nonzero pairing, so their coefficient is nonzero near the identity, and a compact test inside \(G\) giving a nonzero integral in a right half-plane. The actual earlier Hermite theorem, Lemma 3.1 and Theorem 3.2 in Hermite functions, tempered distributions and the Schwartz kernel theorem, proves polynomial-Gaussian Schwartz density after dilation. The continuous normalized bound of Theorem 1.26 therefore approximates that nonzero value by one of (1.32p).

Choose a monic nonzero ideal element \(P_Q\) of least degree. Polynomial division of any other element leaves a smaller-degree remainder in the ideal, which must vanish. Hence \(J_Q=P_Q\mathbb C[s]\). By the definition of its span, this chosen polynomial is an actual finite sum of polynomial multiples of integral values. Thus \[ \sum_{\alpha=1}^r p_\alpha(s)Z_Q(s,\Phi_\alpha,c_\alpha) =P_Q(s)L_{\boldsymbol\chi}(s). \tag{1.32q} \] The actual test operator \(\mathscr S\) in (1.31xa) absorbs each \(p_\alpha\) into \(\Phi'_\alpha=p_\alpha(\mathscr S)\Phi_\alpha\). These are still fixed polynomial Gaussians, and consequently \[ \sum_{\alpha=1}^r Z_Q(s,\Phi'_\alpha,c_\alpha)=L_Q(s). \tag{1.32qa} \] This proves the exact ideal and finite attainment, not an unspecified common meromorphic majorant.

Step 7: Entire division for the whole family; no zeros

The functions \(H=Z_Q/L_{\boldsymbol\chi}\) are entire and jointly continuous with compact-parameter bounds from Step 5 of this argument. For a core coefficient, approximate any Schwartz test by polynomial Gaussians. The resulting \(H\)'s converge locally uniformly. Cauchy's derivative formula preserves every zero multiplicity of \(P_Q\) in the limit, so \(H/P_Q\) is entire. Next approximate both arbitrary smooth coefficient vectors by their core \(A_N\)-approximants. The joint bounds give the same locally uniform convergence, hence the same conclusion for every smooth paired coefficient.

Continuity after division is explicit. Surround the finitely many roots of \(P_Q\) by circles avoiding roots on their boundaries. Cauchy's formula for \(H/P_Q\) bounds it on inner disks by a fixed constant times the controlled \(H\) on those circles. On the complement \(1/P_Q\) is bounded. The same finite Schwartz and quotient-vector seminorms therefore control the entire normalized family on any compact parameter set.

For any \(s_0\), take a nonzero core coefficient and a nonnegative smooth cutoff \(\eta\) in a compact subset of \(G\) on which it is nonzero. The following test, extended by zero, is Schwartz: \[ \Phi(g)=\eta(g)\overline{c(g)} \nu(g)^{-i\operatorname{Im}s_0}. \tag{1.32r} \] Its compact-support integral is entire, and \[ Z_Q(s_0,\Phi,c) =\int_G\eta(g)|c(g)|^2 \nu(g)^{\operatorname{Re}s_0+(n-1)/2}\,dg>0. \tag{1.32s} \] The proved entire normalized quotient would make this value zero if \(L_Q(s_0)=0\). Therefore \(L_Q\) has no zeros. Every root of \(P_Q\) cancels an inducing-product pole, with no excess multiplicity. This does not yet identify which roots occur for every standard-module label.

Step 8: Exact reflection, constant epsilon and conjugation

The raw Fourier equation descended from the actual kernel is \[ \frac{Z_{Q^\vee}(1-s,\widehat\Phi,c^\vee)} {L_{\boldsymbol\chi^{-1}}(1-s)} =\epsilon_{\boldsymbol\chi} \frac{Z_Q(s,\Phi,c)}{L_{\boldsymbol\chi}(s)}. \tag{1.32t} \] Fourier is a bijection of the polynomial-Gaussian space. Coefficient inversion is a bijection of the two core coefficient families by the perfect dual pairing. Taking their polynomial spans and using the automorphism \(s\mapsto1-s\) of \(\mathbb C[s]\) gives \[ P_{Q^\vee}(1-s)\mathbb C[s]=P_Q(s)\mathbb C[s]. \tag{1.32u} \] The units of this ring are nonzero constants. The two monic polynomials have equal degree \(b\); comparison of the reflected leading coefficients gives exactly the constant \((-1)^b\). This proves (1.32d)--(1.32e), including absolute value one of epsilon by the exact scalar phases in (1.31e). The central sign check (1.31ab) survives the correction: the two correction degrees agree and contribute \((-1)^{2b}=1\). The character-change calculation (1.31ac) applies verbatim because the positive and angular central actions descend to \(Q\). Thus \[ \epsilon_Q(s,\psi_{F,a}) =\omega_Q(a)|a|_F^{n(s-1/2)}\epsilon_Q. \tag{1.32ua} \] For a unitary core, the central character has modulus one: its radial Lie scalar is purely imaginary by skew Hermitian invariance, and its compact angular/sign action is unitary by invariance of the positive form. Hence (1.32ua) has modulus one on \(\operatorname{Re}s=1/2\), including every changed basic character.

Conjugation of the actual compact model changes \(t_j\) to \(\bar t_j\), keeps real parity and negates complex angular weights. Conjugate closures and quotients are the actual constructed models. The Euler gamma integral and continuation give \(L_{\overline{\boldsymbol\chi}}(s) =\overline{L_{\boldsymbol\chi}(\bar s)}\). Conjugation preserves the Gaussian polynomial space and conjugates every coefficient. The normalized ideal is consequently the coefficientwise conjugate ideal. Its unique monic generator is \(\overline{P_Q(\bar s)}\), proving (1.32f) with no unknown scalar.

Step 9: Coefficient independence and the unitary normalization

Two constructed models with isomorphic cores and identified dual-core pairings have the same core coefficient functions. Indeed these functions are real analytic by Step 2 of this argument and (1.32o). Every Taylor derivative at the identity is the paired core action of an enveloping-algebra operator, hence agrees under the isomorphism. Analyticity gives equality on a neighborhood and its connected component. Each other component has a representative in \(K\); the isomorphism preserves that action, so the same reasoning covers every component. Thus their unnormalized Gaussian ideals agree. This assertion concerns the constructed principal subquotient models, not comparison with an arbitrary other smooth realization.

An invariant positive Hermitian form on \(Q_0\) identifies \(\bar Q_0\) with its admissible contragredient. It is onto on every finite compact packet by finite-dimensional Hermitian duality, and the direct sum of packets gives the core isomorphism. Skew Lie invariance and compact invariance make it an intertwiner. The preceding coefficient independence therefore shows that \(L_{Q^\vee}\) and \(L_{\bar Q}\) differ by a nonzero constant.

That constant is one. The positive central infinitesimal scalar is \(d\sum t_j\); skew Hermitian invariance forces \(\operatorname{Re}\sum t_j=0\). The inverse-character and conjugate-character inducing products have the same parity/angular shifts, and their norm-exponent sums are \(-\sum t_j=\sum\bar t_j\). Their correcting polynomials have equal degree, by reflection and conjugation, and are monic.

We need only the elementary ratio asymptotic \[ \frac{\Gamma(u+\alpha)}{\Gamma(u)u^\alpha}\longrightarrow1 \quad(u\to+\infty) \tag{1.32v} \] for fixed complex \(\alpha\). To prove it, let \(Y_u\) have density \(e^{-y}y^{u-1}/\Gamma(u)\) on \(y>0\). Gamma recurrence gives mean \(\mathbb E(Y_u/u)=1\) and variance \(1/u\), so \(Y_u/u\to1\) in probability. For an integer \(A>|\operatorname{Re}\alpha|+2\), recurrence also bounds uniformly its positive and negative \(A\)-moments for \(u>A+1\). Outside \([R^{-1},R]\), use the strictly smaller exponent \(\operatorname{Re}\alpha\) and these moments to bound the tails of \((Y_u/u)^\alpha\) by a quantity tending to zero with \(R\). Inside that interval this is a bounded continuous function and convergence in probability gives expectation tending to one. Its expectation is the left side of (1.32v), by the Euler integral. This proves the asymptotic without a quoted Stirling theorem.

Hence \(\Gamma_{\mathbb R}(s+\alpha)/\Gamma_{\mathbb R}(s) \sim(s/(2\pi))^{\alpha/2}\) and \(\Gamma_{\mathbb C}(s+\alpha)/\Gamma_{\mathbb C}(s) \sim(s/(2\pi))^\alpha\) for real \(s\to+\infty\). The equal shift sums make the inverse/conjugate inducing-product ratio tend to one. The equal-degree monic polynomial ratio also tends to one. Their constant ratio is therefore one, proving (1.32g).

Precise unclosed part of the original target

Theorem 1.26a provides actual continuous maps and paired models for every algebraic principal-core subquotient, and Theorem 1.26b proves its exact Gaussian ideal and finite attainment as a finite polynomial correction of the explicit inducing gamma product. That correction is defined by the integrals, not an imported classification statement.

Theorems 1.27–1.31 below prove principal-core occurrence and transfer the Gaussian package to every prescribed actual complete smooth moderate dual-pair realization. Theorems 1.32–1.34 below construct a compatible actual realization for every abstract irreducible admissible core. Explicit identification of \(P_Q\) and its scalar with the intended real character/discrete-series or complex angular standard gamma factors also remains required. LG-LLC-01's unproved classification and the rank-two kernel are not being used to supply them.

The prior programme inputs are the actual scalar Fourier/Tate proofs used in Theorem 1.26 and actual polynomial-Gaussian Schwartz density, Hermite functions, tempered distributions and the Schwartz kernel theorem, Lemma 3.1 and Theorem 3.2. All compact, analytic-core, closure, quotient, annihilator, ideal, reflection and gamma-ratio arguments used here are written above. Goldfeld--Jacquet's free author notes §3, Goldfeld–Jacquet, freely accessible author notes, are a locator for the general target, not a substitute for these proofs or for the still-missing classification.

Finite generation over the minimal nilpotent algebra for real and complex general linear groups

Let \(F=\mathbb R\) or \(\mathbb C\), \(G=GL_n(F)\), \(K=O(n)\) or \(U(n)\), and \(N\) the upper unitriangular group. All enveloping algebras and Harish-Chandra modules below are over \(\mathbb C\). Thus \(\mathfrak g\) denotes the complexification of the real Lie algebra, and \(\mathfrak n,\mathfrak k\) denote the corresponding complexifications. An admissible module means a compatible \((\mathfrak g,K)\)-module with finite-dimensional compact isotypic packets. Compact-finite vectors have finite-dimensional \(K\)-orbits.

Theorem 1.27. Every irreducible admissible \((\mathfrak g,K)\)-module \(V_0\) for \(GL_n(\mathbb R)\) or \(GL_n(\mathbb C)\) is finitely generated over \(U(\mathfrak n)\). Consequently \(V_0/\mathfrak n^qV_0\) is finite-dimensional for every positive integer \(q\). The same statements hold for lower unitriangular \(\mathfrak n\).

This is an algebraic statement about an arbitrary core. No globalization, subrepresentation, Jacquet nonvanishing or classification theorem is used.

Step 1: Finite core generators and the required central operators

Choose a nonzero compact-finite vector \(v\) and let \(W\) be its finite-dimensional \(K\)-orbit span. Then \(U(\mathfrak g)W=V_0\): the former is a nonzero Lie- and compact-stable subspace, because \(\operatorname{Ad}(K)\) preserves every finite-degree part of the enveloping algebra. Irreducibility gives equality.

Use the total-degree filtration on the enveloping algebra. Only the following spanning statement is needed. In a word of Lie generators, exchanging an out-of-order adjacent pair replaces their difference by a single commutator and therefore lowers the word length. Induction first on length and then on the number of inversions expresses each word in ordered monomials. Commutators have smaller length, so the leading symbols commute and there is a surjection \(S(\mathfrak g)\twoheadrightarrow \operatorname{gr}U(\mathfrak g)\). No linear-independence assertion about these ordered monomials is required: this surjection and their spanning suffice for every graded quotient used below.

For a matrix algebra, the invariant polynomial \(\operatorname{tr}(X^r)\) has an invariant symmetrization \(C_r\) in its enveloping algebra. Explicitly, under the trace identification, its symbol is \[ p_r=\sum_{i_1,\ldots,i_r} E_{i_1i_2}E_{i_2i_3}\cdots E_{i_ri_1}. \tag{1.33a} \] Symmetrization averages each degree-\(r\) monomial over all orders. It commutes with every adjoint derivation, since applying a derivation before or after the average gives the same sum. The polynomial \(p_r\) is adjoint-invariant, because differentiating \(\operatorname{tr}(e^{tY}Xe^{-tY})^r\) gives zero. Thus \(C_r\) commutes with every Lie generator and with the full compact group.

For the real group use \(r=1,\ldots,n\) in \(\mathfrak g=\mathfrak{gl}_n(\mathbb C)\). For the complex group \[ \mathfrak g=\mathfrak{gl}_n(\mathbb C)\oplus \mathfrak{gl}_n(\mathbb C),\qquad \mathfrak k=\{(X,-X^t):X\in\mathfrak{gl}_n(\mathbb C)\}; \tag{1.33b} \] use these \(n\) central operators from the first summand. This description follows by complexifying the real map \(Y\mapsto(Y,\bar Y)\); for skew-Hermitian \(Y\), \(\bar Y=-Y^t\).

Each \(C_r\) acts by a scalar \(\lambda_r\) on \(V_0\). Indeed it preserves every finite compact packet. On any nonzero such packet it has an eigenvector over \(\mathbb C\). The kernel of \(C_r-\lambda_r\) is a nonzero Lie- and compact-stable subspace, so is the whole irreducible module. This argument uses neither an infinite-dimensional Schur theorem nor the Harish-Chandra isomorphism.

Filter \(V_0\) by \(V_j=U_j(\mathfrak g)W\). Its associated graded is a finitely generated symmetric-algebra module. The symbols of \(\mathfrak k\) kill its generators: \(\mathfrak kW\subset W\), so this action has degree zero. The symbols \(p_r\) also kill the entire graded module, because \((C_r-\lambda_r)V_0=0\). Commutativity of the symbols now gives a surjection \[ \left(S(\mathfrak g)/ (\mathfrak k,p_1,\ldots,p_n)\right)\otimes W \longrightarrow \operatorname{gr}V_0. \tag{1.33c} \] We prove directly that the algebra on the left is finite over the image of \(S(\mathfrak n)\).

Step 2: The finite commutative calculation

Over the real group, quotienting by \(\mathfrak k\) sets \(E_{ij}=E_{ji}\). Write \(a_i=E_{ii}\), and \(y_{ij}=E_{ij}=E_{ji}\), \(i<j\). The quotient is \[ \mathbb C[y_{ij}:i<j]\,[a_1,\ldots,a_n], \tag{1.33d} \] with the off-diagonal polynomial ring precisely the image of \(S(\mathfrak n)\). The symbols \(p_r\) are traces of powers of the symmetric symbol matrix having diagonal \(a_i\) and off-diagonal entries \(y_{ij}\).

For the complex group, the relations in (1.33b) set \(E_{ij}^{(1)}=E_{ji}^{(2)}\). The upper entries in the first summand and the upper entries in the second supply all off-diagonal entries of one unrestricted symbol matrix. Its diagonal entries \(a_i=E_{ii}^{(1)}=E_{ii}^{(2)}\) supply the remaining variables. Again the quotient has the form (1.33d), now with all ordered off-diagonal entries, and their polynomial ring is the image of \(S(\mathfrak n)\). The selected first-summand \(p_r\)'s are exactly the traces of powers of this unrestricted symbol matrix.

Give diagonal variables degree one and off-diagonal variables degree zero. In either case, \[ p_r=\sum_i a_i^r+\text{terms of diagonal degree less than }r. \tag{1.33e} \] For a term using an off-diagonal matrix entry, at least one factor is off-diagonal, so the asserted strict inequality follows directly from (1.33a).

Newton's identities imply that the ideal \((p_1,\ldots,p_n)\) also contains characteristic-polynomial coefficients \(Q_r\) whose leading diagonal terms are the elementary symmetric polynomials \(e_r(a)\), \(r=1,\ldots,n\). Here the identities can be obtained without any invariant-theory input: for indeterminates \(z_i\), differentiate \(\prod_i(1+z_it)\), divide formally by that product, and compare coefficients with \(\sum_i z_i/(1+z_it)\). This gives \[ r e_r=\sum_{j=1}^r(-1)^{j-1}e_{r-j}\sum_i z_i^j, \quad e_0=1. \tag{1.33f} \] Applying these polynomial formulas to the \(p_j\)'s gives \(Q_r\), and (1.33e) gives \(Q_r=e_r(a)+R_r\) with \(\deg_a R_r<r\).

In the quotient by these relations, use the elementary identity \[ a_i^n=\sum_{r=1}^n(-1)^{r+1}e_r(a)a_i^{n-r} =\sum_{r=1}^n(-1)^rR_r a_i^{n-r}. \tag{1.33g} \] The first equality evaluates \(\prod_j(T-a_j)\) at \(T=a_i\). Every term on the final right side has diagonal degree at most \(n-1\). Whenever a diagonal monomial has an exponent at least \(n\), (1.33g) therefore lowers its total diagonal degree. Iteration terminates and leaves a combination, over the off-diagonal ring, of the finite list \[ a_1^{b_1}\cdots a_n^{b_n},\qquad 0\le b_i<n. \tag{1.33h} \] Thus (1.33c) is finite over \(S(\mathfrak n)\). No assertion about a polynomial gamma majorant or a standard-module classification enters this finite calculation.

Step 3: Lifting generators and finite Jacquet quotients

Lift the finitely many generators (1.33h) times a basis of \(W\) to actual vectors of \(V_0\). They generate \(V_0\) over \(U(\mathfrak n)\): for a vector of filtration degree \(j\), its leading symbol is a combination of the lifted generators with nilpotent-algebra symbols; subtract the corresponding enveloping-algebra combination to lower the filtration degree. Induction on \(j\) finishes the argument.

Let \(I=\mathfrak nU(\mathfrak n)\) be the augmentation ideal. If \(V_0\) has \(r\) nilpotent-algebra generators, there is a surjection \[ (U(\mathfrak n)/I^q)^r\longrightarrow V_0/I^qV_0. \tag{1.33i} \] Ordered monomials of length at least \(q\) belong to \(I^q\); hence the quotient on the left is spanned by the finite list of monomials of length less than \(q\). This proves finite dimension without identifying \(I^q\) with a PBW filtration piece. The notation \(\mathfrak n^qV_0\) means exactly \(I^qV_0\). The quotient is stable under the diagonal Lie algebra and the compact diagonal subgroup, since their adjoint actions preserve \(\mathfrak n\) and every power of its augmentation ideal.

The long permutation matrix in \(K\) exchanges upper and lower unitriangular groups. Transporting the proof by that matrix gives the lower-nilpotent statement and completes Theorem 1.27.

Scope

Theorem 1.27 supplies finite minimal Jacquet quotients for every abstract irreducible admissible core. It does not assert that the first quotient is nonzero: an augmentation ideal is not automatically the Jacobson radical of its enveloping algebra, so a bare use of Nakayama's lemma here would be invalid. Theorem 1.28 below proves that nonvanishing for a supplied actual dual pair. Theorems 1.32–1.34 then prove nonvanishing and construct a compatible complete smooth moderate realization for every abstract irreducible admissible core.

The free primary locator is Casselman's Canonical extensions of Harish-Chandra modules to representations of G, §5, available as the publisher's freely accessible PDF: Casselman, publisher open-access edition. Its cited finite-generation and nonvanishing results are not being imported in place of the proofs written here.

Nonzero minimal Jacquet quotients and actual algebraic Borel occurrence

Let \(G=GL_n(F)\), \(F=\mathbb R\) or \(\mathbb C\), with the upper unitriangular group \(N\), upper triangular group \(B\) and maximal compact group \(K\). Suppose \(E,\widetilde E\) are actual complete Hausdorff locally convex smooth moderate-growth representations, with continuous differentiated actions and a jointly continuous nondegenerate invariant pairing \[ B_E:E\times\widetilde E\longrightarrow\mathbb C . \tag{1.34a} \] Their compact-finite cores are \(V_0,\widetilde V_0\), with \(\widetilde V_0\) the admissible contragredient, and \(V_0\) is irreducible admissible. Moderate growth means that for each continuous seminorm \(p\) there are a continuous seminorm \(q\), \(C\) and \(A\) such that \[ p(\pi(g)v)\le C H(g)^Aq(v),\qquad H(g)=1+\|g\|+\|g^{-1}\| . \tag{1.34b} \] The actions are jointly continuous. No prescribed realization is replaced by an unstated canonical completion.

Theorem 1.28. Under these hypotheses, \(V_0/\mathfrak nV_0\ne0\). It is finite-dimensional by Theorem 1.27. There are smooth diagonal characters \(\chi_1,\ldots,\chi_n\) and an explicit injective \((\mathfrak g,K)\)-map \[ T_0:V_0\longrightarrow I_0(\chi_1,\ldots,\chi_n) \tag{1.34c} \] into the compact-finite core of the full normalized Borel induction. The same assertion holds for \(\widetilde V_0\). Transposing its injection on finite compact packets gives a surjective core map from a full inverse-character Borel induction onto \(V_0\).

This proves occurrence in every rank and for both fields for every core having the stipulated actual realization, including the compatible smooth cuspidal realizations supplied by lesson04 Theorem 1.19. It does not claim existence of those realizations for an arbitrary abstract core. It also does not assert continuity of the algebraic \(T_0\) on \(E\). Removing or restoring a central norm twist does not restrict this scope: multiply the paired actions by that character and its inverse. Their pairing stays invariant, and the norm character and its inverse are bounded by fixed powers of \(H(g)\), so smoothness, completeness and moderate growth persist.

Step 1: Compact smoothing and fixed finite generators

The compact-finite core is dense in each supplied topology. One can check this directly using the polynomial class functions \[ q_K(k)=\frac{1+\operatorname{Re}\operatorname{tr}(k)/n}{2}, \qquad A_m v=\frac{\int_Kq_K(k)^m\pi(k)v\,dk}{\int_Kq_K(k)^m\,dk}. \tag{1.34d} \] They are positive approximate identities: away from an identity neighborhood \(q_K\le c<1\), whereas a smaller neighborhood has \(q_K\ge c'>c\) and positive Haar mass. Its relative exterior mass is at most \((c/c')^m/\operatorname{mass}(U')\). Joint continuity gives \(A_m v\to v\) in every seminorm. The integrals exist by compact Riemann sums and completeness. Under left or right compact translation the polynomial kernel has finite-dimensional orbit, so the output is compact-finite. This proves density without a globalization theorem. Compact character projectors are continuous by the same integration and finite-dimensional Schur orthogonality. Their ranges are exactly the finite isotypic packets in the given core.

The contragredient core is irreducible. If a nonzero proper submodule \(U\subset\widetilde V_0\) existed, some finite compact packet of \(U\) would be proper in the corresponding dual packet. Finite-dimensional duality gives a nonzero vector annihilating \(U\). The annihilator in \(V_0\) is Lie- and compact-stable, hence equals \(V_0\), contradicting \(U\ne0\). Here an entire proper submodule cannot fill every packet, since the core is their algebraic direct sum. Thus both cores are finitely generated over \(U(\mathfrak g)\) by finite compact-stable spaces.

We require the following elementary strengthening in an actual realization. Choose a finite set of compact types whose entire isotypic packets \(W\) contain core generators. Their sum \(W\) is finite-dimensional. Let \(P_W\) be their character projector. Choose nonnegative smooth approximate identities \(h_\epsilon\) on \(G\), supported near the identity, with integral one. Then \[ A=P_W\pi(h_\epsilon)P_W|_W \tag{1.34e} \] is arbitrarily close to identity on the finite space \(W\), hence invertible for small \(\epsilon\). Both compact convolutions convert \(h_\epsilon\) into one bi-compact-finite \(h\in C_c^\infty(G)\); its support lies in the compact set \(K\operatorname{supp} (h_\epsilon)K\). For \(w\in W\), \(w=\pi(h)A^{-1}w\).

Every core vector is a finite sum \(D w\), \(D\in U(\mathfrak g)\), \(w\in W\). Differentiation under the compactly supported convolution and integration by parts express \(D\pi(h)w'\) as \(\pi(h_D)w'\), with \(h_D\in C_c^\infty(G)\). The sign and left/right differentiation are fixed by the identity \(\pi(\exp(tX))\pi(h)=\pi(h(\exp(-tX)\,\cdot))\). Consequently every core vector has a representation \[ \sum_{j=1}^r\pi(f_j)w_j,\qquad f_j\in C_c^\infty(G), \tag{1.34f} \] with a fixed finite list \(w_1,\ldots,w_r\in W\). The functions may depend on the vector. Apply the same construction to the contragredient supplied realization.

Step 2: One coefficient-growth exponent for every dual core vector

There are a continuous seminorm \(Q\) on \(E\) and one exponent \(A\) such that for each \(\widetilde v\in\widetilde V_0\) there is a finite constant \(C_{\widetilde v}\) with \[ |B_E(\pi(g)v,\widetilde v)| \le C_{\widetilde v}H(g)^A Q(v) \quad(v\in E,\ g\in G). \tag{1.34g} \] The exponent and the seminorm do not change when \(\widetilde v\) is differentiated any fixed number of times.

Indeed choose the fixed generators \(\widetilde w_j\) in (1.34f). Joint continuity bounds all \(B_E(v,\widetilde w_j)\) by one continuous seminorm \(p(v)\). Apply (1.34b) to that one \(p\); this gives a single \(Q,A\). For \(\widetilde v=\sum_j\widetilde\pi(f_j)\widetilde w_j\), invariance gives \[ B_E(\pi(g)v,\widetilde v) =\sum_j\int_G f_j(h) B_E(\pi(h^{-1}g)v,\widetilde w_j)\,dh . \tag{1.34h} \] The elementary norm satisfies \(H(h^{-1}g)\le C H(h)H(g)\). Insert the bound just obtained and integrate \(|f_j(h)|H(h)^A\), finite by compact support. This proves (1.34g). Every differentiated dual core vector is still in the same core, so (1.34f) gives its own finite constant and exactly the same \(Q,A\). No unproved uniform asymptotic estimate is being used.

Step 3: A bounded radial finite jet system

We prove the precise fact about compact-finite coefficient functions that will prevent arbitrary exponential decay. Let \(f\) be a smooth function on \(G\), finite under both compact translations, and with a scalar action of the selected matrix central operators. Its finite compact translation span identifies the restriction to \[ a(x)=\operatorname{diag}(e^{x_1},\ldots,e^{x_n}) \tag{1.34i} \] with a vector \(F(x)\) in a fixed finite-dimensional space. Using all finite left/right translates avoids choosing a single possibly vanishing radial component.

Fix a chamber with \(|x_i-x_j|\ge\eta>0\), and its part in which the signs of those differences do not change. Each central trace operator of degree \(r\), normalized as specified below, has radial expression \[ \mathcal C_r F =\sum_i\partial_{x_i}^{\,r}F+L_rF,\qquad \operatorname{ord}_x L_r\le r-1, \tag{1.34j} \] where the coefficients of \(L_r\), and every fixed number of their derivatives, are bounded on this chamber portion. They are matrices on the finite compact translation space.

Here is the coordinate computation, including the bounds. For \(r_i=e^{x_i}\), the right vector field \(E_{ij}\), \(i\ne j\), at \(a\) is a combination of compact left and right fields. Over \(\mathbb R\), take \(K_{ij}=E_{ij}-E_{ji}\). Solving \(aE_{ij}=uK_{ij}a+v aK_{ij}\) gives exactly \[ u=\frac{r_ir_j}{r_j^2-r_i^2} =\frac1{2\sinh(x_j-x_i)},\qquad v=-\frac{r_i^2}{r_j^2-r_i^2} =-\frac1{e^{2(x_j-x_i)}-1}. \tag{1.34k} \] The analogous expression for \(E_{ji}\) uses the same \(u\) and \(v=-r_j^2/(r_j^2-r_i^2)\). These formulas follow by comparing the two off-diagonal entries, so introduce no integration or asymptotic theorem. For \(\mathbb C\), the compact generators \(E_{ij}-E_{ji}\) and \(i(E_{ij}+E_{ji})\) solve the two real off-diagonal systems; the same denominators and bounded ratios occur. Imaginary diagonal generators are compact fields, and real diagonal generators are \(\partial_{x_i}\).

All the displayed coefficients and their derivatives are bounded when the gap has magnitude at least \(\eta\). Derivatives only produce rational expressions in \(e^{x_j-x_i}\) with the same excluded zero denominator; at either infinite endpoint they have finite limits or tend to zero. The local compact-angular coordinate expressions away from identity are obtained by conjugating the Lie generators by compact matrices. Their fixed derivatives are bounded because compact adjoint matrices and their derivatives are bounded. The same two-by-two systems then apply. Diagonal compact redundancy in the complex case is removed by choosing one compact torus factor rather than two; it gives a compact derivative, not a growing radial coefficient.

Applying a product of \(r\) vector fields now gives bounded coefficients on a finite sum of compact derivatives and radial derivatives. The compact derivatives act by fixed matrices on the chosen finite translation space. Only products of the diagonal radial components can contribute radial derivative order \(r\); differentiating any coefficient lowers that order, and an off-diagonal generator has zero radial component at \(a\). The invariant trace symbol restricts on the diagonal to \(\sum_i\xi_i^r\). This proves (1.34j) with every stated bound. Over the complex group use \(2^r C_r^{(1)}\): the first-summand generator \((E_{ii},0)\) has radial component \(\tfrac12\partial_{x_i}\), its other component being compact. Over the real group no such factor is present.

