Proper actions, universal proper spaces and equivariant K-homology

Written by GPT-6.1 Sol (OpenAI). Public domain (CC0).

An equivariant Fredholm cycle needs a space on which the group acts properly. There is no preferred such space for a general group. A universal proper space collects them, and equivariant K-homology with compact orbit support removes the choice of model. We construct the universal comparison space, prove the comparison of its locally compact stages, and give Euclidean, tree and enlarged local-field building models.

Throughout, \(G\) is a second countable locally compact Hausdorff group with a fixed left Haar measure \(dg\). Spaces on which \(C_0\) and \(KK\) are used are locally compact, Hausdorff and second countable. Equivariant cycles and their homotopies are those of Equivariant KK-theory and the Green–Julg theorem, Sections 1–2; the topology and Haar integration tools are proved in Noncompact foundations for equivariant induction, Sections NCF.1 and NCF.3–NCF.5. The root-group prerequisites for the building construction are stated precisely at the beginning of its appendices. Basic freely accessible references are [Lück], [Gómez Aparicio–Julg–Valette] and [Baum–Higson–Schick]. The universal proper-space formulation in this setting is due to Baum, Connes and Higson.

1. Properness and normalized cutoffs

Definition 1.1. A continuous action on \(X\) is proper if \[ G\times X\longrightarrow X\times X,\qquad(g,x)\longmapsto(gx,x) \tag{1.1} \] is proper, meaning that the inverse image of every compact set is compact. A proper space is \(G\)-compact if \(X/G\) is compact. These two properties have different roles: properness controls transporters and stabilizers; \(G\)-compactness controls orbit support.

For compact \(K,L\subset X\), write \[ T(K,L)=\{g\in G:gK\cap L\ne\varnothing\}. \tag{1.2} \] For locally compact Hausdorff \(X\), (1.1) is proper exactly when every \(T(K,L)\) is compact. In one direction project the compact inverse image of \(L\times K\) onto \(G\). Conversely the inverse image of a compact subset of \(X^2\) is closed in \(T(K,L)\times K\), for its two compact coordinate projections \(L,K\). The transporter itself is closed: if group elements converge, their witnessing points in \(K\) have a convergent subnet. In particular every stabilizer \(G_x=T(\{x\},\{x\})\) is compact.

Lemma 1.2 (orbit topology). The quotient \(Y=X/G\) of a proper space in the standing category is locally compact, Hausdorff and second countable. Its quotient map \(\pi\) is open. If \(Y\) is compact, there is a compact \(K\subset X\) with \(GK=X\).

Proof. Saturations of open sets are open. The orbit relation is closed: for a convergent net \((g_i x_i,x_i)\to(y,x)\), put both coordinates eventually in compact neighborhoods of their limits. Properness puts the lifted pairs \((g_i,x_i)\) in a compact set; a subnet gives \(g_i\to g\) and \(y=gx\). The map \(\pi\times\pi\) is an open quotient map, so the closed relation descends to a closed diagonal in \(Y^2\). Thus \(Y\) is Hausdorff. If \(V\subset X\) is precompact and open, \(\pi(V)\) is open and its closure is contained in the compact, hence closed, set \(\pi(\overline V)\). Such sets prove local compactness. Images of a countable base form a countable base. Finally finitely many \(\pi(V_j)\), with each \(V_j\) precompact, cover a compact quotient. Set \(K=\bigcup_j\overline V_j\). \(\square\)

Theorem 1.3 (squared normalization). Every proper \(G\)-compact space \(X\) in the standing category has \(c\in C_c(X)\), \(c\ge0\), with \[ \int_G c(g^{-1}x)^2\,dg=1\qquad(x\in X). \tag{1.3} \] More generally, every proper \(X\) has a continuous \(b:X\to[0,\infty)\) such that \[ \int_G b(g^{-1}x)\,dg=1, \qquad \operatorname{supp}b\cap\pi^{-1}(L) \text{ is compact for every compact }L\subset X/G. \tag{1.4} \] One may take \(c=\sqrt b\); in the cocompact case \(c\) has compact support.

Proof. First suppose \(X/G\) is compact. Take compact \(K\) as in Lemma 1.2 and choose \(f\in C_c(X)\), \(f\ge0\), strictly positive on \(K\). The compact cutoff construction of NCF.1 supplies such an \(f\). Put \[ a(x)=\int_G f(g^{-1}x)^2\,dg. \tag{1.5} \] It is finite and continuous. Indeed, for \(x\) in a compact neighborhood \(Q\), every contributing \(g\) is in \(T(\operatorname{supp}f,Q)\). The integrand is jointly continuous there, and the compact integration domain works throughout that neighborhood. Every orbit meets the positive set of \(f\), so the integrand is positive on a nonempty open subset of \(G\); Haar measure gives \(a(x)>0\). Substitution \(g=sh\) proves \(a(sx)=a(x)\). Thus \(c(x)=f(x)/\sqrt{a(x)}\) is continuous, supported in \(\operatorname{supp}f\), and satisfies (1.3).

For arbitrary \(X\), choose nonnegative \(f_i\in C_c(X)\) whose positive sets \(W_i\) meet every orbit. Let \(V_i=\pi(W_i)\) and \[ h_i(x)=\int_G f_i(g^{-1}x)\,dg. \] The same transporter argument makes \(h_i\) invariant and continuous; its descended function \(\bar h_i\) is positive on \(V_i\). By Lemma 1.2 and the complete partition construction in NCF.1, choose a countable locally finite partition \(\rho_i\) on \(Y\), with compact supports contained in \(V_i\). Repeating a cover label after refinement is allowed. Set \[ b(x)=\sum_i\frac{\rho_i(\pi x)}{\bar h_i(\pi x)}f_i(x). \tag{1.6} \] Extend each summand by zero outside \(\pi^{-1}(V_i)\). This is continuous because \(\operatorname{supp}\rho_i\) is a compact subset of the positive set of \(\bar h_i\). The sum is locally finite on the quotient. Its orbit integral is \(\sum_i\rho_i(\pi x)=1\). Above a compact quotient set, only finitely many partition supports occur. The corresponding support of \(b\) is a closed subset of the union of finitely many \(\operatorname{supp}f_i\), and is therefore compact. This proves (1.4). \(\square\)

The support condition has a useful local consequence. For any compact \(Q\subset X\), all \(g\) with \(b(g^{-1}x)\ne0\) for some \(x\in Q\) lie in one compact set: use the compact support of \(b\) over \(\pi(Q)\), then (1.2). For a discrete group, there are only finitely many such \(g\). This local finiteness will make the Euclidean and tree constructions continuous.

A slice at a point with stabilizer \(H\) means an \(H\)-invariant subspace \(S\) for which the induced map \(G\times_H S\to X\) identifies an invariant neighborhood with that associated space. Slices give useful local descriptions when a slice theorem applies. The proofs of Theorem 1.3 and the models below use transporters and cutoffs directly; no slice-existence assertion is needed in them.

2. An explicit universal comparison space

An ambient model may fail to be locally compact. We will only form \(C_0(Z)\) on its closed, locally compact, second countable, \(G\)-compact stages \(Z\). Say that a proper Hausdorff ambient space \(E\) has the universal property for locally compact sources when every proper space in the standing category has an equivariant map into \(E\), and any two such maps are equivariantly homotopic. We use compactly generated ambient spaces: a subset is closed if its intersection with each compact subset is closed in that compact subset. Metric spaces have this property. The usual full universal proper model has the same mapping and homotopy property on the full chosen category of proper spaces, including ambient models; Section 5 explains the resulting global uniqueness. The stage comparison proved here requires only the explicitly stated locally compact-source property.

Theorem 2.1 (the positive unit sphere). With left translation, the metric space \[ \mathcal U=\{u\in L^2(G,dg):u\text{ is real and nonnegative a.e.},\ \|u\|_2=1\}, \quad (\lambda_su)(g)=u(s^{-1}g), \tag{2.1} \] is proper and has the universal property for locally compact sources. For every compact subgroup \(H\), \(\mathcal U^H\) is nonempty and contractible; no noncompact closed subgroup has a fixed point. The two projections \(\mathcal U\times\mathcal U\to\mathcal U\) are equivariantly homotopic.

Proof. The positive cone is norm closed, so \(\mathcal U\) is a closed subset of the unit sphere. NCF.1 and NCF.3–NCF.5 imply that \(C_c(G)\) is dense in \(L^2(G)\): truncate over a compact exhaustion, approximate by simple functions, use Radon inner and outer approximation, and separate the compact inner set from the complement of its open outer set by a continuous cutoff. A countable relatively compact base and rational simple approximations make \(L^2(G)\) separable. Translation is norm continuous on \(C_c(G)\), by joint continuity on a compact support neighborhood; density and translation isometry give strong continuity on \(L^2(G)\). Thus the action is continuous.

For \(u,v\in C_c(G)\), the coefficient \(\langle\lambda_g u,v\rangle\) vanishes unless \(g\in(\operatorname{supp}v)(\operatorname{supp}u)^{-1}\). Cauchy–Schwarz and approximation give \[ \langle\lambda_gu,v\rangle\longrightarrow0 \quad(g\longrightarrow\infty) \tag{2.2} \] for every \(u,v\in L^2(G)\). Finite norm nets and Cauchy–Schwarz show that this convergence is uniform for \(u,v\) in fixed compact subsets. For compact \(K,L\subset\mathcal U\), the transporter is closed and lies in a compact subset of \(G\), since a matching pair has coefficient one. The inverse image of a compact subset of \(\mathcal U^2\) under the action map is therefore closed in a compact transporter times a compact second projection. This proves properness, including for this ambient space which need not be locally compact.

For compact \(H\), average a nonzero nonnegative \(f\in C_c(G)\) over normalized Haar measure on \(H\): \[ v=\int_H\lambda_hf\,dh. \] This is nonnegative, \(H\)-fixed and compactly supported, and \(\int_Gv=\int_Gf>0\). Normalize it to \(u_H\in\mathcal U^H\). Normalized convex interpolation \[ P(u,v,t)=\frac{(1-t)u+tv}{\|(1-t)u+tv\|_2} \tag{2.3} \] is continuous and equivariant. Its denominator is at least \(1/\sqrt2\), because \(\langle u,v\rangle\ge0\). Taking \(v=u_H\) contracts \(\mathcal U^H\), and taking arbitrary \(u,v\) gives the homotopy of projections. If a noncompact closed subgroup fixed \(u\), its coefficient with itself would be one throughout that subgroup, contrary to (2.2).

For a proper \(X\), use Theorem 1.3 and define \[ F_b(x)(g)=\sqrt{b(g^{-1}x)}. \tag{2.4} \] Equation (1.4) makes its norm one. On a compact neighborhood in \(X\), its integrands are supported in one compact subset of \(G\), as observed after Theorem 1.3. Joint continuity and integration give norm continuity. Substitution gives \(F_b(sx)=\lambda_sF_b(x)\). Any two equivariant maps \(X\to\mathcal U\), whether or not constructed from cutoffs, are homotopic by (2.3). \(\square\)

Lemma 2.2 (locally compact stages). For every compact \(K\subset\mathcal U\), the saturation \(Z_K=GK\) is closed, locally compact, Hausdorff, second countable, proper and \(G\)-compact.

Proof. Fix \(v\in\mathcal U\). By uniform (2.2), there is compact \(C\subset G\) such that \(|\langle\lambda_gk,v\rangle|<1/2\) for \(k\in K\), \(g\notin C\). If \(\|\lambda_gk-v\|<1/2\), then \(\langle\lambda_gk,v\rangle=1-\|\lambda_gk-v\|^2/2>7/8\), so \(g\in C\). Consequently the part of \(GK\) near \(v\) is contained in the compact image \(CK\). Every closure point of \(GK\) lies in such a compact image, hence lies in \(GK\); this proves closedness. At \(v\in GK\), the closure of a sufficiently small relative ball is a closed subset of \(CK\), proving local compactness. Separation and second countability are inherited from the separable metric space \(\mathcal U\). Properness restricts to a closed invariant subspace, and the quotient is the image of \(K\). \(\square\)

Lemma 2.3 (proper maps and their images). If \(X\) is proper, locally compact and \(G\)-compact, and \(T\) is a proper Hausdorff ambient space with compact transporters, every continuous equivariant map \(f:X\to T\) is proper. If \(T\) is compactly generated, its image is closed, locally compact and second countable, with compact orbit space.

Proof. Choose compact \(K\) with \(GK=X\). For compact \(L\subset T\), any \(gk\) mapping into \(L\) has \(g\in T(f(K),L)\). Thus \(f^{-1}(L)\) is closed in the compact image of \(T(f(K),L)\times K\). This is properness.

For closed \(D\subset X\), the intersection \(f(D)\cap L=f(D\cap f^{-1}(L))\) is compact and closed in \(L\). Compact generation of \(T\) makes \(f(D)\) closed. Thus \(f\) is a closed map and \(S=f(X)\) is closed in \(T\). To see that \(S\) is locally compact, cover the compact fiber over \(s\in S\) by finitely many relatively compact open sets of \(X\), with union \(V\). The set \(S\setminus f(X\setminus V)\) is an open neighborhood of \(s\) in \(S\), and its closure is contained in the compact set \(f(\overline V)\). If \(\mathcal B\) is a countable base of \(X\), the sets \(S\setminus f(X\setminus V)\), for finite unions \(V\) of members of \(\mathcal B\), form a countable base: cover each compact fiber inside the inverse image of a prescribed open set by finitely many such members. Finally \(S/G\) is the image of \(f(K)\). \(\square\)

3. Universality by squared-distance minimization

We will use a complete geodesic metric space \(Y\) with the CAT(0) squared-distance inequality \[ d(z,\gamma_{p,q}(t))^2 \le(1-t)d(z,p)^2+t d(z,q)^2-t(1-t)d(p,q)^2. \tag{3.1} \] The geodesic \(\gamma_{p,q}\) is parametrized on \([0,1]\). In a CAT(0) space it is unique. Here the targets are also proper metric spaces, meaning that closed bounded balls are compact. Geodesic interpolation is then continuous directly: for converging endpoints and parameters, the intervening points lie in a common compact ball; every convergent subsequence has the two prescribed endpoint distances and hence is the unique corresponding point on the limiting geodesic.

Lemma 3.1 (barycenters). A compactly supported probability measure \(\mu\) on a complete space satisfying (3.1) has a unique minimizer of \[ E_\mu(y)=\int_Y d(y,z)^2\,d\mu(z). \tag{3.2} \] For a family with a common bounded support and densities continuous in \(L^1\), its minimizer is continuous.

Proof. Choose \(o\in Y\) and \(R\) with the support inside \(\overline B(o,R)\). For \(d(y,o)\ge R\), the triangle inequality gives \(E_\mu(y)\ge(d(y,o)-R)^2\), while \(E_\mu(o)\le R^2\). Put \(m=\inf E_\mu\), and choose \(y_n\) with \(E_\mu(y_n)\le m+1/n\). Integrated midpoint convexity gives \[ m\le E_\mu(\gamma_{y_n,y_j}(1/2)) \le m+\frac1{2n}+\frac1{2j}-\frac14d(y_n,y_j)^2. \] Thus the sequence is Cauchy. Its limit exists by completeness, and continuity of the energy on bounded balls shows it attains the infimum. The same inequality makes two minimizers equal. Notice that compactness of a minimizing ball is not needed for this existence argument.

For continuity, suppose the densities \(w_x\) on a common finite measure space converge in \(L^1\), and their support points all lie in \(\overline B(o,R)\). Every minimizer is in \(\overline B(o,2R)\). On this ball, \[ |E_x(y)-E_{x_0}(y)|\le9R^2\|w_x-w_{x_0}\|_1=:\varepsilon_x. \tag{3.3} \] If \(R=0\), the minimizer is always \(o\). Otherwise minimality gives \(E_{x_0}(m_x)-E_{x_0}(m_{x_0})\le2\varepsilon_x\). Midpoint convexity and minimality for \(E_{x_0}\) give \(d(m_x,m_{x_0})^2\le2(E_{x_0}(m_x)-E_{x_0}(m_{x_0}))\le4\varepsilon_x\). This tends to zero. \(\square\)

Theorem 3.2 (a CAT(0) model). Suppose \(G\) acts continuously, properly and isometrically on a proper complete CAT(0) metric space \(Y\). Then \(Y\) has the universal property for locally compact proper sources. Every compact subgroup has a nonempty contractible fixed set, and the two projections \(Y\times Y\to Y\) are equivariantly homotopic.

Proof. For a proper source \(X\), choose \(b\) from (1.4) and fix \(y_0\in Y\). Put \[ \mu_x=(g\mapsto gy_0)_*\bigl(b(g^{-1}x)\,dg\bigr), \qquad f(x)=\operatorname*{argmin}_{y\in Y} \int_Gd(y,gy_0)^2b(g^{-1}x)\,dg. \tag{3.4} \] These are probability measures. On a compact neighborhood of any \(x_0\), the contributing \(g\)'s lie in one compact subset \(C\subset G\). Its image \(Cy_0\) is compact and bounded; joint continuity makes the densities continuous in \(L^1(C)\). Lemma 3.1 proves existence, uniqueness and continuity of \(f\). For a discrete group the sum has finitely many terms throughout that neighborhood. Substitution \(g=sh\), using left Haar measure, gives \(\mu_{sx}=s_*\mu_x\). Isometry and uniqueness imply \(f(sx)=sf(x)\).

For two equivariant maps \(f_0,f_1\), the homotopy \(\gamma_{f_0(x),f_1(x)}(t)\) is continuous by geodesic interpolation and equivariant by uniqueness. Taking the two coordinate projections gives the projection homotopy. If \(H\) is compact, the measure obtained by averaging \(hy_0\) over normalized Haar measure on \(H\) has compact support and is \(H\)-invariant. Its unique minimizer is \(H\)-fixed. The fixed set is closed and convex; contraction along the geodesics from that fixed minimizer is a contraction through fixed points. \(\square\)

Example 3.3 (lattices). Let \(\mathbb Z^n\) act on \(\mathbb R^n\) by translations. Transporters of compact sets lie in a bounded subset of the discrete lattice, hence are finite. The quotient is compact because \([0,1]^n\) meets every orbit. Euclidean space is proper and complete, and (3.1) is the squared-norm identity. Theorem 3.2 proves it is a universal proper model in the stated category. Its map (3.4) is the explicit affine average \[ f(x)=\sum_{m\in\mathbb Z^n}b(m^{-1}x)m. \tag{3.5} \] Here \(y_0=0\). The weights sum to one and are locally finite, so \(f(sx)=s+f(x)\). In this example the geodesic homotopy is straight-line interpolation.

Example 3.4 (free groups). Let \(F_n\), \(n\ge1\), be the group of reduced words in \(a_1^{\pm1},\ldots,a_n^{\pm1}\). Its Cayley graph joins \(w\) to \(wa_i\); give each edge length one. Cancellation gives a unique reduced word between two vertices, so there is a unique simple path and no circuit. The geometric realization \(T_n\) is therefore a tree. Its degree is \(2n\), hence every bounded ball lies in a finite subtree. This proves properness of the metric and, by compactness of a ball containing a Cauchy sequence, completeness.

Here is a direct verification of (3.1). For a segment \([p,q]\) of length \(L\), let \(w\) be the branch point where the path from \(z\) meets that segment. Put \(a=d(p,w)\), \(h=d(z,w)\). Then \[ d(z,\gamma_{p,q}(t))=h+|a-tL|, \quad d(z,p)=h+a,\quad d(z,q)=h+L-a. \] The right side of (3.1) equals \[ h^2+(a-tL)^2+2h\bigl((1-t)a+t(L-a)\bigr). \] The last parenthesis is at least \(|a-tL|\): subtracting this absolute value gives \(2t(L-a)\) if \(a\ge tL\), and \(2(1-t)a\) otherwise. This proves the inequality for every point of the metric tree.