Newton's polynomial construction (1.33f) applied to these commuting central operators gives degree-\(r\) central operators whose radial expressions are \[ e_r(\partial_x)F+\mathcal L_rF=\mu_rF,\qquad \operatorname{ord}\mathcal L_r\le r-1, \quad r=1,\ldots,n. \tag{1.34l} \] All coefficient derivatives remain bounded, by finite products and the product rule. No Harish-Chandra isomorphism or radial formula has been assumed.

For each \(i\), the ordinary constant-coefficient identity \[ \partial_{x_i}^{\,n} =\sum_{r=1}^n(-1)^{r+1} \partial_{x_i}^{\,n-r}e_r(\partial_x) \tag{1.34m} \] is the characteristic-polynomial identity used in (1.33g). Apply (1.34l) to \(F\) on its right side. Every resulting term has total radial order at most \(n-1\); its coefficients and fixed derivatives are bounded. Differentiate these equations to reduce every derivative whose multi-index has some component at least \(n\). Each reduction strictly lowers total derivative order, so it terminates. Thus the finite jet \[ J(x)=\big(\partial_x^\alpha F(x)\big)_{0\le\alpha_i<n} \tag{1.34n} \] satisfies a first-order system \[ \partial_{x_i}J=A_i(x)J,\qquad \sup_x\|A_i(x)\|<\infty \tag{1.34o} \] on every such fixed-gap chamber portion. Only finitely many derivatives of the bounded coefficients are needed to form these matrices.

Along a ray \(x(t)=x^{(0)}-tH\) remaining in that portion, \(J'(t)=A(t)J(t)\) with \(\|A(t)\|\le M\). For a nonzero initial jet, the elementary integral inequality for the backward equation gives \[ \|J(t)\|\ge e^{-Mt}\|J(0)\|,\qquad t\ge0. \tag{1.34p} \] To check this without dividing by a possibly zero norm, solve backwards from \(t\) and use the integral inequality \(\|J(s)\|\le \|J(t)\|+\int_s^t M\|J(u)\|du\); iteration of its integral series gives the factor \(e^{M(t-s)}\). If \(J(t)=0\), this argument also forces \(J(0)=0\).

Step 4: Nonvanishing of the Jacquet quotient

For \(n=1\), \(\mathfrak n=0\), so there is nothing to prove. For \(n\ge2\), suppose instead that \(V_0=\mathfrak nV_0\). Then \(V_0=\mathfrak n^qV_0\) for every \(q\). Choose a nonzero paired core coefficient \[ f(g)=B_E(\pi(g)v,\widetilde v); \tag{1.34q} \] one can choose its value at identity nonzero. The function is finite under both compact translations. Each chosen central operator acts by a scalar on the irreducible core, by the finite-packet argument in Theorem 1.27. The radial system in Step 3 of this argument therefore applies.

Choose a regular matrix near the identity where this coefficient is nonzero. Such matrices have pairwise distinct singular values: the discriminant of the characteristic polynomial of \(g^*g\) is a nonzero polynomial in the real entries, so its zero set has empty interior. Elementary diagonalization gives \(g=k_1a(x^{(0)})k_2\), with \(x_1^{(0)}<\cdots<x_n^{(0)}\), after a compact permutation. Including all finite compact translates in \(F\) makes \(F(x^{(0)})\ne0\), hence \(J(x^{(0)})\ne0\).

Take real \(H_1>\cdots>H_n\), put \(\epsilon=\min_i(H_i-H_{i+1})>0\), and use \[ a_t=\operatorname{diag}(e^{x_i^{(0)}-tH_i}). \tag{1.34r} \] The ordered gaps grow and stay bounded away from zero. For every upper-root generator \(X_{ij}\), including its imaginary companion over \(\mathbb C\), \[ \operatorname{Ad}(a_t)X_{ij} =e^{x_i^{(0)}-x_j^{(0)}}e^{-t(H_i-H_j)}X_{ij}. \tag{1.34s} \] Every fixed core vector \(w\), since it belongs to \(\mathfrak n^qV_0\), is a finite sum \(X_1\cdots X_qw'\) with upper-root generators. Apply (1.34s) to \(\pi(a_t)w\) and transfer those Lie generators to the dual vector using invariance of (1.34a). The bound (1.34g) uses the same exponent \(A\) for every resulting dual core vector. Since \(H(a_t)\le C e^{Lt}\) with a fixed \(L\), it gives \[ |B_E(\pi(a_t)w,\widetilde w)| \le C_{q,w,\widetilde w} e^{(AL-q\epsilon)t}. \tag{1.34t} \] Constants may depend on \(q\), but are finite and do not depend on \(t\). This estimate therefore beats every prescribed exponential.

Each component of the finite radial jet (1.34n) is a coefficient of fixed core vectors: radial derivatives differentiate the commuting diagonal action on the original vector, and compact translations remain in the core. Apply (1.34t) to its finitely many components. Choosing \(q\epsilon>AL+M+1\) contradicts (1.34p) as \(t\to\infty\). Thus \(V_0/\mathfrak nV_0\ne0\). This is a complete nonvanishing argument in the stated actual realization scope; an external asymptotic expansion or a source statement is not substituting for it.

Step 5: Explicit algebraic Frobenius map into a full Borel induction

By Theorem 1.27 the nonzero quotient \(V_0/\mathfrak nV_0\) is finite-dimensional. The diagonal real Lie algebra commutes with \(M=B\cap K\) on this quotient, and its operators commute with one another. First split the finite compact diagonal action into its one-dimensional characters. This follows by averaging a Hermitian form and simultaneously diagonalizing the commuting unitary matrices; each invariant eigenspace can be treated in turn. On one nonzero compact-character subspace, commuting complex matrices have a common eigenfunctional: choose an eigenspace for one transpose, restrict the others to it and repeat until a common eigenvector is obtained. Pulling it back gives a nonzero \(\ell_0:V_0\to\mathbb C\) with \[ \ell_0(\mathfrak nV_0)=0,\qquad \ell_0(H_iv)=\beta_i\ell_0(v),\qquad \ell_0(mv)=\sigma(m)\ell_0(v). \tag{1.34u} \] Here \(H_i\) is the real diagonal \(E_{ii}\). The sign weights of \(\sigma\) over \(\mathbb R\) are \(e_i\in\{0,1\}\). Its circle weights over \(\mathbb C\) are integers \(\ell_i\), since the smooth circle character solves a constant differential equation and has period \(2\pi\).

Set \(\rho_i=(n+1)/2-i\) and \(d=[F:\mathbb R]\). Choose \[ t_i=\beta_i/d-\rho_i,\qquad \chi_i(x)=\operatorname{sgn}(x)^{e_i}|x|^{t_i} \quad(\mathbb R),\qquad \chi_i(z)=(z/|z|)^{\ell_i}|z|_{\mathbb C}^{t_i} \quad(\mathbb C). \tag{1.34v} \] Then the infinitesimal Borel character of \(\ell_0\) is exactly \(\delta_B^{1/2}\prod_i\chi_i\), including every factor \(d\). The upper nilpotent action is trivial, and the compact diagonal character matches the selected signs or angles.

In the actual compact model of full normalized induction define \[ (T_0v)(k)=\ell_0(kv),\qquad k\in K. \tag{1.34w} \] The finite compact orbit of \(v\) proves this is smooth and compact-finite, with left \(M\)-covariance \((T_0v)(mk)=\prod_i\chi_i(m_i)(T_0v)(k)\). The action and the complete compact model are constructed in the previous written Theorem 1.26; equivalently extend (1.34w) from \(K\) by the row-QR formula \(g=bk\) and \(f(bk)=\delta_B(b)^{1/2}\boldsymbol\chi(b)f(k)\).

It is a Lie intertwiner, as can be checked in compact coordinates. For a real Lie generator \(X\), differentiate \[ k\exp(tX)=b(t,k)k'(t,k). \] At \(t=0\), its infinitesimal decomposition is \(\operatorname{Ad}(k)X=Y_B+Y_K\), with \(Y_B\) upper triangular with real diagonal, and \(Y_K\in\mathfrak k\). The derivative of the inducing multiplier is its Borel character on \(Y_B\), which equals \(\ell_0(Y_Bkv)/\ell_0(kv)\) where that denominator is nonzero; without division the same linear identity follows from (1.34u). The compact derivative contributes \(\ell_0(Y_Kkv)\). Their sum is \[ (dI(X)T_0v)(k)=\ell_0(\operatorname{Ad}(k)X\,kv) =\ell_0(kXv)=(T_0Xv)(k). \tag{1.34x} \] Compact equivariance follows immediately by right translation. Since \(\ell_0\ne0\), \(T_0\ne0\); its Lie- and compact-stable kernel in an irreducible core is zero. This proves (1.34c).

Repeat for the supplied contragredient realization. Each compact packet of its injection is an injective map of finite spaces. Its transpose is surjective onto the dual packet. The invariant compact pairing identifies the dual core of \(I_0(\boldsymbol\chi)\) with \(I_0(\boldsymbol\chi^{-1})\); the pairing is the explicit compact integral in Theorem 1.26. Summing packet transposes therefore gives an actual surjective \((\mathfrak g,K)\)-map \[ I_0(\boldsymbol\chi^{-1})\twoheadrightarrow V_0. \tag{1.34y} \] This is the dual-core occurrence/quotient interface; no closed-range or continuous quotient theorem is concealed in the word “surjective.”

Step 6: The exact continuity boundary

If a selected algebraic \(\ell_0\) in (1.34u) extends to a continuous functional \(\ell\) on the prescribed \(E\), its covariance integrates to the actual Borel group: along each real diagonal one-parameter group the functional satisfies the scalar differential equation, along each upper unipotent one it is constant, and \(M\)-covariance is already prescribed. The diagonal and elementary unipotent generators generate \(B\). Then \[ (Tv)(g)=\ell(\pi(g)v) \tag{1.34z} \] is a continuous \(G\)-map \(E\to I(\boldsymbol\chi)\). For each compact derivative, differentiation under this formula and compact equicontinuity bound its compact \(C^\infty\) seminorm by a continuous seminorm on \(E\). Compact equicontinuity itself follows from a finite cover of \(K\) and joint continuity. Its kernel is zero: a nonzero closed invariant kernel contains a nonzero compact-finite vector by (1.34d), contradicting the injective core map.

This last statement is an injection, without an asserted continuous inverse or closed image. The proofs in this subsection do not establish that every algebraic Borel eigenfunctional extends continuously, nor that the image in (1.34z) is closed in the compact-smooth topology. Theorem 2.0y below supplies a selected continuous Borel eigenfunctional and the actual smooth injection in this complete moderate dual-pair class, using exact principal synthesis and the paired distribution embedding. The present subsection alone does not give that continuity. Theorems 1.32–1.34 below construct a compatible realization of every abstract irreducible admissible core. A continuous inverse or closed image for comparison with another prescribed topology remains unproved.

The free primary comparison locator is Casselman's freely accessible publisher paper, §§5,7–8: Casselman, publisher open-access edition. The finite-generation, radial, uniform-growth, nonvanishing and core-occurrence proofs actually used above are written here.

Analytic core coefficients and comparison of actual smooth models

Let \(F=\mathbb R\) or \(\mathbb C\), \(G=GL_n(F)\), and let \((E,\widetilde E,B_E)\) be an actual smooth continuous paired realization as in Theorem 1.28. Its compact-finite core \(V_0\) is irreducible admissible, and the second core is its admissible dual. The first assertion below needs smoothness and the core action; moderate growth and completeness are needed only in the subsequent topological applications, and are not substituted for analyticity.

Theorem 1.29. Every paired core coefficient \[ f_{v,\widetilde v}(g)=B_E(\pi(g)v,\widetilde v), \qquad v\in V_0,\quad \widetilde v\in\widetilde V_0, \tag{1.35a} \] is real analytic on \(G\). If two actual smooth paired realizations have identified \((\mathfrak g,K)\)-cores and the same core pairing, then all their paired core coefficient functions are identical on \(G\). Neither assertion identifies their complete topologies or proves automatic continuity of an arbitrary algebraic core map.

Step 1: A scalar elliptic equation for a compact packet

On the real Lie algebra use the invariant nondegenerate symmetric form \[ b(X,Y)=\operatorname{tr}(XY)\quad(F=\mathbb R),\qquad b(X,Y)=\operatorname{Re}\operatorname{tr}(XY) \quad(F=\mathbb C). \tag{1.35b} \] The Cartan decomposition is the direct sum of self-adjoint matrices \(\mathfrak p\) and skew-adjoint matrices \(\mathfrak k_{\mathbb R}\). The form is positive definite on the former and negative definite on the latter; the two are orthogonal. Choose bases \(P_i\) and \(K_j\) orthonormal for \(b\) and \(-b\), respectively. Put \[ C_G=\sum_iP_i^2-\sum_jK_j^2,\qquad C_K=-\sum_jK_j^2,\qquad \Delta=\sum_iP_i^2+\sum_jK_j^2=C_G-2C_K . \tag{1.35c} \] The tensor for \(C_G\) is invariant: in a basis and its \(b\)-dual, the two commutator sums cancel by \(b([X,Y],Z)=-b(Y,[X,Z])\). Applying multiplication into the enveloping algebra gives \([X,C_G]=0\). Conjugation by every element of \(K\) also preserves \(b\), so \(C_G\) commutes with the full compact action. Its scalar action \(\mu\) on \(V_0\) follows from the finite-packet eigenvector and irreducibility argument in Theorem 1.27.

On a finite compact orbit, average any positive Hermitian form over \(K\). The differentiated \(K_j\)'s are skew-Hermitian, and \(C_K\) is Hermitian nonnegative. It is compact-invariant, because the sum of squares of an orthonormal basis is invariant under the compact adjoint action. Decompose the finite orbit into eigenspaces of \(C_K\). A core vector is a finite sum of eigenvectors \(v_\kappa\) with eigenvalues \(\kappa\ge0\). For the coefficient of each \(v_\kappa\), right differentiation \(R_Xf(g)=\frac{d}{dt}f(g\exp(tX))|_{t=0}\) gives \[ (\Delta-\mu+2\kappa)f_{v_\kappa,\widetilde v}=0. \tag{1.35d} \]

In any real matrix-entry chart on \(G\), the vector fields \(R_X\) have real analytic coefficients: matrix multiplication is polynomial, and inversion on \(GL_n\) is analytic. The principal symbol of \(\Delta\) is the sum of the squares of a real tangent basis. It is positive definite. Thus (1.35d) is a scalar elliptic equation with analytic coefficients. The analytic regularity argument needed here is proved next.

Step 2: Interior estimates without an analytic-regularity citation

Consider on a real coordinate ball \(B_R\subset\mathbb R^m\) an operator \[ P=\sum_{i,j}a_{ij}(x)\partial_i\partial_j +\sum_i b_i(x)\partial_i+c(x), \tag{1.35e} \] where \(a\) is real symmetric positive definite, all coefficients are analytic, and \(b,c\) may be complex. Shrink \(R\le1\) so that the ellipticity constant is uniform and \(a(x)\) differs as little as needed from \(a(0)\). All constants in this section are fixed by this ball and its coefficients, and do not depend on a derivative order.

For \(u\in C_c^\infty(B_R)\), Fourier transformation on \(\mathbb R^m\) gives \[ \|D^2u\|_2\le C\|a(0):D^2u\|_2,\qquad \|Du\|_2\le \varepsilon\|D^2u\|_2+C_\varepsilon\|u\|_2 . \tag{1.35f} \] For example, the first inequality is the pointwise symbol bound \(\sum_{i,j}|\xi_i\xi_j|^2\le C|\xi^ta(0)\xi|^2\); the second follows from \(|\xi|^2\le \varepsilon^2|\xi|^4+C_\varepsilon^2\). The normalization of the Fourier transform cancels on the two sides. To justify its \(L^2\) identity directly, insert \(e^{-\epsilon|\xi|^2}\) in the inverse-transform double integral. The scalar Gaussian integral turns that integral into convolution with an approximate identity. Letting \(\epsilon\) tend to zero gives the identity first for compact smooth functions, then for their differentiated functions. No distributional regularity theorem is being used in (1.35f).

Write \(a(0):D^2u=Pu-(a-a(0)):D^2u-b\cdot Du-cu\). Smallness of \(a-a(0)\), followed by (1.35f) with a sufficiently small \(\varepsilon\), absorbs the second- and first-derivative terms. Hence \[ \|D^2u\|_2\le C\big(\|Pu\|_2+\|u\|_2\big). \tag{1.35g} \] For a smooth \(v\) and concentric balls \(B_r\subset B_s\subset B_R\), put \(\delta=s-r\). Cutoffs with derivative bounds \(|D^j\zeta|\le C_j\delta^{-j}\), \(j=1,2\), exist by rescaling one fixed smooth radial cutoff. Apply (1.35g) to a cutoff times \(v\) which equals \(v\) on \(B_r\) and is supported before the midpoint between \(r\) and \(s\). The product rule yields \[ \|D^2v\|_{2,B_r} \le C\big(\|Pv\|_{2,B_s} +\delta^{-1}\|Dv\|_{2,B_{(r+s)/2}} +\delta^{-2}\|v\|_{2,B_s}\big). \tag{1.35h} \]

The gradient term is controlled by a separate cutoff \(\zeta\) which is one on \(B_{(r+s)/2}\) and supported in \(B_s\). Integrate the real part of \(-Pv\,\zeta^2\bar v\). Integration by parts puts the principal term in the form \(\int\zeta^2a_{ij}\partial_i v\,\overline{\partial_jv}\). Terms differentiating \(a\) or \(\zeta\), and the lower terms, are bounded by Cauchy's inequality by half that positive principal term plus \(C\delta^{-2}\|v\|_2^2\). The remaining undifferentiated expression is bounded by \(\|Pv\|_2\|v\|_2\). Thus \[ \|Dv\|_{2,B_{(r+s)/2}} \le C\big((\|Pv\|_{2,B_s}\|v\|_{2,B_s})^{1/2} +\delta^{-1}\|v\|_{2,B_s}\big). \tag{1.35i} \] Insert this into (1.35h) and use \(\delta^{-1}(ab)^{1/2}\le \tfrac12a+\tfrac12\delta^{-2}b\). We obtain the fixed-order interior estimate \[ \boxed{\quad \|D^2v\|_{2,B_r} \le C_0\big(\|Pv\|_{2,B_s}+\delta^{-2}\|v\|_{2,B_s}\big). \quad} \tag{1.35j} \] It applies to every differentiated smooth \(v\) with the same constant \(C_0\).

Step 3: Factorial derivative bounds

Analyticity of the coefficients on a slightly larger ball gives fixed constants \(A_0,A\) such that \[ \sup_{B_R}|D^\beta a_{ij}|,\; \sup_{B_R}|D^\beta b_i|,\; \sup_{B_R}|D^\beta c| \le A_0A^{|\beta|}|\beta|!. \tag{1.35k} \] Here is a direct power-series check. On a polydisc of radius \(r\), write a coefficient as \(\sum_\alpha c_\alpha (x-x_0)^\alpha\) with \(\sum_\alpha|c_\alpha|r^{|\alpha|}\le M_0\). On the half polydisc its \(\beta\)-th derivative is bounded by \[ M_0r^{-|\beta|}\beta!\, \sum_{\gamma\ge0}\binom{\gamma+\beta}{\beta}2^{-|\gamma|} =M_0r^{-|\beta|}\beta!\,2^{|\beta|+m}. \] The identity on the right is the product of the differentiated geometric-series identities in each coordinate. Termwise differentiation is justified by these absolutely convergent majorants. A finite cover of the closed smaller ball makes the constants uniform. Since \(\beta!\le|\beta|!\), this proves (1.35k) directly from convergent real power series.

Suppose \(Pf=0\) and \(f\) is smooth. Let \[ M=\max\big(1,\|f\|_{H^1(B_R)}\big). \tag{1.35l} \] For a fixed integer \(N\ge2\), put \[ r_j=R-\frac{jR}{2N},\qquad 0\le j\le N . \tag{1.35m} \] For a derivative order \(j\), use the maximum \(L^2\) norm of its ordered coordinate derivatives; including all \(m^j\) ordered derivatives separately is unnecessary. We show, with a fixed sufficiently large \(B\), \[ \max_{|\alpha|=j}\|D^\alpha f\|_{2,B_{r_j}} \le M(BN)^j,\qquad 0\le j\le N. \tag{1.35n} \] The cases \(j=0,1\) follow from (1.35l), after taking \(B\ge1\). For \(j\ge2\), apply (1.35j) to an ordered derivative of length \(j-2\), using \(B_{r_j}\subset B_{r_{j-1}}\). Commuting that derivative past \(P\), the \(k\) derivatives which fall on a coefficient leave a derivative of \(f\) of order at most \(j-k\), with \(k\ge1\). Their Leibniz coefficients are bounded by \(\binom{j-2}{k}k!\le N^k\). The fixed sums over \(i,j\) and the choices of differentiated coordinates are absorbed by enlarging \(A_0,A\). Since \(r_{j-k}\ge r_{j-1}\), the already established bounds (1.35n) control all these derivatives on \(B_{r_{j-1}}\). Derivatives of \(f\) of smaller order also obey the same upper bound, since \(BN\ge1\). The resulting induction inequality is \[ \max_{|\alpha|=j}\|D^\alpha f\|_{2,B_{r_j}} \le C_0A_0M\sum_{k=1}^{j-2}(AN)^k(BN)^{j-k} +\frac{4C_0N^2}{R^2}\,M(BN)^{j-2}. \tag{1.35o} \] For \(j=2\) the sum is empty, since \(Pf=0\). Choose \(B>A\) so large that \[ C_0A_0\frac{A/B}{1-A/B}+\frac{4C_0}{B^2R^2}\le1. \tag{1.35p} \] Dividing (1.35o) by \(M(BN)^j\) proves (1.35n). This is a direct factorial estimate, rather than a repeated second-order estimate with uncontrolled constants.

Set \(j=N\). The elementary integral comparison \(\sum_{\ell=1}^N\log\ell\ge\int_1^N\log x\,dx \ge N\log N-N\) implies \(N^N\le e^NN!\). Therefore \[ \max_{|\alpha|=N}\|D^\alpha f\|_{2,B_{R/2}} \le M(Be)^N N!. \tag{1.35q} \] The first two orders are covered by increasing the constant.

To pass to supremum norms, fix an integer \(q>m/2\). For a compact smooth function \(u\), Fourier inversion and Cauchy--Schwarz give \[ \|u\|_\infty\le C_q\sum_{|\gamma|\le q}\|D^\gamma u\|_2: \tag{1.35r} \] the squared weight \((1+|\xi|^2)^{-q}\) is integrable. Gaussian regularization justifies inversion as in the proof of (1.35f). Apply this to a fixed cutoff times \(D^\alpha f\), equal to that derivative on \(B_{R/4}\), supported in \(B_{R/2}\). Only \(q\) additional derivatives and fixed cutoff constants occur. By (1.35q), their factorials are at most \((N+q)!\le N!(N+q)^q q!\). The fixed-degree polynomial in \(N\) is bounded by a fixed constant times \(2^N\). Consequently there is \(C_1\) with \[ \max_{|\alpha|=N}\sup_{B_{R/4}}|D^\alpha f| \le M C_1^{N+1}N!. \tag{1.35s} \] Taylor's formula on a short coordinate segment now has remainder bounded by \(M C_1^{N+2}\|h\|_1^{N+1}\). It tends to zero if \(C_1\|h\|_1<1\) and the segment stays in the ball. The Taylor series therefore represents \(f\) there. This proves real analyticity of every smooth solution of (1.35e) with positive principal symbol and analytic coefficients.

Apply this proved result to (1.35d). Each of the finitely many compact-Casimir components of a coefficient is analytic. Their finite sum is analytic, proving the first assertion of Theorem 1.29 in both fields and every rank.

Step 4: The coefficient comparison

Suppose two actual realizations have identified core actions, compact actions and invariant core pairings. Every derivative at identity of the difference of corresponding core coefficients vanishes: \[ R_{X_1}\cdots R_{X_r}f(e) =B_E(X_1\cdots X_rv,\widetilde v). \tag{1.35t} \] The order convention in the product is fixed by successive right differentiation; either convention gives the same matched jets. At identity, the \(R_X\)'s form a coordinate tangent basis. Induction on derivative order expresses coordinate derivatives through these products and lower-order products, so all coordinate Taylor coefficients agree. Theorem 1.29 gives equality on a neighborhood of identity.

An analytic function vanishing on a neighborhood vanishes on a connected component: along any path, finitely many overlapping analytic coordinate balls propagate the zero Taylor series. The connected component of \(GL_n(\mathbb C)\) is the whole group; one can connect a matrix to its positive diagonal singular-value part through \(U(n)\), and then deform the positive values to one. For \(GL_n(\mathbb R)\) the positive-determinant component is connected by the same singular-value argument and connectedness of \(SO(n)\). Plane rotations connect \(SO(n)\) to identity, and diagonal phases connect \(U(n)\) to identity after unitary diagonalization, so these connectivity facts need no classification. The other real component is the first multiplied by \(\operatorname{diag}(-1,1,\ldots,1)\in K\). Applying the matched compact action to a core vector gives the coefficient equality there as well. This proves the second assertion.

Scope for the principal-series comparison

Corollary 1.29a (uniform weak analyticity for a core vector). This corollary only needs an actual complete smooth realization of an irreducible admissible core; a supplied dual is unnecessary. For a fixed core vector \(v\), the conclusion that \(g\mapsto\lambda(\pi(g)v)\) is analytic holds for every continuous linear functional \(\lambda\) on the supplied model, not only the paired core functionals. On each fixed coordinate ball, its factorial derivative constant is uniform over \(\lambda\)'s bounded by a fixed continuous seminorm. In particular \(v\) has a convergent vector Taylor expansion on a neighborhood whose radius is independent of that seminorm.

Indeed the scalar equation (1.35d) used only the actions of the Casimirs on \(v\), so it holds with an arbitrary \(\lambda\). For the functionals satisfying \(|\lambda(w)|\le p(w)\), compact joint continuity bounds the \(H^1\) quantity in (1.35l) by one finite \(M_p\), using finitely many \(p(\pi(g)Dv)\) on the compact coordinate ball. All constants in (1.35j)--(1.35s) depend on the operator and the coordinate ball, and not on \(\lambda,p\). Thus (1.35s) is uniform with \(M=M_p\). Continuous functionals bounded by \(p\) recover the quotient seminorm \(p\); this follows from the elementary linear-functional extension described in the common-refinement proof below. The vector Taylor partial sums are therefore Cauchy in every seminorm on one fixed smaller ball. Completeness supplies their limit in \(E\). Applying every continuous functional, and using the scalar Taylor identity, identifies that limit with the actual orbit vector. A finite decomposition into compact-Casimir eigenvectors gives the assertion for a general fixed core vector.

Theorem 1.29 proves analytic identity of intrinsic core coefficients between any two supplied actual smooth paired models. It applies in particular to the explicit principal-series subquotient models constructed in Theorem 1.26a and to a prescribed cusp model with the same core. It does not prove that the identity on cores extends continuously in either direction, that a selected Borel eigenfunctional is continuous, or that a dense principal map has closed or surjective range in a prescribed topology. Those assertions require an additional estimate or a proved universal property. No analytic-elliptic, subrepresentation or globalization theorem has been invoked without a written proof.

A complete common refinement of prescribed smooth realizations

Suppose two actual complete Hausdorff locally convex smooth moderate-growth \(G=GL_n(F)\) representations \(E_1,E_2\), \(F=\mathbb R\) or \(\mathbb C\), have identified irreducible admissible \((\mathfrak g,K)\)-cores \(V_0\). Denote the core identification by \(J_0\). No continuity of \(J_0\) is assumed.

Theorem 1.30. In \(E_1\times E_2\), the closed graph \[ R=\overline{\{(v,J_0v):v\in V_0\}} \tag{1.36a} \] is a complete smooth moderate \(G\)-representation. Its exact compact-finite core is the graph of \(J_0\). Both projections \[ p_i:R\longrightarrow E_i \tag{1.36b} \] are continuous injective \(G\)-maps with dense images containing the entire respective core. If matching supplied dual-pair realizations are given, the same construction on their dual cores gives a complete nondegenerate invariant dual pair for \(R\). This theorem does not claim that a projection is onto, has closed image, or has a continuous inverse on its image.

Step 1: Separation by a continuous functional

We recall the elementary locally convex separation needed below, including the extension step. If \(L\) is a closed subspace of a Hausdorff locally convex space \(X\) and \(x\notin L\), there is a continuous seminorm \(p\) and \(\epsilon>0\) for which \(p(x-\ell)\ge\epsilon\) for all \(\ell\in L\): choose a convex balanced neighborhood of zero disjoint from \(x-L\), and take a smaller finite intersection of seminorm balls. The maximum of their normalized seminorms is such a \(p\). On \(L+\mathbb Cx\), define \(\lambda(\ell+zx)=z\epsilon\). Then \(|\lambda(\ell+zx)|\le p(\ell+zx)\), by scaling when \(z\ne0\).