Left multiplication acts by isometries, freely on vertices and without edge inversions. An inversion of \([w,wa_i]\) would imply both \(gw=wa_i\) and \(gwa_i=w\), hence \(a_i^2=1\), which is impossible for reduced words. Thus every point stabilizer is trivial. Compact sets meet finitely many edges; only finitely many translates of either of their finite vertex sets can meet the other. If an edge-interior intersection occurs, their endpoint sets also meet after the same translation. Hence transporters are finite, proving the action proper. The finitely many edges \([1,a_i]\), together with their endpoints, meet every orbit, so it is \(F_n\)-compact. Theorem 3.2 proves that \(T_n\) is a universal proper model. The case \(F_2\) is the four-valent tree.

For a torsion-free discrete group, compact subgroups are finite and consequently trivial. Proper stabilizers are therefore trivial, so the family of proper actions coincides with the family of free proper actions. This is the reason \(\underline E\Gamma\) and the free universal model \(E\Gamma\) coincide in that case. In particular \(F_n\) is torsion free: conjugate a nontrivial reduced word to a cyclically reduced word \(v\); the reduced length of \(v^m\) is \(m|v|\) for every positive \(m\).

Example 3.5 (compact groups). If \(G\) is compact, its action on a point is proper. Every space has exactly one map to that point, so both existence and homotopy uniqueness are immediate. The point is already \(G\)-compact.

Theorem 3.6 (enlarged local-field buildings). Let \(k\) be a nonarchimedean local field and \(\mathbf G\) a connected reductive \(k\)-group. There is an enlarged building \(\mathcal B(\mathbf G,k)\) with a continuous, proper, cocompact action of \(\mathbf G(k)\). It is locally compact and second countable, and its metric is proper, complete and CAT(0). Its apartments retain the entire Euclidean factor belonging to the \(k\)-split centre. Consequently it is a universal proper model for locally compact sources.

Proof. The split integral coordinates and the nonreduced unitary calculations are proved in Sections BF.0–BF.7 below. Sections AB.0–AB.10 prove finite-extension valuations, the quotient topology, the apartment and metric comparisons, and integral unramified descent. Sections GB.0–GB.23 complete the relative-root construction and descent for an arbitrary connected reductive group. More precisely, GB.16 proves torus-cocycle vanishing, GB.17 the rational regular-class representative, GB.18 quasi-splitting over a finite unramified extension, GB.19 the anisotropic centralizer and its full central valuation lattice, GB.20 the exact rational root bounds, GB.21 the residue/coface correspondence, and GB.22 common maximal rational apartments and wall exchange. GB.23 proves the full stabilizers, proper quotient topology, equality with the inherited CAT(0) metric, and compactness of metric balls and of the orbit space. Apply Theorem 3.2 to this proper complete metric. \(\square\)

The enlargement is necessary even for \(\mathbf G=\mathbf G_m\). Its reduced building is a point, on which \(k^\times\) would have a noncompact stabilizer. Its enlarged building is \(\mathbb R\), with translation \(x\mapsto x-\omega(t)\); the kernel \(O_k^\times\) is compact and the valuation quotient is a lattice. Sections AB.10 and GB.19 prove the corresponding assertion for every torus and for the centralizer used in descent.

4. Equivariant K-homology with compact orbit support

Definition 4.1. For a proper locally compact \(G\)-compact space \(X\), put \[ K_i^G(X)=KK_i^G(C_0(X),\mathbb C),\qquad i\in\mathbb Z/2. \tag{4.1} \] A proper equivariant map \(f:X\to Y\) induces \(f_*\) by pulling functions back: \(f^*:C_0(Y)\to C_0(X)\). On a cycle it replaces \(\phi\) by \(\phi\circ f^*\), retaining its Hilbert space, action and operator. The cycle defects, direct sums and interval homotopies are preserved because every tested function is pulled back to \(C_0(X)\). Thus \(f_*\) is a group homomorphism, \((h\circ f)_*=h_*f_*\), and the identity map induces the identity. This describes covariance in spaces explicitly.

Proper equivariant homotopies induce equal maps. If \(H:X\times[0,1]\to Y\) is proper, \(a(H(x,t))\) is continuous and vanishes uniformly at infinity in \(x\): for a compact support of \(a\), properness puts all its preimages in one compact subset of the cylinder. Approximation by compactly supported functions gives the assertion for every \(a\in C_0(Y)\). The representations on the interval Hilbert module therefore give a Kasparov homotopy between the two pullback cycles. Lemma 2.3 makes every equivariant homotopy out of a \(G\)-compact proper cylinder proper into its image stage.

Definition 4.2. For a proper compactly generated ambient model \(E\), let \(\mathscr S(E)\) be its closed invariant, locally compact, second countable, \(G\)-compact subspaces. Set \[ RK_i^G(E)=\varinjlim_{Z\in\mathscr S(E)}K_i^G(Z). \tag{4.2} \] The transition map is induced by the closed inclusion. The stages are directed: the union of two stages is closed and \(G\)-compact; it is locally compact by taking neighborhoods with compact closure in the two closed summands, and second countable by the closed-map countable-base argument of Lemma 2.3 applied to their finite disjoint union. An element of (4.2) is represented on one stage, and two representatives agree exactly when their images agree on a larger stage. The notation refers to equivariant K-homology with \(G\)-compact supports; it is distinct from Kasparov's bivariant \(RKK\) notation for families over a base.

For \(\mathcal U\), the subspaces \(Z_K\) of Lemma 2.2 are cofinal: if \(Z\) is any stage, Lemma 1.2 gives compact \(K\subset Z\) with \(GK=Z\). Hence \[ RK_i^G(\mathcal U)=\varinjlim_{K\subset\mathcal U\ \mathrm{compact}} KK_i^G(C_0(GK),\mathbb C). \tag{4.3} \] Only locally compact spaces occur inside \(C_0\) in this formula.

Theorem 4.3 (canonical model independence). If \(E\) and \(F\) are proper Hausdorff compactly generated ambient spaces with the universal property for locally compact sources, their groups (4.2) are canonically isomorphic. These isomorphisms compose canonically for three models and respect equivariant cycles and proper maps of stages.

Proof. For each stage \(Z\subset E\), choose an equivariant map \(u_Z:Z\to F\). By Lemma 2.3 it is proper, and its image \(W\) is a stage of \(F\). Thus it induces \(K_i^G(Z)\to K_i^G(W)\to RK_i^G(F)\). For two choices \(u_Z,v_Z\), universal homotopy uniqueness gives a homotopy \(Z\times[0,1]\to F\); Lemma 2.3 makes its image a stage containing the two endpoint images, and the preceding proper-homotopy argument identifies the induced maps. The same argument for a stage inclusion proves compatibility with transition maps. Therefore these choices give a well-defined map \(\Phi_{E,F}\) of direct limits.

Choose the reverse maps stage by stage. On an original \(Z\subset E\), their composite is an equivariant map back into \(E\). It is equivariantly homotopic to the original inclusion \(Z\hookrightarrow E\); the homotopy image is again a stage by Lemma 2.3. Thus \(\Phi_{F,E}\Phi_{E,F}\) acts as the identity on every representative, hence on the limit. The other composite is identical by the same reasoning. With three models, the chosen composite stage maps are homotopic to the maps chosen directly; this proves \(\Phi_{F,T}\Phi_{E,F}=\Phi_{E,T}\). All equalities use source pullbacks and homotopies of actual locally compact stages, so they preserve the full Kasparov cycle equivalence relations. \(\square\)

If a locally compact model \(E\) is itself \(G\)-compact, its whole space is a final stage. The examples therefore give \[ RK_i^{\mathbb Z^n}(E\mathbb Z^n)=KK_i^{\mathbb Z^n}(C_0(\mathbb R^n),\mathbb C), \quad RK_i^{F_n}(EF_n)=KK_i^{F_n}(C_0(T_n),\mathbb C), \tag{4.4} \] and, for compact \(G\), \(RK_i^G(EG)=KK_i^G(\mathbb C,\mathbb C)\). These are identifications of the source groups; the assembly homomorphism and its further properties belong to the next lesson.

Proposition 4.4 (finite groups). If \(G\) is finite, then \[ RK_0^G(EG)=R(G),\qquad RK_1^G(EG)=0, \tag{4.5} \] where \(R(G)\) is the Grothendieck ring of finite-dimensional complex representations, with direct sum and tensor product.

Proof. Use the point model. The compact scalar-cycle comparison and its homotopy relations are proved in Equivariant KK-theory and the Green–Julg theorem, Theorem 4.3 and Proposition 6.1. Here is the concrete index description. Replace a scalar cycle by its unital summand, average its operator over \(G\), and take its self-adjoint part. These are equivariant compact perturbations of the represented operator and preserve its class, by Proposition 1.2 of that lesson. An even cycle now has an invariant Fredholm block \(T:H^+\to H^-\), with finite-dimensional invariant kernel and cokernel. On their orthogonal complements its polar unitary differs from \(T\) by a compact operator: the image of \(|T|\) in the Calkin algebra is one. This complementary cycle is degenerate. The class is therefore \[ [\ker T]-[\ker T^*]. \tag{4.6} \] Every finite representation occurs as a cycle on a finite-dimensional even Hilbert space with zero operator.

For completeness, independence and injectivity under module homotopy follow from the invariant compact algebra comparison in that preceding proof, rather than from any assumption that kernel dimensions stay constant. The regular representation contains every irreducible; averaging inner products gives orthogonal decompositions and Schur's lemma gives \[ \mathcal K(\mathcal H_{\mathbb C}^G)^G \cong\bigoplus_{V\in\widehat G}\mathcal K(M_V)\otimes1_V. \tag{4.7} \] There are finitely many \(V\), since the regular representation of the finite group is finite dimensional before adding infinite trivial multiplicity. The scalar-cycle comparison sends \(V\) to the rank-one projection in its multiplicity block. Matrix stability and the compact-operator K-theory calculation give one independent integer for each \(V\) in degree zero and zero in degree one. Thus (4.6) identifies the actual equivariant homotopy group with \(R(G)\). The product of the finite zero-operator cycles for \(V,W\) is the zero-operator cycle on \(V\otimes W\): its creation errors are compact and its positivity test is zero. Hence this is a ring identification. Equivalently, an odd invariant self-adjoint Fredholm operator is a compact perturbation of an invariant involution by spectral calculus, including an arbitrary sign on its finite kernel; that involution is degenerate. \(\square\)

5. Other universal models and geometric cycles

The following assertions give context. They are stated with their exact scope and are not used in the cutoff construction, Theorem 4.3, the examples or the building proof.

Stated, unused: recognition for numerable proper spaces. A space is numerable for the family of compact subgroups if it has an invariant open cover admitting equivariant maps to homogeneous spaces \(G/H\), \(H\) compact, and an invariant subordinate partition of unity. Within that category, a space \(E\) is a full universal proper model if and only if every compact \(H\) fixes a point and the two projections \(E\times E\to E\) are equivariantly homotopic. This is [Lück, Definition 2.1 and Theorem 2.5(ii)]. Its numerability hypothesis is part of the statement. In this lesson, existence of maps into \(\mathcal U\), Euclidean spaces, trees and the local-field building has been proved directly, rather than inferred from this recognition assertion.

Stated, unused: homogeneous and Rips models. If \(G\) is almost connected, meaning \(G/G^0\) is compact, it has a maximal compact subgroup \(K\), unique up to conjugacy, and \(G/K\) is a numerable universal proper model [Lück, Theorem 4.3]. This includes connected real reductive Lie groups and their symmetric-space models. If a finitely generated discrete group \(\Gamma\), with finite symmetric generating set \(S\), satisfies \[ d(x,y)+d(z,t)\le\max\{d(x,z)+d(y,t),d(x,t)+d(y,z)\}+2\delta \] for its word metric, the barycentric subdivision of its Rips complex \(P_d(\Gamma,S)\) is a universal proper \(\Gamma\)-CW model whenever the integer \(d\ge16\delta+8\) [Lück, Theorem 4.16]. Its simplices are finite sets of vertices of pairwise distance at most \(d\). Here “finite model” means finitely many cell orbits; the group itself can be infinite. The existence of maximal compact subgroups and the Rips contractibility argument are not proved here.

Stated, unused: the geometric comparison. For a finite CW-pair \((X,A)\), a Baum–Douglas cycle consists of a smooth compact spin\({}^c\) manifold \(M\) with boundary, a smooth Hermitian complex vector bundle \(V\) on \(M\), and a continuous map \(f:M\to X\) with \(f(\partial M)\subset A\). The relations are direct sum/disjoint union, spin\({}^c\) bordism and modification by an even-dimensional spin\({}^c\) vector bundle. Parity is the dimension of each component, with disjoint union separating even and odd components. The Dirac construction induces a natural isomorphism from this geometric group to analytic K-homology for finite CW-pairs [Baum–Higson–Schick, Definitions 5.1–5.7 and Theorem 6.2]. The free preprint proves this precise nonequivariant theorem. It is not an assertion here about arbitrary equivariant spaces. No part of the comparison theorem is needed to define (4.1)–(4.3).

6. Exercises and solutions

Exercise 6.1. Construct a squared-normalized cutoff for the translation action of \(\mathbb Z\) on \(\mathbb R\).

Solution. Put \(h(x)=\max(1-|x|,0)\) and \(c(x)=\sqrt{h(x)}\). It is continuous, nonnegative and supported in \([-1,1]\). Write \(x=k+t\), \(k\in\mathbb Z\), \(0\le t<1\). If \(0<t<1\), the only positive summands in \(\sum_m h(x-m)\) are \(1-t\) and \(t\); at \(t=0\) the only positive summand is one. Thus \(\sum_m c(x-m)^2=1\). Counting measure is the chosen Haar normalization. For \(\mathbb Z^n\), the product \(\prod_jc(x_j)\) has the same property because the finite orbit sum factors.

Exercise 6.2. Verify the two compact-fixed-point and projection-homotopy properties for the Cayley tree of \(F_2\), and prove the mapping property for proper locally compact sources.

Solution. The reduced-word argument in Example 3.4 proves \(F_2\) torsion free, so its only compact subgroup is the trivial group. The tree is nonempty and contracts along the unique paths to any chosen vertex. For the diagonal action, \((p,q,t)\mapsto\gamma_{p,q}(t)\) is a continuous equivariant homotopy from the first projection to the second. Finally, for any proper source \(X\), (3.4) is the unique minimizer of a finite sum of squared tree distances on every compact neighborhood of \(X\). Lemma 3.1 proves its existence and continuity; the change-of-variable argument proves equivariance. The same path homotopy joins any two maps. This verifies universality directly, including the mapping property.

Exercise 6.3. Prove that any two full universal proper models are equivariantly homotopy equivalent. Explain the corresponding statement for the locally compact-stage formulation.

Solution. Let \(E,F\) be models in the category on which their full universal property is imposed. Existence gives equivariant maps \(u:E\to F\), \(v:F\to E\). Homotopy uniqueness in \(E\) gives \(vu\simeq_G\mathrm{id}_E\); uniqueness in \(F\) gives \(uv\simeq_G\mathrm{id}_F\). Hence they are equivariantly homotopy equivalent. In particular this argument applies to the locally compact models whenever both are objects of the standing proper-space category. If an ambient model is only specified by its universal property for locally compact sources, global maps from that ambient space are not part of that weaker definition. Theorem 4.3 instead proves the canonical equivalence of all locally compact-stage K-homology limits, using precisely the map and homotopy properties available. Neither conclusion applies \(C_0\) to a non-locally-compact ambient space.

Exercise 6.4. For a finite \(G\), identify both groups \(RK_i^G(EG)\) and the ring multiplication in degree zero.

Solution. The point is a final \(G\)-compact stage. Thus the groups are \(KK_i^G(\mathbb C,\mathbb C)\). After invariant averaging, an even cycle has the index (4.6). The compact comparison (4.7) proves that irreducible representation classes form an independent integer basis and that every class has this form. Tensoring the finite zero-operator cycles gives the product, so \(RK_0^G(EG)=R(G)\) as rings. The degree-one group of (4.7) is zero; alternatively, the invariant involution replacement at the end of Proposition 4.4 makes every odd cycle degenerate. Hence \(RK_1^G(EG)=0\).

Algebraic and metric foundations for the building model

The following sections supply the proof of Theorem 3.6. Their order is by dependency: split root coordinates and compact point groups, affine chamber geometry and valuation descent, then rational relative roots and arbitrary-group descent. The complete central factor is retained at each stage. The labels BF, AB and GB distinguish the three sets of mathematical formulas and proof locators.

Split root filtrations and compact chamber groups

BF.0. Scope and exact algebraic inputs

Let k be any nonarchimedean local field, O its valuation ring, \(\pi\) a uniformizer, and \(\omega\) ( \(\pi\) )=1. Put |c|= \(q^{-\omega(c)}\) , where q is the finite residue-field cardinality. Thus O and its fractional ideals are compact, O is complete, and k has its valuation topology. These are part of the local-field hypothesis. The compactness can also be seen by choosing residue representatives: compatible finite truncations modulo \(\pi^n\) identify O with their inverse limit; the valuation metric gives the same topology. A product of finitely many O-balls is compact by a diagonal subsequence argument.

Let G be a connected split reductive k-group, T a split maximal torus, X= \(X^*\) (T), \(X^\vee\) = \(X_{*}\) (T), \(\Phi\) its reduced root system and \(a^\vee\) its coroots. The exact earlier programme inputs are:

• Roots and reductive groups of rank one, Lemma 3.1, Theorem 4.1 and Theorem 7.1: the integral big-cell open immersion, additive root parametrizations, the perfect opposite-root pairing, the rank-one \(SL_{2}\) homomorphism and its Gauss identity.
• Root data, Weyl chambers and the Bruhat decomposition, Theorem 2.1, Sections 3–5 and Theorem 4.1: the reduced root datum, finite reflection Weyl group W= \(N_{G}\) (T)/T with root-reflection representatives, and scheme coordinates on each positive unipotent group in every root order; multiplication U^- times T times U^+ is an open immersion. Section 5 proves the torus-weight commutator formula over arbitrary rings.
• Pinnings and the classification of split reductive groups, Theorem 5.1 and Theorem 10.1: the pinned group is obtained from a pinned split reductive group \(G_{Z}\) over Z. Theorem 5.1 proves the pinning-preserving comparison; its proof also proves root transport and normalizer conjugation. The opposite-root pairs in these sections are paired by the rank-one perfect pairing.
• Group schemes, actions and Hopf algebras, Theorem 2.1, Theorem 4.1 and Lemma 5.1: the affine Hopf dictionary, character weight decomposition over a ring, and finite subcomodules over a field.
• The earlier commutative-algebra programme supplies the elementary ring inputs: Noetherian and Artinian rings, Theorem 2.1, proves finite-type and localization Noetherianity; Regular sequences, depth and Cohen–Macaulay modules, Theorem 6.1, proves that regular local rings are domains; Smooth algebras over a field and the Jacobian criterion, Theorem 2.1, proves that smooth local rings are regular. The short DVR and finite-lattice arguments actually needed are also given in BF.4.

The root-valuation conventions are those of Bruhat–Tits I, Definition 6.2.1 and Examples 6.2.3(a)–(b). Integral root coordinates are also discussed in Demazure, SGA 3, Exposé XXIII, Sections 1.1–1.2, and Conrad, Reductive group schemes, Lemma 6.2.8. The preceding lessons named above prove the algebraic inputs; the filtration, compactness and chamber arguments are proved below.

Choose root coordinates \(x_{a}\) : \(\mathbf G_{a} \to U_{a}\) from integral root frames of \(G_{Z}\) , with each negative frame paired with its positive frame. Changing an integral frame only multiplies it by a sign. Consequently a Weyl representative \(n_{w}\) formed from the integral rank-one representatives carries \(x_{a}\) (u) to \(x_{wa}\) ( \(\epsilon\) u), \(\epsilon\) =±1: it is an isomorphism of integral root lines, hence its scalar is a unit of Z. The earlier pinned comparison transports these coordinates to G.

Write V= \(X^\vee\) tensor R. Its central part is \(V_{Z}\) =intersection ker(a), and its derived part is the span \(V_{\mathrm{der}}\) of the coroots. These form a direct sum. Indeed average any inner product over the finite W. Each reflection \(s_{a}\) fixes ker(a) and has minus-one line R \(a^\vee\) , so \(a^\vee\) is perpendicular to ker(a). Therefore \(V_{\mathrm{der}}\) is perpendicular to \(V_{Z}\) ; their dimensions are complementary since the roots span the dual of the coroot space. W acts trivially on \(V_{Z}\) . A point of the enlarged apartment is x=(v,z) in \(V_{\mathrm{der}}\) plus \(V_{Z}\) .

BF.1. Split valuations, all residue characteristics

For v in \(V_{\mathrm{der}}\) define

and for any real r,

The values of nonidentity elements are Z+a(v). Additivity of \(x_{a}\) and the nonarchimedean triangle inequality show that every upper level set is a subgroup. Only the identity has infinite value. The displayed fractional ideal is compact and open in k; \(x_{a}\) is an algebraic coordinate isomorphism, so these groups are compact and open in \(U_{a}\) and shrink to its identity as r tends to infinity.

Here and below “open root subgroup” means open in its root group, not open in G.

For linearly independent roots a,b choose a positive system containing both. In integral positive-root coordinates the commutator has the form

To recall why the exact integrality needed here follows from the earlier coordinate proof, form the commutator morphism over Z. Its root coordinates are polynomials in u,t. Under torus conjugation a monomial \(u^p\) t^q has character pa+qb. Setting either variable to zero makes the commutator the identity, so p,q are positive. Independence of a,b gives at most one exponent pair for each root. Coefficients of this Z-morphism are integers. This proves BF.2 without assuming they are units or dividing by any root-string constant. If u and t have respective values at least r and s, then

Thus the commutator is contained in the subgroup generated by \(U_{pa+qb,v,pr+qs}\) . A coefficient which vanishes or has positive valuation only improves this bound. This includes types B,C,F in residue characteristic two and type \(G_{2}\) in residue characteristics two and three.

The rank-one matrix calculation is

because its \(SL_{2}\) matrix is

Both opposite factors have value - \(\omega\) (u)-a(v)=- \(\varphi_{v,a}\) ( \(x_{a}\) (u)). These are also the only two opposite factors which can complete \(x_{a}\) (u) to a reflection representative: in the \(SL_{2}\) product the two diagonal entries must vanish. The rank-one central quotient does not change that condition or the root coordinates, by Roots and reductive groups of rank one Theorem 7.1.

Every normalizer element with Weyl image \(s_{a}\) is t \(n_{a}\) (1). Conjugation carries \(x_{-a}\) (c) to \(x_{a}\) (-a(t)c); the change between its two valuations is constant in c. Finally

so torus conjugation adds \(\omega\) (a(t)) to the value. These prove all the reduced valuation axioms of Bruhat–Tits I Definition 6.2.1: at least three values, subgroup levels, constant opposite-reflection change, the commutator bound, and the opposite-factor condition. The doubled-root axiom is absent for a reduced root system. No characteristic was excluded.

BF.2. The apartment, origins and the sign of translations

The family { \(\varphi_{v}\) :v in \(V_{\mathrm{der}}\) } is an affine space: the difference \(\varphi_{v'}\) - \(\varphi_{v}\) on \(U_{a}\) minus its identity is a(v'-v), and roots separate \(V_{\mathrm{der}}\) . Let w be the Weyl image of n. Define

Translating \(v\) changes the value by \(a(wv)\). Thus \(n(\varphi_v)=n(\varphi_0)+wv\). For \(t\in T(k)\), let \(\widetilde\nu(t)\in V\) satisfy

and let \(\nu\) (t) be its derived projection. BF.4, using \(t^{-1}\) rather than t, gives

In particular BF.3 gives

Its fixed wall is a(v)+ \(\omega\) (u)=0. This also directly checks the sign of the affine reflection.

The affine family is independent of pinning. To prove this, prescribe the ratios \(c_{\alpha}\) in \(k^*\) of new to old simple frames. The simple roots are a basis of the root lattice Q, so they define a homomorphism c:Q \(\to k^*\) . Over a finite field extension there is t in T with \(\alpha\) (t)= \(c_{\alpha}\) for all simple \(\alpha\) : put the inclusion Q \(\subset\) X into integer Smith normal form and adjoin the finitely many roots required to solve the resulting monomial equations. This permits inseparable roots; no separability is asserted or needed. Torus conjugation and the uniqueness of the pinned comparison imply that the ratio for any root a is c(a), up to a harmless integral sign arising from the integral frame choices. Hence \(\omega\) (c(a))=a(d) for one d in \(V_{\mathrm{der}}\) . The new coordinates give \(\varphi'_v\)= \(\varphi_{v-d}\) as functions on the same root groups. Their entire affine families are equal; only their origins differ.

BF.3. The enlarged normalizer apartment is universal proper

On V= \(V_{\mathrm{der}}\) plus \(V_{Z}\) use the action

This is a group action. Products of the integral Weyl representatives have the same finite Weyl effect, and their residual torus factor is integral with integral inverse, hence has zero valuation. Torus conjugation transforms \(\widetilde\nu\) equivariantly. Thus the action identifies

For detail, a basis of X identifies T(k) with ( \(k^*\) )^rank X. Coordinate valuations are arbitrary integers, their kernel is ( \(O^*\) )^rank X=T(O), and each cocharacter \(\lambda\) is realized by \(\lambda\) ( \(\pi\) ). Every normalizer element is t \(n_{w}\) , by the earlier Weyl comparison. A normalizer element acts trivially in BF.7 exactly when w=1 and its torus valuations vanish. This proves BF.8 including its kernel assertion. T(O) is compact and open in N; the quotient is discrete.

Choose a W-invariant Euclidean norm on V. For compact sets K,L \(\subset\) V, any affine element \(\lambda\) w with ( \(\lambda\) w)K intersecting L has \(\lambda\) in the bounded set L-WK. A lattice has finitely many points in a bounded set: in a lattice basis each integer coordinate is bounded. The transporter in N is therefore a closed subset of a finite union of compact T(O)-cosets and is compact. This is properness of the N-action. Its orbit space is compact because the full translation lattice \(X^\vee\) has a compact fundamental parallelepiped. A point stabilizer projects to a finite subgroup of the discrete affine group and is a finite union of T(O)-cosets; it is compact and open.

A compact subgroup H \(\subset\) N has finite image in the discrete quotient. Average any point over that finite affine group. The result is H-fixed. The entire fixed set is a nonempty affine subspace and is contractible by straight interpolation. The two projections V times V \(\to\) V are N-homotopic by the same interpolation.

Here is a direct universal-property proof. A numerable proper N-space has an invariant locally finite partition of unity ( \(\lambda_{i}\) ) with support( \(\lambda_{i}\) ) contained in invariant open sets \(Y_{i}\) , each with an equivariant map \(p_{i}\) : \(Y_{i} \to\) N/ \(H_{i}\) for a compact \(H_{i}\) . Choose \(z_{i} \in V^{H_{i}}\) . The map \(nH_{i} \to nz_{i}\) is well defined and continuous. Then

is a continuous N-map: a point outside \(Y_{i}\) has a neighborhood disjoint from support( \(\lambda_{i}\) ), so its term extends continuously by zero, and the sum is locally finite, and affine maps preserve a convex combination whose weights sum to one. Two such maps are N-homotopic by straight interpolation, as are any two N-maps to V. This proves universality in the numerable proper category. For proper N-CW spaces the same conclusion follows cell by cell: each compact-isotropy cell N/H times \(D^m\) maps into the nonempty contractible affine space \(V^H\) ; radial extension in that affine space extends boundary maps and homotopies. Thus V is the usual enlarged model for the universal proper normalizer space, with its full central translation directions retained.

BF.4. A local integral representation lemma

The following representation lemma supplies the linear algebra for compactness of the point groups.

Lemma. A split reductive group scheme H over the complete local-field DVR O admits a closed immersion into GL(L) for a finite free O-module L. Every specified split torus in H acts on L through an O-direct sum of character-weight modules.

Proof. Let A=O[H]. It is a finitely generated flat O-algebra. Its generic and special fibres are geometrically integral. Over an algebraically closed field, smoothness gives regular local rings, and the exact earlier ring proofs in BF.0 make these domains. Distinct irreducible components therefore cannot intersect: at an intersection the local ring would have two minimal primes. There are finitely many components, so each is open and closed. Connectedness leaves one component, and the local domain assertion also gives reducedness. Thus a smooth connected affine algebraic group is integral. Reductivity supplies precisely smoothness and geometric connectedness here. Flatness embeds A into its integral generic fibre, so A is a domain, and A/ \(\pi\) A is a domain. Consequently the localization \(A_{(\pi)}\) is a Noetherian local domain with maximal ideal generated by \(\pi\) . It is a DVR. Indeed its intersection of powers of \(\pi\) is zero: their intersection J is an ideal, hence finitely generated, and J= \(\pi\) J by division. For generators of J write this equality as v= \(\pi\) Mv. The determinant of I- \(\pi\) M is a unit in the local ring; its adjugate then forces every generator to be zero. Thus J=0. Repeated division then assigns every nonzero element a finite \(\pi\) -order, and a remainder outside the maximal ideal is a unit. This defines a valuation extending \(\omega\) on k.

Choose finitely many algebra generators of A. By the earlier finite-comodule lemma Group schemes, actions and Hopf algebras, Lemma 5.1 they lie in a finite-dimensional k-subcomodule E \(\subset\) A[ \(\pi^{-1}\) ]; include 1. Set L=E \(\cap\) A. It is an O-lattice in E. Here is the needed boundedness proof: the DVR valuation just defined gives a genuine norm on E. In a k-basis, the triangle inequality gives an upper bound by a constant times the coordinate maximum norm. It also gives continuity. The coordinate unit sphere is compact, by compactness of O, so that positive continuous norm has a positive minimum there. Thus its unit ball is contained in a bounded coordinate lattice. Since A lies in that valuation's unit ball, L is bounded; clearing the finitely many basis denominators makes it full. It is therefore finite free over the DVR.

L is saturated in A. Both A and A/L are torsion-free, hence flat over O: each is a filtered union of its finitely generated torsion-free submodules, which are free by DVR row elimination; tensoring an injection remains injective first on each free submodule and then in the filtered union. For l \(\in\) L, its coaction belongs to A tensor A, and after inverting \(\pi\) its first factor belongs to E=L tensor k. Its image in (A/L) tensor A is therefore \(\pi\) -torsion and zero, since that tensor product is flat. Exactness after tensoring proves Delta(L) \(\subset\) L tensor A. Thus L is an integral H-comodule.

In an O-basis write Delta( \(l_{j}\) )= \(\sum_{i} l_{i}\) tensor \(a_{ij}\) . Counit and antipode give an invertible coefficient matrix and a group-scheme map H \(\to\) GL(L). Applying counit in the first factor writes every \(l_{j}\) as an O-linear combination of the \(a_{ij}\) . All chosen algebra generators of A belong to L, so the matrix-coefficient algebra surjects onto A. This is a closed immersion. Finally the split-torus weight projections are obtained by comparing Laurent monomials in the coaction, precisely as in Group schemes, actions and Hopf algebras, Theorem 4.1. They split L into finitely many projective, hence free, weight modules. \(\square\)

BF.5. Compact point groups and their normalizer intersections

For x=(v,z) \(\in\) V let \(P_{v}\) be generated by T(O) and \(U_{a,v,0}\) for every root. Let \(N_{x}\) be the stabilizer for BF.7 and put \(K_{x}\) = \(P_{v} N_{x}\) . Then \(P_{v}\) and \(K_{x}\) are compact open subgroups of G(k), with

Proof. Apply BF.4 to the integral pinned model over O. Choose an O-basis of each character-weight module \(L_{\lambda}\) . On L tensor k define

The expansion rho( \(x_{a}\) (u))= \(\sum_{i\ge 0} u^i R_{i}\) has integral matrices and \(R_{0}\) =1. Torus equivariance says that \(R_{i}\) maps the \(\lambda\) -weight module to the ( \(\lambda\) +ia)-weight module. This follows by comparing Laurent characters over O, rather than by differentiating; it remains true in positive characteristic. If \(\omega\) (u)+a(v) \(\ge\) 0, each term has norm bound

because a(z)=0. The inverse is \(x_{a}\) (-u), with the same bound. Thus every generating root element is an isometry of BF.10. T(O) is an isometry as well. The isometry group of this finite-dimensional norm is compact: entries of an operator and its inverse are bounded by fixed constants in the chosen basis; the pairs (M, \(M^{-1}\) ) form a closed subset of a finite product of compact matrix balls. The closed immersion in BF.4 is a topological embedding on k-points, since its inverse coordinate functions are regular on its image. Its intersection with this compact isometry group is compact.

\(P_{v}\) contains an open neighborhood of the identity. The big-cell coordinates identify a sufficiently small product of root neighborhoods and a torus neighborhood contained in T(O) with such a neighborhood; all root levels here contain sufficiently small fractional ideals. Therefore \(P_{v}\) is an open subgroup. An open subgroup is closed, since its complement is a union of open cosets. \(P_{v}\) is consequently a closed subset of the compact isometry intersection and is compact.

Write n=t \(n_{w}\) . The integral map rho( \(n_{w}\) ) identifies the weight lattices by integral invertible matrices, so it preserves their coordinate maximum norms. On a vector of weight \(\lambda\) , rho(n) sends that lattice isometrically to its w- \(\lambda\) lattice and multiplies it by (w \(\lambda\) )(t). The resulting BF.10 norm ratio is

If n \(\in P_{v}\) , it is an isometry, so this ratio is 1 for every occurring weight \(\lambda\) . Equivalently \(\lambda\) (x)= \(\lambda\) ( \(w^{-1}\) (x+ \(\widetilde\nu\) (t))). The occurring weights span X tensor R: otherwise a positive-dimensional subtorus would act trivially in the closed faithful representation. This forces wx- \(\widetilde\nu\) (t)=x, hence n \(\in N_{x}\) . Notice that this proves the full central condition, not merely the derived condition.

BF.1–BF.2 show that normalizer conjugation transports every level group and T(O) to the corresponding level at nx, proving the first equality of BF.9. In particular \(N_{x}\) normalizes \(P_{v}\) . Thus \(K_{x}\) is a subgroup. The compactness of \(N_{x}\) was proved in BF.3; explicitly, in the Weyl component t \(n_{w}\) its fixed-point equation is \(\widetilde\nu\) (t)=wx-x. Each such fibre is empty or a coset of the compact kernel T(O). There are finitely many w, so \(N_{x}\) is compact. Consequently \(K_{x}\) is compact as the product \(P_{v} N_{x}\) and is open because it contains \(P_{v}\) . Normalizer conjugation gives its covariance. If p \(n_{0} \in K_{x} \cap\) N with \(n_{0} \in N_{x}\) , then p \(\in P_{v} \cap\) N \(\subset N_{x}\) ; hence p \(n_{0} \in N_{x}\) . The reverse inclusion is immediate. Pinning independence follows because the generating root level sets are functions of the valuation point, as proved in BF.2. \(\square\)

BF.6. Split chamber factorization in every root order

Remove the affine hyperplanes a(v)+m=0, m \(\in\) Z, from \(V_{\mathrm{der}}\) . Let C be one of the resulting open chambers. The integer

is constant for v \(\in\) C: it can change only on a removed hyperplane. For every root, \(f_{C}\) (a)+a(v)>0 and \(f_{C}\) (a)+ \(f_{C}\) (-a)=1. Define \(I_{C}\) to be generated by T(O) and \(x_{a}\) ( \(\pi^{f_{C}(a)}\) O). For any positive system and any orders of its positive and negative roots, multiplication is a homeomorphism

In particular \(I_{C}\) is contained in the big cell; its root-block factors are exactly its intersections with U^- and U^+.

Proof. We first prove that every finite word in the prescribed root generators lies in the big cell with precisely the required root bounds.

Fix roots \(a_{1}\) ,…, \(a_{m}\) , with repetitions permitted, and form over Z the word morphism

Let the split integral torus act on \(t_{i}\) with character \(a_{i}\) . F is equivariant for conjugation. Let Omega=U^- T U^+ be the integral big cell. Its inverse root coordinates have their corresponding root characters, while torus characters on its T-factor are conjugation-invariant. Let J be the radical ideal of the closed complement of \(F^{-1}\) (Omega) in R=Z[ \(t_{1}\) ,…, \(t_{m}\) ]. This ideal is homogeneous. For a completely algebraic check, the action on the product with the split torus is the Laurent-ring automorphism \(t_{i} \mapsto t_{i} X^{a_{i}}\) of R[ \(X_{1}^{±1}\) ,…, \(X_{d}^{±1}\) ]. Equivariance of F and conjugation invariance of Omega make that closed complement invariant. Its extended ideal is radical, because the Laurent polynomial ring over R/J is reduced, so invariance of the closed set preserves that ideal. For f \(\in\) J its image is \(\sum_{\chi} f_{\chi} X^\chi\) . Unique Laurent coefficients imply \(f_{\chi} \in\) J for every character \(\chi\) . This proves homogeneity without division or averaging. The zero section lies in \(F^{-1}\) (Omega). Pulling J back along that section gives the unit ideal of Z, so J contains a polynomial D with D(0)=1. Taking its weight-zero part retains membership in J and the value 1. Thus choose D of weight zero. The principal open D(D) lies in \(F^{-1}\) (Omega).

For the a-root coordinate, pullback along F on that principal open has form \(N_{a}\) / \(D^h\) , with \(N_{a}\) an integral polynomial of character a and \(N_{a}\) (0)=0. To justify homogeneity, use the character grading of Z[ \(t_{i}\) , \(D^{-1}\) ], where D has degree zero; the homogeneous component of any representative of the specified character is another representative. A torus character \(\chi\) and its inverse pull back similarly with weight zero and value1 at zero.

Now substitute \(u_{i}\) with \(\omega\) ( \(u_{i}\) )+ \(a_{i}\) (v)>0. Each nonconstant weight-zero monomial has valuation

Consequently D(u) \(\in\) 1+ \(\pi\) O and is a unit. F(u) therefore lies in Omega(k). Each monomial of \(N_{a}\) has weight \(\sum_{i} j_{i} a_{i}\) =a, and hence

Integral coefficients have nonnegative valuation; the denominator is a unit. The nonarchimedean inequality gives the same strict bound for the root coordinate. Its valuation is an integer, so that bound is exactly \(\omega\) (root coordinate) \(\ge\) ceil(-a(v))= \(f_{C}\) (a). For a torus character and its inverse, the constant term is1 and every other term has positive valuation. Thus all torus coordinates are units, so its torus factor belongs to T(O).