Here is the linear-functional extension principle in this setting. For a real linear functional \(a\) with \(a\le p\), extension to one additional real direction \(y\) requires choosing \(a(y)\) between \[ \sup_{u\in D}\big(a(u)-p(u-y)\big) \quad\hbox{and}\quad \inf_{w\in D}\big(p(w+y)-a(w)\big). \tag{1.36c} \] The left endpoint is no greater than the right: by subadditivity, \(a(u)+a(w)=a(u+w)\le p(u+w) \le p(u-y)+p(w+y)\). The formulas obtained by dividing the two possible signs of the new real coefficient verify \(a(u+ty)\le p(u+ty)\). A maximal compatible extension, obtained from the union on each chain of extension domains, therefore has the whole real space as domain; otherwise (1.36c) extends it further. Since \(p(-z)=p(z)\), it obeys \(|a|\le p\). For a complex functional, first extend its real part by this argument and define \(\Lambda(z)=a(z)-i\,a(iz)\). It is complex linear. Rotating \(z\) by a phase which makes \(\Lambda(z)\) positive real shows \(|\Lambda(z)|\le p(z)\), because \(p\) is balanced. This proves the required continuous extension and separation. Applied to a single vector modulo the kernel of a seminorm, the same argument shows \[ p(x)=\sup_{\Lambda:\,|\Lambda|\le p}|\Lambda(x)|. \tag{1.36d} \] The only set-theoretic extension step is the maximal-chain principle; no representation-theoretic continuity theorem occurs.

Step 2: The graph is group-stable

The core graph is Lie- and compact-stable. Let \(\Lambda\) be a continuous functional on \(E_1\times E_2\) which annihilates its closure \(R\). For a core graph vector \(w=(v,J_0v)\), the function \[ g\longmapsto\Lambda((\pi_1(g),\pi_2(g))w) \tag{1.36e} \] is analytic, by Corollary 1.29a applied in each factor. Every derivative at identity vanishes, since the Lie algebra preserves the core graph. It is therefore zero on the identity component by the proved analytic identity principle in Theorem 1.29. The compact action preserves the core graph and meets the other real component, so it is zero on all of \(G\). The separation in Step 1 of this argument implies \((\pi_1(g),\pi_2(g))w\in R\). By continuity of the action and density of the core graph in its closure, this holds for every \(w\in R\). Thus \(R\) is group-stable.

It is complete because it is closed in the complete product. Orbit derivatives in the product belong to \(R\), since it is closed and invariant; restrictions of the continuous differentiated maps give the actual smooth differentiated action in the subspace topology. The orbit maps have their full smooth derivatives there: a remainder tends to zero in the induced seminorms exactly when it does so in the ambient product. Joint continuity is inherited. Every continuous seminorm on \(R\) is bounded by the restriction of a finite sum of continuous seminorms on the two factors, by the definition of the subspace topology. Apply their moderate inequalities, take the larger exponent and sum the controlling seminorms. This proves moderate growth on \(R\) without an open-mapping or globalization theorem.

Step 3: Exact compact packets and the two projections

For each compact type \(\tau\), its character projector is a continuous operator on the product and on \(R\). The projection of the core graph is exactly the finite graph packet \[ R_\tau=\{(v,J_0v):v\in(V_0)_\tau\}. \tag{1.36f} \] A finite-dimensional subspace in a Hausdorff locally convex space is closed: select continuous functionals separating a basis, use their finite invertible coordinate matrix, and reduce closure to closure in a finite Euclidean space. Projecting the approximating core graph vectors thus shows that the entire \(\tau\)-packet of \(R\) is (1.36f). Every compact-finite vector lies in finitely many such packets, so the exact core of \(R\) is the core graph.

If the kernel of \(p_i\) were nonzero, it would be a nonzero closed compact-stable subspace of \(R\). The polynomial compact approximate identities (1.34d) preserve it, converge to identity on it, and have compact-finite values. Some approximant of a nonzero vector would therefore be a nonzero core vector in that kernel. This is impossible on the core graph. Both projections are consequently injective. They are continuous equivariant maps by construction and contain the respective cores in their images. Core density (1.34d) makes both images dense.

This identifies an actual complete common refinement of the prescribed topologies and constructs both continuous comparison maps. It does not promote dense images to onto or closed images. A complete space can map continuously and injectively onto a proper dense subspace of another complete space; completeness alone supplies no missing inverse estimate.

Step 4: The paired version

Apply the same construction to the prescribed admissible dual cores and actual dual models, obtaining \(\widetilde R\). Restrict the first given pairing to \[ B_R(r,\widetilde r) =B_1(p_1r,\widetilde p_1\widetilde r). \tag{1.36g} \] It is jointly continuous and invariant. It equals the second restricted pairing: both agree on the core graphs by the specified core pairing, and each core graph is dense in its complete refinement. First fix a dual core vector and extend in \(r\); then fix \(r\) and extend in \(\widetilde r\). Separate continuity suffices for this equality. If \(B_R(r,\widetilde R)=0\), density of \(\widetilde p_1(\widetilde R)\) and nondegeneracy of \(B_1\) give \(p_1r=0\). Injectivity then gives \(r=0\). The other radical is zero in the same way. Thus \((R,\widetilde R,B_R)\) is an actual compatible complete smooth moderate dual pair with the identified core and dual core.

Remaining comparison obligation

Apply Theorem 1.30 to a prescribed cusp model and the explicit complete principal-series subquotient model of Theorem 1.26a, once their cores are identified using Theorem 1.28. It gives actual injective dense continuous maps from their complete common refinement to each model. To identify the prescribed model itself with that subquotient, one still needs an inverse seminorm estimate or surjectivity of these comparison maps. The existence of a full principal-core quotient in 1.34y does not establish that estimate. Theorems 1.32–1.34 below prove occurrence and construct a compatible actual realization for every abstract irreducible admissible core. No assertion here identifies the residual correction polynomial with the intended standard real/complex gamma factors.

The canonical Gaussian package in every prescribed actual dual-pair model

Use \(F=\mathbb R\) or \(\mathbb C\), the positive trace Fourier argument, the basic programme characters \(\psi_{\mathbb R}(x)=e^{-2\pi ix}\) and \(\psi_{\mathbb C}(z)=e^{-2\pi i(z+\bar z)}\), and their self-dual measures. Let \((E,\widetilde E,B_E)\) satisfy the actual complete smooth moderate dual-pair hypotheses of Theorem 1.28, with irreducible admissible core \(V_0\). Let \(\mathcal S_0\) denote the polynomial multiples of \(G_{\mathbb R}(X)=e^{-\pi\operatorname{tr}(XX^t)}\), or of \(G_{\mathbb C}(X)=e^{-2\pi\operatorname{tr}(XX^*)}\). Write \(\nu(g)=|\det g|_F\) and \[ Z_E(s,\Phi;v,\widetilde v)= \int_G\Phi(g)B_E(\pi(g)v,\widetilde v) \nu(g)^{s+(n-1)/2}\,dg. \tag{1.37a} \] Its initial convergence and continuation in the prescribed topologies follow from the written continuation proof of Theorem 1.21, with the locally convex extension supplied in Step 3 below.

Theorem 1.31. In every such prescribed model, in every rank and both fields, the Gaussian coefficient span is exactly \[ \operatorname{span}_{\mathbb C} \{Z_E(s,\Phi;v,\widetilde v): \Phi\in\mathcal S_0,\ v\in V_0,\ \widetilde v\in\widetilde V_0\} =L_E(s)\mathbb C[s]. \tag{1.37b} \] One can choose full Borel inducing characters by Theorem 1.28 and a nonzero monic polynomial \(P\) for which \[ L_E(s)=P(s)L_{\boldsymbol\chi}(s),\qquad L_{\boldsymbol\chi}(s)= \begin{cases} \prod_i\Gamma_{\mathbb R}(s+t_i+e_i),&F=\mathbb R,\\ \prod_i\Gamma_{\mathbb C}(s+t_i+|\ell_i|/2),&F=\mathbb C. \end{cases} \tag{1.37c} \] There is an actual finite attaining family \[ \sum_{\alpha=1}^r Z_E(s,\Phi_\alpha;v_\alpha,\widetilde v_\alpha)=L_E(s). \tag{1.37d} \] For every Schwartz test and every pair of smooth vectors, \(Z_E/L_E\) is entire with compact-parameter joint test/vector seminorm bounds. The generator has no zeros. For its compatible paired dual normalization, \[ P_{\widetilde E}(1-s)=(-1)^bP(s),\qquad b=\deg P, \quad \epsilon_E=(-1)^b \begin{cases} (-i)^{\sum e_i},&F=\mathbb R,\\ (-i)^{\sum|\ell_i|},&F=\mathbb C, \end{cases} \tag{1.37e} \] and the full scalar Fourier equation is \[ Z_{\widetilde E}(1-s,\widehat\Phi;\widetilde v,v) =\epsilon_E \frac{L_{\widetilde E}(1-s)}{L_E(s)} Z_E(s,\Phi;v,\widetilde v). \tag{1.37f} \] Conjugation, and compatible unitary duality, have the exact normalizations proved in Theorem 1.26b. The polynomial in (1.37c) is the integral-defined correction. This theorem does not yet compute its roots and scalar from every intended real discrete-series, parity or complex angular standard-factor label.

Step 1: Occurrence is now proved in the required actual scope

Theorem 1.28 constructs an actual algebraic injection \(T_0:V_0\hookrightarrow I_0(\boldsymbol\chi)\). It does so from finite \(U(\mathfrak n)\)-generation Theorem 1.27, the nonzero Jacquet proof in Steps 1–4 of Theorem 1.28 and the explicit compact map 1.34w. No source statement supplies an unproved occurrence premise.

Apply the written closure theorem Theorem 1.26a with \(W_0=0\) and the submodule \(T_0V_0\) in that principal core. It constructs the complete smooth moderate principal submodel \(Q=\overline{T_0V_0}\) and its actual paired dual \[ Q^\vee=I(\boldsymbol\chi^{-1})/ \operatorname{Ann}(Q). \tag{1.37g} \] Their exact cores and compact pairing identify with \(V_0,\widetilde V_0,B_E|_{V_0\times\widetilde V_0}\). The finite-packet transpose in 1.34y supplies this dual core identification; the smooth paired construction is Theorem 1.26a. These are actual constructed models. No continuous map between the entire original \(E\) and \(Q\) is being assumed.

Step 2: Core coefficients identify the ideals and the attaining family

Theorem 1.29 proves \[ B_E(\pi(g)v,\widetilde v) =B_Q(\pi_Q(g)T_0v,\widetilde T_0\widetilde v) \qquad(g\in G) \tag{1.37h} \] for every pair of core vectors, where \(\widetilde T_0\) denotes the exact dual core identification in (1.37g). Both actual realizations have the same core Lie/compact actions and pairing, so their analytic coefficient jets agree at identity and propagate over the whole group. This is the proved Theorem 1.29 comparison, rather than an asserted uniqueness of globalizations.

Consequently the original integrals for these coefficients are identical on an initial convergence half-plane and their meromorphic continuations are identical. The exact Gaussian ideal theorem Theorem 1.26b for \(Q\) proves (1.37b)--(1.37c) for \(E\). Its finite attaining tests and core vectors, pulled back through the two algebraic core identifications, prove (1.37d) in the original model. Every polynomial multiplier used in constructing the attaining family has already been absorbed into a polynomial-Gaussian test by the Euler operator in Theorem 1.26b. Thus (1.37d) is an actual finite family of allowed integrals, not a formal polynomial gcd. This proves the genuine Gaussian-ideal premise in the prescribed model; a common gamma majorant was not substituted for it.

Step 3: Full Schwartz division and its continuity

Theorem 1.21 was stated for complete Fréchet models. Here is the precise extension of its argument to the complete locally convex models allowed in Theorem 1.28. Its central determinant operator proof in the preceding dense-core and central-determinant subsections is algebraic after density of the core. Replace its countable compact-density argument by (1.34d); continuity of differentiation then extends the same scalar central identities to the whole supplied model. Its far-right integral bound and scalar integration by parts, in the preceding bounds and integration-by-parts subsection, use joint pairing seminorms, moderate growth and finitely many derivative seminorms, and no countable topology assumption. Thus the same proved differential recurrence holds: \[ B_E(s)Z_E(s,\Phi)=Z_E(s+a,\mathscr D\Phi),\qquad \deg B_E=2n,\quad B_E\text{ monic}, \tag{1.37ha} \] where \(a=2,\mathscr D=D^2\) over \(\mathbb R\) and \(a=1,\mathscr D=D_+D_-\) over \(\mathbb C\). Every coordinate coefficient derivative in its integration-by-parts proof is bounded by a finite negative determinant power and a finite list of continuous seminorms. Taking the real part of \(s\) large makes its extension across the singular set \(C^{2n}\). This is exactly the scalar line-by-line extension verified there; it uses no vector-valued integration across that set.

If \(B_E(s)=\prod_j(s-\beta_j)\), put \(H_E(s)=\prod_j\Gamma((s-\beta_j)/a)\). The proved scalar gamma recurrence gives \(H_E(s+a)=a^{-2n}B_E(s)H_E(s)\). For a compact parameter set choose \(N\) so that its translate by \(Na\) lies beyond the integral convergence threshold. Define \[ F_E(s,\Phi;v,\widetilde v) =a^{-2nN} \frac{Z_E(s+Na,\mathscr D^N\Phi;v,\widetilde v)} {H_E(s+Na)} . \tag{1.37hb} \] The recurrence shows independence of larger \(N\). These formulas patch to an entire family with the far-right joint seminorm bounds, since \(\mathscr D^N\) is a continuous Schwartz operator and the scalar reciprocal gamma product is entire and bounded on the shifted compact set. This writes the extension required for the broader topology, rather than silently importing the Fréchet theorem with enlarged hypotheses. In the Fréchet scope it is Theorem 1.21. In either scope it gives the entire continuous family \[ F_E(s,\Phi;v,\widetilde v)=Z_E(s,\Phi;v,\widetilde v)/H_E(s), \tag{1.37i} \] with joint test/vector seminorm bounds on every compact parameter set. Its proof uses the written central differential identity and a smooth-model coefficient bound. It does not require a principal-series topological comparison. The finite identity (1.37d) shows that \[ h_E(s)=L_E(s)/H_E(s) \tag{1.37j} \] is an entire nonzero function. For a polynomial-Gaussian core integral, (1.37b) says \(F_E=h_E p\) with \(p\in\mathbb C[s]\).

The core is dense in each prescribed topology by (1.34d). Polynomial-Gaussian tests are dense in the full Schwartz space: the actual earlier programme proof is Hermite functions, tempered distributions and the Schwartz kernel theorem, Lemma 3.1 and Theorem 3.2. Its Hermite truncations converge in every Schwartz seminorm; a scalar dilation changes its fixed Gaussian to \(G_F\). Over \(\mathbb C\), polynomials in \(Z,\bar Z\) are the full real coordinate polynomial algebra, so no complex-angular tests are discarded.

Approximate a given \(\Phi,v,\widetilde v\) in these topologies. In a general locally convex topology this may be a net; the joint bounds in (1.37i) imply uniform convergence on every compact parameter set, regardless. At a zero of \(h_E\) of order \(m\), the first \(m\) jets of every approximating function vanish. Cauchy's formula on a fixed small circle gives convergence of the jets and hence their vanishing in the limit. Thus \(F_E/h_E=Z_E/L_E\) is entire.

The same circle proves a joint seminorm bound for the division: inside it the quotient is its Cauchy integral, with denominator \(h_E\) nonzero on the circle. Bound its numerator by (1.37i). Away from these isolated zeros divide directly. Finitely many circles and neighborhoods cover a compact parameter set. This gives full continuity with the original prescribed vector seminorms, rather than seminorms of an unproved equivalent model. The absence of zeros of \(L_E\) follows from Theorem 1.26b or independently by the compact-support test in the proof of Proposition 1.21b: at a prescribed \(s_0\), use a bump times the conjugate of a nonzero coefficient and the inverse imaginary norm phase. Its integral is strictly positive. An entire division by a zero of \(L_E\) would force it to vanish. This proves that no correction root can create an excess zero.

Step 4: The Fourier equation on every test and smooth coefficient

Theorem 1.26b proves (1.37e)--(1.37f) on \(Q,Q^\vee\). For core coefficients, (1.37h) transfers it to \(E,\widetilde E\). First take polynomial-Gaussian tests. Their Fourier transforms remain polynomial-Gaussian: differentiating the exact scalar Gaussian transform gives the transformed polynomial, and the Gaussian is self-dual with the specified real/complex measures. These facts are proved in Additive characters, self-dual measures and Poisson summation on the adèles, Proposition 5.2 and Lemma 5.4A.

Approximate arbitrary Schwartz tests and both smooth vectors as in Step 3 of this argument. Fourier transformation is continuous on the whole Schwartz space by that same written Lemma5.4A. For both the representation and its dual, divide first by the entire-continuation majorants in (1.37i). At parameter points avoiding the discrete poles of the fixed scalar gamma ratio, joint compact convergence passes the Fourier identity to the limit. The identity of meromorphic continuations gives it everywhere. This proves (1.37f) for all smooth coefficients without any continuous lifting of them to the principal model. It also identifies the already proved general scalar in Theorem B.5 / lesson04 Theorem1.22 with (1.37e)--(1.37f).

Step 5: Duality, conjugation and the unitary phase

The paired principal model for \(Q^\vee\) uses the inverse inducing characters. Theorem 1.26b proves its exact polynomial reflection \(P_{Q^\vee}(1-s)=(-1)^bP_Q(s)\) and the sign \(\epsilon_Q=(-1)^b\epsilon_{\boldsymbol\chi}\). Equation (1.37h), applied also to the swapped actual dual pair, transfers those identities to the prescribed dual normalization. Under conjugation, real parity remains \(e_i\), complex angular weight becomes \(-\ell_i\), and \(t_i\) becomes \(\bar t_i\). Its product gamma factor is the complex conjugate at \(\bar s\). The monic-polynomial normalization in Theorem 1.26b then gives exactly \[ L_{\bar E}(s)=\overline{L_E(\bar s)}. \tag{1.37k} \] If the supplied core pairing comes from an invariant positive Hermitian form, core conjugation identifies its conjugate with the admissible dual. The Theorem 1.26b core-unitary argument proves, with no unidentified scalar in this paired normalization, \[ L_{\widetilde E}(s)=\overline{L_E(\bar s)}. \tag{1.37l} \] In particular the basic scalar in (1.37e) has absolute value one. Changing the basic character to \(\psi_a(x)=\psi(ax)\) gives the exact factor \[ \epsilon_E(s,\psi_a)= \omega_E(a)|a|_F^{n(s-1/2)}\epsilon_E(s,\psi). \tag{1.37m} \] To check it directly, the self-dual additive matrix measure changes by \(|a|_F^{n^2/2}\), Fourier substitution is \(Y\mapsto aY\), and changing \(g\mapsto a^{-1}g\) in the dual zeta integral contributes the determinant power \(|a|_F^{-n(1-s+(n-1)/2)}\) and the dual central coefficient \(\omega_E(a)\). The net exponent is \(n(s-1/2)\). For unitary \(\omega_E\), its modulus is one on the critical line.

Exact scope and the remaining target

Theorem 1.31 proves a genuine Gaussian ideal, finite attainment, full Schwartz division with continuity, full scalar Fourier equation and paired dual/conjugation normalization in every prescribed actual complete smooth moderate dual-pair realization of an irreducible admissible real/complex \(GL_n\) core. Theorem 1.19 supplies those models in the full automorphic cusp scope, so this removes the abstract occurrence/comparison premise from the Gaussian-ideal existence argument in that scope. Theorem 1.30 gives further actual continuous dense comparison maps, but a topological isomorphism with the original model is not needed for the coefficient-and-density proof above.

Theorems 1.32–1.34 below prove nonzero Jacquet quotient and construct a compatible realization for every abstract irreducible admissible core. Full continuous inverse/closed-image comparison for arbitrary prescribed topologies also remains unproved. Finally, (1.37c) is monic relative to the inducing product chosen by Theorem 1.28. Computing \(P\) and the resulting scalar from all intended standard-factor labels, including real discrete-series blocks and complex angular data, remains a precise mathematical obligation. The free classification statements in Goldfeld–Jacquet do not supply that proof.

Analytic coefficient germs of an arbitrary admissible general linear core

Let \(F=\mathbb R\) or \(\mathbb C\), \(G=GL_n(F)\), and \(K=O(n)\) or \(U(n)\). Complexify the real Lie algebra of \(G\). Let \(V\) be an arbitrary irreducible admissible \((\mathfrak g,K)\)-module. No actual noncompact action or topology on \(V\) is assumed. Its algebraic admissible contragredient \(V^\vee\) is the direct sum of the duals of its finite compact packets, with the usual contragredient compact and Lie actions. Write \[ B(v,w)=w(v),\qquad B(Xv,w)=-B(v,Xw). \tag{1.38a} \] The second action preserves compact-finiteness. Indeed a functional supported in a finite set of compact types can, after a Lie generator is applied, meet only types in their tensor products with the finite compact representation \(\mathfrak g^*\). Those tensor products are finite-dimensional and have finite irreducible decompositions. Each compact packet remains finite-dimensional. The contragredient is irreducible: a nonzero proper submodule has a proper part in some finite dual packet, whose finite-dimensional annihilator gives a nonzero submodule of \(V\); irreducibility and the perfect packet pairing contradict its being proper.

Theorem 1.27 above applies to both cores. In particular they are generated over \(U(\mathfrak g)\) by finite compact-stable spaces, and the selected invariant central trace operators act by scalars. These are written algebraic proofs, not assumptions about classification or globalization.

Theorem 1.32. For every \(v\in V,\;w\in V^\vee\) there is a unique real analytic germ \(c_{v,w}\) on a neighborhood of \(K\) with the compact covariance in (1.38c) and the identity jets \[ (R_Dc_{v,w})(1)=B(Dv,w),\qquad D\in U(\mathfrak g). \tag{1.38b} \] The germs are bilinear, compact-finite under left and right translations, and satisfy wherever both sides are defined \[ R_Xc_{v,w}=c_{Xv,w},\qquad L_Xc_{v,w}=-c_{v,Xw},\qquad c_{v,w}(k_1gk_2)=c_{k_2v,k_1^{-1}w}(g). \tag{1.38c} \] Here \(R_Xf(g)=\frac d{dt}f(g\exp(tX))|_{0}\) and \(L_Xf(g)=\frac d{dt}f(\exp(tX)g)|_{0}\). The neighborhood may depend on the finite compact orbits of \(v,w\). This theorem asserts germs of scalar functions; it does not assume or construct a noncompact action on an abstract vector.

Step 1: The formal germ and its identities

We first justify the enveloping-algebra jet dictionary needed here, rather than importing an analytic-vector theorem. Choose a real matrix Lie basis \(X_1,\ldots,X_d\) and local analytic coordinates at identity. Their invariant vector fields have independent values there. Commuting an out-of-order pair of invariant fields introduces a single commutator and lowers total order. Therefore ordered monomials span the enveloping algebra. They are also independent as differential operators: the highest-order symbol at identity of the monomial of multi-index \(\alpha\) is the distinct commutative monomial in the basis of cotangent variables dual to \(X_i\). A relation of highest total order \(m\) has zero degree-\(m\) symbol only when every coefficient of that order is zero; induction removes all orders. This proves the required ordered-monomial independence directly in the actual matrix group.

Consequently the evaluations at identity of invariant monomials of order at most \(m\) are a triangular invertible linear change from ordinary coordinate derivatives of order at most \(m\). The leading diagonal blocks are the change of symmetric basis just described. Specifying a linear functional on \(U(\mathfrak g)\) therefore specifies a unique compatible formal Taylor germ, and every formal germ specifies that functional. Apply this to \(D\mapsto B(Dv,w)\). Right differentiation gives the first identity of (1.38c). Left fields commute with right fields; at identity \[ (R_D L_Xc)(1)=B(XDv,w)=-B(Dv,Xw), \] which proves the second identity on formal jets. Compact changes of variables give the third identity on jets by compatibility of the given \(K\)-action with its Lie action. These computations also give bilinearity and the scalar central equations.

These identities may be pulled back through any actual analytic matrix-coordinate map. They hold as formal identities there; after multiplying by the determinants of its differential, they also hold for rational lifts on a generically invertible coordinate map. This assertion is elementary: the chain rule for each finite jet is a polynomial identity in finitely many derivatives of the coordinate map. Its inverse uses the adjugate divided by its determinant, so multiplication clears its denominators. No convergence is used at this stage.

On the diagonal \(a(x)=\operatorname{diag}(e^{x_i})\) the formal restriction is explicitly \[ F_{v,w}(x)= \sum_{\alpha\in\mathbb N^n} B(H_1^{\alpha_1}\cdots H_n^{\alpha_n}v,w) \frac{x^\alpha}{\alpha!},\qquad H_i=E_{ii}. \tag{1.38d} \] For a fixed pair, take the vector of all its finite left and right compact translates. It is a formal power series with values in one finite-dimensional space. All compact derivatives act there by fixed matrices.

Step 2: A pole-order refinement of the written radial system

The radial computation in Theorem 1.28, Step 3, equations (1.34j)–(1.34o), is a calculation with matrix vector fields. It applies to formal jets as well as to actual functions by the preceding chain rule. We need the following refinement at the origin.

Fix a compact set of real diagonal directions \(h\) with all \(|h_i-h_j|\ge\eta>0\), and bounded \(|h_i|\). Then a coefficient of a radial derivative of order \(k\) in a central operator of degree \(r\), restricted to \(x=th\), has pole order at most \(r-k\) at \(t=0\). After that power of \(t\) is multiplied in, it is holomorphic in a disk whose radius and bounds are uniform on this compact set of \(h\).

Here is the additional calculation. At \(a(x)\), an off-diagonal right field is a sum of left and right compact fields, with coefficients \[ \frac{1}{2\sinh(x_j-x_i)},\quad -\frac{1}{e^{2(x_j-x_i)}-1},\quad -\frac{e^{2(x_j-x_i)}}{e^{2(x_j-x_i)}-1}. \tag{1.38e} \] Each has a simple pole along \(x_i=x_j\). Real diagonal fields are \(\partial_{x_i}\); imaginary diagonal fields for the complex group are compact fields. At \(k_1a(x)k_2\), first conjugate the Lie generator by \(k_2\) and use the same formulas. All angular coefficients and their angular derivatives are analytic and bounded on the compact factors. Off-diagonal compact coefficients have one pole, and a radial derivative of any such coefficient adds at most one pole. An angular derivative adds none.

Induction on the number of fields in a word proves the assertion: creating a compact differentiation uses one field and at most one pole; creating a radial differentiation uses one field and raises \(k\) by one; differentiating an existing coefficient uses one field and raises its pole order by at most one. Thus the sum of radial derivative order and pole order is at most the word length. The finite compact derivatives can then be replaced by their matrices. Symmetrizing invariant trace words preserves this bound. Newton's finite polynomial construction and the product rule preserve it as well.

More precisely, choose the normalizations used in Theorem 1.28: the real \(C_r\), or \(2^rC_r^{(1)}\) for the complex group. The factor \(2^r\) compensates exactly for the radial \(\frac12\partial_{x_i}\) in the first complexified summand. The resulting elementary-symmetric central equations are \[ e_r(\partial_x)F=\mu_rF-\mathcal L_rF,\qquad \operatorname{ord}\mathcal L_r\le r-1. \tag{1.38f} \] A coefficient of a derivative \(\partial^\beta\) on the right has pole order at most \(r-|\beta|\); the scalar term has order zero. Every fixed radial derivative of a coefficient adds at most its own order to this bound. These statements hold in the localized formal ring obtained by inverting the nonzero root linear forms \(x_j-x_i\). They follow from (1.38e), the matrix chain rule and the invariant trace symbol, not a general radial-part theorem.

Use the constant-coefficient identity \[ \partial_{x_i}^{\,n} =\sum_{r=1}^n(-1)^{r+1} \partial_{x_i}^{\,n-r}e_r(\partial_x). \tag{1.38g} \] Replacing \(e_r(\partial_x)F\) by (1.38f) expresses its left side as derivatives of total order at most \(n-1\), with a coefficient for \(\partial^\beta F\) of pole order at most \(n-|\beta|\). Differentiate and repeat to reduce every derivative having an index at least \(n\). Each replacement lowers total derivative order. The pole bound is preserved because the loss of derivative order pays for all coefficient derivatives and all subsequent substitutions.

For \[ J_\alpha=\partial_x^\alpha F,\qquad 0\le\alpha_i<n, \] this gives \[ \partial_{x_i}J_\alpha =\sum_\beta (A_i)_{\alpha\beta}(x)J_\beta, \qquad \operatorname{pole}_{t=0} (A_i)_{\alpha\beta}(th) \le |\alpha|+1-|\beta|. \tag{1.38h} \] An entry with a negative upper bound is zero: the reduction never increases total derivative order. The entries with \(|\beta|=|\alpha|+1\) are the unreduced constant jet shifts. The formulas are those of the fixed finite compact array. Only finitely many differentiations and substitutions are required, so the holomorphic bounds along \(th\) are uniform on the fixed compact set of directions.

Set \[ Y_\alpha(t)=t^{|\alpha|}J_\alpha(th). \] Equation (1.38h) gives a genuine Fuchsian formal system \[ tY'(t)=A_h(t)Y(t), \tag{1.38i} \] where \[ (A_h)_{\alpha\beta} =|\alpha|\delta_{\alpha\beta} +\sum_i h_i\,t^{1+|\alpha|-|\beta|} (A_i)_{\alpha\beta}(th). \] Every entry is holomorphic at zero, uniformly for the selected directions. \(Y\) itself is an ordinary formal power series, since every \(J_\alpha\) comes from (1.38d).