This argument uses a finite polynomial on a finite word; no convergence, perfect-residue assumption, general Iwahori theorem, or ramified-extension descent is hidden in it. A torus-unit factor in a word can be moved to its left using torus conjugation: this only multiplies root parameters by units. Inverses of root generators are \(x_{a}\) (-u), which have the same bound. Therefore the entire generated group \(I_{C}\) has exactly the ordered factorization asserted in BF.11. Conversely every such factor is one of its generators. Uniqueness follows from the earlier integral big-cell open immersion and the root-coordinate isomorphisms in every order. Both maps are continuous, since the inverse is given by regular big-cell coordinates. Thus BF.11 is a homeomorphism.

If \(v_{0} \in\) closure C, then \(f_{C}\) (a)+a( \(v_{0}\) ) \(\ge\) 0 by taking limits. Hence every \(I_{C}\) generator lies in \(P_{v_{0}}\) , so \(I_{C} \subset P_{v_{0}}\) . By BF.5 \(P_{v_{0}}\) is compact. \(I_{C}\) is open: its coordinate product contains a neighborhood of the identity by the big-cell chart. The quotient \(P_{v_{0}}\) / \(I_{C}\) is therefore both compact and discrete and is finite. This proves the claimed finite index, including points on several walls. \(\square\)

BF.7. The nonreduced unitary rank-one valuation

Let L/k be any separable quadratic extension, with involution u \(\mapsto\) bar(u). Extend \(\omega\) to L with its restriction to k normalized; its value group is \(e^{-1}\) Z, where e is the ramification index. The following proof covers ramified and unramified extensions, residue characteristic two and field characteristic two.

First, \(\omega\) (bar(u))= \(\omega\) (u). To supply the field fact used here, take a k-basis of L. Its coordinate maximum norm bounds |u| from above by the ultrametric inequality. Thus |u| is continuous in those coordinates; on the compact coordinate unit sphere it is positive and has a positive minimum. Scaling by powers of \(\pi\) proves the reverse bound. Every k-linear map on L is consequently bounded. In particular |bar(u)| \(\le\) B|u| for some B. Apply this to \(u^n\) and take nth roots, then let n tend to infinity; this gives |bar(u)| \(\le\) |u|. Applying the same argument to bar(u) gives equality. The norm comparison also proves that all closed valuation balls in L are compact and have their usual coordinate topology.

Choose \(\lambda\) with \(\lambda \ne\) bar( \(\lambda\) ), put \(\delta\) = \(\lambda\) -bar( \(\lambda\) ), and theta= \(\lambda\) / \(\delta\) . Then

Indeed the trace is onto since Tr(theta)=1; its kernel is one-dimensional and contains \(\delta\) . These statements remain valid in characteristic two, where \(\delta\) is fixed by the involution and \(L^0\) =k. The pairs satisfying v+bar(v)=-u bar(u) are exactly

This is also a coordinate homeomorphism L times k \(\to\) H. It uses division by the nonzero \(\delta\) , not division by two.

Put

Write a(s)=s. For (u,v) \(\in\) H use exactly the coordinates

Matrix multiplication verifies their unitary equations. Conversely, the unitary equation on a triangular unipotent matrix forces precisely these coordinates and the trace relation. Thus \(U_{a}\) = \(x_{+}\) (H), \(U_{-}\) a= \(x_{-}\) (H), \(U_{2a}\) = \(x_{+}\) (0, \(L^0\) ), and \(U_{-}\) 2a= \(x_{-}\) (0, \(L^0\) ). Over L, the algebra L \(\otimes_{k}\) L is L times L with the involution exchanging the factors; the unitary equation identifies the second matrix with J times the inverse transpose of the first times J. The determinant condition therefore identifies the base-changed group with \(SL_{3}\) . In particular this is the smooth connected reductive unitary group.

For clarity its k-split rank is one: characters of any k-split torus acting faithfully on \(L^3\) pair under the invariant nondegenerate Hermitian form with their negatives. In dimension three there is at most one nonzero pair, so these characters span a space of dimension at most one. S attains that rank. Its weights are a,0,-a; the upper u tangent coordinates have weight a and the trace-zero v coordinates weight 2a, both nonzero. This assertion compares torus characters, not their derivatives in k, so characteristic two does not identify a with 2a. The negative groups give their negatives, and differences of the three representation weights give no other nonzero root. Hence the relative root system is {a,2a,-a,-2a}.

For either sign, direct multiplication gives

where the commutator is \(ghg^{-1} h^{-1}\) . Expanding the trace verifies closure. The doubled-root group is central; it is the entire centre since for u \(\ne\) 0 choosing w=u \(\lambda\) makes the last parameter nonzero. It is also the derived group: with u=1, the map w \(\mapsto\) bar(w)-w is a nonzero k-linear map onto \(L^0\) , and BF.13 supplies a root element for every w. The quotient by that centre is the additive group of L.

If v=0, the trace relation forces u=0. For every pair in H,

At the identity both sides have their extended infinite values. Define

with infinity only at the identity. A short-root level r is the set \(\omega\) (v) \(\ge\) 2r. BF.16 then gives \(\omega\) (u) \(\ge\) r. The cross term in BF.15 has valuation at least 2r for two such elements, proving that this is a subgroup. Inversion preserves the value. Long-root levels are additive balls in \(L^0\) , so are subgroups as well. The short-root level is the closed subset H of the product of the two compact balls \(\omega\) (u) \(\ge\) r and \(\omega\) (v) \(\ge\) 2r; it is compact and open in H. As r increases both coordinates tend to zero, so these sets give exactly the coordinate neighborhood basis. The long-root assertion follows identically in \(L^0\) . Nonidentity finite value sets are discrete; they need not be all of (2e)^{-1}Z. The elements x_±(0, \(\delta \pi^n\) ) give infinitely many values, while the exact long-root set is \(\omega\) ( \(\delta\) )+Z.

The doubled-root identity is exactly \(\varphi_{2a}\) =2varphi_a on \(U_{2a}\) , and similarly negatively. BF.15–BF.16 also give

Commutators with the corresponding doubled-root group vanish. These exhaust the valuation commutator axiom: opposite-ray pairs are excluded in that axiom, and a+a=2a is its only possible positive-integer root sum on one ray, apart from the negative counterpart.

We now prove the opposite-factor axiom for every possible normalizer factorization, including its uniqueness. The S-weight lines show that its centralizer is diagonal and its normalizer either fixes or exchanges the two outer lines. An automorphism of the one-dimensional split torus sends its character to ±a: an invertible group morphism on k[t, \(t^{-1}\) ] has t \(\mapsto t^{±1}\) . Solving the determinant and unitary equations consequently gives every normalizer matrix as one of

The outer column pairings of M(v) are 1, its middle column has norm 1, and its determinant is 1, verifying membership directly.

For \(x_{+}\) (u,v) \(\ne\) 1 put V=bar(v). The two opposite factors are

They belong to H because 1/V+1/v=-u bar(u)/(vV). Direct multiplication first gives

and then y' \(x_{+}\) (u,v)y''=M(v). Both factors have value - \(\omega\) (v)/2=- \(\varphi_{a}\) ( \(x_{+}\) (u,v)). They are the only possible factors even if their product is merely required to belong to N: in \(x_{-}\) (p,q) \(x_{+}\) (u,v) \(x_{-}\) (r,s), the (1,3) entry is v \(\ne\) 0, so a normalizer product must be antidiagonal. Vanishing of (1,2) forces r=-u/V, then (1,1) forces s=1/V, (2,3) forces p=-u/v and (3,3) forces q=1/V. This proves the axiom for every permissible product. For a long-root input u=0, V=-v; both opposite factors are \(x_{-}\) (0,-1/v), with value - \(\omega\) (v). Conjugation by J proves both negative-root cases. None of these computations divides by two.

The normalizer conjugations, with all parameters shown, are

Every right-hand pair satisfies the trace relation: the norm multiplier on its first coordinate is exactly the multiplier on its second. The short-root valuation changes for D(t) are respectively + \(\omega\) (t) and - \(\omega\) (t); for M(v) they are respectively - \(\omega\) (v) and + \(\omega\) (v). Long-root changes are twice these. Thus the required normalizer difference is constant on each nonidentity root group.

On the real apartment with a(h)=h, the action is

BF.21 verifies filtration covariance directly. This is a group action: matrix multiplication gives

and applying \(\omega\) checks all four affine compositions. For the reflection in BF.20 put \(r=\varphi_a(x_+(u,v))\). Its action is \(h\mapsto-h-2r\), with fixed wall \(a(h)+r=0\). The doubled-root wall is \(2a(h)+\varphi_{2a}=0\) and gives the same reflection. For a negative root the wall is \(-a(h)+\varphi_{-a}=0\), with the analogous doubled-root formula.

At any other origin add a(h)=h to the positive short-root valuation,2h to its double, and their negatives to the opposite roots. Subgroup and topology assertions merely relabel their levels. In BF.18 the two short-root shifts add to the doubled-root shift. In BF.20 the opposite shifts are exactly the negative of the input shift. The normalizer differences remain constant. Thus all valuation axioms and topology hold at every real origin, with no ramification or characteristic restriction. The real factor 1/2 in BF.17 never represents division by two in k or L. \(\square\)

Affine apartments and unramified descent

AB.0. Finite separable extensions of a local field

Theorem. Let \(k\) be a nonarchimedean local field, with complete discrete valuation ring \(O\), uniformizer \(\pi\), and normalized valuation \(\omega_k\). Every finite separable extension \(L/k\) has a unique real-valued valuation extending \(\omega_k\). Its valuation ring is the integral closure \(B\) of \(O\) in \(L\). This ring is a complete DVR, finite free of rank \([L:k]\) over \(O\). If \(\Pi\) is its uniformizer and \(\pi=\varepsilon\Pi^e\) with \(\varepsilon\in B^*\), then

Writing \(f=[B/(\Pi):O/(\pi)]\), one has \([L:k]=ef\). The extended valuation topology is the finite-dimensional \(k\)-vector topology. In particular \(L\) is complete, its valuation balls are compact, and every \(k\)-automorphism of \(L\) preserves \(\omega_L\).

Proof. We first prove the finiteness assertion without assuming a valuation on \(L\). Elements integral over a ring form a ring: if \(x,y\) satisfy monic equations, the powers of \(x,y\) below those two degrees span a finite module stable under both elements. The determinant identity for the corresponding multiplication operators gives monic equations for \(x+y\) and \(xy\). The same argument proves transitivity of integrality by adjoining the finitely many coefficients of a monic equation first. The ring \(O\) is integrally closed in \(k\): a monic equation for an element of negative valuation has its leading term of strictly smaller valuation than every other term, which cannot sum to zero.

Choose a \(k\)-basis \(b_1,\ldots,b_n\) of \(L\) consisting of integral elements. Such a basis is obtained by multiplying an arbitrary basis by sufficiently large powers of \(\pi\), clearing the coefficients of its finitely many minimal polynomials. The trace pairing on \(L\) is nondegenerate. Here is the field argument: since \(k\) is infinite and the extension is separable, a linear combination of a basis can be chosen outside the finitely many hyperplanes on which two distinct embeddings agree. It has \(n\) distinct conjugates and hence generates \(L\). For its power basis, the matrix of evaluations at these conjugates is a Vandermonde matrix with nonzero determinant; the trace matrix is its transpose times itself and is invertible.

If \(x\in B\), then \(\operatorname{Tr}_{L/k}(xb_i)\in O\). Indeed each conjugate of an integral element is integral, their sum is integral, and a trace lies in \(k\), where integral elements belong to \(O\). The nondegenerate trace pairing therefore places \(B\) inside the finite free trace-dual lattice of \(\sum_i O b_i\), while that latter lattice is contained in \(B\). A submodule of a finite free module over the DVR is finite free, without assuming a generator list. Project a submodule of \(O^n\) to its first coordinate. Its nonzero image ideal has a generator of minimum valuation; lift that generator to the submodule. Its free rank-one span splits off, since the first coordinate of every other element is a unique multiple of the chosen generator. The remaining kernel lies in O^(n \(-\) 1), so induction proves finite freeness. If the image ideal is zero, apply the induction directly to that kernel. Thus \(B\) is finite free of rank \(n\), and is complete for its \(\pi\)-adic topology. It is a domain and is integrally closed in \(L\), by transitivity of integrality. Also \(B[\pi^{-1}]=L\).

We next show that \(B\) is local. Every maximal ideal contains \(\pi\). Otherwise its nonzero residue field, a finite \(O\)-module, would satisfy \(M=\pi M\). On finite generators this gives a matrix equation with coefficient matrix \(1-\pi A\); its determinant is an \(O\)-unit and annihilates \(M\), a contradiction. The finite ring \(B/\pi B\) is an Artinian algebra over the finite residue field. If it had two maximal ideals, it would have a nontrivial idempotent. Explicitly, its nilradical is nilpotent: choose finitely many nilpotent generators and use the pigeonhole principle on products. Chinese remainders for powers of the finitely many distinct maximal ideals then express the ring as a product of two nonzero rings and supply that idempotent.

Every idempotent modulo \(\pi B\) lifts to \(B\). If \(z^2-z\in\pi B\), then \(2z-1\) is invertible: its square is \(1+4(z^2-z)\), and \(1+\pi B\) consists of units by the convergent geometric series. Newton's update

squares the order of the error. Completeness gives an idempotent with the specified reduction. This argument is valid in characteristic two. A domain has only the idempotents zero and one, so \(B/\pi B\), and therefore \(B\), is local.

For completeness we prove the DVR assertion too. A nonzero prime of \(B\) contains \(\pi\): otherwise its localization would be a prime of the field \(B[\pi^{-1}]\), and injectivity of that localization would make it zero. Primes containing \(\pi\) are maximal, since the quotient is a finite-dimensional residue algebra. Thus the sole nonzero prime is the maximal ideal \(\mathfrak m\). Choose \(0\ne a\in\mathfrak m\). Then \(\sqrt{(a)}=\mathfrak m\); finitely many generators give \(\mathfrak m^d\subset(a)\) for some \(d\). For minimal \(d\), take \(b\in\mathfrak m^{d-1}\setminus(a)\) and put \(u=b/a\). Then \(u\notin B\) but \(u\mathfrak m\subset B\). If \(u\mathfrak m\subset\mathfrak m\), the determinant identity on this finite nonzero ideal makes \(u\) integral over \(B\), contrary to normality. Hence \(u\mathfrak m=B\), and \(\mathfrak m=u^{-1}B\) is principal. Write its generator as \(\Pi\). The intersection \(J=\bigcap_d\Pi^dB\) satisfies \(J=\Pi J\) by cancellation in the domain. It is finite, so the same determinant identity, with unit determinant modulo \(\mathfrak m\), makes it zero. Repeated division by \(\Pi\) now writes every nonzero element uniquely as a unit times a nonnegative power of \(\Pi\). This is precisely a DVR.

Since \(\pi\) is a nonzero nonunit, it is \(\varepsilon\Pi^e\) for a positive integer \(e\). Define \(\omega_L\) to be the \(\Pi\)-order divided by \(e\). Every \(O\)-unit remains a \(B\)-unit, so its restriction is \(\omega_k\). Any other extending valuation is nonnegative on every integral element, by the same leading-term argument; it has value zero on \(B^*\) and therefore value \(1/e\) on \(\Pi\). It coincides with \(\omega_L\), proving uniqueness.

The filtrations \(\pi^dB\) and \(\Pi^dB\) are cofinal. In the finite free \(O\)-basis the former is exactly the coordinate topology. Completeness and compactness follow from completeness and compactness of \(O\). Finally the \(\Pi\)-filtration of \(B/\pi B\) has \(e\) quotients, each its residue field of dimension \(f\) over \(O/\pi O\). On the other hand its dimension is \(n\), by freeness. Thus \(n=ef\). An automorphism pulls the valuation back to an extending valuation; uniqueness proves invariance. This establishes the exact extension prerequisite used in BF.7. \(\square\)

The primary arithmetic source consulted is J. S. Milne, Algebraic Number Theory, the free author-hosted text, Proposition 2.29 and Theorem 7.38. The proof above supplies the lattice, idempotent, DVR, normalization and topology arguments locally; its valuation assertion is not merely a citation to that theorem.

AB.1. The split quotient and its proper topology

Use the split local-field data of BF.0–BF.6, including the enlarged apartment \(V=V_{\mathrm{der}}\oplus V_Z\), normalizer \(N\), compact open groups \(K_x\), and their covariance and normalizer intersection. Define

Theorem. This relation is an equivalence relation. Its quotient \(\mathcal B\) has a continuous transporter-proper \(G\)-action and is a second countable locally compact Hausdorff space with compact orbit space. The map \(j(x)=[1,x]\) is injective and restricts to a homeomorphism on every compact subset of \(V\). Its point stabilizer is \(G_{[g,x]}=gK_xg^{-1}\). A root element \(x_a(u)\) fixes every \(x\) with \(a(x)+\omega(u)\ge0\).

Proof of the set assertions. Reflexivity uses \(n=1\). If \(k=g^{-1}hn\in K_x\) and \(y=nx\), then \(h^{-1}gn^{-1}=nk^{-1}n^{-1}\in K_y\), proving symmetry. For a second relation, \(z=my\) and \(h^{-1}\ell m=k'\in K_y\), the required element is \(mn\), because

Covariance makes the last membership valid. Left translation preserves the relation. If \([1,x]=[1,y]\), its normalizer witness belongs to \(K_x\cap N=N_x\); hence \(y=x\). The same computation, with the apartment coordinate unchanged, gives the displayed stabilizer. A root element of nonnegative value belongs to \(P_x\subset K_x\), proving the half-apartment assertion.

Here are the topology details; closedness of a relation alone would not suffice. First the family \(K_x\) has the following local property:

For each of the finitely many roots, the integer \(\lceil-a(x)\rceil\) is locally constant unless \(a(x_0)\) is integral; at such a point its nearby values are its value there or that value plus one. Thus all nearby root level groups lie in their groups at \(x_0\), and \(P_x\subset P_{x_0}\). By BF.3 only finitely many affine normalizer labels can fix a point of a fixed compact neighborhood of \(x_0\). Each label's fixed locus is closed. Shrink the neighborhood to exclude those which do not fix \(x_0\); then \(N_x\subset N_{x_0}\). This proves AB.3. In particular the graph of \(K_x\) is closed. Over a compact set \(D\subset V\), the union \(K_D=\bigcup_{x\in D}K_x\) is compact: AB.3 first bounds it in finitely many compact groups, and the closed graph over \(D\) then proves that this union is closed in that bound.

Choose a reduced closed chamber \(\overline C\) and a compact parallelepiped \(D_Z\) for the lattice \(X_*(T)\cap V_Z\); put \(D=\overline C\times D_Z\). The chamber and the lattice translates cover \(V\) locally finitely, and \(\overline C\) is compact. The complete elementary chamber proof is AB.2 below. The central lattice has full rank in \(V_Z\), since that subspace is defined by integer root equations. By BF.6 the open compact group \(I=I_C\) fixes every point of \(D\). Consequently the set quotient is also the quotient of

This is a countable disjoint union of compact metrizable sets. Countability follows because \(G\) has a countable base and its disjoint open \(I\)-cosets each contain a different basis member.

The equivalence relation on \(S\) is closed and has proper projections. Indeed only finitely many affine normalizer labels move a point of \(D\) into \(D\), by BF.3. Their torus-unit kernel is already in every \(K_x\), so choose finitely many representatives \(n\). For a fixed representative \(g\) of a \(G/I\)-coordinate, all equivalent coordinates \(hI\) have

This is a finite union of compact sets, which meets only finitely many open \(I\)-cosets. Closedness now follows from the closed graph just proved, checking the finitely many normalizer representatives. The preimage under either relation projection of a compact subset of \(S\) is a closed subset of finitely many coordinate copies of \(D\times D\), hence compact. The fibers are finite, because both the possible coset coordinates and the affine images of the specified point are finite.