Step 3: Convergence of the formal Fuchsian solution

We prove exactly the elementary ODE fact used here. Suppose \(A(t)=\sum_{k\ge0}A_kt^k\) is holomorphic on a disk, with \(\|A_k\|\le C R^{-k}\), and \(Y=\sum_{m\ge0}Y_mt^m\) formally satisfies \(tY'=AY\). Then \[ (mI-A_0)Y_m=\sum_{k=1}^m A_kY_{m-k}. \tag{1.38j} \] For \(m>2\|A_0\|\), the geometric series gives \(\|(mI-A_0)^{-1}\|\le2/m\). Choose an integer \(m_0>2\|A_0\|+4C+1\) and \(S\ge2/R\). Choose \(M\) so that \(\|Y_m\|\le MS^m\) for \(m\le m_0\). For larger \(m\), induction and (1.38j) give \[ \|Y_m\| \le \frac{2C}{m}MS^m \sum_{k\ge1}(RS)^{-k} \le MS^m. \tag{1.38k} \] Thus the series converges. Resonant small integers cause no problem: the finitely many initial coefficients already satisfy the formal equation. The same choices work uniformly for a compact family of \(A_h\) whose finite initial coefficients are bounded.

In (1.38i) the initial coefficients are fixed finite jets from (1.38d), hence polynomials in \(h\), with bounded compact matrices on the chosen array. They are uniformly bounded. In particular \[ \left|\frac{B((\sum h_iH_i)^m v,w)}{m!}\right| \le M S^m \tag{1.38l} \] uniformly for separated bounded diagonal \(h\), and for \(v,w\) in bounded parts of the fixed compact orbit spaces. The same conclusion holds in rank one directly from its scalar central equation.

Finite-dimensional diagonalization

Lemma 1.32a. A real symmetric matrix has an orthonormal real eigenbasis, and a complex Hermitian matrix has an orthonormal complex eigenbasis. A commuting family of complex Hermitian matrices has a common orthonormal eigenbasis. Consequently every unitary matrix, and every commuting family of unitary matrices, can be diagonalized by a unitary change of basis.

Proof. For a symmetric or Hermitian matrix \(A\), the real function \(\langle Ax,x\rangle\) attains its maximum on the unit sphere. Let \(x\) be a maximizing unit vector. For every unit \(y\perp x\), differentiate this function at \(x\cos t+y\sin t\) when \(t=0\). The derivative is \(2\operatorname{Re}\langle Ax,y\rangle\), hence vanishes. Over \(\mathbb C\) replace \(y\) by \(iy\) as well, so \(\langle Ax,y\rangle=0\). Therefore \(Ax=\lambda x\), with \(\lambda=\langle Ax,x\rangle\) real. The orthogonal complement of \(x\) is invariant, since \(\langle Az,x\rangle=\langle z,Ax\rangle=0\) for \(z\perp x\). Induction on dimension supplies the asserted eigenbasis.

For a commuting Hermitian family, every eigenspace of one member is invariant under all the others. Choose a nonscalar member, decompose into its mutually orthogonal proper eigenspaces, and apply induction on their dimensions. If every member is scalar there is nothing to do. This argument applies to an arbitrary family, since each nonscalar choice strictly lowers dimension.

For a unitary matrix \(U\), its Hermitian real and imaginary parts \[ H=(U+U^*)/2,\qquad J=(U-U^*)/(2i) \tag{1.38ak} \] commute: use \(U^*=U^{-1}\). A common eigenbasis for \(H,J\) diagonalizes \(U=H+iJ\). A commuting family of unitary matrices commutes also with every inverse, so all its Hermitian real and imaginary parts commute. The same common-eigenbasis argument diagonalizes the whole family. \(\square\)

This proof also gives the singular-value factorizations used below: diagonalize the positive definite matrix \(g^*g\), take the positive square roots of its eigenvalues, and divide the corresponding columns of \(g\) by those square roots. Their orthonormality follows by taking inner products. Every positive-definite eigenspace has positive eigenvalue since \(\langle g^*gx,x\rangle=\|gx\|^2>0\) for \(x\ne0\).

Step 4: From separated diagonal directions to a full local germ

Let \(\mathfrak p_{\mathbb R}\) be the real symmetric or Hermitian matrices. For each \(m\) the expression \[ p_m(P)=B(P^m v,w)/m! \tag{1.38m} \] is an ordinary homogeneous polynomial of degree \(m\) on this finite-dimensional real vector space. Choose a small closed cube \(Q\) about a matrix with simple eigenvalues, so that every matrix in \(Q\) has eigenvalue gaps at least \(\eta\), with bounded eigenvalues. Lemma 1.32a gives \(P=k(\sum h_iH_i)k^{-1}\). All \(k\)-translates of \(v,w\) stay in the fixed bounded compact orbit spaces, so (1.38l) gives \(\sup_Q|p_m|\le MS^m\). No smooth choice of eigenvectors is needed for this bound.

We give the polynomial extrapolation which removes the gap restriction. After an affine change sending \(Q\) to \([-1,1]^b\), expand any polynomial \(p\) of total degree at most \(m\) in products of Chebyshev polynomials \(T_j(\cos\theta)=\cos(j\theta)\). Orthogonality on the product of circles bounds each coefficient by \(2^b\sup_Q|p|\). The triangular leading terms ensure that only multi-indices of total degree at most \(m\) occur. On any fixed complex coordinate box, the transformed coordinates have magnitude at most \(A\); the recurrence \(T_{j+1}(z)=2zT_j(z)-T_{j-1}(z)\) gives \(|T_j(z)|\le(2A+1)^j\). There are at most \((m+1)^b\le2^{bm}\) terms for \(m\ge1\). Therefore, on that fixed complex box, \[ \sup|p_m|\le M_1S_1^m. \tag{1.38n} \] Because the original \(p_m\) is homogeneous, on a sufficiently small complex neighborhood of zero the series \(\sum_mp_m(P)\) converges normally. Cauchy's elementary one-variable integral estimate applied successively in the coordinates controls all derivatives on smaller boxes. In this use it follows directly by integrating a polynomial on a circle: the integral of \(z^j/z^{k+1}\) is zero unless \(j=k\), when it is \(2\pi i\). Normal convergence passes this coefficient identity to the series; a smaller circle gives the geometric bounds for every fixed differentiated series. The limit is analytic and its Taylor series is the given formal series.

For \(g\) near \(K\), polar coordinates are analytic: \[ P=\tfrac12\log(g^*g),\qquad k=g\exp(-P),\qquad g=k\exp P. \] Near \(g^*g=I\), the convergent matrix logarithm series proves this directly; its adjoint entries are analytic real-coordinate functions. Compact finite-dimensional representations are analytic: in coordinates formed by finite products of small one-parameter exponentials their matrices are the ordinary finite matrix exponentials, by the finite-dimensional constant-coefficient differential equation. Both components of \(O(n)\) are included by its given compact action. Define \[ c_{v,w}(k\exp P)=\sum_{m\ge0}B(P^mv,k^{-1}w)/m!. \tag{1.38o} \] The bounds above are uniform for the compact translates in this formula. It is therefore an actual analytic germ near \(K\). Its formal jets agree with the germ of Step 1: the pullback through \(k\exp P\) is exactly the iterated formal Lie exponential, and the invariant-jet dictionary makes this equality unique. All identities (1.38c) and the central equations have identical formal jets on their common neighborhoods. Analyticity therefore proves those identities there, and proves uniqueness. This completes Theorem 1.32 without a supplied realization.

Proof dependencies

The algebraic finite-generation and central-scalar input is the proof of Theorem 1.27; the exact root-frame and Newton radial calculation is the radial computation in Theorem 1.28, Step 3. The pole refinement, formal-germ dictionary, Fuchsian convergence and polynomial extrapolation are proved here. Neither a source's subrepresentation statement, a globalization theorem, an asymptotic expansion, nor an abstract analytic-vector theorem is used.

The nonzero minimal Jacquet quotient without a supplied realization

Use the notation of Theorem 1.32 above. In particular \(V\) is an arbitrary irreducible admissible real or complex general linear Harish-Chandra core, \(V^\vee\) is its algebraic admissible contragredient, and its coefficient germs have been constructed and proved convergent without any actual noncompact representation.

Theorem 1.33. For every \(n\ge1\), and \(F=\mathbb R\) or \(\mathbb C\), \[ V/\mathfrak n V\ne0, \tag{1.38p} \] where \(\mathfrak n\) is the complexified upper unitriangular Lie algebra. The quotient is finite-dimensional by Theorem 1.27. The same conclusion holds for the lower unitriangular algebra.

The proof first extends the scalar germs only to matrices with distinct singular values. It never assumes extension across the singular walls, or that those scalar functions already come from an actual representation.

Step 1: Analytic continuation on the regular chamber

Let \[ C=\{x\in\mathbb R^n:x_1<\cdots<x_n\}. \tag{1.38q} \] For one fixed finite compact coefficient array, the analytic germ of Theorem 1.32 gives the jet \(J\) of (1.34n) on a neighborhood of zero intersected with \(C\). The matrix system \(\partial_iJ=A_i(x)J\) of (1.34o) has analytic coefficients everywhere in \(C\). The explicit coefficients consist of finite products, derivatives and quotients in (1.38e), whose denominators do not vanish there.

For \(x\in C\) choose \(\delta>0\) small enough that \(\delta x\) lies in that germ neighborhood, and solve the ordinary linear equation \[ \frac{d}{dt}J(tx)= \left(\sum_i x_iA_i(tx)\right)J(tx), \qquad \delta\le t\le1, \tag{1.38r} \] with its germ value at \(\delta x\). There is a unique solution on this compact interval. For completeness, on a small interval the iterated integral series has its \(m\)-th increment bounded by \(M^m|t-\delta|^m/m!\) times the initial norm. It converges uniformly, differentiates to the equation, and the same integral iteration proves uniqueness. Finitely many such intervals cover the path. Choosing a smaller admissible \(\delta\) gives the same solution, because the known germ already solves (1.38r) on their overlap.

This construction is analytic in \(x\). Around any fixed path, the explicit matrix coefficients extend holomorphically to a small complex neighborhood of its compact parameter set, with uniform bounds. The initial germ is holomorphic there as well. The same iterated integral series converges normally in \(x,t\) on successive small intervals, so the endpoint is holomorphic in \(x\). Use a fixed sufficiently small \(\delta\) for each such parameter neighborhood; independence of \(\delta\) patches the results into an analytic \(J\) on all \(C\).

The differences \(\partial_iJ-A_iJ\) and \(J_\alpha-\partial^\alpha J_0\) are analytic on \(C\). They are zero on its nonempty germ portion, so they are zero everywhere. Here the elementary identity principle requires no additional PDE theorem: an analytic function zero on a ball has all its Taylor derivatives zero there; overlapping analytic balls continue this fact along any path. The convex set \(C\) is connected. Thus the continued \(J\) is the genuine full finite jet of its zeroth component and solves every radial equation, not just the particular ray equation used to construct it.

Step 2: Scalar functions on the regular part of the group

Every matrix in \[ G_{\mathrm{reg}} =\{g:g^*g\text{ has }n\text{ distinct eigenvalues}\} \] has an ordered singular-value factorization \(g=k_1a(x)k_2\), \(x\in C\), \(k_i\in K\). Lemma 1.32a supplies this factorization directly: diagonalize \(g^*g\), take positive square roots of the eigenvalues and normalize the resulting columns of \(g\). Define \[ c_{v,w}(k_1a(x)k_2) =F_{k_2v,k_1^{-1}w}(x), \tag{1.38s} \] where \(F\) is the continued zeroth radial component. The required compact translates form finite arrays, so this formula is well-defined by the preceding construction. Changing to a larger finite array gives exactly the same components: the analytic continuations agree on the initial germ portion of \(C\), hence everywhere in \(C\). The same argument preserves all bilinear relations between different pairs. Thus these separately constructed finite arrays define one consistent bilinear family.

The only ambiguity with ordered distinct singular values is \((k_1,k_2)\mapsto(k_1m,m^{-1}k_2)\), where \(m\) is a diagonal sign matrix or diagonal unitary matrix. To check this, compare two factorizations and square the positive diagonal factor. A compact matrix commuting with a diagonal matrix of distinct eigenvalues is diagonal; its entries are signs or phases. Invariance of \(B\), and commutation of this \(m\) with all \(H_i\), give \[ F_{m^{-1}v,m^{-1}w}(x)=F_{v,w}(x) \] on the formal germ, then on all \(C\) by analyticity. This proves independence of the ambiguity in (1.38s).

The functions are analytic locally on \(G_{\mathrm{reg}}\). Here a local analytic singular-value factorization can also be verified without a spectral perturbation theorem. At a simple eigenvalue, the characteristic polynomial has nonzero derivative. Its nearby root is given by the uniformly convergent Newton iteration on a sufficiently small complex coordinate neighborhood: its derivative stays bounded away from zero and the Newton error squares at each step. This gives an analytic root. The projectors \(\prod_{j\ne i}(g^*g-\lambda_j I)/(\lambda_i-\lambda_j)\) are analytic. Apply each to a fixed eigenvector which remains nonzero locally, and divide by the positive square root of its squared norm, using the branch positive at the starting point. These are analytic orthonormal eigenvectors in real coordinates. The left singular vectors are obtained by division by the nonzero singular values. This supplies the desired local coordinates.

Each determinant component of \(G_{\mathrm{reg}}\) is connected. For the real group the factorization is the image of \(SO(n)\times C\times SO(n)\), or of the same space with one fixed compact reflection inserted. A simultaneous diagonal sign in the two compact factors changes both compact determinants, so these images cover the corresponding determinant component. \(SO(n)\) is connected: successive rotations in coordinate planes reduce an orthogonal matrix of determinant one to identity, and reversing those finite rotations gives a path. \(U(n)\) is connected by diagonalizing a unitary matrix and replacing its eigenphases by their multiples of a parameter between zero and one. Its unitary diagonalization is proved in Lemma 1.32a. Rank one is treated separately below.

Consequently the Lie and compact identities (1.38c) extend from the germ to all \(G_{\mathrm{reg}}\). For a Lie identity, both sides are analytic on this open set, including the derivative of the analytic left side. They agree on a nonempty germ portion of each component, so the identity principle applies. Compact covariance also follows directly from (1.38s). The scalar central equations extend in the same way.

Step 3: One exponential bound for all abstract core pairs

This is the step that replaces the supplied moderate dual-pair topology used in Theorem 1.28, Step 2. Choose finite compact-stable algebraic generating spaces \(W\subset V\), \(U\subset V^\vee\), so that \[ U(\mathfrak g)W=V,\qquad U(\mathfrak g)U=V^\vee. \tag{1.38t} \] Such spaces exist by irreducibility, exactly as in Theorem 1.27, Step 1. Take the one finite coefficient array consisting of all pairs in fixed bases of \(W,U\). It contains every left and right compact translate of these pairs. Let \(J_{\mathrm{base}}\) be its central radial finite jet.

Choose \(x^{(0)}\in C\) and \(H_1>\cdots>H_n\), and put \[ x(t)=x^{(0)}-tH,\qquad a_t=a(x(t)),\qquad t\ge0. \tag{1.38u} \] All gaps remain at least \(\eta=\min_i(x^{(0)}_{i+1}-x^{(0)}_i)>0\). The explicit 1.34o system is bounded on this fixed-gap chamber portion. Therefore its ordinary integral iteration gives \[ \|J_{\mathrm{base}}(x(t))\| \le \|J_{\mathrm{base}}(x^{(0)})\|e^{Mt} \tag{1.38v} \] for one finite \(M\), depending on this base array and ray.

Every fixed radial derivative of every base component is a linear combination of that same finite jet with bounded coefficients along the ray. This follows by differentiating (1.38g)–(1.38h) and reducing total order, as in Theorem 1.28, Step 3: every required fixed derivative of the coefficients in (1.38e) is bounded when the gaps are at least \(\eta\).

The same assertion holds for an arbitrary fixed left and right Lie differential word on a base component. To make its angular uniformity precise, lift right fields to \(K\times C\times K\). For \(g=k_1a(x)k_2\), conjugate a right generator by \(k_2\) and write \(a(x)\operatorname{Ad}(k_2)X\) as radial, left compact and right compact fields using (1.38e). The real diagonal is radial and the imaginary diagonal is compact. This is a global differential lift; the compact diagonal redundancy is harmless because the pulled-back function is invariant under it. Left fields are obtained by the analogous decomposition, or by the same computation on the inverse matrix. All angular coefficients are bounded with all fixed angular derivatives on the compact factors. Composition of finitely many fields and the product rule therefore give a finite sum of radial derivatives and compact derivatives with bounded coefficients on the fixed-gap portion. Compact derivatives act by matrices on the same base array; radial derivatives reduce to its same finite jet. Hence for every fixed \(D,E\), \[ |L_D R_E c_{w,u}(a_t)|\le C_{D,E,w,u}e^{Mt}, \quad w\in W,\ u\in U, \tag{1.38w} \] with exactly the \(M\) of (1.38v). The constants may depend on the words; the exponential rate does not.

By (1.38t) and the globally valid Lie identities, every \(c_{v,w}\) is a finite linear combination of such differentiated base components. We have proved \[ |c_{v,w}(a_t)|\le C_{v,w}e^{Mt} \quad(v\in V,\ w\in V^\vee) \tag{1.38x} \] with one \(M\) for all pairs, including all differentiated pairs. This is a bound on the specified regular ray. No global-growth theorem, topology, continuity of an abstract functional or classification is being presumed.

Step 4: The contracting-root contradiction

Suppose \(V=\mathfrak nV\). Induction then gives \(V=\mathfrak n^qV\) for every positive integer \(q\). Every core vector is a finite sum \(X_1\cdots X_qv'\), with \(X_j\) upper-root generators, including their imaginary companions for the complex field. For the ray (1.38u), \[ \operatorname{Ad}(a_t)X_{ij} =e^{x^{(0)}_i-x^{(0)}_j} e^{-t(H_i-H_j)}X_{ij}. \tag{1.38y} \] Put \(\epsilon=\min_i(H_i-H_{i+1})>0\). The identity between left and right tangent vectors at a fixed \(g\), combined with (1.38c), gives \[ c_{Xv,w}(g)=-c_{v,\operatorname{Ad}(g)X\,w}(g). \tag{1.38z} \] Iterate this pointwise identity; \(g\) stays fixed while the pairs are replaced, so no derivative of the coefficient \(\operatorname{Ad}(g)\) is introduced. For a word the dual order is reversed: \[ c_{X_1\cdots X_qv',w}(g) =(-1)^q c_{v', \operatorname{Ad}(g)X_q\cdots \operatorname{Ad}(g)X_1w}(g). \] Equations (1.38x)–(1.38y) imply for every pair \[ |c_{v,w}(a_t)| \le C_{q,v,w}e^{(M-q\epsilon)t}. \tag{1.38aa} \] The constant may depend on \(q\); the \(M\) does not.

Choose \(v,w\) with \(B(v,w)\ne0\). Its analytic diagonal germ is nonzero at identity, so choose a sufficiently small \(x^{(0)}\in C\) with \(F_{v,w}(x^{(0)})\ne0\). The finite compact array of this pair has nonzero initial jet \(J_0\). Its own bounded radial system along the ray gives, by the backwards integral iteration, \[ \|J(t)\|\ge e^{-M_0t}\|J_0\|,\qquad t\ge0. \tag{1.38ab} \] Each component is a coefficient of fixed core pairs: compact translation stays in the core, and radial derivatives apply commuting diagonal Lie generators to the first member of the pair. Apply (1.38aa) to these finitely many components. Choose \(q\epsilon>M+M_0+1\). The resulting upper bound contradicts (1.38ab) as \(t\) tends to infinity. This proves (1.38p).

For \(n=1\) there is no nilpotent algebra and \(V/\mathfrak nV=V\ne0\). The central Lie algebra and compact action make the irreducible admissible core one-dimensional by the same finite-packet eigenvector argument as in Theorem 1.27. No chamber argument is needed. Conjugating upper roots by the compact long permutation proves the lower version.

Scope

Theorem 1.33 is purely about arbitrary abstract irreducible admissible cores, in every rank for both archimedean fields. The auxiliary scalar functions have been proved analytic only on a neighborhood of \(K\) and on \(G_{\mathrm{reg}}\). Their use establishes a nonzero finite Jacquet quotient; it does not silently declare them matrix coefficients of an already globalized core. The compatible actual completion will instead be constructed from the algebraic Borel occurrence and the already written closed-principal-model proof.

Full Borel occurrence and an actual compatible realization of every core

Let \(V\) be any irreducible admissible Harish-Chandra core of \(GL_n(\mathbb R)\) or \(GL_n(\mathbb C)\), in every rank. Theorems 1.27 and 1.33 prove that its upper minimal Jacquet quotient is finite-dimensional and nonzero. Those proofs require no pre-existing realization.

Theorem 1.34. There is an explicit injective core map \[ T_0:V\hookrightarrow I_0(\chi_1,\ldots,\chi_n) \tag{1.38ac} \] into a full normalized Borel induction, and a full inverse Borel core surjecting onto \(V\). The closure of (1.38ac) in the actual compact smooth induced model has exactly core \(V\). With the paired principal annihilator quotient it is a complete Fréchet smooth moderate-growth continuous dual-pair realization of \(V,V^\vee\), with jointly continuous nondegenerate invariant bilinear pairing.

This asserts existence of a compatible actual realization for an arbitrary core, rather than identification with every independently prescribed smooth topology. The principal-core quotient is onto algebraically; no onto range for its extension to another topology is inferred.

Step 1: The algebraic occurrence map

Put \(d=[F:\mathbb R]\), \(M=K\cap B\), and \(\rho_i^0=(n+1)/2-i\). On the nonzero finite quotient \(V/\mathfrak nV\), the diagonal real Lie operators commute with one another and with the compact diagonal group \(M\). Average a positive Hermitian form over \(M\); by Lemma 1.32a, the commuting unitary compact matrices split the quotient into character spaces. On one nonzero such space, the commuting transposed diagonal operators have a common eigenvector: choose an eigenspace for one transpose and continue restricting to it when a remaining operator is not scalar. Its dimension strictly drops at such a step, so the procedure terminates. Pull it back to a nonzero algebraic functional \(\ell_0\) on \(V\): \[ \ell_0(\mathfrak nV)=0,\quad \ell_0(H_iv)=\beta_i\ell_0(v),\quad \ell_0(mv)=\sigma(m)\ell_0(v). \tag{1.38ad} \] This is the same finite-dimensional construction as (1.34u), now with the nonzero quotient justified for an arbitrary core by Theorem 1.33.

The character \(\sigma\) has sign exponents \(e_i\in\{0,1\}\) in the real case and integer circle exponents \(l_i\) in the complex case. The circle statement follows directly from the finite-dimensional smooth character differential equation and period \(2\pi\). Define \[ t_i=\beta_i/d-\rho_i^0,\qquad \chi_i(x)=\operatorname{sgn}(x)^{e_i}|x|^{t_i} \quad(F=\mathbb R), \quad \chi_i(z)=(z/|z|)^{l_i}|z|_{\mathbb C}^{t_i} \quad(F=\mathbb C). \tag{1.38ae} \] Thus the inducing character \(\delta_B^{1/2}\prod_i\chi_i\) has exactly the infinitesimal diagonal and compact characters of \(\ell_0\), including the factor \(d\). The actual full compact model \(I(\boldsymbol\chi)\) and its smooth action are the row-QR model constructed in Theorem 1.26 above.

Set \[ (T_0v)(k)=\ell_0(kv),\qquad k\in K. \tag{1.38af} \] Its compact orbit is finite; it is smooth and has the prescribed left \(M\)-covariance. For a real Lie generator \(X\), differentiate the actual factorization \(k\exp(tX)=b(t,k)k'(t,k)\). Its infinitesimal decomposition is \(\operatorname{Ad}(k)X=Y_B+Y_K\), with \(Y_B\) upper triangular with real diagonal and \(Y_K\) compact. The derivative of its multiplier equals the eigencharacter of \(\ell_0\) on \(Y_B\); the compact derivative contributes \(\ell_0(Y_Kkv)\). Therefore \[ (dI(X)T_0v)(k) =\ell_0((Y_B+Y_K)kv) =\ell_0(kXv)=(T_0Xv)(k). \tag{1.38ag} \] This is a linear identity even where \(\ell_0(kv)=0\); no division is taken. Right compact translation gives the compact intertwining. \(T_0\) is nonzero because evaluation at identity is \(\ell_0\); its Lie- and compact-stable kernel is zero by irreducibility.

Apply the same construction to the abstract \(V^\vee\). The compact integral pairing between full induction with characters \(\boldsymbol\eta\) and \(\boldsymbol\eta^{-1}\), proved in Theorem 1.26, pairs their finite packets perfectly. Transposing the injection of \(V^\vee\) on each finite packet gives an onto map of that packet onto its dual packet in \(V\). The algebraic direct sum of these maps is the onto core map \[ I_0(\boldsymbol\eta^{-1})\twoheadrightarrow V. \tag{1.38ah} \] No global topological quotient is needed to justify this finite-packet transposition. It is precisely the interface (1.34y), now available for every core.

Step 2: A concrete complete smooth paired model

Let \(S_0=T_0(V)\subset I_0(\boldsymbol\chi)\), and put \[ E=\overline{S_0}^{\,C^\infty(K)},\qquad E^\vee= I(\boldsymbol\chi^{-1})/ \operatorname{Ann}(E). \tag{1.38ai} \] The closure and annihilator are taken in the actual compact smooth induced spaces, with their countable derivative seminorms and the compact integral pairing. This is the special case \(W_0=0,V_0=S_0\) of the closure construction of Theorem 1.26a above. We state its concrete verified conclusions and the mechanisms relevant to this construction.

First, compact-finite vectors in a full compact model are coordinate polynomials. This is proved using the positive polynomial compact approximate identities (1.32h), finite-dimensional compact decomposition and Schur orthogonality. Row-QR applied to \(k\exp(tX)\) gives a holomorphic exponential expansion in a disk whose radius is independent of that coordinate polynomial: its row-Gram pivots stay near one uniformly on compact \(K\). Thus its compact-smooth Taylor series converges on a common disk. Every Taylor coefficient of a vector in \(S_0\) stays in \(S_0\), by Lie stability. Continuity, the common radius, finite iteration and polar generation of \(G\) make its closure \(E\) an actual closed \(G\)-subrepresentation. This is the group-stability proof in Theorem 1.26a, Steps 2–3.

Second, every compact projector has finite-dimensional range. Projecting the closure \(E\) puts its packet in the closure of the packet of \(S_0\), which is already a closed finite-dimensional subspace. Hence the core of \(E\) is exactly \(S_0\), and is identified with \(V\) through (1.38ac). The annihilator is closed and group-stable by the invariant continuous pairing. Finite packet duality gives exactly the core \(V^\vee\) in the quotient in (1.38ai), as proved in Theorem 1.26a, Steps 4–5. The quotient is Fréchet and complete: Theorem 1.26a gives the explicit summable lifting argument for a Cauchy sequence in a closed Fréchet quotient, not an assumed comparison theorem.

Third, the full induced actions and all their derivatives are continuous and moderate, with polynomial matrix-height bounds on each compact derivative seminorm from Theorem 1.26. Restriction to a closed subspace and the quotient seminorm bounds preserve these properties; quotient smoothness and differentiation are verified by the actual smooth lift in Theorem 1.26a. The compact integral gives the jointly continuous invariant pairing \[ E\times E^\vee\longrightarrow\mathbb C. \tag{1.38aj} \] It is nondegenerate in both variables: a nonzero compact packet has a detecting paired packet by finite-dimensional duality, and the compact polynomial approximate identities detect a nonzero vector. The same identities prove density of both cores. These are actual maps and actual complete spaces.

Finally, core irreducibility makes \(E\) and \(E^\vee\) topologically irreducible. A nonzero closed invariant subspace has a nonzero compact-finite vector by the positive compact approximate identities. Its core is then the whole irreducible core, whose density makes that subspace the whole model. This completes Theorem 1.34.

Step 3: Consequences and comparison

Every irreducible admissible abstract core therefore has a compatible actual complete smooth moderate dual pair. Theorem 1.31 applies to this constructed pair and proves its attained Gaussian ideal, entire Schwartz division and paired Fourier equation.

For a separately prescribed actual complete smooth moderate dual pair, Theorems 1.29–1.31 give coefficient comparison, a complete common refinement and the Gaussian package in that original topology. Theorems 2.0x–2.0y below extend a principal-core quotient continuously with dense range and embed the entire continuous dual. None of these dense-range statements proves surjectivity or an inverse seminorm estimate for a comparison with an independently prescribed topology.

The integral-defined polynomial in Theorem 1.31 also requires a separate computation to identify it with every intended real discrete-series, parity and complex angular standard-factor label.

The remaining canonical archimedean input

Theorem 1.21 proves continuation, entire division by a common pole majorant and rapid strip bounds for every smooth coefficient and Schwartz test in the stated actual dual-pair models. Theorem 1.22 proves their scalar Fourier equation, including reflection and character scaling. Proposition 1.21b and Corollary 1.22e prove full-Schwartz canonical division and constant epsilon once the genuine Gaussian ideals have been established; Corollary 1.22f proves the scalar critical-line phase in compatible unitary models. Theorem 1.19 and Theorems 1.11–1.13 furnish the actual smooth pair for cuspidal automorphic constituents.

Theorem 1.26 proves the exact Gaussian product and one-pair attainment for every full real/complex Borel induction in the actual compact model, including reducible inductions. Theorems 1.26a–1.26b construct actual closed subquotient/dual models for every supplied principal-core subquotient and prove its exact monic polynomial correction, finite attaining family, entire Schwartz division and Fourier equation. Theorem 1.28 proves all-rank principal-core occurrence for every prescribed actual smooth moderate dual pair; Theorems 1.29–1.31 prove coefficient comparison, a complete common refinement, and the full Gaussian package in that original realization. Theorems 1.32–1.34 prove abstract-core existence and full Borel occurrence. Full onto/closed-image comparison with separately prescribed topologies remains open, together with explicit standard-factor identification of the correction polynomial and scalar. The generator normalization in Theorem 1.26b is monic relative to its supplied inducing product. Theorem 1.31 proves the paired dual and unitary conjugation normalization of these integral-defined factors. For every abstract irreducible admissible core, Theorem 1.34 supplies such an actual smooth dual pair. Propositions 1.7 and 1.18 supply the complete canonical package for determinant characters over both real and complex fields. The majorant example (1.25t) shows why polynomial-module stability and an entire common-denominator quotient cannot replace the Gaussian-ideal proof.

The freely accessible Goldfeld–Jacquet author notes, §3, Theorem 3.5 and Lemma 3.6, PDF pages 14–19 describe the canonical archimedean package. Their induced-representation reduction and generator statements do not supply the missing full proof. The continuation, distributional descent and Fourier arguments above are proved here.

Matrix theta estimates and singular orbits

Take cusp vectors \(\phi\in\pi_0\) and \(\widetilde\phi\in\widetilde\pi_0\), paired by

For a Schwartz–Bruhat function \(\Phi\) on \(M_n(\mathbb A)\), define

We first justify the sums and integrations.

Lemma 1.1d (matrix lattice bounds). For \(x=b\omega_1\), \(y=b'\omega_2\) in the Siegel sets of 1.1a, there are polynomial height bounds

The same estimates hold after any number of logarithmic \(t\)-derivatives.

Proof. The compact \(\omega_i\) put the transformed tests in a bounded Schwartz family with a common finite support. Rational matrices in that support lie in a fixed full lattice \(\mathcal M\) in \(M_n(F_\infty)\), of real dimension \(D=n^2d\). Set \(\lambda=t^{1/(nd)}\). A matrix height \(H\ge1\) bounds the inverse operator norm of \(X\mapsto b^{-1}Xb'\), so Schwartz bounds reduce the sum to \(\sum_{X\in\mathcal M\setminus\{0\}}(1+\lambda H^{-1}\|X\|)^{-N}\).

A lattice has \(O((1+R)^D)\) points in a ball of radius \(R\), and a positive minimum length. Dyadic summation therefore bounds this sum by \(C_N\eta^{-D}\) for \(0<\eta=\lambda/H\le1\), and by \(C_N\eta^{-N}\) for \(\eta\ge1\), when \(N>D\). For \(t\le1\), this is \(C H^D t^{-n}\). For \(t\ge1\), choose \(N>D+ndM\). In the region \(\lambda\le H\), the first bound is at most \(C H^N\lambda^{-ndM}\); in the region \(\lambda\ge H\), the second has the same upper bound. This is the second estimate. A logarithmic derivative replaces \(\Phi\) by a constant multiple of its archimedean Euler derivative, which is again Schwartz. \(\square\)

Multiplying (1.5r) by \(|\widetilde\phi(x)\phi(y)|\) and using Lemma 1.1b makes both height factors integrable. Hence \(K_\Phi(t)=O(t^{-n})\) at zero and \(O_M(t^{-M})\) at infinity. The same proof works for the sum over matrices of any fixed positive rank. All these statements also hold for \(\widehat\Phi\).

For \(0<r<n\), matrices of rank \(r\) form one \(G(F)\times G(F)\)-orbit of \(e_r=\operatorname{diag}(I_r,0)\). Its stabilizer consists of pairs \(p,q\) satisfying \(p^{-1}e_rq=e_r\), or

Unfold the rank-\(r\) sum in the double quotient integral. This is legitimate by the absolute estimate just proved. The stabilizer contains the upper and lower block radicals

Their adelic quotients by rational points are compact copies of \((F\backslash\mathbb A)^{r(n-r)}\). Disintegrate the unfolded integral first over these two normal unipotent subgroups. The matrix kernel is unchanged, since \(U_+^{-1}e_r=e_r=e_rU_-\). Its two inner averages are therefore the \(U_+\)-constant term of \(\widetilde\phi\) and the \(U_-\)-constant term of \(\phi\). Both are zero by cuspidality. Thus every positive singular-rank contribution vanishes. Norm-one restrictions do not affect these inner averages, since unipotent determinants are one.

The zero matrix also contributes zero. To justify this separately, integration on the finite-volume quotient gives a bounded \(G^1\)-invariant functional on the unitary space of \(\pi_0\). Restriction from \(G(\mathbb A)=a_{\mathbb R_{>0}}G^1\) to \(G^1\) remains irreducible, because the removed central group acts by scalars. A nonzero invariant functional would therefore make this restriction trivial. Its forms would be constant on \(G^1\), contradicting a proper-parabolic zero constant term when \(n\ge2\). Hence \(\int_{[G]^1}\phi=\int_{[G]^1}\widetilde\phi=0\), and the rank-zero double period is zero.

Poisson summation and the Mellin transform

Additive self-duality, covolume one and Poisson summation for \(F\backslash\mathbb A\) are proved in Additive characters, self-dual measures and Poisson summation, Theorems 5.4–5.5. Their finite Cartesian product applies to \(M_n\). The trace pairing exchanges the \((i,j)\) and \((j,i)\) coordinates, so the annihilator of \(M_n(F)\) is again \(M_n(F)\), with covolume one. Whole-block Schwartz inversion is the same real-vector-space calculation as Lemma 5.4A there; lattice periodization is justified by the same dyadic argument, in dimension \(n^2d\). Thus this application includes Schwartz functions whose archimedean matrix entries do not separate.

The linear map \(T(X)=x^{-1}a_tXy\) has additive modulus \(t^n\), since \(\nu(x)=\nu(y)=1\) and scalar multiplication by \(a_t\) on \(n^2\) entries has modulus \(t^n\). Its trace adjoint is \(T^*(Y)=a_tyYx^{-1}\). Poisson gives

Integrate against \(\widetilde\phi(x)\phi(y)\). Every singular-rank period on both sides vanishes by (1.5s)–(1.5t) and the zero-rank argument. Therefore, denoting the swapped cusp vectors on the right explicitly,

The large-\(t\) bound in Lemma 1.1d now shows that \(K_\Phi\) also decreases faster than every power at zero: choose the exponent on the right larger than \(n\) plus any prescribed power. This conclusion holds for every logarithmic derivative; differentiate (1.5v), using the derivative bounds already proved at infinity.

For \(\operatorname{Re}s>(n+1)/2\), absolute unfolding of the matrix-coefficient integral gives

One first inserts (1.5p), changes \(g\) to \(x^{-1}g\), then writes \(g=a_t\gamma y\) with \(y\in[G]^1\). The absolute integral of the unfolded expression is finite: (1.5r) and rapid cusp decay bound its small-\(t\) part by a constant times \(\int_0^1t^{\operatorname{Re}u-n}dt/t\), and its large-\(t\) part by an arbitrarily fast decaying power. Haar measures in (1.5w) are chosen compatibly with (1.5a).

Both ends of the Mellin integral are now rapidly decreasing. For every compact set of \(s\), the integral and all its \(s\)-derivatives converge uniformly, since derivatives insert powers of \(\log t\). It defines an entire function. More precisely, put \(v=\log t\). On every closed strip the function \(K_\Phi(e^v)e^{(\operatorname{Re}s+(n-1)/2)v}\), together with every \(v\)-derivative, is integrable uniformly in \(\operatorname{Re}s\). Repeated integration by parts, with zero boundary terms, proves

for every \(N\). This proves the required strip control without a Phragmén–Lindelöf argument.

Changing \(t\) to \(t^{-1}\) in (1.5w) and using (1.5v) changes the exponent to \(n-u=(1-s)+(n-1)/2\). The swapped vectors give \(c^\vee\). Hence the global integral functional equation is

The shift in (1.5) is exactly what turns \(u\mapsto n-u\) into \(s\mapsto1-s\).

Cuspidal Hilbert realizations and products of local pairings

Let \(F\) be a number field, \(d=[F:\mathbb Q]\), \(G=GL_n\), and \(\nu(g)=|\det g|_{\mathbb A}\). Put \(G^1=\ker\nu\), \(\Gamma=G(F)\), and \(X=\Gamma\backslash G^1\). For \(t>0\), take \(a_t\) to have scalar entry \(t^{1/(nd)}\) at every infinite embedding and entry \(1\) at the finite places. Then \(\nu(a_t)=t\), and \(G(\mathbb A)=a_{\mathbb R_{>0}}\times G^1\). We fix probability Haar measure on the norm-one idèle class group and a compatible quotient Haar measure \(dx\) on \(X\). Inner products are linear in the first argument.

The statements below concern an actual admissible cuspidal Hilbert constituent, and also give a reverse comparison under an explicit square-integrability hypothesis. They prove the product pairing and the compatible unitary tensor realization. Theorem 1.19 below proves existence and admissibility of every cuspidal constituent by elliptic estimates, a cusp-tail bound and a spectral decomposition.

Central normalization. An irreducible admissible automorphic module has a continuous scalar central character. Its restriction to the norm-one idèle classes is unitary. Its positive-central action is \(a_t\mapsto t^\mu\) for some \(\mu\in\mathbb C\). Consequently \(\pi_0=\pi\otimes\nu^{-\operatorname{Re}\mu}\) has unitary central character, and its positive-central action is \(t^{i\operatorname{Im}\mu}\). This normalization needs no L-function theorem.

To verify this even for the algebraic subquotient definition, the central Lie generators act on every automorphic form through finite-dimensional spaces, by enveloping-center finiteness. On each such space, solving the constant-coefficient ordinary differential equation for central real translation gives the matrix exponential of the Lie action. Thus these translations preserve algebraic mixed submodules and their quotients. Finite-place central elements and the compact real center already act in the mixed category. Every central translation operator commutes with the mixed action; the finite-corner Schur argument makes it scalar on an irreducible admissible module. The positive-central generator is likewise scalar \(\mu\), giving \(t^\mu\). Continuity follows from this exponential formula, the compact real action, and finite-place smoothness on a nonzero finite-level vector. Scalar rational translations are trivial by left automorphy. Finally \(F^\times\backslash\mathbb A^1\) is compact by Idèles and the idèle class group, Theorem 3.3; a continuous character of a compact group has modulus one, since its modulus has compact image in \(\mathbb R_{>0}\), whose only compact subgroup is \(\{1\}\). Twisting by the indicated power of \(\nu\) removes just the real part of \(\mu\). All Hilbert assertions below apply to \(\pi_0\); the original module is recovered by the inverse twist.

Lemma 1.8 (the cuspidal Hilbert space). The space \(L^2(X)\) carries a strongly continuous unitary right action of \(G^1\). For each unitary character \(\omega^1\) of \(F^\times\backslash\mathbb A^1\), its central \(\omega^1\)-space is a closed invariant subspace. The simultaneous kernel of all proper-parabolic constant terms, interpreted as distributions on the group, is closed and right invariant. Fixing a real number \(\tau\) extends this representation to \(G(\mathbb A)\) by letting \(a_t\) act as \(t^{i\tau}\).

Proof. The matrix coordinate embedding and The adèle ring of a number field, Theorem 2.2 make \(\Gamma\) closed and discrete. The product formula proved in Places of number fields in extensions and the product formula, Theorem 5.2 puts it in \(G^1\). At each place \(|\det g|_v^{-n}d g\), with \(d g\) additive matrix measure, is both left and right Haar measure: either multiplication has additive Jacobian \(|\det g|_v^n\). The restricted product is therefore unimodular, and its central direct-product decomposition makes \(G^1\) unimodular as well.

A quotient measure can be constructed by choosing a measurable fundamental set. Indeed, the group is second countable, so a countable collection of open coordinate sets whose rational translates are disjoint covers the quotient. Enumerate those sets and assign a coset to its first available chart. Restricting Haar measure to the resulting disjoint measurable section defines \(dx\). Left invariance makes the definition independent of the chosen section and gives

for compactly supported continuous \(h\), first for nonnegative \(h\) and then by linearity. Right Haar invariance now proves right invariance of \(dx\). Thus right translations are unitary. They are strongly continuous: on continuous compactly supported quotient functions, uniform continuity on the compact union of nearby translated supports proves convergence in \(L^2\); regularity of the quotient measure and approximation by such functions give it for every \(L^2\) vector.

The norm-one scalar center acts through \(F^\times\backslash\mathbb A^1\), compact by Idèles and the idèle class group, Theorem 3.3. Its character projection is

Character orthogonality makes this a self-adjoint idempotent with the asserted range. Centrality makes the range invariant.

For a proper parabolic with radical \(N_P\), its rational adelic quotient is compact by Automorphic representations and automorphic L-functions, Lemma 1.0. Choose compact representatives \(D_P\). On any compact set \(B\subset G^1\), pullback from \(X\) has local \(L^2\)-norm at most a fixed constant times the quotient norm: only finitely many rational translates of a compact lift can meet that lift, because their ratios lie in a compact set intersected with the closed discrete \(\Gamma\). Jensen's inequality and left translation on the group therefore give

This also defines the integral as a distribution if a representative has not been chosen. The kernels of these bounded local maps, over all \(B\) and \(P\), have closed intersection. Right translations commute with constant terms, so the intersection is invariant. Finally define, for \(g\in G(\mathbb A)\),

Centrality of \(a_t\) makes this a strongly continuous unitary representation extending the given one. On smooth representatives it is the right action on the extension \(u(a_t x)=t^{i\tau}u(x)\). \(\square\)

Lemma 1.9 (finite vectors in an admissible unitary representation). Let \(H\) be an irreducible strongly continuous unitary representation of \(G(\mathbb A)\). Assume that every simultaneous compact-open finite-place fixed space and finite packet of \(K_\infty\)-types is finite dimensional. Then its mixed finite-vector space \(H_{\mathrm{fin}}\) is smooth at infinity, dense, admissible and irreducible as a \((\mathfrak g,K_\infty)\times G(\mathbb A_f)\)-module. It is a common operator core for every real Lie generator. At each finite level it is dense in the smooth-vector topology

Proof. Finite-place compact-open averages tend strongly to the identity. Compact matrix-coefficient averages do the same at infinity, by the uniform density proved in Matrix coefficients and the Peter–Weyl theorem, Theorem 4.1: first average against a continuous probability kernel concentrated near the identity and then approximate that kernel uniformly by a finite sum of coefficients. These averages have finite compact-type range. The two kinds of averaging commute, proving density of \(H_{\mathrm{fin}}\).

Fix one finite-dimensional joint corner \(E=e_Ee_UH\). Average a real smooth compactly supported approximate-identity kernel under conjugation by \(K_\infty\). Its convolution commutes with the compact-type projector, and with \(e_U\), so it preserves \(E\). On \(E\) its strong convergence to the identity is convergence of finite matrices. It is eventually invertible. Its image is smooth, since all derivatives can be transferred to its smooth compact kernel. Thus every vector of \(E\) is smooth. Lie differentiation preserves finite compact type because \(D\) has a finite-dimensional adjoint \(K_\infty\)-orbit in each bounded enveloping degree; it also preserves finite level. We have obtained the mixed module.

The same compact approximation works in every seminorm (1.11). Indeed,

and the adjoint orbit of \(D\) lies in a finite-dimensional space. Its coefficients vary continuously over the compact group. The orbit of a smooth vector is therefore continuous in all the seminorms, and its compact supremum in any fixed seminorm is finite. Uniform kernel approximation and the approximate-identity argument prove the asserted smooth density, retaining the same finite level.

We give the operator-core argument, since norm density alone would not justify integration of the Lie action. Let \(A_X\) be the generator of \(R(\exp(tX))\). It is closed: if \(u_j\to u\) and \(A_Xu_j\to v\), pass to the limit in \(R(\exp(tX))u_j-u_j=\int_0^t R(\exp(sX))A_Xu_j\,ds\); differentiation of the resulting identity puts \(u\) in its domain with derivative \(v\). Smooth convolution supplies a dense domain. For a smooth compact kernel \(f\) on the real group, let \(L_Xf(g)\) and \(R_Xf(g)\) mean differentiation of \(f(\exp(-tX)g)\) and \(f(g\exp(-tX))\), respectively. Changes of variable in convolution give, for \(u\in\operatorname{Dom}A_X\),

Take mass-one kernels \(f_\epsilon\) in a coordinate ball of radius \(O(\epsilon)\). The coefficients of \(L_X-R_X\) vanish at the identity and are \(O(\epsilon)\) on that ball; the first derivatives of the scaled kernel have \(L^1\)-norm \(O(\epsilon^{-1})\). Hence \(\|(L_X-R_X)f_\epsilon\|_1\) is bounded uniformly. Its integral is zero, because both invariant vector fields have zero divergence for the bi-invariant Haar measure. Strong continuity gives

Equation (1.12) proves convergence of \(R(f_\epsilon)u\) to \(u\) in the graph norm of \(A_X\). These convolved vectors are real-smooth. Finite-place compact-open averages commute with \(A_X\) and approximate both \(u\) and \(A_Xu\); the real compact smooth approximation just proved then approximates the resulting smooth finite-level vectors in graph norm by mixed finite vectors. Thus \(H_{\mathrm{fin}}\) is a core for \(A_X\).

Let \(M\) be a nonzero algebraic mixed submodule and let \(P\) be orthogonal projection onto its Hilbert closure. This closure is invariant under the compact group and under the finite adelic group, so \(P\) commutes with their averages. For a joint corner \(e\), the subspace \(eM\subset eH\) is finite dimensional and closed. Density gives \(e\overline M=eM\). Therefore \(P\) maps \(H_{\mathrm{fin}}\) into \(M\). On finite vectors, \(M\) and its orthogonal complement are both Lie invariant: for \(u\perp M\), \(\langle dR(X)u,m\rangle=-\langle u,dR(X)m\rangle=0\). Hence \(P A_X=A_XP\) on the common core. Closedness extends this identity to \(\operatorname{Dom}A_X\).

It then holds for the actual one-parameter group. For \(u\) in that domain, differentiate \(R(\exp(-tX))P R(\exp(tX))u\); the derivative is zero. Density proves the commutation identity on all of \(H\). Exponentials generate the identity component, and \(K_\infty\) meets every real component of the general linear group, so \(P\) commutes with the whole adelic representation. Irreducibility gives \(\overline M=H\). Applying each finite-dimensional joint corner again gives \(eM=eH\), and their union is \(H_{\mathrm{fin}}\). Thus \(M=H_{\mathrm{fin}}\). \(\square\)

The smooth space at a fixed finite level, with seminorms (1.11), is Fréchet and complete. To verify completeness, take a sequence for which all word derivatives are Cauchy. Closedness of each one-parameter generator first identifies the limits of its first derivatives and then, successively, the limits of every word derivative. Coordinate products of exponentials and the fundamental theorem of calculus identify the resulting derivatives with derivatives of the orbit map. Thus the limit is a smooth vector. The displayed adjoint identity also proves moderate growth of this smooth representation: coefficients of \(\operatorname{Ad}(g)D\) are polynomially bounded in the entries of \(g\) and \(g^{-1}\).

Lemma 1.10 (a polynomial quotient Sobolev bound). At a fixed compact-open finite level \(U\), every real-smooth vector in \(L^2(X)^U\) has a unique smooth representative. Write

where infinite norms are ordinary operator norms, and

For \(m=dn^2-1>0\) and every right differential operator \(D\), there is a constant \(C_{U,D}\) and a finite list of Lie words of length at most \(m\) such that

For \(m=0\) use the corresponding finite-level point-evaluation norm bound. In particular the inclusion of smooth vectors into smooth functions is continuous on compact sets, for every derivative, and all such vectors and derivatives have polynomial growth.

Proof. Use a small exponential coordinate box in the real Lie group \(G_\infty^1\), of dimension \(m\). There is a constant \(c_U>0\) such that

Here is an arithmetic verification. If two points in that set differ by \(\gamma\in\Gamma\), then \(g^{-1}\gamma g\) has infinite entries \(O(\epsilon)\) away from the identity and finite component in \(U\). For any nonzero entry \(a\) of \(\gamma-I_n\), finite-place matrix multiplication gives

The finite constant is a finite product, since \(U\) is integral outside finitely many places. The product formula makes the product of these two bounds at least one. For \(\epsilon<c_U\mathcal H(g)^{-4}\), with \(c_U\) small enough, their product is less than one. Thus \(\gamma=I_n\), proving (1.14). This uses no Siegel covering or finite-volume theorem.

On the injective chart, \(U\)-invariance gives

and the same inequality for every right derivative. Chart Haar density is bounded above and below by fixed positive constants. Coordinate derivatives are linear combinations of right Lie words with uniformly bounded smooth coefficients on a fixed small chart.

Choose a smooth cutoff equal to one near zero and supported in the coordinate box, scaled by \(\epsilon\). For a smooth function \(h\) so cut off, iterating the one-variable fundamental theorem of calculus gives

The integral has support of volume \(O(\epsilon^m)\). Cauchy–Schwarz and the product rule bound it by a constant times \(\epsilon^{-m/2}\) times the sum of the local \(L^2\)-norms of derivatives of order at most \(m\). Take \(\epsilon\) a fixed multiple of \(\mathcal H(g)^{-4}\) and use the chart inequalities. This proves (1.13).

For an \(L^2\) smooth vector, its group derivatives are weak coordinate derivatives locally: this follows by differentiating translation against a compact coordinate test function. Mollify in those coordinates. Weak derivatives converge in local \(L^2\), and the just proved inequality, applied also after arbitrary derivatives on a smaller box, makes the mollified functions Cauchy in every compact smooth seminorm. Their limit is a smooth representative of the original class. Such representatives agree on overlapping charts, since their difference is continuous and zero almost everywhere. The same argument proves (1.13) for the representative. If \(m=0\), the infinite chart is a single point and its finite \(U\)-fibre has positive measure, giving the asserted norm bound directly. \(\square\)

Theorem 1.11 (comparison inside an admissible Hilbert cusp constituent). Let \(H\ne0\) be a closed irreducible subrepresentation of the cuspidal central-character space in Lemma 1.8, and assume its joint compact corners are finite dimensional. Then \(\pi=H_{\mathrm{fin}}\) is an irreducible admissible module of automorphic cusp forms. Moreover

Its finite-level smooth-vector space has exactly the smooth-realization properties required for continuous archimedean Whittaker functionals.

Proof. Lemma 1.9 gives smoothness, irreducibility and admissibility. An endomorphism of this simple module is scalar: restrict it to a nonzero finite-dimensional joint corner, take an eigenvector there, and use simplicity on the kernel of the endomorphism minus that eigenvalue. For \(GL_n\), the enveloping center commutes with the compact action as well as the Lie and finite-place actions; its elements therefore act by scalars. Compact components also fix that center, since their actions are inner on each factor of the complexified general linear Lie algebra. Thus every finite vector is enveloping-center finite.

Lemma 1.10 gives its smooth representative and moderate growth, including every derivative. The positive-central extension has polynomial height as well: \(\nu(g)\) and \(\nu(g)^{-1}\) are bounded by powers of \(\mathcal H(g)\), so \(\mathcal H(a_{\nu(g)}^{-1}g)\) is polynomially bounded in \(\mathcal H(g)\); the factor \(\nu(g)^{i\tau}\) has modulus one. The distributional constant terms defining the Hilbert cusp space now coincide with smooth compact-domain averages. They vanish everywhere. Hence the finite vectors are automorphic cusp forms. The reverse inclusion in (1.15) follows immediately from the finite-level and compact-finiteness requirements in the definition of an automorphic form. Density is Lemma 1.9.

At fixed finite level the topology (1.11) is complete, smooth and of moderate growth, as proved after Lemma 1.9. Lemma 1.10 makes its function-space inclusion continuous in every compact derivative seminorm. Compact averaging and Lie differentiation are continuous. Its mixed finite core is \(\pi\). These are precisely the required realization properties. \(\square\)

Lemma 1.12 (factorization of an invariant positive form). Suppose \(S,T\) are simple admissible modules for algebras \(A,B\) with directed self-adjoint local units, and \(S\otimes T\) has a positive Hermitian form satisfying

Then the form is a product \(h_S\otimes h_T\) of positive invariant forms. These factors are unique up to multiplying one by a positive scalar and the other by its inverse.

Proof. Fix \(0\ne t_0\in T\), and set \(h_S(s,s')=\langle s\otimes t_0,s'\otimes t_0\rangle\). It is positive and invariant. For arbitrary \(t,t'\), consider \(b_{t,t'}(s,s')=\langle s\otimes t,s'\otimes t'\rangle\). If a self-adjoint unit \(e\) fixes \(s\), invariance gives \(b_{t,t'}(s,z)=b_{t,t'}(s,ez)\). Finite-dimensional Riesz representation on \(eS\) therefore gives a vector \(Ts\in eS\) with

Larger units give the same vector, since the positive form separates vectors. Thus \(T\) is a well-defined linear operator on \(S\). Invariance of both forms gives \(Ta=aT\). The finite-corner Schur lemma proved in Restricted tensor products and the tensor product theorem, Lemma 3.2 implies \(T\) is scalar. Choose \(s_0\) of \(h_S\)-norm one and put \(h_T(t,t')=\langle s_0\otimes t,s_0\otimes t'\rangle\). The scalar just obtained is \(h_T(t,t')\), proving the product identity. Positivity and \(B\)-invariance follow from this formula. Comparing with any other product form on nonzero fixed vectors proves the claimed scalar freedom. \(\square\)

Theorem 1.13 (unitary restricted tensor realization and local pairings). In the situation of Theorem 1.11, take the algebraic restricted factorization proved in Automorphic representations and automorphic L-functions, Theorem 1.1. There are positive invariant local Hermitian forms \(h_v\), local irreducible unitary Hilbert representations \(H_v\) with finite cores \(\pi_v\), and spherical reference vectors \(\xi_v\) of norm one outside a finite set, such that

The Hilbert isomorphism integrates the actual full adelic action. The local contragredient finite core is the conjugate core through \(\overline y\mapsto[x\mapsto h_v(x,y)]\). In the automorphic realization, if \(\phi_x,\phi_y\) are the pure tensor cusp forms corresponding to \(x,y\), then

Thus the bilinear automorphic pairing with the dual cusp form \(\widetilde\phi_y=\overline{\phi_y}\) is exactly the product of local dual pairings. Neither local Whittaker uniqueness nor any standard or Rankin–Selberg analytic theorem is used.

Proof. The global Hilbert form is invariant for the convolution involution at finite places, for compact distributions, and for \(X^*=-X\) on real Lie generators. The compact-open and compact-type local units are self-adjoint orthogonal projections. The algebraic local modules are simple admissible modules by the already proved factorization theorem. Apply Lemma 1.12 at each finite tensor stage.

Here is a compatible normalization, avoiding an infinite product of arbitrary constants. Choose a nonzero vector \(\xi_v\) in every local factor, using the specified fixed reference line at all but finitely many places, and let \(x_0=\bigotimes_v\xi_v\). Put \(C=\|x_0\|^2>0\). Define the form in factor \(v\) by replacing just that factor in \(x_0\) and dividing the resulting form by \(C\). It gives \(\xi_v\) norm one. Lemma 1.12 on a finite stage shows that its global form is \(C\) times the product of these normalized local forms: evaluate at its reference tensor to determine the one remaining scalar. The forms obtained at different stages agree, since their definitions use the same fixed \(x_0\). Absorb \(C\) into the form at one infinite place. Now (1.16)'s product identity holds on the whole algebraic restricted tensor product, and every tail reference vector still has norm one. Its finite-stage embeddings are isometric. Completing gives an isometry onto \(H\), since the finite core is dense.

We must show that the forms give actual unitary local groups also at infinity. Regroup the algebraic core as \(S\otimes T=\pi_v\otimes\pi^v\) and its form as \(h_v\otimes h^v\). For a nonzero finite tensor \(t\in T\), the operator

is an orthogonal projection of norm one for the product form, so it extends to \(H\). It preserves the mixed core and commutes there with every local Lie operator and local compact operation at \(v\), and with every finite-place group element when \(v\) is finite. Lemma 1.9's common operator core and the one-parameter commutation argument extend the infinitesimal commutation at infinity to the actual \(G(F_v)\)-action. Thus \(P_tH\) is a reducing local Hilbert subspace. Its identification with the completion of \(\pi_v\), with the harmless fixed factor \(\|t\|\), defines \(H_v\) and its full unitary local action.