We record why this gives the required quotient topology. A relation with proper projections has closed saturation of every closed set. To check this here, a convergent sequence of saturated points lies in a compact set after adding its limit; properness supplies convergent related witnesses, and closedness supplies the limit relation. Thus the quotient map \(q:S\to\mathcal B\) is closed. Its fibers are compact. Two distinct fibers have disjoint neighborhoods with no related pair, by closedness of the relation and compactness of their product. Replacing each neighborhood \(U\) by the saturated open set \(S\setminus q^{-1}q(S\setminus U)\) separates their quotient points. Thus the quotient is Hausdorff.

A closed map with compact fibers pulls compact sets back to compact sets: for an open cover of the preimage, cover each fiber by finitely many members; closedness gives a quotient neighborhood whose entire preimage is in that finite union, and compactness of the target set reduces these neighborhoods to finitely many. Every fiber here lies in finitely many open coordinate copies of \(D\), a compact set. The same saturated-open construction gives a quotient neighborhood with compact closure, proving local compactness. Finally the sets \(\mathcal B\setminus q(S\setminus U)\), where \(U\) ranges over finite unions of a fixed countable base of \(S\), form a countable quotient base: a compact fiber inside an open preimage has such a finite base cover. This proves second countability.

The topology from AB.4 is exactly the topology from \(G\times V\). In one direction the map from \(G\times D\) factors through the open coset map \(G\to G/I\). In the other direction, restrict to each \(G\times nD\) in the locally finite closed normalizer cover of \(G\times V\). On that set the map is, after replacing \(x\) by \(n^{-1}x\), the continuous map \((g,x)\mapsto q(gnI,n^{-1}x)\). A subset is closed if its restrictions to a locally finite closed cover are closed, so the map from \(G\times V\) is continuous for the AB.4 topology. The two quotient topologies coincide.

The action is jointly continuous. At any quotient point, closedness of \(q\) gives a neighborhood all of whose lifts lie in finitely many prescribed coordinate copies of \(D\). An open subgroup of \(G\) contained in the stabilizers of these finitely many \(G/I\)-coordinates fixes that entire neighborhood. This proves continuity near \((1,b)\); translation proves it everywhere. If compact subsets \(A,B\subset\mathcal B\) have lifts with coset representatives \(a_i,b_j\), then an element \(g\) with \(gA\cap B\ne\varnothing\) satisfies

This is compact. The transporter is closed, because witnesses in the compact sets have convergent subnet limits and the action is continuous. It is therefore compact. Every orbit meets \(j(D)\), proving compactness of the orbit space. The continuous injection \(j\) on any compact apartment set is a homeomorphism onto its image by Hausdorffness. \(\square\)

AB.2. Affine reflection geometry from the root datum

Let \(\Phi\) be the reduced root system of the split group in BF, on the derived apartment \(V_D\). For \(a\in\Phi\) and \(m\in\mathbb Z\), set

Choose an invariant Euclidean inner product. Its existence follows by averaging any inner product over the finite Weyl group, whose finiteness and root-datum reflection action are proved in Root data, Weyl chambers and the Bruhat decomposition, §§2–3. The displayed map is the orthogonal reflection in its wall. The arrangement is locally finite, since there are finitely many roots and only finitely many integers \(m\) whose walls meet a given bounded set. Its chambers are open convex intersections of wall half-spaces. A chamber is bounded: for every root \(a\), its values on a chamber lie between two consecutive integers, and the roots span \(V_D^*\). Its closure is consequently a compact convex polytope. In rank zero take the chamber to be the one-point apartment.

Lemma. If \(C\) is a chamber, the reflections in its walls generate \(W_{\mathrm{aff}}=Q^\vee\rtimes W\). This group acts simply transitively on chambers. Those reflections, denoted \(S\), have exactly the Coxeter relations \(s^2=1\) and \((st)^{m(s,t)}=1\) for the finite pair orders; an infinite pair order supplies no relation. For \(w\in W_{\mathrm{aff}}\), its length is the number of walls separating \(C\) from \(wC\). If \(v\in\overline C\), its stabilizer is generated by \(J_v=\{s\in S:s(v)=v\}\). No two distinct points of \(\overline C\) are in the same \(W_{\mathrm{aff}}\)-orbit.

Proof. The reflection preserves the entire arrangement, because

and the Cartan integer is integral. At a wall of \(C\) it interchanges the two adjacent chambers. A generic polygonal path between interior points crosses finitely many walls, crosses them one at a time, and misses their pairwise intersections. Starting at \(C\), reflect across each crossed wall. At the current image \(wC\), that reflection is \(wsw^{-1}\) for a wall reflection \(s\) of \(C\). Thus these reflections reach every chamber.

We supply the elementary gallery argument that also proves freeness and the presentation. Realize a closed chamber gallery by a polygonal loop crossing each prescribed common wall in its relative interior. Fill the loop by a polygonal disc in \(V_D\). Only finitely many arrangement hyperplanes meet its compact image. Subdivide and perturb the interior vertices, keeping the boundary crossings, so that the disc misses intersections of codimension at least three, meets codimension-two intersections at finitely many interior points, and is transverse to individual walls away from those points. These perturbations exist by avoiding the finitely many proper polynomial conditions on the vertices; subdivisions permit small perturbations that preserve the boundary. For one-dimensional \(V_D\), an interval path gives just backtracking cancellations. In higher dimensions, sweeping the disc by polygonal arcs from the boundary loop to a constant arc changes the crossed-wall word in only two ways. A tangency inserts or removes two consecutive crossings of the same wall. Passing a codimension-two intersection replaces one route around that intersection by the other route. This description can also be obtained by cutting small disjoint discs around the intersection points: the remaining wall curves are disjoint arcs, across which consecutive crossings cancel.

At a codimension-two intersection, use its perpendicular two-dimensional plane. The walls through the intersection form a finite reflection arrangement: the corresponding reflections fix that point, and their linear parts are in the finite group \(W\). Consecutive wall lines bound a sector. If their angle is \(\theta\), their reflections generate a finite dihedral group, and its lines divide the plane into sectors of angle \(\pi/m\). Since the two original lines were consecutive in the full invariant arrangement, no generated line lies inside their sector; hence \(\theta=\pi/m\). Crossing around the intersection gives precisely the alternating braid relation of length \(m\), or equivalently \((st)^m=1\). Pulling the current chamber back to \(C\) expresses this in its two wall generators. Thus every closed gallery word reduces by the stated relations. These relations hold as actual reflections. An element preserving \(C\) therefore has a closed gallery word and is the identity; the same argument shows that every relation is generated by the listed ones. This proves both simple transitivity and the Coxeter presentation, without assuming them as an external building theorem.

Every wall is a wall of some chamber, so its reflection is now a conjugate of a generator. All arrangement reflections generate exactly \(Q^\vee\rtimes W\): their products satisfy \(s_{a,m}s_{a,0}=t_{-ma^\vee}\), giving all coroot translations, while the reflections with \(m=0\) generate \(W\); conversely each displayed reflection is in this semidirect product.

A gallery must cross every wall separating its endpoints. A generic straight segment crosses each separating wall once and crosses no other wall, so the minimal gallery length is exactly their number. This proves the length assertion, including \(\ell(sw)=\ell(w)\pm1\).

The chambers whose closures contain a fixed point \(y\) form its local star. In a sufficiently small ball around \(y\), only the walls through \(y\) occur. Their finitely many sectors are connected by crossing consecutive walls, so any two star chambers are joined by a gallery crossing only those walls. If \(w\) fixes \(v\in\overline C\), apply this to \(C,wC\). Pulling successive crossings back to \(C\) gives only generators in \(J_v\). Simple transitivity then gives \(w\in\langle J_v\rangle\); the reverse containment is immediate. If \(wv=y\) with \(v,y\in\overline C\), the same local-star argument supplies an element \(u\) fixing \(y\) and sending \(C\) to \(wC\). Freeness gives \(u=w\), whence \(v=y\). Point stabilizers are finite: for each finite linear Weyl element there is at most one translating vector making it fix that point. \(\square\)

AB.3. Split affine wall exchange and the generated group

Keep the pinned split group, the valuation and the enlarged apartment of BF. Write \(T^b=T(O)\), let \(I=I_C\) be BF.6's compact open chamber group, and let \(N_{\mathrm{aff}}\) be the inverse image of \(W_{\mathrm{aff}}\) in \(N_G(T)(k)\). Thus \(N_{\mathrm{aff}}/T^b=W_{\mathrm{aff}}\). For a wall \(\alpha=a+m\) of \(C\), positive on \(C\), choose \(\dot s=n_a(\pi^m)\), whose action is \(s_{a,m}\). Its square is \(a^\vee(-1)\in T^b\).

Lemma. For every \(n\in N_G(T)(k)\),

Moreover \(I\cap N_G(T)(k)=T^b\), \(\dot s I\dot s^{-1}\ne I\), and

is an open normal subgroup of \(G(k)\). The full group is \(T(k)G^0\). Its core under normalizer conjugation is \(\bigcap_{n\in N}nIn^{-1}=T^b\).

Proof. On \(C\) the affine root \(\alpha\) takes values between zero and one. BF.6 therefore gives the levels \(f_C(a)=m\), \(f_C(-a)=1-m\). Crossing this wall changes only these two root levels: a wall for any other root does not separate the two adjacent chambers. Choose a positive root system containing \(a\), and order its positive factors with \(a\) last. The arbitrary-order factorization in BF.6 writes

Indeed the torus and every factor except \(U_{a,m}\) have levels that remain admissible after reflection. Consequently it suffices to examine \(\dot s x_a(u)n\), with \(\omega(u)\ge m\). If \(\omega(u)>m\), conjugation by \(\dot s\) puts this root factor in \(U_{-a,1-m}\subset I\), giving the first cell in (AB.6). If \(\omega(u)=m\) and \(n^{-1}\alpha\) is positive on \(C\), conjugation by \(n^{-1}\) puts \(x_a(u)\) in \(I\), giving that same cell. An affine root has constant sign on \(C\), so the remaining case is that \(n^{-1}\alpha\) is negative.

Put \(c=\pi^m\). Multiplication of the two-by-two matrices in the pinned rank-one map gives the identity

This is valid in every residue characteristic, including two. The first parameter has valuation \(m\), the torus parameter is a unit, and the last factor has the level of \(-\alpha\). In the remaining case \(n^{-1}(-\alpha)\) is positive, so that last factor conjugates into \(I\). The result lies in \(InI\). This proves (AB.6) with its signs and integer thresholds.

BF.5 gives \(K_v\cap N=N_v\). Every element of \(I\) fixes every \(v\in C\times V_Z\). An element of \(I\cap N\) consequently has affine action equal to the identity on an open set and hence everywhere. BF.3 identifies its kernel as \(T^b\). Conversely \(T^b\subset I\). Reflection conjugation carries \(U_{a,m}\) to \(U_{-a,-m}\); the latter has a parameter of valuation \(-m\) and is not in \(I\), whose opposite-root level is \(1-m\), by BF.6's uniqueness and root intersections. Thus the conjugated groups differ.

Let \(R=\langle T^b,U_a(k)\rangle\). Every \(n_a(u)\) is a product of opposite root elements by the pinned rank-one formula, so \(I,N_{\mathrm{aff}}\subset R\). Conversely conjugating the root tails in \(I\) by powers of a coroot translation lowers their levels without bound, since \(\langle a,a^\vee\rangle=2\). Those tails exhaust \(U_a(k)\), proving \(R=G^0\). It is open since it contains \(I\). The preceding programme proof Root data, Weyl chambers and the Bruhat decomposition, Theorem 6.2, writes every element of the split group in a finite Bruhat cell. Its unipotent coordinates are root elements and its Weyl representative is a product of \(n_a(1)\). Therefore \(G=T(k)G^0\). The torus normalizes \(R\), and the root generators belong to it, so it is normal in \(G\).

Finally an element of the intersection of all normalizer conjugates belongs to \(I\). Write its unique BF.6 negative-root, torus and positive-root coordinates. Torus conjugation preserves those coordinates and scales each root coordinate by the corresponding root character. For each nonzero parameter, a power of a cocharacter with nonzero pairing lowers its valuation below the fixed chamber threshold, contradicting membership in all torus conjugates of \(I\). Thus every root parameter is zero. The element lies in \(T^b\), and this subgroup is normal in \(N\), giving equality. \(\square\)

AB.4. Double cosets, proved from the wall exchange

We prove the group-theoretic argument needed for the building, instead of citing a BN-pair theorem. Suppose a group \(H\) is generated by subgroups \(B,N_0\), with \(T=B\cap N_0\) normal in \(N_0\). Suppose \(W_0=N_0/T\) is generated by nonidentity involutions \(S\), and, for representatives of \(s\in S\),

The double coset \(D_w=B\dot wB\) is independent of its representative. Denote word length in \(S\) by \(\ell\).

Lemma. The double cosets \(D_w\) partition \(H\), their labels are distinct, and

The length changes by exactly one. Products along a reduced word equal the double coset of its label. For \(J\subset S\),

Proof. First \(D_s^2\subset B\cup D_s\), and it contains \(B\), because \(\dot s^{-1}\in D_s\). If it were contained in \(B\), then \(\dot sB\dot s^{-1}\subset B\); since \(\dot s^2\in T\), conjugating again yields the reverse inclusion. This contradicts (AB.9). Hence \(D_s^2=B\cup D_s\). The union of all \(D_w\) is closed under multiplication by the generators \(B\) and \(D_s\), using (AB.9), and under inverses. It is therefore \(H\).

For distinctness, suppose \(D_v=D_w\), and induct on \(\min(\ell(v),\ell(w))\). If this is zero, equality with \(B\) places the normalizer representative in \(B\cap N_0=T\), so both labels are the identity. Otherwise choose the shorter label \(v=sv'\), with \(\ell(v')=\ell(v)-1\). Since \(\dot s^{-1}\in D_s\), \(D_{v'}\subset D_sD_v=D_sD_w\subset D_w\cup D_{sw}\). Double cosets either coincide or are disjoint, so \(D_{v'}\) equals one of the two terms. Induction gives \(v'=w\) or \(v'=sw\). The first contradicts the choice of the shorter length; the second gives \(v=w\).

We next prove the ascending product rule by induction on \(\ell(w)\), assuming only \(\ell(sw)\ge\ell(w)\). The identity case is immediate. Write \(w=w't\) reduced, with \(t\in S\). Since \(\ell(sw)\le\ell(sw')+1\), one has \(\ell(sw')\ge\ell(w')\). Induction, followed by the right-hand version of (AB.9), obtained by inversion, gives

It also lies in \(D_w\cup D_{sw}\). The labels \(sw\) and \(w\) differ. Also \(sw'\ne w\): equality would give \(sw=w'\), contradicting \(\ell(sw)\ge\ell(w)\). Distinctness therefore leaves precisely \(D_{sw}\), which is contained in the product by multiplication of representatives. This proves the ascending rule.

If \(\ell(sw)\le\ell(w)\), the ascending rule applied to \(sw\) gives \(D_sD_{sw}=D_w\). Multiplying once more gives \(D_sD_w=(B\cup D_s)D_{sw}=D_{sw}\cup D_w\). Equality of the two lengths would give both formulas and contradict distinctness. Since a generator changes word length by at most one, its change is exactly one. Right-hand products follow by inversion; repeated ascending products prove the reduced-word assertion. The union of labels in \(W_J\) is a subgroup by the product formulas and equals the generated \(P_J\). Iterating the right product formula shows that \(D_wP_J\) is contained in the union in (AB.11); conversely multiplication of representatives puts each \(D_{wu}\) in that product. \(\square\)

Apply this lemma with \(H=G^0,B=I,N_0=N_{\mathrm{aff}}\), using AB.2–AB.3. It gives a bijection \(W_{\mathrm{aff}}\to I\backslash G^0/I\). For the full normalizer quotient \(\widetilde W=N/T^b\), put \(\Omega=\{w\in\widetilde W:wC=C\}\), where stabilization refers to the reduced chamber, allowing central translations. Each label is uniquely \(\omega w\), \(\omega\in\Omega,w\in W_{\mathrm{aff}}\), by AB.2's simple transitivity. Such \(\omega\) preserves all chamber root levels, so its representative normalizes \(I\); \(W_{\mathrm{aff}}\) is normal in \(\widetilde W\) by the lattice and finite-root action. Since \(G=T(k)G^0\), every element is in a cell \(InI\).

We verify distinctness also for these extended labels; this is needed to identify \(G^0\cap N\). The preceding distinctness induction applies to a comparison of an affine label \(v\) with any label \(w\in\widetilde W\): induct on \(\ell(v)\), using (AB.6) for the possibly nonaffine label. The identity case still uses \(I\cap N=T^b\). In the inductive step equality gives \(v'=w\) or \(v'=sw\). In the first case \(w=v'\) is already affine, so the proved affine distinctness contradicts the shorter length; in the second case \(w=sv'=v\). Thus equality forces \(w=v\). Thus \(G^0\cap N=N_{\mathrm{aff}}\). If \(D_{\omega u}=D_{\omega'v}\), left multiplication by \(\omega^{-1}\), which normalizes \(I\), compares the affine label \(u\) with \(\omega^{-1}\omega'v\). The comparison just proved forces \(\omega^{-1}\omega'\in W_{\mathrm{aff}}\). It also belongs to \(\Omega\), whose intersection with \(W_{\mathrm{aff}}\) is trivial by AB.2. Thus \(\omega=\omega'\) and \(u=v\). We have proved

This argument uses the freely available primary author exposition of Todd Trimble on double-coset exchange as a mathematical source. The entire argument actually used is written above; its omitted Coxeter-criterion proof is not used, since AB.2 proves the required geometry directly.

AB.5. Common apartments, face groups and the enlarged product

Proposition. Any two points of the split quotient of AB.1 lie in an apartment. Any two chambers do also. For \(v\in\overline C\),

There is a homeomorphism

where \((gI,v)\sim(hI,v)\) exactly when \(g^{-1}h\in P_v\). Each reduced apartment has its Euclidean affine-Coxeter tiling.

Proof. Translate point coordinates into \(\overline C\times V_Z\) using \(N_{\mathrm{aff}}\). If their group coordinates are \(g,h\), write \(g^{-1}h=i_1ni_2\) by (AB.12). The group \(I\) fixes both point coordinates. Therefore the two points are in the apartment \(gi_1V\): the first is \([gi_1,x]\), the second \([gi_1,ny]\). The identical argument with chamber interiors proves common apartments for chambers.

By definition \(P_v\) is generated by \(T^b\) and root elements of nonnegative value, so \(I\subset P_v\subset G^0\). If \(p\in P_v\) lies in the affine Bruhat cell \(I\dot wI\), then \(\dot w\in P_v\), and BF.5 implies \(w(v)=v\). AB.2 identifies such labels as \(W_v=\langle J_v\rangle\). Conversely each wall representative for \(J_v\) is a product of opposite root elements of value zero at \(v\), by BF.1's rank-one formula. It lies in \(P_v\). This proves (AB.13).