Its finite core is exactly \(\pi_v\). A local compact-open fixed vector at a finite \(v\), or a local compact-finite vector at an infinite \(v\), combined with the fixed other-place tensor \(t\), is a global mixed finite vector: the other components of \(t\) already have finite level and finite compact type. It therefore belongs to the global core, and hence to \(\pi_v\otimes\mathbb Ct\). Conversely all vectors of that local core have the required properties. They are dense by compact averaging. A nonzero closed local invariant subspace contains a nonzero such vector; simplicity of \(\pi_v\) forces it to contain the whole core, and density forces it to be \(H_v\). This proves local irreducibility and admissibility. The operator-core proof of Lemma 1.9 applies to this local representation with its local compact projectors, so \(\pi_v\) is a core for every local real Lie generator.

The full tensor action equals the given global action. At a finite place this already holds on the invariant algebraic core and extends by continuity. At infinity the two actions have the same Lie generators on the core, and this core is a common generator core on both sides. For the tensor side, approximate in the other Hilbert factors by finite-dimensional orthogonal projections and in the active factor by its generator core; this approximates both the vector and its derivative in graph norm. Closed generators therefore agree under the isometry. Their one-parameter groups agree by the differentiated identity used in Lemma 1.9, and compact components agree on the core. This proves full adelic equivariance. The restricted tensor action is strongly continuous because the compact tail fixes every pure reference-tail tensor, while only finitely many active local factors remain; density and unitarity extend continuity to the completion.

For a local finite vector \(y\), the functional \(x\mapsto h_v(x,y)\) has finite compact support: a self-adjoint finite compact projector fixing \(y\) can be transferred to \(x\). Conversely every functional in the restricted compact-finite dual is supported on one finite-dimensional admissible corner, where Riesz representation supplies such a \(y\). This identifies the conjugate core with the contragredient core. The group identity \(h_v(x,\pi_v(g)y)=h_v(\pi_v(g^{-1})x,y)\) proves equivariance. Complex conjugation of the automorphic functions supplies the dual central character and realizes this dual pairing as the integral in (1.17). That integral is absolutely convergent by Cauchy–Schwarz. The tensor isometry gives its product identity; outside a finite set \(g_v\in K_v\) and both vectors equal the unit \(\xi_v\), so every omitted factor is exactly one. Cauchy–Schwarz and unitarity give the stated uniform bound. \(\square\)

Proposition 1.14 (reverse comparison under square integrability). Let \(\pi\) be an irreducible admissible cuspidal submodule of automorphic forms with unitary central character. Suppose that every \(R(D)\phi\), for \(\phi\in\pi\) and every real Lie word \(D\), belongs to \(L^2(X)\). Then its closure in \(L^2(X)\) is an irreducible unitary cuspidal Hilbert constituent, its mixed finite core is exactly \(\pi\), and Theorem 1.13 applies to it.

Proof. The functions and their derivatives define real-smooth Hilbert vectors. For a real generator \(X\), the pointwise fundamental theorem of calculus gives

The right side is an \(L^2\) integral. Strong continuity makes its norm derivative at zero equal to \(R(X)\phi\). Repeating with all Lie words and using coordinate products of exponentials proves Hilbert smoothness. Thus the restricted integral form is positive and infinitesimally invariant.

We give the integration argument rather than assume that the algebraic module is already invariant under arbitrary real translates. On the real Lie algebra of the product of general linear groups, take the Cartan involution \(\theta X=-X^*\) and the invariant real trace form, scaled on each factor so that it is negative on \(\mathfrak k\) and positive on \(\mathfrak p\). Let \(K_i,P_j\) be corresponding orthonormal bases. The element

is central: invariance of the trace form makes the commutator with its dual-basis quadratic sum zero. The finite-corner Schur argument makes it scalar \(\lambda\) on \(\pi\). Infinitesimal unitarity makes \(\lambda\) real and gives

Fix a finite compact-orbit space \(E\) containing \(w\). The vectors obtained by Lie words of length at most \(r\) are a compact-equivariant image of \((\mathbb C\oplus\mathfrak g)^{\otimes r}\otimes E\). On that finite-dimensional compact tensor representation, the norm of any \(K_i\) is at most \(rM+M_E\), by the tensor derivative formula. This remains a bound on its image: compact unitarity makes each \(K_i\) normal, and its eigenvalues on a quotient are among those on the tensor representation. Equation (1.18) therefore bounds the norm of every real generator on such an image by \(C(r+1+M_E)\).

Iteration, followed by expansion of a fixed \(X\) in the chosen basis, yields

The constant \(A_X\) is independent of the initial compact packet: \(\prod_{j=1}^r(j+c)/r!\le e^c r^c\), so its size affects only \(C_w,b_w\). Taylor's integral remainder for the actual unitary one-parameter group is bounded by \(|t|^r\|R(X)^rw\|/r!\). For \(|t|A_X<1\), its Lie-polynomial partial sums therefore converge to \(R(\exp(tX))w\). All those sums lie in \(\pi\). The common radius and density prove that the Hilbert closure is invariant under these near-identity exponentials, hence all exponentials. Compact components and finite adelic translations already preserve \(\pi\), so its closure is fully invariant.

Each joint compact corner of this closure is the closure of the corresponding corner of \(\pi\). The latter is finite dimensional by admissibility, so it is already closed. Thus the closure is admissible and its finite core equals \(\pi\). Any nonzero closed invariant subspace contains a nonzero finite vector by compact averaging; irreducibility of \(\pi\) and density make that subspace the whole closure. The closure remains cuspidal by Lemma 1.8. This proves every assertion. \(\square\)

Lemma 1.15 (smoothing before unitarity). Let an irreducible admissible cuspidal mixed module \(\pi\) with unitary central character be embedded in a full \(G(\mathbb A)\)-invariant space \(V\) of smooth, finite-level, moderate-growth cuspidal functions. Assume that each fixed-finite-level space of \(V\) is complete Fréchet, that the right action is continuous and smooth at infinity, that its inclusion in \(C^\infty_{\mathrm{loc}}\) is continuous, and that its mixed finite core is exactly \(\pi\). No Hilbert norm is assumed. Every \(\phi\in\pi\) is reproduced by a smooth compactly supported convolution kernel. All its right derivatives have a common polynomial growth exponent. Consequently the rapid-decay argument in Lemma 1.1b applies with its Hilbert assumption replaced by these hypotheses. In conjunction with the adelic reduction Proposition 1.1a, it gives the square-integrability hypothesis of Proposition 1.14, and hence the compatible unitary realization and product pairings of Theorem 1.13.

Proof. Choose a finite compact-type packet and finite-place level containing \(\phi\), and let \(P\) be their averaging projector. Then \(E=PV\) is precisely the corresponding joint corner of \(\pi\), so it is finite dimensional. Take a smooth real approximate identity, with a fixed finite-place compact-open average small enough to fix \(E\). Convolution tends to the identity on each vector of \(E\) by continuity of the action. Hence

in any finite-dimensional operator norm. It is eventually invertible. If \(p(z)\) is its characteristic polynomial, then \(p(0)\ne0\) and \(p(T_\epsilon)=0\) expresses \(I_E\) as a polynomial in \(T_\epsilon\) having zero constant term. The compact projectors are compactly supported distributions, real convolution with \(h_\epsilon\) makes their product kernel smooth, and its positive convolution powers are again smooth and compactly supported. Their indicated linear combination supplies a kernel \(f\) with

For derivatives in the positive central direction the asserted unitary character acts by a scalar, so it suffices to work on \(G^1\). Differentiation in (1.19) transfers every right derivative to a derivative of \(f\); all these kernels have a common compact support. If \(|\phi(g)|\le C\mathcal H(g)^q\), submultiplicativity of the matrix height on that compact support gives

for every \(D\), with the same exponent \(q\). All differentiations and integrations are legitimate on compact sets by the stated smooth inclusion. Right differentiation preserves cuspidality because the unipotent averaging domains are compact.

The remaining proof of Lemma 1.1b uses only (1.19), this common exponent, vanishing constant terms, compact unipotent quotients, additive Poisson summation, and the lattice bounds; it uses no positive Hermitian form. The general reduction and Iwasawa density therefore make \(R(D)\phi\) square integrable for every \(D\). Apply Proposition 1.14. \(\square\)

Lemma 1.15 is a comparison route once \(V\) is given. The general construction in Theorem 1.19 below supplies an actual admissible smooth realization for every abstract cuspidal subquotient. Its proof treats the algebraic subquotient through the discrete Hilbert spectrum and a positive-central filtration; it does not assume that arbitrary real convolution preserves an algebraic mixed submodule.

The complete general comparison is stated in Getz–Hahn, freely accessible author draft of 22 April 2022, Theorem 6.5.1, printed page 150/PDF page 166, and §9.7, printed pages 233–236/PDF pages 249–252. The free account of unitary tensor products and cusp spaces is Goldfeld–Jacquet, Automorphic representations and L-functions for GL(n), §4, printed/PDF pages 19–21, and §8, printed/PDF pages 31–39. Theorem 1.19 now supplies the general comparison, with its analytic and spectral steps proved below.

The general cuspidal realization

Let \(F\) be a number field, \(d=[F:\mathbb Q]\), \(G=GL_n\), \(\nu=|\det|_{\mathbb A}\), \(G^1=\ker\nu\), and \(X=G(F)\backslash G^1\). We use the quotient measure, closed cuspidal Hilbert space, positive-central splitting and matrix height \(\mathcal H\) already constructed in L-functions for GL_n: Godement–Jacquet and Rankin–Selberg, Lemmas 1.8–1.10. The general covering in Proposition 1.1a is \[ S=\Omega_N A(c)\Omega_TK,\qquad G^1=G(F)S, \] where \(b=\operatorname{diag}(b_i)\in A(c)\) has the same positive entries at every infinite place, finite entries one, \(\prod b_i=1\), and \(\rho_i=b_i/b_{i+1}\ge c>0\). Decrease \(c\) if needed so \(c\le1\). Each diagonal entry of \(\Omega_T\) has idèle norm one. Write \[ B(b)=\max\left(1,\beta(b)\right),\qquad \beta(b)=\prod_{i=1}^{n-1}\rho_i^d . \] The already proved covering also gives \(\mathcal H(b)\le C B(b)^a\) for a fixed \(a\), and finite quotient volume. We treat \(n\ge2\) first.

On \(\mathfrak g_\infty^1\), restrict the invariant real trace form with weights one at real places and two at complex places. The omitted positive scalar direction is its orthogonal complement, so this restriction is nondegenerate. The Cartan involution is \(X\mapsto-X^*\). Choose orthonormal bases \(K_j\) of its negative compact part and \(P_j\) of its positive part. Put \[ \Omega=\sum_jP_j^2-\sum_jK_j^2,\qquad C_K=-\sum_jK_j^2,\qquad L=-\sum_jP_j^2-\sum_jK_j^2=-\Omega+2C_K . \tag{1.23a} \] Invariance of the trace form proves that \(\Omega\) is central, by commuting a generator with the dual-basis quadratic sum. The operator \(L\) is elliptic on the real group: its principal symbol is the sum of squares of a real tangent frame. The scalar \(C_K\) on an irreducible compact type \(\sigma\) will be denoted \(c_\sigma\ge0\). Scalarity follows from compact Schur; nonnegativity follows by integration of the squares.

Lemma 1.19a (the local estimates used below). On fixed relatively compact nested real coordinate boxes, a smooth solution of a scalar elliptic differential equation \(Pu=0\), of order \(2r\) with principal symbol whose modulus is bounded below by a positive multiple of \(|\xi|^{2r}\), satisfies \[ \|u\|_{C^j(B_0)}\le C_j\|u\|_{L^2(B_1)} \tag{1.23b} \] for every \(j\), where \(B_0\Subset B_1\). For a second-order smooth operator with positive real principal quadratic form, an \(L^2_{\rm loc}\) distributional eigenfunction is smooth. These assertions apply to \(L\) and its polynomials on a fixed group chart. Left-translating the chart leaves their constants unchanged.

Proof. Here are the Fourier estimates and the regularity argument. Define the integer Sobolev norms by the Fourier weight \((1+|\xi|^2)^{s/2}\), after inserting a compact coordinate cutoff. For a constant principal operator \(P_0\), ellipticity gives \[ \|v\|_{H^{s+2r}}\le C\bigl(\|P_0v\|_{H^s}+\|v\|_2\bigr). \] This is the pointwise inequality between its Fourier symbol and the displayed weights, integrated against \(|\widehat v|^2\). On a sufficiently small box freeze the highest coefficients at its centre. The difference of the highest coefficients contributes at most \(\epsilon C\|v\|_{H^{s+2r}}\); differentiating a coefficient, and every lower-order term, contributes at most \(C_s\|v\|_{H^{s+2r-1}}\). The elementary Fourier inequality \[ \|v\|_{H^{k-1}}\le\eta\|v\|_{H^k}+C_{k,\eta}\|v\|_2 \] follows by splitting \(|\xi|\) below and above a sufficiently large threshold. Absorb the first term with \(\epsilon,\eta\) small. For compactly supported \(v\) in that box we obtain \[ \|v\|_{H^{s+2r}}\le C_s\bigl(\|Pv\|_{H^s}+\|v\|_2\bigr). \tag{1.23c} \] For a solution, apply this to \(\chi u\). The commutator \([P,\chi]\) has order at most \(2r-1\). On boxes \(B_t\Subset B_T\), choose cutoffs whose \(l\)-th derivatives are bounded by constants times \((T-t)^{-l}\). Apply the preceding interpolation to a second cutoff equal to one on the first one's support. For any chosen \(0<\theta<1\), (1.23c) gives \[ \|u\|_{H^{s+2r}(B_t)} \le\theta\|u\|_{H^{s+2r}(B_T)} +C_{s,\theta}(T-t)^{-N_s}\|u\|_{L^2(B_T)} . \] There are only finitely many cutoff derivatives, so \(N_s\) is finite. Iterate on radii approaching a fixed larger box with gaps proportional to \(2^{-j}\), choosing \(\theta<2^{-N_s-1}\). The resulting geometric series converges; its last term tends to zero because the smooth function has a finite Sobolev norm on that fixed larger box. Thus all interior Sobolev norms are bounded by the larger-box \(L^2\) norm. Fourier Cauchy–Schwarz bounds \(C^j\) by \(H^s\) when \(s>j+\dim(B_1)/2\), proving (1.23b). A finite chart cover gives the same assertion without the small-box restriction.

For clarity, weak regularity does not require an assumed smooth representative. In divergence coordinates a second-order elliptic operator has leading term \(-\partial_i(a_{ij}\partial_j)\). Integration by parts and positivity give, for compactly supported smooth \(v\), \[ \|v\|_{H^1}\le C\bigl(\|Pv\|_{H^{-1}}+\|v\|_2\bigr). \] Mollify a localized weak solution with kernels \(J_\epsilon\). The commutator \([P,J_\epsilon]\) maps \(L^2\) to \(H^{-1}\) with a bound independent of \(\epsilon\). To check this, the leading commutator is \(-\partial_i([a_{ij},J_\epsilon]\partial_j u)\). Transfer the derivative of \(u\) to its integral kernel. The difference \(a(x)-a(y)\) is \(O(\epsilon)\) on that kernel's support, while a kernel derivative has \(L^1\) norm \(O(\epsilon^{-1})\); the remaining coefficient derivatives have bounded kernels. Thus the expression in parentheses is bounded in \(L^2\). Lower terms and cutoff commutators obey the same, or an easier, estimate. Young's inequality gives the asserted bounds directly. The displayed estimate makes the mollifications bounded in local \(H^1\); weak compactness and their \(L^2\) convergence give \(u\in H^1_{\rm loc}\).

Now \([P,J_\epsilon]\) maps \(H^1\) to \(L^2\) uniformly. In nondivergence form, transfer one of the two derivatives to the kernel: the same coefficient difference times a kernel derivative is bounded, and the remaining derivative of \(u\) is in \(L^2\). Estimate (1.23c) with \(s=0,r=1\), including a nested cutoff, gives local \(H^2\) bounds and hence \(H^2_{\rm loc}\) regularity. If \(u\in H^k_{\rm loc}\) satisfies \(Pu=\lambda u\), commute a derivative of order \(k-1\) with \(P\). Its commutator uses derivatives of \(u\) of order at most \(k\), already in \(L^2\). Applying the \(H^2\) result to that derivative gives \(H^{k+1}_{\rm loc}\). Induction and Fourier Cauchy–Schwarz prove smoothness. All coefficient bounds are fixed in a left-translated group chart because the differential fields in (1.23a) are left invariant. \(\square\)

Lemma 1.19b (rapid decay without a realization assumption). Every moderate-growth, finite-level, compact-finite, enveloping-center-finite cusp function with unitary central character, restricted to \(G^1\), and every one of its right derivatives, satisfies \[ |R(D)\phi(b\omega)|\le C_{D,M}B(b)^{-M} \tag{1.23d} \] for every \(M\), with \(\omega\) in a fixed compact set and \(b\in A(c)\). Consequently every such derivative belongs to \(L^2(X)\).

Proof. Compact projection decomposes \(\phi\) into finitely many compact types. Enveloping-center finiteness supplies a nonzero polynomial \(p\) with \(p(\Omega)\phi=0\); a nonzero constant polynomial forces \(\phi=0\), so suppose its degree is positive. On type \(\sigma\) this becomes \[ p(-L+2c_\sigma)\phi_\sigma=0 . \] This is an elliptic scalar equation with constant nonzero leading coefficient. Apply Lemma 1.19a on \(gB_1\). Moderate growth and \(\mathcal H(gh)\le C_{B_1}\mathcal H(g)\) give \[ |R(D)\phi(g)|\le C_D\mathcal H(g)^q \tag{1.23e} \] for every \(D\), with one exponent \(q\). The compact projections have preserved that exponent. This step takes place on the group and needs no injectivity radius in the quotient.

For a cut \(i\), the radical \(V_i\) of the maximal upper block parabolic is the abelian group of matrices with entries in rows \(r\le i\), columns \(s>i\). Put \(f_i(v)=R(D)\phi(vb\omega)\). Its average over \(V_i(F)\backslash V_i(\mathbb A)\) is zero. The finite components of \(b\) are one and \(\omega_f\) ranges in a compact set. Finite level therefore gives one compact open additive subgroup \(W_i\subset V_i(\mathbb A_f)\) that fixes every such \(f_i\): choose it so that \(\omega_f^{-1}W_i\omega_f\) lies in the fixed level throughout that compact family. Existence follows by a finite subcover of the continuous conjugation map near the identity.

Strong approximation and the lattice construction in The adèle ring, Theorems 2.2–2.3 identify the resulting quotient with \(V_i(F_\infty)/\Lambda_i\), where \(\Lambda_i=V_i(F)\cap W_i\) is a full real lattice. Indeed projection of strong approximation to the finite coordinates makes \(F\) dense in \(\mathbb A_f\), so every class has an infinite representative modulo \(F+W_i\). Its kernel is \(\Lambda_i\). A compact open \(W_i\) contains a product \(M\widehat{\mathcal O}_F\) and is contained in a product \(M^{-1}\widehat{\mathcal O}_F\) for some positive rational integer \(M\). Its rational intersection is therefore a full finite-index lattice between the corresponding Minkowski integer lattices. Compactness makes the resulting continuous torus bijection a homeomorphism. Its Fourier frequencies form a dual lattice, with a positive nonzero minimum and a summable \(\sum_{\lambda\ne0}|\lambda|^{-2r}\) for \(2r>\dim_\mathbb R V_i(F_\infty)\). These facts follow by an invertible lattice-basis change from the corresponding integer-lattice facts. Character completeness is Matrix coefficients and the Peter–Weyl theorem, Theorem 4.1; on a torus the characters are precisely the dual-lattice exponentials.

For an upper root \(E_{rs}\) in \(V_i\), \(b^{-1}E_{rs}b=(b_s/b_r)E_{rs}\), and \[ b_r/b_s=\prod_{\ell=r}^{s-1}\rho_\ell\ge c^{n}\rho_i . \] Abelianity makes differentiation of \(f_i\) transfer to the constant right field \(\operatorname{Ad}(\omega^{-1}b^{-1})E_{rs}\). Its coefficients are \(O(\rho_i^{-1})\). Compact representatives of the torus and (1.23e) thus bound every \(2r\)-th derivative by \(C_{D,r}\rho_i^{-2r}B(b)^{aq}\). Integrating by parts in its Fourier coefficient extracts \((4\pi^2|\lambda|^2)^{-r}\). The zero coefficient is zero. Absolute summation gives \[ |R(D)\phi(b\omega)|\le C_{D,r}\rho_i^{-2r}B(b)^{aq}. \] Choose a largest \(\rho_i\). It is at least \(\beta(b)^{1/(d(n-1))}\), and is bounded below independently of \(b\) even when \(\beta<1\). Increasing \(r\) proves (1.23d). All differentiations preserve cusp averages, whose domains are compact by Automorphic representations and automorphic L-functions, Lemma 1.0. The finite-volume Iwasawa density and covering then give square integrability, also after arbitrary polynomial height weights. No compact convolution was assumed to preserve an algebraic mixed submodule. \(\square\)

Lemma 1.19c (finite intersection for the proved covering). The set \[ \{\gamma\in G(F):\gamma S\cap S\ne\varnothing\} \] is finite.

Proof. We give the flag and matrix argument; this assertion is not inferred just from a covering. For decomposable \(0\ne w\in\bigwedge^rF^n\), define \(h_g(w)\) as the product of the exterior Euclidean norms at infinity, squared at complex places, and the exterior coordinate maximum norms at finite places. It is invariant under rational rescaling. Let \(m_r(g)\) be its infimum over such \(w\), with \(m_0=1\). Rational left multiplication permutes this set, so \(m_r(\gamma g)=m_r(g)\).

For \(g=ubhk\in S\), \(b^{-1}ub\), \(h\), \(k\) and their inverses range in fixed compact sets. Indeed upper entries of the first factor are multiplied by \(b_j/b_i\le c^{-(j-i)}\). Thus \(g=b\omega\) with \(\omega\) in a fixed compact set. For any rational exterior vector, a nonzero rational coordinate and the product formula proved in Places of number fields in extensions and the product formula, Theorem 5.2 give \(h_1(w)\ge1\). The least exterior weight of \(b\) is at least a fixed multiple of \(\prod_{j=n-r+1}^n b_j\), because every \(\rho_i\ge c\). Compact exterior operator bounds therefore give \[ C^{-1}\left(\prod_{j=n-r+1}^n b_j\right)^d \le m_r(g)\le \left(\prod_{j=n-r+1}^n b_j\right)^d . \tag{1.23f} \] For the upper bound take the last \(r\) standard rows: upper unipotence fixes their exterior vector, \(k\) preserves the chosen local norms, and the product of the corresponding \(h_j\)-norms is one.

Suppose \(g'=\gamma g\), with \(g=ubhk\) and \(g'=u'b'h'k'\) in \(S\). Applying (1.23f) to every \(r\), and dividing adjacent last-row products, bounds all \(b'_j/b_j\) above and below by one fixed constant. The finite components of \(\gamma=g'g^{-1}\) lie in a fixed compact set, so its entries belong to one fixed fractional ideal of \(F\). The full Minkowski embedding of this ideal is a lattice, by Lattices, Minkowski's theorem and the Minkowski embedding, Proposition 7.2. A nonzero entry therefore has full infinite Euclidean norm at least \(\eta>0\).

At infinity write \[ \gamma=u'b'Mb^{-1}u^{-1},\qquad M=h'k'k^{-1}h^{-1}. \] All entries of \(u,u^{-1},u',u'^{-1},M\) are uniformly bounded. If \(a>b\), the \((a,b)\)-entry of \(\gamma\) is a sum of terms indexed by \(r\ge a\), \(s\le b\), each bounded by a constant times \(b'_r/b_s\). For a cut \(\ell\) with \(b\le\ell<a\), this is at most \(C/\rho_\ell\). Take a fixed threshold \(T\) so large that the full infinite bound is less than \(\eta\) whenever \(\rho_\ell>T\). Rationality forces every entry below that cut to be zero.

Partition the indices into blocks at those large cuts. Then \(\gamma\), and hence \(M\), is block upper triangular. Diagonal \(h,h'\) preserve that condition, so \(k'k^{-1}\) is block upper triangular at every infinite place. A real orthogonal or complex unitary block upper triangular matrix is block diagonal: its last block row is orthogonal to all preceding rows, forcing the corresponding last block column to vanish above its diagonal; repeat on the preceding block. Thus \(M\) is block diagonal at infinity.

Inside each block the adjacent ratios are at most \(T\) and at least \(c\). Together with \(b'_j/b_j\) bounded, this bounds every entry of the block diagonal \(b'Mb^{-1}\). Consequently \(\gamma=u'(b'Mb^{-1})u^{-1}\) has uniformly bounded infinite entries. Its finite entries were already uniformly bounded. A compact adelic matrix set contains only finitely many rational matrices, by discreteness of \(F\) in the adèles proved in The adèle ring, Theorem 2.2. This proves finiteness.

In particular, Haar integration over \(S\) has bounded multiplicity over \(X\). If two lifts of a coset lie in \(S\), their ratio is one of this finite list; the number of lifts is therefore at most its cardinality. The Iwasawa parameterization only introduces its usual compact fibres of fixed Haar volume. Compact sets of torus representatives may be thickened within the compact norm-one torus coordinates when necessary. Hence, for nonnegative quotient functions \(f\), \[ \int_{S}f(g)\,dg\le C_S\int_X f(x)\,dx . \tag{1.23g} \] The same bound holds for the standard Iwasawa parameter integrals restricted to the compact factors of the covering.

We may also arrange that these parameter integrals dominate integration on the quotient when its representative is chosen in the original covering. Here is the compact-fibre detail. At infinity an upper triangular unitary matrix is diagonal, so the Iwasawa fibre changes only compact diagonal phases. At a finite place it changes the diagonal by units and the unipotent coordinate by a member of the integral unipotent group conjugated by the finite diagonal. The finite diagonal lies in the fixed compact \(\Omega_{T,f}\). Enlarge \(\Omega_T\) once by its compact diagonal phases and units, and enlarge \(\Omega_N\) once by the compact set of these conjugated integral unipotents. Then every full Iwasawa fibre over a point in the original covering is contained in the enlarged parameter set, with its probability Haar measure. These fibre changes preserve every \(b_i\) and \(\beta\). The enlarged compact factors still satisfy the finite-intersection proof. This gives both the domination needed for quotient tails and the upper bound (1.23g) for their parameter integrals. \(\square\)

Lemma 1.19d (compact cusp form embedding). Fix a finite level \(U\) and a unitary norm-one central character. On that entire closed cuspidal Hilbert space, the form \[ q(u)=\sum_j\|R(P_j)u\|_2^2+\sum_j\|R(K_j)u\|_2^2 \tag{1.23h} \] is densely defined and closed. Its form-domain embedding into the Hilbert space is compact. The same holds after projection to a finite packet of compact types.

Proof. Derivatives mean domains of the actual closed right-translation generators. Their graph intersections make the form closed. Smooth real convolution approximates every vector and gives a dense domain; the kernels preserve finite level and the closed cusp space. For a form-domain vector it converges also in the form norm: differentiation under translation uses the continuously varying finite matrix \(\operatorname{Ad}(g^{-1})\), so its orbit is continuous in all first-derivative norms. The Cartan inner product is invariant under \(K_\infty\), so compact-type projection reduces this form; finite-level and central projections also do. Such projections can therefore be retained in this approximation whenever a compact packet is prescribed.

We prove the uniform tail estimate. For the maximal radical \(V_i\), write the full upper unipotent group \(N=V_i\rtimes N_i\), where \(N_i\) is block diagonal upper unipotent. Quotient integration is integration first over \(V_i(F)\backslash V_i(\mathbb A)\), then over \(N_i(F)\backslash N_i(\mathbb A)\). Choose compact representatives of the second quotient. Conjugation by \(N_i\) preserves additive Haar measure on \(V_i\), since it is triangular with diagonal entries one, so these fibre identifications preserve the probability measures.

At \(n_i b\omega\), the function \(v\mapsto u(vn_i b\omega)\) has zero average. The same uniform finite-level argument as in Lemma 1.19b gives a fixed real torus on which to apply the zero-mean Fourier Poincaré inequality \[ \int |f|^2\le C\sum_\alpha\int|\partial_\alpha f|^2 . \] It follows by dividing nonzero Fourier coefficients by the positive minimum frequency norm; it holds for weak \(H^1\) functions by Fourier truncation. Conjugating a \(V_i\)-direction successively by \(n_i^{-1}\), \(b^{-1}\) and \(\omega^{-1}\) gives a right-field linear combination with coefficients \(O(\rho_i^{-1})\): the first conjugation stays in that radical and is bounded on the chosen compact representatives; all its roots cross cut \(i\). Hence \[ \int_{N(F)\backslash N(\mathbb A)}|u(nb\omega)|^2\,dn \le C\rho_i^{-2} \int_{N(F)\backslash N(\mathbb A)} \sum_X|R(X)u(nb\omega)|^2\,dn , \tag{1.23i} \] where \(X\) ranges over the full fixed real frame in (1.23h). This holds first for smooth vectors and then on the form domain by approximation. Distributional cuspidality gives the zero mean almost everywhere in these fibre integrals.