Every point of \(\mathcal B\) is represented by \((g^0,v,z)\) with \(g^0\in G^0,v\in\overline C,z\in V_Z\), since the full group is \(\Omega G^0\) and \(\Omega\) preserves the reduced chamber. Consider two such representatives. The equivalence relation gives an \(n\in N\) carrying \((v,z)\) to \((v',z')\), with \((g^0)^{-1}h^0n=p n_0\), \(p\in P_v,n_0\in N_{(v,z)}\). Hence \(n n_0^{-1}\in G^0\cap N=N_{\mathrm{aff}}\). Its action carries \((v,z)\) to \((v',z')\), so AB.2 gives \(v=v'\), while its central translation is zero and gives \(z=z'\). Its finite stabilizer label is in \(W_v\), whose representatives lie in \(P_v\) by (AB.13). Thus the group-coordinate relation is precisely \((g^0)^{-1}h^0\in P_v\). The converse is immediate. This proves the set bijection (AB.14).

For topology, \(P_v\) is compact, contains the open group \(I\), and has finite index over it. The level description and finite stabilizers give the same local upper-containment and closed-relation argument as AB.1 for the quotient defining \(\mathcal B^0\). It is a closed quotient with finite fibers, locally compact Hausdorff and second countable. The map (AB.14) is continuous. It is proper: a compact set in \(\mathcal B\) has, by AB.1, lifts in finitely many \(G/I\)-coordinate copies of \(D\). For each of their finitely many representatives write \(g=\omega g^0\). Conjugation by \(\omega\) preserves \(I,G^0\), preserves \(\overline C\), and translates \(V_Z\) by one fixed vector. Thus all the corresponding preimages lie in finitely many chamber copies times one bounded closed central set, a compact set. The preimage is closed there. A proper continuous bijection between these locally compact Hausdorff spaces is a homeomorphism: images of closed sets are closed, checked on compact neighborhoods. This proves (AB.14).

The map from \(G\) to the central translations factors through \(G/G^0=N/N_{\mathrm{aff}}\). It is the central projection of the torus valuation translation of BF.3; the Weyl group acts trivially on \(V_Z\). Consequently the full group acts on (AB.14) by its chamber action on \(\mathcal B^0\) and these central translations. \(\square\)

AB.6. The metric from affine chamber data

This paragraph is a general geometric theorem, with explicit inputs. Let \(W_{\mathrm{aff}}\) act on a finite-dimensional Euclidean apartment with the locally finite alcove tiling proved in AB.2. Let \((H,B,N_0)\) satisfy the exchange hypotheses of AB.4, with this affine Weyl group, and let its face groups be the subgroups \(P_{J_v}\) in (AB.11). Form \(X=((H/B)\times\overline C)/\sim\), using those face groups. No residue field, compactness or valuation hypothesis is assumed in this paragraph. Apartments are the translates of the chamber-tiling apartment supplied by \(N_0\). These inputs have all been proved for \(\mathcal B^0\) above.

Every two chambers lie in an apartment: if \(g^{-1}h=b_1nb_2\), both are in \(gb_1A\), just as in AB.5. Face equivalence and (AB.11) also prove that the base apartment is embedded. More explicitly, for a fixed \(v\in\overline C\), labels \(w,w'\) give the same face point exactly when \(w^{-1}w'\in W_{J_v}\), which is exactly the equality of the Euclidean points \(wv,w'v\), by AB.2.

Define a chain distance: take the infimum of the sum of Euclidean lengths of finitely many successive segments, each lying in a closed chamber. There is always such a chain because two points lie in a common apartment and a straight segment meets finitely many chambers. To show that this is a metric and that apartments retain their Euclidean distance, we prove the required retraction rather than presupposing a building metric.

For the base chamber, a chamber \(gB\) has a unique Bruhat label \(w\). Send its point with chamber coordinate \(v\) to the Euclidean point \(wv\) in the base apartment. This is independent of all face representatives: by (AB.11), the possible labels of \(gP_{J_v}\) are exactly \(wW_{J_v}\), and each of these labels gives the same point \(wv\). This map \(r_C\) is an isometry on each closed chamber and therefore does not increase chain lengths. It fixes the base apartment. The same construction, conjugated, gives a retraction centered at every chamber onto every apartment containing that chamber. Indeed any apartment containing the base chamber is \(bA\) for some \(b\in B\): in \(gA\), that chamber has coordinate \(nB\), hence \(gn\in B\). Bruhat labels are unchanged by left multiplication by \(b\), so the retraction restricts to an isometry on every such apartment.

If \(x,y\) are in an apartment \(A\), choose a center chamber in \(A\). Applying its retraction to any chain gives a length at least \(|x-y|\). The straight segment in \(A\) gives the reverse inequality. Thus

Any distinct two points lie in a common apartment and have positive distance there. Thus the chain distance is a metric. Straight apartment segments are geodesics. All the retractions are 1-Lipschitz in this metric. We shall need the following stronger equality. If \(p\) lies in the closed center chamber, and \(z\) is any point, take an apartment containing that chamber and a chamber containing \(z\). The retraction is an isometry there and fixes \(p\). Hence

All assertions here follow from the displayed affine data and the locally proved double-coset identities. There is no additional apartment axiom assumed implicitly.

AB.7. CAT(0), completeness and comparison of the two topologies

Theorem. The metric in AB.6 is geodesic and CAT(0). It is complete even when the number of chambers at a panel is infinite. For the split group over a local field it is proper and induces precisely the quotient topology in AB.5. The enlarged product with the Euclidean central factor has all these properties, and the full group acts isometrically.

Proof. Let \(c(t)\), \(0\le t\le1\), be the straight segment from \(x\) to \(y\) in a common apartment \(A\). For each \(t\), choose a chamber of \(A\) whose closure contains \(c(t)\), and retract onto \(A\) with that chamber as center. Write \(r\) for this retraction. By (AB.16), \(d(z,c(t))=|rz-c(t)|\). The Euclidean squared-distance identity and 1-Lipschitz property give

This is the argument of the freely accessible original Bruhat–Tits I, §3.2.1, with the distance equality required for its center proved explicitly in AB.6. It works at walls and vertices as well as interior points.

Any midpoint of another geodesic from \(x\) to \(y\) has distances \(d(x,y)/2\) to both endpoints. Substituting it for \(z\) in (AB.17), with \(t=1/2\), shows that it coincides with \(c(1/2)\). Repeating on dyadic subsegments and using continuity proves uniqueness of the geodesic. Apartment intersections are convex, since both apartments contain their identical unique segment between any two common points.

For completeness of the curvature assertion, let \(p=c_{xy}(t)\), \(q=c_{xz}(s)\), and put \(A=d(x,y), B=d(x,z), C=d(y,z)\). Apply (AB.17) first along \(xy\), then along \(xz\) to bound \(d(y,q)^2\), while \(d(x,q)=sB\). It yields

The right side is exactly the squared distance between the corresponding points in the Euclidean comparison triangle, by its cosine identity. Any two boundary points are on two edges with a common vertex; relabeling proves the full CAT(0) triangle comparison, including degenerate triangles.

Here is a completeness proof which makes no local-finiteness assumption on the chamber multiplicities. The chamber coordinate defines a type map \(\lambda:X\to\overline C\). It is well defined at faces and is an isometry on each closed chamber, hence is 1-Lipschitz for the chain metric. Fix \(y\in\overline C\). In the standard apartment there are finitely many walls meeting the closed unit ball about \(y\). Set

For a point \(p\) of type \(y\), take a common apartment for a chamber containing \(p\) and a chamber containing any \(q\) with \(d(p,q)\le r(y)\). Formula (AB.15) identifies this distance with the Euclidean distance there. The ball crosses no wall not passing through \(p\). Thus \(q\) lies in the closure of a chamber containing the minimal face of \(p\): its signs at every wall not through \(p\) are those of that face, while the walls through \(p\) only select a neighboring chamber. Each such chamber contains \(p\), and its type map is injective. In particular distinct points of type \(y\) are at distance greater than \(r(y)\).

If \(p_n\) is Cauchy, then \(y_n=\lambda(p_n)\) converges to some \(y\) in the compact polytope \(\overline C\). Choose a closed chamber containing \(p_n\), and in that same chamber choose its point \(q_n\) of type \(y\). One has \(d(p_n,q_n)=|y_n-y|\to0\). The sequence \(q_n\) is Cauchy, and its uniform separation just proved makes it eventually constant. Hence \(p_n\) converges. This proves completeness with arbitrary panel multiplicity; it does not assert that affine chamber data for a particular nonsplit group have already been constructed.

For the split local-field construction, every wall group \(P_{\{s\}}=I\cup I\dot sI\) is contained in the compact group \(P_v\) at a point of that wall, so it has finitely many right \(I\)-cosets. A reduced product in (AB.10) therefore shows that each \(I\dot wI/I\) is finite. Fix \(p\) in the closed base chamber. Any chamber meeting a metric ball of radius \(R\) about \(p\) lies with the base chamber in an apartment. Retracting onto the base apartment sends it isometrically to \(w\overline C\), meeting that same ball. The diameter of \(\overline C\) is finite and the Euclidean tiling is locally finite, so there are only finitely many such labels \(w\). For each label there are only finitely many chambers, by the finite coset assertion. Thus each bounded ball is covered by finitely many closed chambers. Those chambers are compact for the metric, by (AB.15), so a closed ball is compact. This proves properness directly.

The quotient-to-metric identity is continuous, since its restrictions to the chamber-coordinate copies are isometries and the quotient has that defining topology. A neighborhood ball of radius one is contained in a finite union \(K\) of closed chambers. This \(K\) is compact for the quotient topology and Hausdorff for the metric, so the continuous identity on \(K\) is a homeomorphism. At its center, metric convergence is therefore quotient convergence. These neighborhoods prove the reverse continuity everywhere, establishing equality of the two topologies.

For the enlarged product define

Product segments are geodesics. Adding (AB.17) to its Euclidean equality gives the same strong convexity inequality, and (AB.18) proves CAT(0). Completeness and properness follow from those of the two factors; their metric topology is the product topology of (AB.14). The group preserves apartment Euclidean distances: on the derived factor this follows from its chamber action and the normalizer's affine Weyl action, and on the central factor from BF.3's translations. The chain metric, and therefore (AB.20), is invariant. \(\square\)

AB.8. The split enlarged building is a universal proper space

The action is proper and cocompact by AB.1 and AB.14, and AB.7 supplies the compatible proper CAT(0) metric. We give the fixed-set and universal-map argument, since contractibility of the building alone would not prove universality.

Proposition. For every compact subgroup \(H\subset G\), \(\mathcal B^H\) is nonempty and contractible. For every proper \(G\)-space carrying an invariant partition of unity subordinate to slice maps to \(G/H_i\), with \(H_i\) compact, there is a \(G\)-map to \(\mathcal B\), unique up to \(G\)-homotopy. Thus \(\mathcal B\) is a model for the numerable universal proper \(G\)-space.

Proof. A compact subgroup has a bounded orbit \(E\). The function \(F(b)=\sup_{e\in E}d(b,e)^2\) is continuous and tends to infinity when \(b\) tends to infinity, since \(E\) is bounded. Properness gives a minimum. Formula (AB.17), at the midpoint, makes its minimizer unique. The function is \(H\)-invariant, so the minimizer is fixed. The fixed set is closed and convex, because isometries preserve the unique segments. Contract it along the unique segments to one fixed point.

The segment operation is continuous. Indeed CAT(0) comparison for two segments with the same initial point gives distance at fraction \(t\) at most \(t\) times the distance of their endpoints. Inserting the intermediate segment with one endpoint from each of two segments and applying the triangle inequality yields \(d(c_{xy}(t),c_{x'y'}(t))\le(1-t)d(x,x')+t d(y,y')\). Together with the constant-speed formula this gives joint continuity in endpoints and parameter.

The model itself is numerable. Point stabilizers \(K_p\) are compact open. Its orbit \(Gp\) is discrete and closed: properness makes the map \(G/K_p\to\mathcal B\) proper, and a compact subset of the discrete space \(G/K_p\) is finite. Thus choose \(\delta>0\) with no other orbit point within distance \(\delta\) of \(p\). If \(\varepsilon<\delta/3\), the translates of \(B(p,\varepsilon)\) intersect only when their translating cosets in \(G/K_p\) agree, since an intersection implies \(d(p,gp)<2\varepsilon\). This gives an equivariant slice map from its saturated open set to \(G/K_p\). The orbit space has the metric \(\bar d(Gx,Gy)=\inf_gd(x,gy)\): distinct closed orbits have positive distance, since a sequence \(gy\to x\) in a bounded ball has a convergent transporter subsequence by properness; its zero-distance limit would belong to that orbit. Metric balls are precisely the images of saturated unions of ordinary balls, so this metric gives the quotient topology. This quotient, denoted \(Q\), is compact by AB.1. Finitely many slice neighborhoods cover it; denote their smaller neighborhoods by \(U_i\). Shrink them to a finite open cover with closures still inside the original neighborhoods: a sufficiently small uniform ball radius does this on a compact metric space. The functions \(\min(1,\operatorname{dist}(x,Q\setminus U_i))\), with value one if that complement is empty, divided by their positive sum, give a finite partition of unity whose closed supports are in the original ones. Pulling it back gives the invariant subordinate partition. Thus the model meets the numerability requirement as well as properness.

For the universal map, choose a point \(b_i\in\mathcal B^{H_i}\) for each slice map \(p_i:Y_i\to G/H_i\). It defines a continuous equivariant point map \(z_i:Y_i\to\mathcal B\) by \(gH_i\mapsto gb_i\). Let \(\lambda_i\) be the subordinate invariant locally finite partition of unity, with support contained in \(Y_i\). At \(y\), minimize \(\sum_i\lambda_i(y)d(b,z_i(y))^2\). There are finitely many active terms. Their total weight is one; boundedness of their point set gives coercivity, so a minimum exists by properness. The midpoint inequality gives uniqueness. At a fixed point shrink a neighborhood to exclude the finitely many nearby closed supports not containing that point. Each remaining support is inside its map domain, so the corresponding finitely many point maps are defined and bounded on a further neighborhood. A fixed comparison point bounds the minimum values. The inequality \(d(b,z_i)\ge d(b,o)-d(o,z_i)\), summed with total weight one, places all minimizers in one compact ball. Any convergent subnet of minimizers minimizes the limiting finite sum, by continuity of its weights and active point maps; zero-weight terms tend to zero on this bounded ball. Uniqueness forces the limit, proving continuity. Invariance of the weights and uniqueness of the minimizer prove equivariance. Two equivariant maps are homotopic by the continuous unique-segment operation. \(\square\)

AB.0–AB.8 prove the full split local-field block, including the enlarged central factor required for properness. AB.6–AB.7 also provide an abstract geometric theorem usable over an infinite-residue valued field once its exact affine chamber data have separately been proved. They do not supply nonsplit root filtrations or connected facet models by assertion.

AB.9. Unramified integral descent

Here k is a nonarchimedean local field, R its complete DVR, and f= \(F_{q}\) its residue field. AB.0 proves the finite-extension arithmetic. The regular-domain and local Jacobian criteria used below are proved in the preceding lessons named in BF.0: Regular sequences, depth and Cohen–Macaulay modules, Theorem 6.1, and Smooth algebras over a field and the Jacobian criterion, Lemma 1.1 and Theorem 2.1. All lifting, openness and descent arguments used here are supplied below. References are Conrad, Lang’s theorem and unirationality, Section 2, and Bruhat–Tits II, Sections 5.1.5–5.1.9.

AB.9.1. FINITE UNRAMIFIED EXTENSIONS AND INTEGRAL TRACE

Let E/k be finite unramified (a finite separable extension with ramification index one) and B= \(R_{E}\) . Write m=[E:k]. AB.0 gives that \(\pi\) is also a uniformizer of B, B is complete and finite free, and B/ \(\pi\) B= \(F_{q^m}\) . Let bar( \(\alpha\) ) generate \(F_{q^m}\) over f, and let bar(F) be its monic minimal polynomial, of degree m. Such a generator exists: the finite multiplicative group of a field is cyclic. To check the latter claim, in each primary part of a finite abelian subgroup choose an element of maximal order and multiply these elements. Its order is the exponent h of the group. Every element is a root of \(X^h\) -1, so the root bound gives the group order at most h; equality follows. A generator of the multiplicative group generates the field as well.

Lift bar(F) to a monic F in R[T]. It is irreducible over k. Indeed if F=PQ with monic P,Q in k[T], multiply P and Q by the smallest nonnegative powers \(\pi^r\) , \(\pi^s\) making their coefficients integral and primitive. Their reductions have nonzero product over f, so their product is primitive. The content of \(\pi^{r+s}\) F is \(\pi^{r+s}\) ; consequently r+s=0. The factors were already integral and monic, and their reductions would contradict irreducibility of bar(F).

The residue polynomial is separable because finite fields are perfect: the q-power map is bijective, and an irreducible polynomial with zero derivative would be a pth power. Choose an arbitrary lift \(a_{0}\) of bar( \(\alpha\) ) in B. Then F( \(a_{0}\) ) is in \(\pi\) B and F'( \(a_{0}\) ) is a unit. Newton's iteration a -> a-F(a)/F'(a) at least doubles the \(\pi\) -order of the error and keeps the derivative a unit. Completeness gives a root \(\alpha\) in B with the specified reduction. A second root with that reduction would have difference times a unit equal to zero, by polynomial division and the derivative test, so the lift is unique.

Since deg(F)=m=[E:k], E=k( \(\alpha\) ). In fact B=R[ \(\alpha\) ]. For x in B, write its residue as a polynomial of degree less than m in bar( \(\alpha\) ), subtract the corresponding element of R[ \(\alpha\) ], and divide by \(\pi\) . Repeating gives x as a \(\pi\) -adic sum of elements in the free R-module with basis 1, \(\alpha\) ,..., \(\alpha^{m-1}\) . That module is complete and is contained in B; the partial sums converge to x in B. Thus B = R[T]/(F), F'( \(\alpha\) ) in \(B^*\) . This finite free algebra is etale: in a square-zero lifting problem the equation F(a)=0 has a unique correction because F'(a) is invertible. Iterating square-zero steps handles nilpotent ideals; this is formal etaleness, and the displayed finite presentation is the etale criterion.

Each of the m residue conjugates bar( \(\alpha\) )^{ \(q^i\) } has a unique root lift \(\alpha_{i}\) of F in B. Hence E is the splitting field of a separable polynomial of degree m and E/k is Galois. Reduction identifies its group \(\Gamma\) with Gal( \(F_{q^m}\) / \(F_{q}\) ), generated by q-Frobenius. These m Frobenius powers are distinct: if a power \(q^i\) with 0<i<m fixed the entire field, \(X^{q^i}\) -X would have \(q^m\) roots. Thus the group is cyclic of order m. The differences \(\alpha_{i}\) - \(\alpha_{j}\) are units, so the Chinese remainder theorem proves the Galois-cover identity B \(\otimes_{R}\) B -> \(\prod_{\gamma\in \Gamma}\) B, x tensor y |-> (x \(\gamma\) (y))_ \(\gamma\) , is an isomorphism. B is faithfully flat over R because it is free of positive rank. This proves the complete finite etale Galois-cover claim.

The trace reduces to the residue trace. One can see this from the Galois sum Tr(x)= \(\sum_{\gamma} \gamma\) (x), or from a multiplication matrix in the displayed integral basis. The residue trace is onto f: its polynomial is T(X)=X+ \(X^q\) +...+X^{ \(q^{m-1}\) }. It is a nonzero polynomial of degree less than \(q^m\) and therefore cannot vanish on all of \(F_{q^m}\) . It is f-linear and has values in f, so its nonzero image is all of the one-dimensional f-space f. Choose bar(c) of trace one and any lift \(c_{0}\) in B. The element t=Tr( \(c_{0}\) ) lies in R and is congruent to 1 modulo \(\pi\) , so t is a unit. Then c= \(c_{0}\) /t satisfies Tr(c)=1. This never divides by m. In particular it remains valid when p divides m.

AB.9.2. ADDITIVE SEMILINEAR COCYCLES

Let M be any B-module, with \(\gamma\) (bu)= \(\gamma\) (b) \(\gamma\) (u). No finiteness, flatness or freeness of M is required. For \(z_{\sigma \tau}\) = \(z_{\sigma}\) + \(\sigma\) ( \(z_{\tau}\) ), put v=- \(\sum_{\gamma\in \Gamma} \gamma\) (c) \(z_{\gamma}\) , where c is from AB.9.1. Reindex \(\delta\) = \(\sigma \gamma\) and use \(\sigma\) ( \(z_{\gamma}\) )= \(z_{\sigma \gamma}\) - \(z_{\sigma}\) . Directly, \(\sigma\) (v) =- \(\sum_{\delta} \delta\) (c)( \(z_{\delta}\) - \(z_{\sigma}\) ) =v+( \(\sum_{\delta} \delta\) (c)) \(z_{\sigma}\) =v+ \(z_{\sigma}\) . Thus \(z_{\sigma}\) = \(\sigma\) (v)-v with exactly the stated sign. Applying the same calculation to \(F_{q^m}\) / \(F_{q}\) and a residue trace-one element proves the residue-module assertion, including degrees divisible by p.

AB.9.3. SMOOTH INTEGRAL POINTS AND CONGRUENCE QUOTIENTS

Let H be a smooth affine group scheme of finite presentation over R. Fix a point modulo \(\pi^j\) , j>=1. Its reduction modulo \(\pi\) lies in a standard smooth chart. Choose arbitrary lifts of the chart coordinates modulo \(\pi^{j+1}\) ; the equations have errors in \(\pi^j\) / \(\pi^{j+1}\) . Correct the c coordinates indexed by an invertible Jacobian minor by solving the c linear equations for the negative error, leaving the other coordinates fixed. Quadratic correction terms vanish because ( \(\pi^j\) / \(\pi^{j+1}\) )^2=0. The chart's localization denominators remain units because they were units at the residue point. This constructs a point modulo \(\pi^{j+1}\) lifting the given point. Starting from any specified point modulo \(\pi^n\) and repeating constructs compatible points at every higher level.

If H=Spec(A), compatible maps A -> R/ \(\pi^j\) define a map A -> \(\lim_{j}\) R/ \(\pi^j\) =R by taking the value on each a in A. Addition, multiplication and constants are respected at every level, hence in R; uniqueness follows from \(\bigcap_{j} \pi^j\) R=0. Consequently H(R)= \(\lim_{j}\) H(R/ \(\pi^j\) ), and the constructed compatible lifts show that every reduction map is onto. This is an actual lifting proof, not an assumption that smooth points lift over an arbitrary residue field.

Set \(K_{n}\) =ker(H(R)->H(R/ \(\pi^n\) )), n>=1, and I= \(\pi^n\) R/ \(\pi^{n+1}\) R. A point of H(R/ \(\pi^{n+1}\) ) reducing to the identity modulo \(\pi^n\) is a map \(\epsilon\) +d, where \(\epsilon\) is the identity's evaluation and d:A -> I satisfies d(ab)=bar( \(\epsilon\) (a))d(b)+bar( \(\epsilon\) (b))d(a), d(R)=0. Conversely every such derivation gives a point: \(I^2\) =0 makes the displayed equation exactly the multiplicativity requirement. Since \(\pi\) I=0, this is the tangent space at the identity of \(H_{f}\) with coefficients I, canonically Lie( \(H_{f}\) ) \(\otimes_{f}\) I. The comultiplication giving the group law has linear term \(d_{1}\) + \(d_{2}\) ; all products of their values vanish in \(I^2\) . Thus this kernel is the additive group of that vector space. The surjectivity just proved identifies it with \(K_{n}\) / \(K_{n+1}\) . The derivation description uses only identity evaluation and ring maps, so it commutes with base change and ring automorphisms. In particular for B/R it is the semilinear \(F_{q^m}\) -module Lie( \(H_{f}\) ) \(\otimes_{f}\) ( \(\pi^n\) B/ \(\pi^{n+1}\) B).

AB.9.4. THE FINITE-FIELD COCYCLE STEP, WITH ITS LANG PROOF

A smooth connected affine group J over a finite field is geometrically connected. Here is the needed descent check. Over the algebraic closure its identity component is open and closed: smooth local rings are regular domains, so different irreducible components cannot meet; there are only finitely many components. The component containing the rational identity is invariant under every field automorphism. Its defining idempotent in the affine coordinate ring descends, since the coefficients of an invariant element of A \(\otimes_{f}\) bar(f) belong to f (expand in a finite linearly independent set of elements of A). A proper identity component would therefore disconnect J over f. Connectedness excludes this.

Write F for q-Frobenius on \(J_{bar}\) (f). It has zero differential. For any x define the right twisted action \(T_{g}\) (x)= \(g^{-1}\) xF(g). Its orbit map \(\psi_{x}\) sends the identity to x; after left translation by \(x^{-1}\) its differential there is -Ad( \(x^{-1}\) ), hence invertible. The identity \(\psi_{x}\) (g h)= \(\psi_{\psi_{x}(g)}\) (h) shows that its differential is invertible everywhere. The Jacobian criterion for morphisms between smooth varieties of equal dimension makes these orbit maps etale, hence open. On closed points their images are the disjoint orbits and cover \(J_{bar}\) (f). Distinct open images cannot intersect, since a nonempty intersection has a closed point; their union is all, since its closed complement would otherwise have a closed point. Geometric connectedness forces a single orbit. In particular g |-> \(g^{-1}\) F(g) is onto on bar(f)-points. This proves precisely the Lang assertion used here.

For clarity, openness in this argument is the elementary etale-chart fact. A standard etale chart is a localization of C[t]/(P), with P monic and P' invertible there. The algebra before localization is finite free. For a principal open D(u) in such an algebra, a fibre meets D(u) exactly when multiplication by u on that finite-dimensional fibre is not nilpotent. Equivalently at least one nonleading coefficient of its characteristic polynomial is nonzero, by Cayley--Hamilton. Its image is therefore the union of the principal opens defined by these coefficients. Thus these charts, and consequently etale morphisms, are open. The tangent criterion is the same local Jacobian criterion: in smooth local coordinates an invertible differential gives the dimension-zero invertible-Jacobian presentation.

Now let a be a multiplicative \(\Gamma\) -cocycle with values J( \(F_{q^m}\) ). For the Frobenius generator \(\sigma\) , put A= \(a_{\sigma}\) . The cocycle identity gives A F(A)... \(F^{m-1}\) (A)=1. Lang supplies bar(b) in J(bar(f)) with A=bar(b)^{-1}F(bar(b)). Telescoping the preceding product gives bar(b)^{-1} \(F^m\) (bar(b))=1. Therefore bar(b) is actually in J( \(F_{q^m}\) ). The values on all powers of \(\sigma\) are determined by A, so \(a_{\gamma}\) =bar(b)^{-1} \(\gamma\) (bar(b)) for every \(\gamma\) . No passage to a larger residue field is needed at the end, and no restriction on m was used.

AB.9.5. UNRAMIFIED INTEGRAL \(H^1\) , INCLUDING p-DIVISIBLE DEGREES

Let H/R be smooth affine of finite presentation with connected special fibre. Let \(a_{\sigma \tau}\) = \(a_{\sigma} \sigma\) ( \(a_{\tau}\) ) take values in H(B). Its residue cocycle is trivial by AB.9.4. By AB.9.3 lift a residue solution to \(b_{1}\) in H(B), so \(r^{(1)}_{\sigma}\) = \(b_{1} a_{\sigma} \sigma\) ( \(b_{1}\) )^{-1} is in \(K_{1}\) . These r-values still form a cocycle for the original, untwisted \(\Gamma\) action, as direct multiplication shows.

Inductively suppose \(b_{n}\) has \(r^{(n)}_{\sigma}\) in \(K_{n}\) . In the quotient \(K_{n}\) / \(K_{n+1}\) , AB.9.3 turns the cocycle into an additive semilinear cocycle \(z_{\sigma}\) . By AB.9.2 choose \(v_{n}\) with \(z_{\sigma}\) = \(\sigma\) ( \(v_{n}\) )- \(v_{n}\) . Lift \(v_{n}\) to \(h_{n}\) in \(K_{n}\) and set \(b_{n+1}\) = \(h_{n} b_{n}\) . Then \(r^{(n+1)}_{\sigma}\) = \(h_{n} r^{(n)}_{\sigma} \sigma\) ( \(h_{n}\) )^{-1}, whose additive class is \(v_{n}\) + \(z_{\sigma}\) - \(\sigma\) ( \(v_{n}\) )=0. Thus its values lie in \(K_{n+1}\) . Also \(b_{n+1}\) and \(b_{n}\) have the same image modulo \(\pi^n\) . The compatible images define b in H(B), by the inverse-limit equality of AB.9.3. For each \(\sigma\) and every n, b \(a_{\sigma} \sigma\) (b)^{-1}=1 mod \(\pi^n\) . Separatedness of B makes the equality exact. Consequently \(a_{\sigma}\) = \(b^{-1} \sigma\) (b), and \(H^1\) ( \(\Gamma\) ,H(B)) consists of one element. The trace-one computation, rather than division by | \(\Gamma\) |, is the entire congruence step. Hence the proof includes every finite unramified degree, also when p divides it.

There is also an infinite-unramified version for discrete continuous cohomology. Let \(R_{\mathrm{ur}}\) be the union of all finite unramified integer rings and \(\Gamma_{\mathrm{ur}}\) =Gal( \(k^ur\) /k). Every point of H( \(R_{\mathrm{ur}}\) ) is defined over a finite subextension because H is finitely presented. A continuous cocycle from the compact \(\Gamma_{\mathrm{ur}}\) into this discrete group has finite image. Its value is 1 on some open normal subgroup, by continuity at the identity. Choose one finite unramified E containing the definitions of all image points and with \(\Gamma_{E}\) inside that subgroup. The cocycle factors through \(\Gamma_{\mathrm{ur}}\) / \(\Gamma_{E}\) , takes values in H( \(R_{E}\) ), and the finite result supplies an integral solution. This also proves the asserted continuous \(H^1\) vanishing; compactness here is used only for finite image.

AB.9.6. FIXED COSETS

Suppose H has identified generic fibre G. The affine coordinate ring of H is torsion-free and embeds into its localization k[G], so its integral points embed into G(k). Put \(P_{E}\) =H(B), P=H(R). If \(xP_{E}\) is \(\Gamma\) -fixed, then \(a_{\sigma}\) = \(x^{-1} \sigma\) (x) is in \(P_{E}\) and is a cocycle. AB.9.5 gives \(a_{\sigma}\) = \(b^{-1} \sigma\) (b) with b in \(P_{E}\) . The representative x \(b^{-1}\) is fixed by \(\Gamma\) , since \(\sigma\) (x \(b^{-1}\) )=x \(a_{\sigma} \sigma\) (b)^{-1}=x \(b^{-1}\) . It therefore belongs to G(k); this last assertion follows by applying \(\Gamma\) -invariance to the finitely many affine coordinates of G. Thus G(k)/P -> (G(E)/ \(P_{E}\) )^ \(\Gamma\) is onto. If g,h in G(k) have equal images, every H-coordinate of \(g^{-1}\) h lies in B intersection k=R, so \(g^{-1}\) h lies in H(R). This proves injectivity. The construction commutes with left G(k)-translation.

The quotient topologies are discrete. Choose finitely many R-algebra generators of the affine ring of H. Inside G(k), integrality of each of their values is an open condition, because R is open in k; hence P is open. H(R) is also a closed subset of \(R^d\) cut out by its equations, so it is compact. The same applies to \(P_{E}\) . A bijection of the indicated discrete quotients is a homeomorphism.

For \(k^ur\) choose a finite unramified Galois E containing the finitely many coordinates of x in G( \(k^ur\) ). If its coset is fixed, then for \(\sigma\) in Gal(E/k) the element \(x^{-1} \sigma\) (x) lies in G(E) intersection H( \(R_{\mathrm{ur}}\) )=H( \(R_{E}\) ): this equality follows coordinate by coordinate from \(R_{\mathrm{ur}}\) intersection E= \(R_{E}\) and AB.0's extension valuation. Apply the finite argument. Injection is the same intersection argument over k. The target is given its discrete coset topology (also obtained from the valuation topology, since the integer ring of \(k^ur\) is open).

AB.9.7. AFFINE SEMILINEAR DESCENT WITHOUT INVOKING A DESCENT BLACK BOX

Let D be a B-algebra with semilinear \(\Gamma\) -action. Set A= \(D^\Gamma\) . First prove module descent explicitly, since it will also descend all the group maps. Choose the basis \(x_{i}\) = \(\alpha^{i-1}\) of B and its trace-dual basis \(y_{i}\) . The latter is integral: the residue trace pairing is nondegenerate (if u is nonzero, pair it with bar(c)/u), so the integral trace Gram determinant is a unit. These bases satisfy \(\sum_{i} x_{i} \gamma\) ( \(y_{i}\) )= \(\delta_{\gamma,1}\) . For example let U=( \(\gamma\) ( \(x_{i}\) )) and V=( \(\gamma\) ( \(y_{i}\) )). The unit Vandermonde determinant makes U invertible over B, and trace duality says \(U^t\) V=I. Thus U \(V^t\) =I, whose identity-row entries are the formula.

For any semilinear B-module M put \(M_{0}\) = \(M^\Gamma\) . The natural map \(\Phi\) :B \(\otimes_{R} M_{0}\) -> M, b tensor u |-> bu is an isomorphism. Its explicit inverse as an R-linear map is Psi(m)= \(\sum_{i} x_{i}\) tensor ( \(\sum_{\gamma} \gamma\) ( \(y_{i}\) m)). The second factors are invariant. The displayed \(\delta\) identity gives \(\Phi\) Psi(m)=m. For an invariant u, trace duality gives Psi \(\Phi\) (b tensor u)= \(\sum_{i} x_{i}\) tensor Tr( \(y_{i}\) b)u=b tensor u. Both identities hold for arbitrary modules, without assuming they are finite, free or torsion-free. Since \(\Phi\) is B-linear, its inverse is B-linear too. Applied to D this proves B \(\otimes_{R}\) A = D as algebras.

One further useful identity removes any exactness ambiguity. For a semilinear module the operator \(P_{c}\) (m)= \(\sum_{\gamma} \gamma\) (c m) is an R-linear projection onto its invariants: on an invariant element it is multiplication by Tr(c)=1. If M=B \(\otimes_{R}\) N with action only on B, its image lies exactly in 1 tensor N, because \(P_{c}\) ( \(\sum_{j} b_{j}\) tensor \(n_{j}\) )=1 tensor \(\sum_{j}\) Tr(c \(b_{j}\) ) \(n_{j}\) . Hence (B \(\otimes_{R}\) N)^ \(\Gamma\) =N for every R-module N.

Assume now D is of finite presentation over B. Take finitely many B-algebra generators \(d_{j}\) and write each as \(\sum_{i} x_{i} a_{ij}\) , with \(a_{ij}\) in A, using Psi. Let \(A_{0}\) be the R-algebra generated by these finitely many \(a_{ij}\) . Then B tensor \(A_{0}\) -> D is onto, and its map into B tensor A is injective because B is free. Faithful freeness annihilates no nonzero quotient, so A/ \(A_{0}\) =0. Thus A is of finite type. Since R is noetherian, the polynomial presentation of A has finitely generated kernel, making A finitely presented. Here the elementary Hilbert basis argument is enough: leading coefficients of ideals in one polynomial variable generate a finite ideal in the coefficient ring; subtract finitely many chosen leading terms to reduce degrees, then use induction in the bounded-degree coefficient module. Repeat for the finitely many variables.

If D is smooth over B, it is B-flat and hence R-flat. A is an R-direct summand by \(P_{c}\) , so A is R-flat. Equivalently in this DVR situation it is torsion-free, and a torsion-free R-module is a filtered union of its finite free submodules; tensoring shows it is flat. The generic and special fibres of Spec(A) become the smooth fibres of Spec(D) after the respective field extensions E/k and \(f_{E}\) /f. Over an algebraic closure they are the same schemes, so both are geometrically smooth.

Here is the DVR fibre-to-total-space step explicitly. Present A=R[ \(t_{1}\) ,..., \(t_{d}\) ]/I. At a point of the special fibre choose equations bar( \(F_{1}\) ),...,bar( \(F_{c}\) ) generating the localized fibre ideal with an invertible Jacobian minor; the fibre is smooth. Flatness gives I intersection \(\pi\) R[t]= \(\pi\) I, so these equations lift from I/ \(\pi\) I to equations \(F_{i}\) in I. In the local polynomial ring at this point, the finite module I/( \(F_{1}\) ,..., \(F_{c}\) ) is its own \(\pi\) -multiple. Nakayama's determinant argument forces it to vanish: on finite generators the matrix 1- \(\pi\) C has unit determinant and annihilates the module. After shrinking, I is generated by these \(F_{i}\) and their Jacobian minor is invertible. This is a standard smooth chart over R. At a generic-fibre point the same chart argument takes place after inverting \(\pi\) . These are all points of Spec(A), proving smoothness.

Finally let Spec(D) be a B-group scheme and require the semilinear action to respect its group law, identity and inverse. Under D=B tensor A one has D \(\otimes_{B}\) D = B \(\otimes_{R}\) (A \(\otimes_{R}\) A). The projection identity above identifies its invariants with A \(\otimes_{R}\) A. Equivariance therefore restricts comultiplication, counit and antipode to maps A -> A tensor A, A -> R, A -> A. Their Hopf identities hold because they hold after the faithfully free extension to B. Hence Spec(A) is a smooth affine finitely presented R-group scheme descending the given scheme and action. The construction also proves uniqueness: invariant maps recover exactly the maps before base change.

If the original special fibre is connected, so is the descended one: a nontrivial idempotent in A/ \(\pi\) A would remain nontrivial after the faithful field extension \(f_{E}\) /f, contradicting connectedness of D/ \(\pi\) D. In the smooth-group case it is geometrically connected by the identity component argument of AB.9.4. Since \(\pi\) is \(\Gamma\) -fixed and D is torsion-free, invariants commute with inverting \(\pi\) . Thus if the generic semilinear descent is G, the generic fibre of Spec(A) is G; the scheme is an integral model of G.

AB.10. The valuation lattice of an arbitrary local-field torus

Proposition. Let \(T\) be any \(k\)-torus and \(L/k\) a finite Galois splitting field. Put \(\Lambda=X_*(T_L)\), \(\Gamma=\operatorname{Gal}(L/k)\), and \(e=e(L/k)\). The valuation vector

lies in \(\Lambda^\Gamma\otimes\mathbb R\), and

Its kernel \(T^b\) is compact open. The invariant lattice is the cocharacter lattice of the maximal split subtorus \(S\subset T\). Thus the image is a full discrete lattice in \(V(S)\), including all split central directions. These assertions impose no condition on the ramification or residue characteristic.

Proof. AB.0 provides the unique extending valuation and its value group. The characters are a free lattice; multiplicativity of their evaluations makes (AB.21) a homomorphism from that lattice to \(\mathbb R\), hence a vector in \(\Lambda\otimes\mathbb R\). Choose a character basis. Its coordinates have valuations in \(e^{-1}\mathbb Z\), so the vector is in \(e^{-1}\Lambda\). Galois conjugation of the character evaluation does not change its valuation, by AB.0. Since \(t\) is rational, the vector is fixed by \(\Gamma\), giving the second containment in (AB.22).