Partition the diagonal region according to a largest \(\rho_i\). Integrate (1.23i) against its Iwasawa density and the compact torus and compact-group factors. On \(\beta>R\), \(\rho_i^{-2}\le R^{-2/(d(n-1))}\). Let \(C_R\subset X\) be the image of the covering parameters with \(\beta\le R\); it is compact since all diagonal ratios and entries are then bounded. Every point outside \(C_R\) has a covering representative with \(\beta>R\). Bounded multiplicity (1.23g) gives \[ \|u\|_{L^2(X\setminus C_R)}^2 \le C R^{-2/(d(n-1))}q(u). \tag{1.23j} \]

On a compact part of \(X\), a fixed finite level gives finitely many real coordinate charts with positive finite-fibre volume. Injective charts follow from the arithmetic chart construction in Lemma 1.10 of this lesson, or directly from discreteness on a compact group set. Their local \(H^1\) norms are bounded by \(\|u\|_2^2+q(u)\). Insert chart cutoffs and expand on fixed boxes in Fourier series. The bound controls the sum of squared coefficients weighted by \(1+|k|^2\); the \(L^2\) tail beyond \(|k|>A\) is \(O(A^{-2})\). The finitely many remaining coefficients have convergent subsequences. A finite chart cover and then (1.23j) prove precompactness globally.

For completeness, only finitely many real quotient components occur at fixed \(U\): the finite part of the proved covering is compact and has finitely many cosets modulo \(U\). Each component is a quotient of \(G_\infty^1\) by a discrete group. Its discreteness follows because a compact real set times its fixed compact finite stabilizer meets \(G(F)\) only finitely. Thus the real charts just used are genuine group-quotient charts. No cusp compactness or finite-intersection statement was assumed. \(\square\)

Lemma 1.19e (finite-dimensional cusp spaces). Fix \(U\), finitely many compact types, a unitary norm-one central character, and a nonzero polynomial \(p\). The space of smooth moderate-growth cusp functions on \(X\) satisfying \[ p(\Omega)\phi=0 \tag{1.23k} \] with those level and compact-type constraints is finite dimensional. In particular the fixed-level, fixed-compact-type, fixed-cofinite-infinitesimal-character space is finite dimensional.

Proof. Lemma 1.19b applies using just (1.23k), so all these functions and their derivatives lie in \(L^2\). On a single compact type, use the closed form in Lemma 1.19d. For \(f\) in its Hilbert space, the complete form norm gives a unique \(Tf\) with \[ q(Tf,v)+\langle Tf,v\rangle=\langle f,v\rangle . \] The representing-vector assertion follows directly from the Hilbert projection argument: minimize the distance from a vector outside a bounded functional's closed kernel; the parallelogram identity makes a minimizing sequence Cauchy, and the closest-point difference is orthogonal to the kernel and represents the functional after scaling.

Thus \(T\) maps a Hilbert unit ball into a bounded form ball. It is compact by Lemma 1.19d, self-adjoint by the displayed identity, positive, injective and has dense range; injectivity follows from density of the form domain. The compact self-adjoint argument gives an orthonormal eigenbasis with positive eigenvalues tending to zero and finite multiplicities. To recall its mechanism, a bounded maximizing sequence has a weakly convergent subsequence: diagonalize its coefficients in a countable orthonormal basis, use Bessel's inequality for the limit, and approximate each test vector by its finite coefficient sums. Compactness then makes its images converge in norm, so a maximal positive Rayleigh value is attained. Varying the vector proves its eigenvector equation. Restrict to its orthogonal complement and repeat. An infinite orthogonal family whose eigenvalues stay away from zero contradicts compactness; a remaining orthogonal complement has zero operator and is zero by injectivity. This also proves finite multiplicities.

Define \(L=T^{-1}-1\). Its eigenvalues \(\kappa\ge0\) tend to infinity. It is the weak differential operator in (1.23a). Indeed start with the same derivative form on the whole ambient \(L^2(X)\). The closed cusp projection commutes with every right group action and hence with its generators. Finite-level projection has the same property for the real generators. Compact and central projections reduce the form because it is invariant under those compact groups. The Riesz resolvent therefore reduces to the just constructed restricted resolvent. Its equation against unrestricted compact coordinate tests is exactly the distributional differential equation for \(L\).

A function in (1.23k) has all Lie derivatives in \(L^2\). The pointwise fundamental theorem of calculus and strong right continuity identify them with its Hilbert derivatives, as in Proposition 1.14 of this lesson. Skew-adjoint generator identities then give \[ q(\phi,v)=\left\langle-\sum_XR(X)^2\phi,v\right\rangle \] for every form-domain \(v\); the identity extends from smooth vectors by form approximation. Hence \(\phi\) belongs to the operator domain, and repeating gives every power of \(L\). On type \(\sigma\), \(\Omega=-L+2c_\sigma\). Its expansion in the eigenbasis of \(L\) can therefore involve only eigenvalues solving \(p(-\kappa+2c_\sigma)=0\). There are finitely many roots and each eigenspace is finite dimensional. Sum over the finite compact packet.

Finally a cofinite ideal in the enveloping centre supplies a nonzero polynomial in \(\Omega\), because its powers are linearly dependent modulo that ideal. This proves the last assertion. Notice that the whole cusp space at a fixed compact type was not asserted to be finite dimensional. \(\square\)

Corollary 1.19f (reproducing kernels for algebraic cusp forms). Every automorphic cusp form is reproduced by a smooth compactly supported convolution kernel on the central-normalized group.

Proof. For a fixed positive-central character, its central-normalized restriction, annihilating polynomial \(p(\Omega)\), finite level, compact packet and norm-one central character put it in the finite-dimensional actual function space \(E\) of Lemma 1.19e. For a general central-finite form use all the finitely many coefficient functions in its positive-central exponential-polynomial expansion (1.23m), and all its finitely many norm-one characters; their construction is proved below and does not use this corollary. Choose the finite direct sum of constrained spaces containing those coefficients as \(E\). Smooth compact real convolution, and a fixed finite compact-open average, preserve \(p(\Omega)\) and each norm-one central character. The polynomial is preserved because the general linear adjoint action fixes the centre of the complexified enveloping algebra. Convolution preserves moderate growth and cuspidality. Applying the compact packet projector \(P\) makes \[ T_\epsilon=P R(h_\epsilon)P \] an operator on \(E\), converging to the identity there. These facts concern the entire proved finite-dimensional constrained function space, not an assumed real-group-invariant algebraic mixed submodule.

Invertibility for small \(\epsilon\), followed by the characteristic-polynomial identity, writes the identity on \(E\) as a polynomial in \(T_\epsilon\) with zero constant term. Compact projectors are compactly supported distributions; convolution with the smooth \(h_\epsilon\) and all its positive convolution powers are smooth and compactly supported. Their linear combination is the required reproducing kernel. Since its action on \(G^1\) fixes every coefficient in (1.23m), it fixes the original form on all positive-central translates as well. \(\square\)

Lemma 1.19g (compact cuspidal convolution and discrete spectrum). Every smooth compactly supported convolution on a fixed norm-one central-character cuspidal Hilbert space is compact. This space is a Hilbert direct sum of irreducible unitary representations, each with finite multiplicity.

Proof. A finite-place smooth compact kernel is bi-invariant under some compact open \(U\); this follows by covering its compact finite support by finitely many locally constant coordinate neighbourhoods. Its operator has image in that \(U\)-level. Derivatives transfer to its smooth compact kernel, so Young's inequality gives \[ \|R(X)R(f)u\|_2\le C_{f,X}\|u\|_2 . \] Its unit-ball image is bounded in the form domain of Lemma 1.19d, which was proved for the whole \(U\)-level, with no compact-type restriction. The operator is therefore compact. This does not assert finite dimensionality of that level.

Here is the representation decomposition argument; no type-I theorem is needed. Smooth compact approximate identities \(h_j\) tend strongly to the identity by strong continuity. The operators \(T_j=R(h_j)^*R(h_j)\) are positive compact and also tend strongly to the identity. On a nonzero closed invariant subspace \(W\), some \(T_j\) is nonzero and has a positive finite-dimensional eigenspace \(E\subset W\). Every invariant orthogonal projection commutes with \(T_j\). Among invariant subspaces of \(W\) meeting \(E\), choose one \(V\) minimizing the positive integer \(\dim(E\cap V)\), and replace it by the closed group span of that intersection; the dimension is unchanged.

If \(V\) had a proper nonzero invariant subspace, its orthogonal decomposition would split \(E\cap V\). Two nonzero parts contradict minimality. A zero part forces the corresponding subspace to be zero, because \(V\) is generated by \(E\cap V\). Thus \(V\) is irreducible. A maximal orthogonal family of these irreducible subspaces exhausts the Hilbert space: any nonzero orthogonal complement has the same construction. The Hilbert space is separable, from its countable quotient charts, so the family is countable.

For an irreducible representation that occurs, choose a \(T_j\) nonzero on it and a positive eigenvalue there. Every equivalent copy has that same eigenvalue. Infinitely many copies would give an infinite-dimensional eigenspace of the compact global \(T_j\), which is impossible. This proves finite multiplicity. \(\square\)

Lemma 1.19h (admissibility of the cuspidal Hilbert constituents). Every irreducible Hilbert constituent just obtained has finite joint level-and-compact-type corners. Its mixed finite core is an irreducible admissible module of actual cusp forms.

Proof. Let \(H_i\) be a constituent. Compact averaging and finite-place averaging give a nonzero vector in some \(U\)-level and compact type \(\sigma\). Projection onto \(H_i\) commutes with the closed real generators and reduces their form, and commutes with the compact and finite projections. Thus the compact resolvent constructed in Lemma 1.19e restricts to this nonzero corner. It has an eigenvector \(u\ne0\) there. The ambient weak equation is \(Lu=\kappa u\), as proved there. Lemma 1.19a supplies a smooth representative. On type \(\sigma\), this gives \(\Omega u=\lambda u\), where \(\lambda=2c_\sigma-\kappa\) is real.

We establish the global derivative estimates needed to integrate its algebraic span. A fixed finite level has finitely many real group-quotient components. Give them the complete metric whose orthonormal fields are the frame in (1.23a). Completeness and proper distance balls can be seen directly: a path of length \(R\) bounds a matrix and its inverse by \(e^{CR}\), by the differential equation \(g'=gX\). Bounded closed real matrix sets are compact. Paths lift to the group quotient, so quotient distance balls are compact images of such balls. Lipschitz cutoffs \(\eta_R\), equal to one on larger balls, can therefore be chosen with compact support and \(\sum_X|X\eta_R|^2\le C/R^2\). Lipschitz cutoffs are legitimate weak energy tests by local mollification.

For any \(L^2\) smooth compact-finite distributional \(\Omega\)-eigenvector \(v\), project to its finitely many compact types. On each type it satisfies \(Lv_\sigma=(2c_\sigma-\lambda)v_\sigma\). Test this equation against \(\eta_R^2v_\sigma\). Integration by parts and \(2ab\le a^2/2+2b^2\) bound \[ \int\eta_R^2\sum_X|R(X)v_\sigma|^2 \le 2|2c_\sigma-\lambda|\|v_\sigma\|_2^2+ \frac{C}{R^2}\|v_\sigma\|_2^2 . \] Letting \(R\) increase proves that every first derivative is in \(L^2\). Its right derivatives still satisfy the same \(\Omega\)-eigen-equation, and their compact packets are contained in the original packet tensored with the finite adjoint representation. Repeat the argument to prove all word derivatives belong to \(L^2\). The fundamental theorem of calculus identifies these with Hilbert derivatives. In particular the quotient Sobolev bound in Lemma 1.10 of this lesson gives polynomial growth of every derivative.

Let \(M\) be the mixed algebraic module generated by \(u\). It is made of smooth moderate-growth cusp functions with \(\Omega=\lambda\). Lemma 1.19e shows every joint corner of \(M\) is finite dimensional. The algebraic span is thus admissible, without assuming it is already simple.

Its Hilbert closure is invariant under the real group. We recall the uniform-radius argument to make this step explicit. For its finite vectors, \[ \sum_j\|P_jw\|^2+\sum_j\|K_jw\|^2 =-\lambda\|w\|^2+2\sum_j\|K_jw\|^2 . \] A Lie word of length at most \(r\) from a compact-orbit space \(E\) has compact representation contained in the image of \((\mathbb C\oplus\mathfrak g)^{\otimes r}\otimes E\). Norms of compact generators there are at most \(rA+A_E\). This follows from the tensor derivative formula; on a quotient their eigenvalues are a subset, and compact unitarity identifies operator norms with maximal eigenvalue moduli. The displayed energy identity bounds every real generator on that image by \(C(r+1+A_E)\). Iteration gives \[ \|R(Y)^rw\|\le C_w A_Y^r r!\,r^{b_w}, \] where \(A_Y\) is independent of the starting compact packet: use \(\prod_{j=1}^r(1+a/j)\le e^a r^a\). Taylor's integral remainder for the actual unitary one-parameter group is bounded by \(|t|^r\|R(Y)^rw\|/r!\). On a common neighbourhood it therefore converges to its Lie-polynomial series in \(M\). Density then makes \(\overline M\) invariant under these exponentials, hence the whole connected real group. Compact components meet every real component of \(GL_n\), and the finite adelic action already preserves \(M\). Thus \(\overline M\) is fully invariant. Irreducibility of \(H_i\) gives \(\overline M=H_i\).

For each joint compact projector \(e\), \(eM\) is finite dimensional by Lemma 1.19e, hence closed. Projecting the dense \(M\) gives \(eH_i=eM\). All joint corners of \(H_i\) are consequently finite dimensional and its mixed finite core is \(M\). The actual-Hilbert comparison in L-functions for GL_n: Godement–Jacquet and Rankin–Selberg, Theorem 1.11 now applies and makes this core simple and enveloping-center finite, as well as an actual cusp module. There is no reliance on an admissibility theorem for arbitrary local unitary representations. \(\square\)

Theorem 1.19 (every abstract cuspidal representation has an actual realization). Every irreducible admissible cuspidal automorphic subquotient for \(GL_n/F\) is isomorphic to an embedded smooth cusp module. After removal of the real positive-central norm exponent it is the mixed finite core of an actual irreducible admissible cuspidal Hilbert constituent. Its finite-level complete smooth realization and compatible unitary local tensor pairings are those constructed in Theorems 1.11 and 1.13 of this lesson. This holds for every number field and every \(n\ge1\).

Proof. First fix a unitary norm-one central character and trivial positive-central action. Let \(\mathcal C_\omega\) be the algebraic space of cusp forms with these data. Lemma 1.19b places every vector and all its derivatives in the closed Hilbert cusp space. Lemma 1.19g decomposes that Hilbert space, with finite multiplicities, into \(H_i\). Lemma 1.19h and the actual-Hilbert comparison put every \(H_{i,\mathrm{fin}}\) in \(\mathcal C_\omega\).

Conversely take \(\phi\in\mathcal C_\omega\), fixed by \(U\), in a finite compact packet, and killed by \(p(\Omega)\). Orthogonal constituent projections commute with real generators and their domains, since the constituents reduce the full unitary action. They therefore retain those three constraints. Every nonzero projected vector is a finite vector of its admissible constituent and hence an actual automorphic cusp form. All these vectors are pairwise orthogonal and lie in the finite-dimensional constrained space of Lemma 1.19e. Only finitely many can be nonzero. Consequently \[ \mathcal C_\omega=\bigoplus_i H_{i,\mathrm{fin}} \tag{1.23l} \] as an algebraic direct sum of simple admissible mixed modules.

We now justify the passage from general algebraic subquotients, including nonunitary positive-central exponents and possible central Jordan blocks. Let \(H_0\) denote differentiation of the positive splitting \(a_{e^t}\). It is a central Lie operator. Every automorphic form has a finite-dimensional orbit under the real central Lie algebra, because that algebra lies in the enveloping centre. Solving its constant-coefficient ordinary differential equations gives the actual matrix-exponential central translations on this finite-dimensional orbit. Thus every algebraic mixed submodule is preserved by those translations; they are not being assumed to extend a noncentral real action.

The norm-one scalar idèle-class group is compact, by Idèles and the idèle class group, Theorem 3.3. Its orbit on any given finite-level, central-finite automorphic form is finite dimensional. To see this explicitly, let \(W\) be its finite-dimensional real-central Lie orbit. The connected archimedean scalar group acts on \(W\) by those matrix exponentials. Scalar finite elements in the fixed finite level act trivially. The image of their product with the connected archimedean norm-one scalar group is an open subgroup of the norm-one idèle-class group: it is the image of the corresponding open subgroup of \(\mathbb A^1\). Compactness gives finite index. Finitely many coset representatives therefore put the entire orbit in a finite sum of translates of \(W\). A finite-dimensional continuous representation of a compact abelian group splits into characters: average an inner product, simultaneously diagonalize its commuting unitary matrices, and use their common eigenspaces. Those characters have modulus one. Their projections are finite linear combinations on the finite orbit and preserve every algebraic mixed submodule.

For a fixed norm-one character \(\omega\), the locally finite operator \(H_0\) gives a direct sum of generalized eigenvalue spaces. In its generalized \(\mu\)-space every function has a finite expansion \[ \phi(a_{e^t}x)=e^{\mu t}\sum_{j=0}^r t^j f_j(x), \qquad x\in G^1 . \tag{1.23m} \] The coefficients are smooth cusp functions, at finite level, compact finite and central finite on \(G_\infty^1\), with polynomial growth. For a verification, evaluations at finitely many real \(t\)'s recover these coefficients by an invertible evaluation matrix: point evaluations span the dual of the finite-dimensional exponential-polynomial solution space, since a function killed by every evaluation is zero. Evaluations at those fixed central translates commute with compact actions, finite levels and central differential operators. They preserve polynomial growth. Linear independence in \(t\) and compact-domain integration show that every cusp constant term of every coefficient is zero. Thus \(f_j\in\mathcal C_\omega\).

These observations also prove full cuspidal arithmetic finiteness when a central character has not been specified separately. At fixed \(U\), the open subgroup made from connected archimedean norm-one scalars and scalar finite level has a fixed finite index in the compact idèle-class group. A cofinite ideal in the enveloping centre supplies polynomial equations for a basis of the central Lie algebra, as well as for \(H_0\) and \(\Omega\). There are finitely many joint Lie-character choices. They determine the character on the connected archimedean scalar subgroup, because that subgroup is generated by its exponentials; the finite remaining quotient permits only finitely many extensions. The polynomial in \(H_0\) bounds both its possible \(\mu\)'s and every Jordan degree \(r\). The polynomial in \(\Omega\) annihilates all coefficient functions in (1.23m). Lemma 1.19e makes each of their constrained spaces finite dimensional. Recovering the original function from its finitely many coefficients therefore proves finite dimensionality at fixed finite level, compact packet and cofinite infinitesimal-character ideal without an extra central-character restriction.

Filter this generalized space by the degree \(r\) in (1.23m). The filtration is invariant under the full mixed action. The leading-coefficient map identifies its degree-\(r\) quotient with \[ \mathcal C_\omega\otimes\nu^\mu . \tag{1.23n} \] Indeed right translation by \(g\) replaces \(t\) by \(t+\log\nu(g)\), multiplies the leading coefficient by \(\nu(g)^\mu\), and translates \(x\) by \(g a_{\nu(g)}^{-1}\). This also verifies the assertion infinitesimally. Every \(f\in\mathcal C_\omega\) occurs as a leading coefficient: \(e^{\mu t}t^r f(x)\) is a legitimate automorphic cusp function. It has moderate growth because \(\nu^{\pm1}\), the normalized representative height and powers of \(\log\nu\) are bounded by fixed powers of the original matrix height. Central finiteness follows from the finite polynomial degree in \(t\) and central finiteness of \(f\); the Lie algebra is the direct sum of its norm-one algebra and the central \(H_0\), so its enveloping algebra is the corresponding tensor product.

Let now \(\pi=A_1/A_0\) be an abstract irreducible admissible cuspidal subquotient. Finite-corner Schur makes its real central Lie action scalar, say \(H_0=\mu\), and makes its norm-one central action a scalar character \(\omega\). The finite-orbit compact argument makes that character unitary. Choose a vector of \(A_1\) with nonzero image. Its finite central orbit can be projected to the matching \(\omega\) and generalized \(\mu\) pieces; the other pieces have zero image. For the latter assertion, \(H_0-\mu\) is invertible on a finite orbit of any other eigenvalue, whereas it is zero on \(\pi\). These projections preserve \(A_1,A_0\) and supply a nonzero preimage with the desired data.

Choose the least degree \(r\) for which \(A_1\) intersected with that filtered space has nonzero image in \(\pi\). Its image is all of \(\pi\) by irreducibility; the preceding degree has zero image. Consequently \(\pi\) is a simple subquotient of (1.23n). Equation (1.23l) makes (1.23n) semisimple. One can check the needed elementary assertion without a categorical theorem: a cyclic preimage has support in finitely many simple summands; a submodule of a finite direct sum of simples is a sum of simples, by induction, projecting to the first summand and splitting off its graph after choosing the inductive complement to the kernel. Its quotient is again a sum of simples. Thus every simple subquotient is isomorphic to one of the original simple summands. We obtain \[ \pi\simeq H_{i,\mathrm{fin}}\otimes\nu^\mu . \tag{1.23o} \]

Remove \(\operatorname{Re}\mu\). The remaining imaginary twist is an actual unitary central extension of \(H_i\); it is therefore an irreducible admissible cuspidal Hilbert realization of \(\pi\otimes\nu^{-\operatorname{Re}\mu}\). Its complete smooth-vector spaces and continuous automorphic inclusion are supplied by the already proved actual-Hilbert comparison. Restoring the real determinant twist gives the full invariant smooth moderate-growth embedded cusp realization of \(\pi\) itself. The local unitary tensor factors and product pairings apply to its unitary normalization, with the prescribed norm-twist bookkeeping.

For \(n=1\), \(X=F^\times\backslash\mathbb A^1\) is compact and is the whole norm-one centre. A fixed central character function is a scalar multiple of that character: \(\phi(xz)=\omega(z)\phi(x)\) determines it from its value at one. Its Hilbert space is one dimensional. The finite central-orbit decomposition, positive-central expansion and filtration argument just given therefore apply directly. There are no proper-parabolic conditions or cusp tails in this case.

This proves the theorem in the subquotient definition used in Automorphic representations and automorphic L-functions. In particular the realization hypothesis of its archimedean genericity Corollary 1.4 and of the global product pairing is now furnished by an actual construction. \(\square\)

Further readings are Getz–Hahn, freely accessible author draft of 22 April 2022, Theorem 6.5.1, printed page 150/PDF page 166, and §9.7, printed pages 233–236/PDF pages 249–252. The tensor and cusp-space discussion is Goldfeld–Jacquet, Automorphic representations and L-functions for GL(n), §§4 and 8, printed/PDF pages 19–21 and 31–39. The preceding proof establishes the realization and admissibility assertions. No standard or Rankin–Selberg analytic continuation theorem, local Whittaker uniqueness theorem, or general type-I classification is used.

Recovering the complete Euler product

The algebraic restricted tensor decomposition is proved in Automorphic representations and automorphic L-functions, Theorem 1.1. Theorem 1.13 now proves the compatible unitary restricted tensor realization and product pairings for the actual admissible Hilbert cusp constituents of Theorem 1.11. Proposition 1.14 and Lemma 1.15 extend this conclusion to the specified embedded smooth cuspidal realizations. We apply the integral argument to these realizations: their finite vectors are the smooth moderate-growth cusp forms used above. Theorem 1.19 places every abstract cuspidal subquotient in such a realization after its specified norm normalization, so this integral argument applies to that definition as well.

In this realization pure tensor cusp vectors and their duals, with spherical unit vectors at almost every place, have matrix coefficient \(c(g)=\prod_vc_v(g_v)\). Choose \(\Phi=\bigotimes_v\Phi_v\), with the good-place tests of (1.5n) outside a finite set \(S\) containing infinity, ramification and the different.

The absolute estimates (1.5o) and the exceptional-place estimate justify product integration in the initial half-plane. Thus

Only finitely many normalized factors occur on the right. At each \(v\in S\), local input (1.5j) says \(1=\sum_jp_{v,j}(s)H_{v,j}(s)\). Multiply these finite identities and use (1.5z) for each tensor choice. This writes

Every coefficient in (1.6) is entire, with at most polynomial growth on a closed strip: \(q_v^{\pm s}\) has bounded modulus there and the infinite-place coefficients are polynomials. The global integrals are entire and decrease faster than every power by (1.5x). Consequently (1.6) proves that the complete standard product is entire and bounded on every closed strip.

For one tensor choice whose normalized product is not identically zero, apply the local equation (1.5k) to (1.5z) and compare with (1.5y). This gives

Cancel the nonzero meromorphic product. Such a choice exists, since the finite identity (1.6) and the nonzero Euler product in its initial half-plane preclude every choice being zero. Equation (1.5l) now gives the first equation in (1.4), with root constant \(w(\pi_0)\).

Finally, unitarity gives \(\widetilde\pi_0\simeq\overline{\pi_0}\), by the invariant Hermitian pairing as in Proposition 2.3 below. The conjugation normalization in local input 1.1c therefore gives \(\mathcal L(s,\widetilde\pi_0)=\overline{\mathcal L(\overline s,\pi_0)}\). The entire product is not identically zero, so some point \(s=1/2+iT\) has nonzero value. At that point the functional equation has the form \(\mathcal L(s,\pi_0)=w(\pi_0)A(\pi_0)^{-iT}\overline{\mathcal L(s,\pi_0)}\). Taking absolute values proves \(|w(\pi_0)|=1\). Multiplying by \(A^{s/2}\) yields the conductor-normalized equation.

To restore the removed central character, the local twist identity gives \(\mathcal L(s,\pi)=\mathcal L(s+i\tau,\pi_0)\) and \(\mathcal L(1-s,\widetilde\pi)=\mathcal L(1-s-i\tau,\widetilde\pi_0)\). The conductor is unchanged and \(\varepsilon(\pi)=w(\pi_0)A(\pi)^{-i\tau}\), still of absolute value one. Entireness and strip bounds are unchanged by this imaginary shift. This completes the global deduction for all \(n\ge2\) and all number fields, conditional on the remaining general conductor/factor identification and general archimedean local inputs. Theorem 1.19 supplies the realization comparison for every abstract cuspidal subquotient, and Theorem 1.13 supplies the compatible product pairings.

For \(\pi=|\cdot|^{it}\) in degree one, \(\mathcal L(s,\pi)=\mathcal L(s+it,1)\). The complete zeta function has poles at zero and one, so the shift gives the two poles in Theorem 1.1. The finite Dedekind zeta function has no pole at zero; this is why the finite product cannot replace the completion in that assertion.

2. Rankin–Selberg factors and their reciprocals

Let \(\pi\) and \(\pi'\) be unitary cuspidal automorphic representations of \(GL_n(\mathbb A)\) and \(GL_m(\mathbb A)\). The nonzero global Whittaker transform for every nonzero smooth cuspidal function, over every number field and in every rank, is proved in Automorphic representations and automorphic L-functions, Theorem 1.3. Corollary 1.4 there proves local Whittaker existence for an irreducible admissible cusp module with its specified smooth cuspidal realization: the full adelic right action, a finite-level Fréchet smooth topology of moderate growth, continuous inclusion for all derivatives on compact sets, the given \(K_\infty\)-finite core, and continuous differentiation and compact averaging. At infinity the resulting functional is continuous. Theorem 1.19 supplies this smooth realization for every abstract cuspidal subquotient. Thus Corollary 1.4 of that earlier lesson gives a continuous archimedean Whittaker functional for every such local factor. Theorem 2.0 below proves local uniqueness at every finite place in every rank; Theorem 2.0n proves the archimedean distributional-principal-series bound; Theorems 2.0x–2.0z below construct the full continuous-dual embedding and prove uniqueness in every supplied actual complete smooth moderate dual-pair model. Theorem 1.34 supplies a compatible pair for every abstract irreducible admissible core.

For a fixed nontrivial additive character \(\psi:F\backslash\mathbb A\to\mathbb C^\times\), local Rankin–Selberg integrals pair a \(\psi_v\)-Whittaker function with a \(\psi_v^{-1}\)-Whittaker function. For equal ranks a Schwartz function also enters. The opposite characters ensure that the integrand descends to the unipotent quotient. The defining integrals and their analytic factor construction are [Getz–Hahn, 22 April 2022 draft, §11.5]. Local Whittaker uniqueness, local factor identification and the general integral construction remain additional assertions, rather than consequences of the existence proofs.

The resulting factor \(L_v(s,\pi_v\times\pi'_v)\) has degree at most \(nm\) at a finite place and degree \(nm\) when both representations are unramified. Define

where \(S\) is finite and contains infinity and the ramified places. The products initially mean products in a sufficiently far right half-plane.

Local Whittaker uniqueness over every nonarchimedean local field

This proof concerns complex smooth admissible representations. The field \(F\) is an arbitrary nondiscrete nonarchimedean local field, of either characteristic, and \(n\geq1\). Set \(G=GL_n(F)\), let \(U\) be its upper unitriangular subgroup, and let \[ \Psi(u)=\psi\!\left(\sum_{i=1}^{n-1}u_{i,i+1}\right), \qquad \tau(g)=w_0\,{}^tg\,w_0, \qquad \theta(g)=\tau(g)^{-1}, \tag{2.6a} \] where \(\psi\) is a nontrivial continuous additive character and \(w_0\) has ones on the antidiagonal. Thus \(\tau\) is an involutive antiautomorphism, \(\theta\) is an involutive automorphism, \(\tau(U)=U\), \(\Psi\circ\tau=\Psi\), and \(\Psi\circ\theta=\Psi^{-1}\).

We prove the theorem for every nondegenerate continuous character of \(U\), not only the normalization in (2.6a). All the distribution, duality and multiplicity arguments needed for the theorem are included below. The classical comparison is Bernstein–Zelevinsky's freely accessible author-hosted 1976 paper, §§6–7, especially Theorems 6.9–6.13 and 7.3 and the proof of 5.16 on printed/PDF pages 58–60. The localization, duality and multiplicity arguments are proved below.

The local topology and Haar measure used here are the actual earlier proofs in Local fields: classification and Haar measure, Proposition 3.1 and Proposition 6.1. The compact averaging and smooth-dual proofs in Smooth local representations and the Hecke-module dictionary, §§1, 3–5 apply to any totally disconnected group; their necessary arbitrary-rank arguments are also written below. No admissibility theorem for all irreducible representations is assumed: admissibility is a hypothesis of the local theorem.

Test functions, closed support and localization

An \(l\)-space is a locally compact Hausdorff space with a basis of compact open sets. Write \(\mathcal S(X)=C_c^\infty(X)\) for its locally constant compactly supported complex functions. A distribution is a linear functional on \(\mathcal S(X)\). In the spaces below the usual test-function topology is the inductive limit of the finite-dimensional spaces obtained by fixing a compact support and a finite compact-open partition. Thus these algebraic functionals are exactly the usual distributions; no finite-order differential distributions occur in a nonarchimedean direction.

Lemma 2.0a (closed support). If \(Y\subset X\) is closed, restriction gives an exact sequence \[ 0\longrightarrow\mathcal S(X\setminus Y) \longrightarrow\mathcal S(X) \longrightarrow\mathcal S(Y)\longrightarrow0. \tag{2.6b} \] Consequently a distribution supported on \(Y\) is exactly a distribution on \(Y\), composed with restriction. The same statement holds for a locally closed subset after restricting to an open neighbourhood in which it is closed.

Proof. A test function on \(Y\) has a finite partition of its compact support into subsets open and closed in \(Y\), on each of which it is constant. Choose compact open neighbourhoods in \(X\) whose intersections with \(Y\) give these subsets near their compact supports; finitely many suffice. Taking differences and intersections refines them to a disjoint compact-open partition in \(X\), and extending the constants on that partition gives an extension with compact support. A function restricting to zero on \(Y\) has compact support disjoint from \(Y\), so is the extension by zero of a function in \(\mathcal S(X\setminus Y)\). Dualizing the sequence proves the assertion. In particular there are no additional transverse derivatives attached to a closed subset. \(\square\)

Suppose an \(l\)-group \(H\) acts continuously on \(X\), and \(\chi:H\to\mathbb C^\times\) is a continuous character. On test functions use \(L_h f(x)=f(h^{-1}x)\). A \(\chi\)-equivariant distribution satisfies \(D(L_hf)=\chi(h)D(f)\). It is the dual of \[ Q_H^\chi(X)=\mathcal S(X)\big/\operatorname{span} \{L_hf-\chi(h)f:h\in H,\ f\in\mathcal S(X)\}. \tag{2.6c} \]

Lemma 2.0b (localization over an \(l\)-space). Let \(p:X\to B\) be continuous, where \(B\) is an \(l\)-space and \(p(hx)=p(x)\). If every fibre \(X_b\) has no nonzero \(\chi\)-equivariant distribution, neither does \(X\).

Proof. Multiplication by functions pulled back from \(\mathcal S(B)\) commutes with \(H\) and acts on (2.6c). It has local units: for a test function \(f\), choose a compact-open function on \(B\) equal to one on the compact set \(p(\operatorname{supp}f)\). At \(b\in B\), quotient this module by the ideal of functions vanishing at \(b\). The resulting fibre of \(\mathcal S(X)\) is precisely \(\mathcal S(X_b)\). Restriction is onto by Lemma 2.0a. If \(f\) vanishes on \(X_b\), the compact set \(p(\operatorname{supp}f)\) avoids \(b\), and a compact-open function equal to one on that set and zero near \(b\) expresses \(f\) as an element of the stated ideal. Taking the further quotient by the \(H\)-relations commutes with this quotient. Therefore the fibre of \(Q_H^\chi(X)\) is \(Q_H^\chi(X_b)=0\).

For completeness, zero fibres imply a zero module here. Given a class \(q\), its zero class at \(b\) expresses it as a finite sum \(\sum a_jq_j\) with every \(a_j(b)=0\). All \(a_j\) vanish on a common compact-open neighbourhood \(V_b\), so \(1_{V_b}q=0\). A representative of \(q\) has a compact image of its support in \(B\). Cover that image by finitely many \(V_b\), refine to disjoint compact-open pieces, and use a local unit for \(q\). Summing the corresponding zero products gives \(q=0\). A nonzero vector-space quotient has a nonzero linear functional, so the asserted vanishing of distributions on a fibre is equivalent to its quotient being zero. \(\square\)

We will also need localization by orbits without importing an algebraic-group distribution theorem. An action is called constructive here if its orbit relation \[ R_H=\{(x,hx):x\in X,h\in H\}\subset X\times X \tag{2.6d} \] is a finite union of locally closed subsets. This is a condition we verify explicitly for both matrix actions used below.

Lemma 2.0c (localization by constructive orbits). Suppose \(H\) is second countable, \(X\) is an \(l\)-space, and the action is constructive. If every orbit has no nonzero \(\chi\)-equivariant distribution, \(X\) has none. In this assertion each orbit has its subspace topology, which agrees with \(H/H_x\).

Proof. Here are the topological details. For a subset \(M\) which is a union of \(r\) locally closed sets, let \(U(M)\) be the points of \(M\) at which \(M\) is closed in some ambient open neighbourhood, and put \(M^0=M\), \(M^{j+1}=M^j\setminus U(M^j)\). The set \(U(M)\) is locally closed and open in \(M\), and \(M^j\) is closed in \(M\). These constructions commute with restriction to an ambient open set. They terminate with \(M^r=\varnothing\). Indeed, write \(M=\bigcup_{i=1}^r S_i\), with \(S_i\) locally closed. On the complement of \(\overline{S_i}\), induction on \(r\) gives \(M^{r-1}=\varnothing\). Hence \(M^{r-1}\subset\bigcap_i\overline{S_i}\). The subset \(S_i\cap M^{r-1}\) is open in \(M^{r-1}\), because \(S_i\) is open in its closure, and closed in \(S_i\), because \(M^{r-1}\) is closed in \(M\). It is therefore a locally closed open part of \(M^{r-1}\). Its points belong to \(U(M^{r-1})\); these parts cover \(M^{r-1}\), proving the induction. In particular \(U(M)\) is nonempty when \(M\) is nonempty, and is dense in \(M\), by applying the same conclusion on every nonempty ambient open set meeting \(M\).

Let \(Y=X/H\), initially just a quotient topological space, and \(p:X\to Y\) the orbit map. It is open, since \(p^{-1}p(V)=\bigcup_h hV\). The image of an invariant closed subset is closed, by the definition of the quotient topology. An invariant locally closed subset has a locally closed image: express it as its invariant closure intersected with the invariant open complement of its boundary. Apply the preceding intrinsic decomposition to \(R_H\), which is invariant under \(H\times H\). Since \(p\times p\) is an open quotient map, the diagonal of \(Y\times Y\) is a finite union of locally closed sets. At some point \((y,y)\) that diagonal is closed in an open neighbourhood. Choosing a smaller product neighbourhood gives a nonempty open subset \(V\subset Y\) on which the diagonal is closed. Thus \(V\) is Hausdorff. It is an \(l\)-space: if \(x\in p^{-1}V\), a compact-open neighbourhood \(C\subset p^{-1}V\) has compact-open image \(p(C)\subset V\). Lemma 2.0b applies on \(p^{-1}V\).

Every orbit is locally closed. It is constructive by slicing (2.6d); its nonempty locally closed part \(U(Hx)\) is intrinsic and \(H\)-invariant, hence is the whole transitive orbit. To check the topology assertion, a continuous transitive action of a second-countable locally compact group on a locally compact Hausdorff orbit has an open orbit map. Given an identity neighbourhood \(A\subset H\), choose a compact neighbourhood \(C\) with \(C^{-1}C\subset A\). Countably many translates of \(C\) cover \(H\). Their compact orbit images are closed and cover the orbit. The Baire property of a locally compact Hausdorff space implies that some translate of \(Cx\) has interior; translating back and then by an element of \(C^{-1}\) shows that \(Ax\) contains an orbit neighbourhood of \(x\). The Baire property used here follows by placing successively smaller nonempty open sets with compact closures inside the complement of each proposed closed set with empty interior; the nested compact closures have a common point. Therefore \(H/H_x\to Hx\) is a homeomorphism.

Finally, if \(D\ne0\), replace \(X\) by its closed support. The support is \(H\)-invariant, remains an \(l\)-space, and the restricted orbit relation is constructive. The argument just given produces a nonempty invariant open subset with Hausdorff \(l\)-space orbit quotient. Its fibres have zero equivariant distributions, so \(D\) vanishes there by Lemma 2.0b, contradicting the definition of its full support. \(\square\)

The distribution on a single homogeneous orbit

All groups to which the following homogeneous calculation is applied have arbitrarily small compact-open subgroups: for matrix groups use their intersection with \(1+\varpi^rM_n(\mathcal O)\), for unipotent matrix groups use the corresponding integral root coordinates, and for the finite extensions choose these subgroups stable under the specified involution. Haar measures on the unipotent groups and their pattern subgroups are the triangular-coordinate measures proved in §3. Haar measures on \(GL_n\) and on the conjugacy stabilizers are constructed explicitly in §§4–5. Thus the applications do not require an unstated Haar-existence or centralizer-unimodularity theorem.

Lemma 2.0d (homogeneous distributions). Let \(H\) be a second-countable \(l\)-group, \(J\subset H\) closed, and \(X=H/J\).

1. An \(H\)-invariant distribution on \(X\) has dimension at most one. If a nonzero one exists, it is a scalar multiple of a positive invariant quotient Haar measure.
2. If \(H\) and \(J\) are unimodular, a \(\chi\)-equivariant distribution exists precisely when \(\chi|_J=1\), and then has dimension one. It is the weighted quotient measure \[ D_\chi(f)=\int_{H/J} f(hJ)\chi(h)\,d\dot h . \tag{2.6e} \]
3. Suppose an involutive automorphism \(\alpha\) of \(H\) preserves \(J\) and \(\chi\), and let \(s\) be the induced involution of \(H/J\). If \(H,J\) are unimodular, \(s\) fixes the distribution in part 2. If \(\chi=1\), a more general assertion needs neither a fixed point nor unimodularity: any involution normalizing the \(H\)-action and preserving this orbit fixes its invariant distributions.

Proof. We give the descent argument, including uniqueness. Averaging on a right fibre defines \[ qf(hJ)=\int_J f(hj)\,dj,\qquad q:\mathcal S(H)\to\mathcal S(H/J). \tag{2.6f} \] It is onto. To see this without a section theorem, a compact-open subgroup \(C\subset H\) has compact-open orbit \(CJ\) in \(H/J\); the average of \(1_C\) is the constant \(\operatorname{vol}_J(C\cap J)\) on that orbit and zero off it. Its translates, with successively smaller compact-open \(C\), give a basis of compact-open neighbourhoods in \(H/J\). Finite partitions of a compact support therefore give every test function in the image. With a character, use instead \(q_\chi f(hJ)=\int_J f(hj)\chi(hj)^{-1}\,dj\) when \(\chi|_J=1\).

A left-invariant distribution on \(H\) is a scalar multiple of left Haar measure. Indeed, its value on \(1_{hC}\) is independent of \(h\). If \(C'\subset C\) are compact open, their finite coset partition gives \(D(1_C)=[C:C']D(1_{C'})\), exactly the Haar relation. Intersecting two such subgroups compares their constants, and partitioning arbitrary compact-open supports into cosets proves uniqueness. Multiplication by \(\chi\) gives the corresponding statement for \(\chi\)-equivariant distributions on \(H\), with density \(\chi(h)\,dh\).

Pull a distribution on \(H/J\) back by (2.6f). It is therefore a scalar Haar distribution if it is invariant. Right translation by \(j\in J\) scales \(qf\) by the modular character of \(J\), while it scales Haar integration on \(H\) by that of \(H\). Nonzero descent forces \(\Delta_H|_J=\Delta_J\). Under this equality, integrating first along \(J\) defines a positive quotient measure. Here is a kernel calculation that establishes independence of the lift without assuming a local section. Choose a compact-open \(C\subset H\) such that \(f\) is left \(C\)-invariant. It is a finite linear combination of \(1_{Ch_i}\). Group these cosets by their right \(J\)-orbits in \(C\backslash H\). On one such orbit choose a representative \(h\), so its terms are \(1_{Chj_i}\). Their fibre integrals at \(hJ\) are positive constants proportional to the left-Haar masses of \((h^{-1}Ch\cap J)j_i\), and their quotient functions are supported on the same compact-open \(C\)-orbit \(ChJ\). Thus a combination has zero fibre integral exactly when its weighted coefficient sum is zero. Subtracting that sum times the representative \(1_{Ch}\) writes it as a finite sum of right-\(J\) translation differences, with precisely the positive modular factors given by those masses. Under \(\Delta_H|_J=\Delta_J\), Haar integration on \(H\) has the same right-translation factors and annihilates all these differences. This proves descent. It is positive because \(q\) has nonnegative lifts of nonnegative test functions, obtained from the compact-open quotient patches used to prove surjectivity. This proves part 1 and the quotient formula, not merely uniqueness up to an unspecified distribution.

The identical argument after multiplying by \(\chi^{-1}\) proves part 2. If \(\chi\) is nontrivial on \(J\), pullback would have both the fibre transformation law and the incompatible character law on \(J\); with unimodularity these cannot agree. If it is trivial, (2.6e) is well-defined, nonzero, and has the stated transformation law under \(L_h\).

For part 3, \(\alpha\) preserves Haar measure on \(H\) and \(J\): each Haar modulus is positive and its square is one. Hence it preserves quotient measure, and \(\chi\circ\alpha=\chi\) preserves the weight in (2.6e). More generally one can normalize the quotient measure directly: an involution carries a positive invariant quotient measure to a positive scalar multiple of itself, and that scalar has square one, hence equals one. This proves the last assertion when \(\chi=1\). \(\square\)

We use part 3 only with the literal involutive group automorphism just specified. It follows that if every orbit of a constructive action is preserved by an involution and every invariant orbital distribution is fixed, every invariant distribution is fixed. To prove this assertion, subtract its involution translate. The difference is invariant under the original group and transforms by sign under the group extended by the involution. This extended action is constructive: its orbit relation is the union of the original relation and its translate. On each extended orbit its sign-equivariant distribution space is zero by Lemma 2.0d. The original-group orbits are open and closed in that extended orbit because the original group has finite index and its orbit map is open, as proved in Lemma 2.0c. Consequently Lemma 2.0c applies to the sign character and kills the difference.

Bruhat orbits carrying a generic character

We first record all the relevant matrix geometry. For a permutation \(p\in S_n\), let \(w_pe_j=e_{p(j)}\), put \[ U_p=U\cap w_pU^-w_p^{-1}, \qquad C_p=U\,T\,w_p\,U, \tag{2.6g} \] and let \(T\) denote the diagonal torus. Each matrix in \(C_p\) has a unique expression \(a\,t\,w_p\,b\), with \(a\in U_p\), \(t\in T\), \(b\in U\). The coordinates are polynomial in the unipotent parameters and rational in matrix entries on the cell; they give its \(F\)-topological coordinates.

Here is a direct verification. Upper row operations preserve the ranks of the last \(n-i+1\) rows in the first \(j\) columns, and upper column operations preserve those ranks as well. Eliminate successively from the first column, choosing its lowest nonzero entry as pivot, clearing all entries above it by upper row operations and the other entries in its pivot row by upper column operations. Remove that row and column and repeat in the remaining ordered rows and columns. These invertible operations give precisely one permutation pivot position in every row and column. The pivot diagonal scalars give \(t\). Once the permutation is fixed, the permitted entries in \(a\) are exactly \(r<s\) with \(p^{-1}(r)>p^{-1}(s)\); the other upper entries can be moved through \(t w_p\) into \(b\). Both sets of roots are closed under addition, so their ordered elementary products are subgroups, and moving entries in increasing height gives unique coordinates. Alternatively uniqueness follows because \(U_p\cap w_pUw_p^{-1}=\{1\}\), while diagonal entries force equality of \(t\). Every pivot division is by a specified nonzero minor. Thus the cell and its coordinates are described by finitely many polynomial equalities and nonzero inequalities.

In particular, \[ r_{ij}(g)=\operatorname{rank}g_{\{i,\ldots,n\},\{1,\ldots,j\}} =\#\{\ell\leq j:p(\ell)\geq i\}\quad(g\in C_p). \tag{2.6h} \] These rank patterns distinguish the cells, and each cell is locally closed. The orbit relation for the action \(H=U\times U\), \((u,v)g=u g v^{-1}\), is constructive: two matrices are in the same orbit exactly when they have the same permutation \(p\) and the same \(t\) in these coordinates. Equality of the rational coordinates, after their nonzero pivot denominators are specified, is a finite union of locally closed conditions. This verifies the hypothesis of Lemma 2.0c directly, rather than citing a general algebraic orbit theorem.

Use the character \(\chi(u,v)=\Psi(u)\Psi(v)^{-1}\) of \(H\). At \(g=t w_p\) the stabilizer is \[ H_g=\{(gvg^{-1},v):v\in U\cap w_p^{-1}Uw_p\}. \tag{2.6i} \] The groups in (2.6i) are unimodular. Their elementary coordinates are the roots \(i<j\) with \(p(i)<p(j)\); these roots are closed under addition. Translation in coordinates ordered by height is triangular with diagonal one, so product additive Haar measure is left and right invariant. The same proves unimodularity of \(U\) and \(H\).

Lemma 2.0e (relevant orbits). The orbit of \(t w_p\) has a nonzero \(\chi\)-equivariant distribution precisely when the list \(p(1),\ldots,p(n)\) consists of consecutive increasing intervals placed in decreasing order, and \(t\) is constant on each of these image intervals. Every such representative satisfies \(\tau(t w_p)=t w_p\).

Proof. For a root \(i<j\) with \(p(i)<p(j)\), conjugation in (2.6i) gives \[ g(1+xE_{ij})g^{-1} =1+x\frac{t_{p(i)}}{t_{p(j)}}E_{p(i),p(j)}. \tag{2.6j} \] The character on the right-hand root is \(\psi(x)\) exactly when \(j=i+1\), and is trivial otherwise. The conjugated root contributes exactly when \(p(j)=p(i)+1\). Since \(\psi(cx)=\psi(x)\) for all \(x\) forces \(c=1\), stabilizer compatibility is equivalent to

• \(j=i+1\) if and only if \(p(j)=p(i)+1\), whenever \(i<j\) and \(p(i)<p(j)\);
• \(t_{p(i)}=t_{p(j)}\) for these simultaneous simple roots.

Every ascending step in the list \(p(1),\ldots,p(n)\) must therefore rise by exactly one. Split the list at its descending steps. Each run is a consecutive increasing interval of values. Distinct runs are disjoint intervals, and a descending step forces the preceding interval to lie wholly above the following one. Thus the intervals occur in decreasing order. Conversely a list of this form has an increasing pair of consecutive values only within one run, where their positions are consecutive; both conditions follow, provided \(t\) is constant on that run. These root conditions imply compatibility on the whole stabilizer, since the root groups generate it and both sides are characters. Lemma 2.0d proves the existence and uniqueness of its weighted orbit distribution.

The matrix \(t w_p\) now consists of square scalar-identity blocks on the block antidiagonal. Reflection across the full antidiagonal carries each such block to itself and reverses its diagonal entries. Those entries are equal by the condition on \(t\), proving \(\tau(t w_p)=t w_p\). \(\square\)

Theorem 2.0f (generic distribution symmetry). Every distribution \(D\) on \(G\) satisfying \[ D\bigl(f(u^{-1}\,{\cdot}\,v^{-1})\bigr) =\Psi(u)\Psi(v)D(f) \qquad(u,v\in U) \tag{2.6k} \] is fixed by \(\tau\).

Proof. In the \(H\)-action convention above this is \(\chi\)-equivariance. The involution \(\tau\) normalizes that action through the involutive automorphism \[ (u,v)\longmapsto\bigl(\tau(v)^{-1},\tau(u)^{-1}\bigr), \tag{2.6l} \] which preserves \(\chi\). By Lemma 2.0e, any orbit carrying an equivariant distribution has a representative fixed by \(\tau\). Formula (2.6e) and Lemma 2.0d therefore show that its orbital distribution is fixed by \(\tau\): (2.6l) preserves both Haar measures and the character weight. Every other orbit has zero equivariant distribution space. The difference \(D-\tau D\) is \(\chi\)-equivariant and transforms by sign under \(\tau\). The finite extension by (2.6l) is constructive. On each extended orbit its sign-equivariant distribution space is zero, by the preceding orbital calculation; original \(H\)-orbits are open and closed there. Lemma 2.0c proves that the difference is zero. This argument treats distributions on all Bruhat boundaries, through closed support and localization, not only functions on the open cell. \(\square\)

Transpose and conjugation-invariant distributions

The representation-theoretic duality needed below must be established separately. A product multiplicity bound by itself would not prove uniqueness for a representation whose dual had not been shown to be generic.

Lemma 2.0g (constructiveness of conjugacy). The conjugacy orbit relation on \(M_n(F)\), and on its open subset \(G\), is a finite union of locally closed sets. Every matrix is similar over \(F\) to its transpose.

Proof. We supply the polynomial-algebra verification. Regard \(F^n\), with \(z\) acting by \(A\), as an \(F[z]\)-module. It is presented by the polynomial matrix \(zI-A\): the map \(\sum z^rv_r\mapsto\sum A^rv_r\) is onto, and repeatedly subtracting multiples of \(zI-A\) reduces any polynomial vector uniquely to a constant vector. That constant is its image, so the kernel is precisely the image of \(zI-A\). Elementary polynomial row and column operations put any polynomial matrix into Smith form \(\operatorname{diag}(d_1,\ldots,d_n)\), with monic \(d_i\mid d_{i+1}\). Here is the algorithm: choose a nonzero entry of smallest degree; divide other entries in its row and column by it and subtract the multiples. A nonzero remainder has smaller degree and becomes the new pivot. Once the pivot divides its row and column, clear them. If it fails to divide an entry of the remaining block, add that entry into the pivot row and perform division again, decreasing the pivot degree. Thus that stage terminates. Remove the cleared row and column and repeat. The resulting pivots divide their successors because, before removal, each pivot divides the remaining block. Multiplication by a scalar makes each pivot monic. For \(zI-A\) no pivot is zero, since its determinant is monic of degree \(n\).

Elementary operations preserve the ideals generated by the \(k\)-minors. Their monic generators are \(\Delta_k=d_1\cdots d_k\), with \(\Delta_0=1\); hence the Smith factors are unique. The presentation decomposes into \(\bigoplus F[z]/(d_i)\), omitting unit factors. In the power basis of each nonconstant quotient the action of \(z\) is its companion matrix. Thus two matrices over \(F\) are similar if and only if they have the same \(\Delta_1,\ldots,\Delta_n\). Transposing a matrix permutes the minors of \(zI-A\), so preserves all these generators and proves similarity to the transpose. This works for inseparable polynomials and in characteristic two as well.

To verify constructiveness, compute each \(\Delta_k\) as the monic gcd of the finitely many \(k\)-minors, polynomials of degree at most \(n\). A polynomial's degree is found by testing its finitely many coefficients, and division by its nonzero leading coefficient is rational in those coefficients. The Euclidean algorithm decreases degree at each nonzero remainder, so a gcd of two such polynomials has at most \(n+1\) division stages. Compute the gcd of a finite list by successive pairs, ignoring zero polynomials. There are consequently only finitely many possible branches. On each branch the output coefficients are rational functions of the entries of \(A\), with denominators required to be nonzero. The branch itself is given by polynomial equalities and nonzero inequalities. For a pair \(A,B\), choose branches for both, require equal degrees and equal output coefficients for each \(\Delta_k\), and clear the specified nonzero denominators. Each condition is locally closed; the finite union of them is exactly the similarity relation. No algebraic orbit constructibility theorem has been used. \(\square\)

Here are explicit Haar measures for this conjugacy calculation. On \(G\), the density \(dg=|\det g|^{-n}d^{n^2}g\) is both left and right invariant: multiplication by a matrix \(a\) scales additive matrix measure by \(|\det a|^n\), which is cancelled by the density. For the stabilizer of \(A\), put \(\mathcal C_A=\{X\in M_n(F):XA=AX\}\). This is a finite-dimensional \(F\)-algebra, and \(G_A=\mathcal C_A^\times\) is its open unit set. Indeed a matrix inverse of a commuting invertible matrix still commutes with \(A\), and invertibility of its left multiplication \(L_x\) on \(\mathcal C_A\) is equivalent to having such an inverse: surjectivity supplies \(xy=I\).

Choose additive product Haar measure \(dx\) in a basis of \(\mathcal C_A\). Then \[ d\mu_A(x)=\frac{dx}{|\det_F L_x|_F} \qquad(x\in\mathcal C_A^\times) \tag{2.6ha} \] is a left Haar measure. It is positive, finite on compact subsets and locally nonzero. Left multiplication by \(j\) changes the numerator by \(|\det L_j|_F\), while \(L_{jx}=L_jL_x\) changes the denominator by the same factor. Right multiplication instead has the positive factor \(|\det R_j|_F/|\det L_j|_F\), giving its modular character explicitly. Thus the closed stabilizer's Haar measure exists without an external theorem; no assertion that this character is trivial is needed. An invariant distribution, if it exists on the orbit, forces the compatible quotient measure by Lemma 2.0d.

Theorem 2.0h (transpose symmetry for conjugacy distributions). Every conjugation-invariant distribution on \(G\) is invariant under \(g\mapsto{}^tg\).

Proof. Transpose normalizes conjugation by \(h\mapsto{}^th^{-1}\), has order two, and preserves each conjugacy orbit by Lemma 2.0g. On any such orbit an invariant distribution is either zero or a scalar positive quotient Haar measure by Lemma 2.0d. Transpose fixes that measure: it carries it to a positive scalar multiple of itself, and the scalar has square one. The conjugation action is constructive by Lemma 2.0g. The sign-equivariant localization argument following Lemma 2.0d therefore applies. \(\square\)

Smooth duality and identification by traces

Give \(G\) Haar measure \(dg=|\det g|^{-n}\,d^{n^2}g\), up to a positive constant. Left or right multiplication by \(a\) scales additive matrix measure by \(|\det a|^n\), cancelling the determinant density. Thus \(G\) is unimodular. Transpose preserves this measure. Inversion preserves it as well: it carries left Haar to right Haar and has positive modulus whose square is one. Inner conjugation and \(\tau,\theta\) therefore preserve Haar measure.

Let \(\mathcal H=\mathcal S(G)\) with convolution. A test function is left and right invariant under a sufficiently small common compact-open subgroup \(K\). The averages \(e_K=\operatorname{vol}(K)^{-1}1_K\) are local units. For a smooth representation \(V\), the integral \(\pi(f)v\) is a finite sum: on the compact support of \(f\), intersect its finite locally constant partition with the open stabilizer cosets of \(v\). The convolution action and group action determine each other; a group translate of a smooth vector equals its Hecke translate by \(\delta_g*e_K\) when \(K\) fixes it.

The smooth contragredient is the open-stabilizer part \(V^\vee\) of the algebraic dual, with action \(\pi^\vee(g)\ell=\ell\circ\pi(g^{-1})\). For every compact open \(K\), restriction and averaging give \[ (V^\vee)^K=(V^K)^*,\qquad \ell\longleftrightarrow \ell|_{V^K},\quad a\longmapsto a\circ e_K. \tag{2.6m} \] If \(V\) is admissible, the dual is admissible and evaluation identifies \(V\) with its smooth bidual, since this is ordinary finite-dimensional biduality on every fixed space. If \(V\) is irreducible, so is \(V^\vee\): the annihilator in \(V\) of a nonzero invariant subspace \(W\subset V^\vee\) is proper and invariant, hence zero. Averaging shows that the annihilator of \(W^K\) in \(V^K\) is zero; finite dimension yields \(W^K=(V^\vee)^K\), and the union over \(K\) gives \(W=V^\vee\). A commuting endomorphism of an irreducible admissible module is scalar: restrict it to a nonzero finite-dimensional fixed space, choose an eigenvalue, and use the invariant nonzero kernel of the difference from that scalar.

For an admissible \(V\), \(\pi(f)\) has finite rank, because it has image in \(V^K\) and factors through \(e_K\) for any common bi-invariance subgroup of \(f\). Define the trace distribution \(\Theta_V(f)=\operatorname{tr}\pi(f)\). It is conjugation-invariant by finite-rank trace invariance under conjugation. It is a genuine distribution because it is a linear functional on \(\mathcal H\).

Lemma 2.0i (traces determine irreducible admissible modules). Two irreducible admissible representations with the same trace distribution are isomorphic.

Proof. Choose \(K\) with \(V^K\ne0\). Equality of \(\Theta(e_K)\) gives \(\dim V^K=\dim W^K>0\). The corners are simple modules for the unital algebra \(A=e_K\mathcal H e_K\). Indeed, if \(0\ne E\subset e_KV\) is \(A\)-invariant, \(\mathcal H E=V\) by simplicity, and \(e_K\mathcal H E=e_K\mathcal H e_KE=E\), proving \(E=e_KV\).