For clarity the invariant-lattice assertion does not require a rational splitting of the torus. A cocharacter of \(T_L\) is a monomial homomorphism from \(\mathbf G_{m,L}\); its coordinate formulas descend to \(k\) exactly when that homomorphism is Galois invariant. One may check descent coefficient by coefficient in the affine coordinate rings: invariants of \(A\otimes_k L\) are \(A\), by expanding in a finite linearly independent set of elements of \(A\), and the invariance of a morphism makes its ring map take values there. The invariant lattice is saturated, because if \(n\lambda\) is invariant then \(n(\gamma\lambda-\lambda)=0\) in the torsion-free lattice. Choose a basis for this saturated sublattice and extend it to a basis of \(\Lambda\): Euclidean row and column operations reduce a lattice inclusion to Smith diagonal form, and saturation forces every nonzero diagonal factor to be one. The associated monomial map from a split torus to \(T_L\) is a closed immersion, its coordinate map being surjective on characters. It descends by the preceding coefficient argument and gives a split subtorus \(S\) with that cocharacter lattice. Every split subtorus has invariant cocharacters and is contained in it, so it is maximal. For each invariant cocharacter \(\lambda\), its value \(\lambda(\pi)\in T(k)\) has valuation vector \(\lambda\), proving the first containment in (AB.22). The middle group is consequently a finite-index over-lattice of \(\Lambda^\Gamma\), and is discrete and full.

In splitting coordinates, \(T^b\) is exactly \(T(k)\cap(O_L^*)^{\operatorname{rank}T}\). The coordinate isomorphism \(T(L)\simeq(L^*)^r\) is a homeomorphism, with inverse given by Laurent monomials. The subset \(T(k)\) is closed: it is the fixed subset of the finitely many continuous Galois automorphisms on \(T(L)\); its topology is the original \(k\)-point topology, since affine coordinates and their inverse formulas give the finite-dimensional subspace topology of AB.0. Thus this intersection is compact and open in \(T(k)\). In particular an anisotropic local-field torus is compact. \(\square\)

The proposition supplies the torus lattice and compact kernel. Relative roots and reflection representatives are proved in GB.1–GB.5. For the original nonsplit group, the centralizer lattice and the full rational normalizer action are proved in GB.19 and GB.22–GB.23.

AB.11. Passage to arbitrary reductive groups

The two-cell identity and the quasi-split root-coordinate construction are proved in GB.0–GB.6. Smooth root models and connected facet models are constructed in GB.7–GB.10. GB.11–GB.13 give the exact component-cocycle test, torus lifting and stable apartments. The arithmetic of the unramified field is GB.9 and GB.14; its torus vanishing, regular-class representative and quasi-splitting are proved in GB.16–GB.18. For descent to the original field, GB.19 proves the anisotropic centralizer and central lattice, GB.20 the rational filtration and half-apartment bounds, GB.21 the residue/coface correspondence, GB.22 the maximal rational apartment and wall-exchange assertions, and GB.23 the full stabilizers, topology, metric and properness. These statements provide the hypotheses for the affine geometry above, while retaining the full Euclidean centre and the finite component obstruction.

Relative roots and arbitrary-group descent

GB.0. The complete filtered rank-one two-cell identity

Use the valuation extended from the original local field, without renormalizing intermediate fields. Let \(E/k\) be finite separable. In the first case take \(H=\operatorname{SL}_2(E)\), with its usual upper and lower root coordinates and \(\varphi_\pm(x_\pm(u))=\omega(u)\). In the second case let \(F/E\) be separable quadratic, take \(H=\operatorname{SU}_3(F/E)\), and use exactly BF.7's matrices

In this second case \(\varphi_\pm(x_\pm(u,v))=\omega(v)/2\); the doubled-root group is already included in each short-root group. The diagonal bounded torus \(T_0^b\) consists respectively of \(\operatorname{diag}(t,t^{-1})\) with \(\omega(t)=0\), and \(D(t)=\operatorname{diag}(t,\bar t/t,1/\bar t)\) with \(\omega(t)=0\). Let \(r\) be an attained nonidentity value, and put

Choose \(u_0\in A\setminus A^+\) and its BF rank-one normalizer completion \(s=m(u_0)\). In SL \(_2\) this is \(n(c_0)\), with \(\omega(c_0)=r\); in SU \(_3\) it is \(M(v_0)\), with \(\omega(v_0)=2r\), where

Theorem. In both cases,

is a compact subgroup. Its normalizer element acts as reflection in the wall \(a(h)+r=0\) and has square in \(T_0^b\). The statement includes every attained short-root level of every ramified SU \(_3\) chart in every residue characteristic.

Proof. For real weights \(w_i\), set \(\|z\|_w=\max_i\exp(-\omega(z_i)-w_i)\). A matrix and its inverse preserve this norm exactly when both have the entry inequalities \(\omega(g_{ij})\ge w_j-w_i\): necessity follows by testing basis vectors, sufficiency by the nonarchimedean triangle inequality. In SL \(_2\) use weights \((-r/2,r/2)\); in SU \(_3\) use \((-r,0,r)\). The unitary inverse is \(J\bar g^{\mathsf T}J\), where \(J\) is the antidiagonal form matrix of BF.7, so the same inequalities for \(g\) imply those for its inverse. The SL \(_2\) inverse formula gives the identical conclusion. Thus the set of matrices in \(H\) satisfying these inequalities is a norm-isometry subgroup, denoted \(K\). It is compact: each entry is in a fixed compact valuation ball by AB.0, and the determinant and unitary equations cut out a closed subset of their finite product. The inequalities are

They place \(T_0^b,A,D\) in \(K\); the trace inequality \(\omega(u)\ge\omega(v)/2\) is BF.7's locally proved identity. The opposite factors completing \(u_0\) have value \(-r\) by BF.7, or the pinned SL \(_2\) formula, so \(s\) is in the generated subgroup. We now prove that every element of \(K\) has the displayed two-cell expression. This also proves subgroup closure of (GB.4), rather than assuming it from the union formula.

First take SL \(_2\) and write \(g=\begin{pmatrix}a&b\\c&d\end{pmatrix}\). If \(\omega(c)>-r\), then \(\omega(bc)>0\); the equation \(ad-bc=1\) forces \(a,d\) to be units. Direct Gaussian multiplication gives

If \(\omega(c)=-r\), left multiplication by \(x_+(-a/c)\in A\) gives \(\begin{pmatrix}0&-1/c\\c&d\end{pmatrix}=n(-1/c)x_+(d/c)\). The last parameter has valuation at least \(r\), and \(n(-1/c)s^{-1}\in T_0^b\), since both normalizer parameters have valuation \(r\). Thus \(g\in A sT_0^b A\). These are all cases.

Now take SU \(_3\) and write its first column as \((a,b,c)^{\mathsf T}\). Its unitary isotropy equation is

If \(\omega(c)>-2r\), then \(a\) must be a unit. Indeed, if \(\omega(a)>0\), (GB.7) forces \(\omega(b)>-r\). All three first-column entries would then have strictly greater valuation than their entry bounds. The weighted norm of \(ge_1\) would be strictly smaller than that of \(e_1\), contradicting norm preservation. With \(a\) a unit, (GB.7) again gives \(\omega(b)>-r\). The lower pair \((p,q)=(b/a,c/a)\) satisfies \(q+\bar q=-p\bar p\) and has value greater than \(-r\). Multiplication by \(x_-(p,q)^{-1}\) kills the two lower first-column entries. The unitary equations then force the result to be upper triangular, with diagonal \(D(a)\), and after removing that diagonal its upper coordinates are \(u=-\overline{g_{12}/a}\), \(v=g_{13}/a\). Both follow from the unchanged first row of that multiplication. The entry bounds imply \(\omega(u)\ge r\), \(\omega(v)\ge2r\). Hence \(g\in B T_0^b A\).

In the remaining case \(\omega(c)=-2r\), put

The inequalities give \(\omega(u)\ge r\), \(\omega(v)\ge2r\). Equation (GB.7), divided by \(c\bar c\), gives \(a/c+\bar a/\bar c=-b\bar b/(c\bar c)\). Consequently

This equality is an identity in the field and never divides by two. Thus \(x_+(u,v)\in A\). It sends the first column to \((0,0,c)^{\mathsf T}\). The unitary pairings with this column and the determinant-one condition force the resulting matrix to have the form

where \(u'=e/(-\bar c/c)\), \(v'=j/c\); its remaining unitary equations give \(f=-c\bar u'\) and \(v'+\bar v'=-u'\bar u'\). Since \(x_+(u,v)\) is a norm isometry, \(g'\) still satisfies (GB.5), so \(\omega(u')\ge r\), \(\omega(v')\ge2r\). Also \(\omega(1/\bar c)=2r=\omega(v_0)\), so \(M(1/\bar c)s^{-1}=D((1/\bar c)/v_0)\in T_0^b\). After moving this unit diagonal across \(s\), we obtain \(g\in A sT_0^b A\). This completes all cases and proves (GB.4).

Finally the square formulas are \(n(c_0)^2=\operatorname{diag}(-1,-1)\) and \(M(v_0)^2=D(v_0/\bar v_0)\), whose parameters are units. The apartment reflection formula is respectively BF.2 or BF.7, with \(\omega(c_0)=r\) or \(\omega(v_0)=2r\). It is exactly reflection in \(a(h)+r=0\). \(\square\)

Ambient torus extension. Suppose a compact bounded torus group \(T^b\) in the ambient group contains the displayed rank-one unit torus and conjugates the two root charts by absolute-character factors of valuation zero. It normalizes \(A,D,B\), and hence their subgroup \(K_r\). The product \(T^bK_r\) is a compact subgroup. Moving its torus factors across the root groups and across the reflection changes them only within \(T^b\), so (GB.4) becomes exactly \(A sT^b A\cup B T^b A=\langle T^b,A,D\rangle\). AB.10 supplies the zero absolute-character valuations of the bounded torus. Thus once the relative chart has been identified in the ambient group, this gives the full torus form of the filtered two-cell statement.

The actual primary reading behind the rank-one chart is free Bruhat–Tits II §§4.1.9–4.1.12, with the sign conversion already bound in BF.7. The new proof does not cite the source's omitted matrix computation: (GB.7)–(GB.10) contain the computation used here.

GB.1. Relative roots and their rank-one charts

Let \(G/k\) be connected reductive and quasi-split. The preceding programme proof Automorphisms, forms and parabolic subgroups, Proposition 5.1, gives a finite Galois splitting field \(L/k\) and a pinning on \(G_L\) for which \(\Gamma=\operatorname{Gal}(L/k)\) acts by based-datum automorphisms, permuting the simple frames. Central directions and isomorphic components may be permuted. Let \(T,B\) be the descended torus and Borel, let \(S\) be the maximal split subtorus of \(T\), and let \(\widetilde\Phi,\widetilde\Delta\) be the absolute roots and simple roots. AB.10 identifies the real apartment of \(S\) with the invariant absolute cocharacter space. All statements below concern this chosen rational Borel pair.

Theorem. No absolute root restricts to zero on \(S\), and \(Z_G(S)=T\). Thus \(S\) is a maximal split torus of \(G\). The nondivisible relative roots form a reduced crystallographic root system. Its finite Weyl group is \(N_G(S)(k)/T(k)\). For each nondivisible relative root \(a\), the rank-one derived subgroup is centrally covered by exactly one of

with \(E/k\) finite separable and \(F/E\) separable quadratic. These covers induce isomorphisms of the positive and negative unipotent groups, including doubled-root subgroups. Every relative Weyl element has a rational representative which permutes absolute integral root coordinates with signs.

Proof. Choose an invariant Euclidean inner product on the absolute derived root space by averaging over the finite Weyl and diagram groups. Choose a point in the absolute positive chamber and average its finite \(\Gamma\)-orbit. All simple-root values of the average are positive, so it is an invariant regular point. Every positive absolute root has positive value there; consequently no absolute root restricts to zero. The restrictions of the simple roots belonging to one \(\Gamma\)-orbit are equal. Restrictions from distinct orbits are linearly independent: averaging in the absolute simple-root basis makes them the independent orbit sums, divided by their orbit sizes. Thus they form the simple inequalities of the positive chamber in the invariant space.

We first determine the subsystem supported on one simple orbit \(O\), without appealing to the classification list of Dynkin diagrams. The graph of the simple roots has no cycle. Normalize the simple vectors to length one. On an edge their inner product is at most \(-1/2\), since the product of the two nonzero Cartan integers is at least one; other inner products are nonpositive. If a cycle had \(m\) vertices, the squared norm of their sum would therefore be at most \(m-m=0\), contradicting positive definiteness. The induced graph on \(O\) is a finite forest and \(\Gamma\) acts transitively on its vertices, so every vertex has the same degree. A finite regular forest has degree zero or one: each nonempty tree has a leaf, and a regular tree of degree one is a single edge. Thus the subsystem is a product of \(A_1\)'s or a product of \(A_2\)'s. In the edge case both vertices have equal length, so their equal off-diagonal Cartan integers must be \(-1\) by positive definiteness of the two-by-two matrix. This also verifies the asserted \(A_2\) identification directly.

Let \(w_O\) be the product of the longest Weyl elements of these components. In the isolated case it is the product of their commuting simple reflections. In the edge case its factor is \(s_i s_j s_i=s_j s_i s_j\). It commutes with \(\Gamma\), fixes pointwise the invariant hyperplane where the common restriction \(a\) vanishes, and sends \(a\) to \(-a\). It is therefore the orthogonal reflection in that hyperplane. Its relative coroot is

Indeed the reflection formulas on an invariant vector have equal simple-root values within the orbit. In the edge case the longest element is reflection in \(\alpha_i+\alpha_j\), whose value there is \(2a\). In either case \(a(a^\vee_{\mathrm{rel}})=2\), and the coroot is an invariant integral cocharacter.

The absolute-root hyperplanes restricted to the invariant space form a finite arrangement. A regular invariant point lies in one absolute chamber, which is \(\Gamma\)-stable. Conversely, averaging an interior point of a stable absolute chamber produces an invariant interior point. Thus relative chambers are exactly the intersections with stable absolute chambers. Reflection in an orbit wall sends the base relative chamber to its neighbor: its longest component element reverses the roots supported on that orbit while retaining positivity of every root with another simple-root coefficient, because that coefficient is unchanged. A generic finite wall-crossing path now shows that the group generated by the \(w_O\) is transitive on relative chambers. It acts freely: if an absolute Weyl element preserving the invariant space stabilizes the relative positive chamber, it sends a regular positive invariant point into the same absolute positive chamber, and the actual absolute chamber theorem of Root data, Weyl chambers and the Bruhat decomposition, §3, forces it to be the identity. The same argument shows that every \(\Gamma\)-fixed absolute Weyl element is in this generated group, by first matching its relative chamber and then using freeness.

Any relative-root hyperplane is a wall of a relative chamber. Moving that chamber to the base by the generated group shows that the absolute roots on its normal ray are conjugate to the subsystem supported on one simple orbit. The support has only the roots of \(A_1\) or \(A_2\), so their restrictions are exactly \(\pm a\), or \(\pm a,\pm2a\). This proves that the nondivisible relative roots are reduced. Formula (GB.12), transported by the generated Weyl group, gives integral Cartan pairings with all other restricted roots. Finiteness, spanning, reflection invariance and these pairings prove the crystallographic root-system axioms.

The absolute Bruhat-cell coordinates also prove the centralizer assertion scheme-theoretically. Over a splitting field, a point centralizing \(S\) maps in \(G/B\) to an \(S\)-fixed point. In each absolute Bruhat chart all its affine coordinates have nonzero restricted weights, so a fixed point has every unipotent coordinate zero and is a Weyl-labelled point. The remaining Weyl label must fix \(S\) pointwise. It fixes the regular invariant point above, hence is the identity by the absolute chamber theorem. On closed geometric points the centralizer is therefore \(T\). Near \(T\) the integral big-cell chart has root coordinates of nonzero \(S\)-weight; requiring commutation as a group-scheme identity sets them all to zero. This excludes infinitesimal directions as well: a coordinate multiplied by a nontrivial Laurent character can be fixed only when it is zero, over any coefficient algebra. Hence the entire finite-type centralizer scheme is \(T\). A split torus containing \(S\) would lie in \(T\), contrary to maximality there proved in AB.10. Thus \(S\) is maximal in \(G\).

The pinned normalizer representatives of the orbit reflections are rational. For isolated components take the product of \(n_i(1)\). For an edge take \(n_i(1)n_j(1)n_i(1)\). Its equality with the reversed braid can be checked in the pinned \(\operatorname{SL}_3\) model: both matrices are \(\begin{pmatrix}0&0&1\\0&-1&0\\1&0&0\end{pmatrix}\). This computation and the commuting isolated factors make the product fixed by every diagram permutation. Each representative acts on all absolute integral root lines by signs, by the actual pinned comparison theorem and integral normalizer tables in Pinnings and the classification of split reductive groups. Since the orbit reflections generate the fixed Weyl group, every element has such a rational representative. Conversely a rational normalizer of \(S\) normalizes \(Z_G(S)=T\), and its absolute Weyl image is \(\Gamma\)-fixed. We have proved exactly \(N_G(S)(k)/T(k)\) equals the relative reflection Weyl group, including surjectivity on rational representatives.

It remains to identify the rank-one groups as groups, not only their root diagrams. For a simple orbit its supported semisimple group is centrally covered over \(L\) by the product of the matrix groups \(\operatorname{SL}_2\) or \(\operatorname{SL}_3\) just found. This is an actual pinned root-datum argument: quotient the matrix group's centre to match the intermediate character lattice of the supported derived group, then use Root data, Weyl chambers and the Bruhat decomposition Theorem 7.1 and Lesson 05 Theorem 5.1. Those earlier proofs construct the central quotient and pinned isomorphism over arbitrary characteristics. Their big-cell quotient proof preserves each root group and each positive or negative root-coordinate product, so the central cover induces actual unipotent isomorphisms, including when its centre is non-smooth.

The Galois group permutes these components transitively. For the stabilizer of one component, let \(E\) be its fixed field. In the \(A_1\) case the based diagram has no nontrivial automorphism, and its pinning descends to \(\operatorname{SL}_{2,E}\). The transitive product is its restriction of scalars: over \(L\) this consists of its factors indexed by the embeddings of \(E\), and invariance determines every component from the chosen one, coefficient by coefficient. In the \(A_2\) case the component stabilizer exchanges its two simple roots; otherwise the original orbit would not contain both vertices of an edge. The kernel of that action fixes a separable quadratic field \(F/E\). Its pinned twist is exactly the special unitary matrix group in BF.7: over \(F\), the nontrivial action is \(g\mapsto J(g^{\mathsf T})^{-1}J\). Choose the integral simple frames \(x_{12}(t)\) and \(x_{23}(-t)\); this automorphism exchanges those frames exactly. Pinned uniqueness identifies this model with the given diagram action. The fixed equation is \(g^{\mathsf T}J\bar g=J\), with determinant one. The triangular fixed equations are precisely (GB.1). Thus its transitive product descends to \(\operatorname{Res}_{E/k}\operatorname{SU}_3(F/E)\), with the long root its trace-zero central coordinate. Transporting by a rational relative Weyl representative handles every relative root. This proves (GB.11), its exact root-group assertions, and all stated generalities. \(\square\)

The free original Bruhat–Tits II §§4.1.1–4.1.12 supplies the primary rank-one/orbit problem and matrix conventions. The finite-forest argument above replaces its reference to the diagram classification; its pinned quotient and coordinate identifications are bound to the preceding proofs just specified.

GB.2. Weighted relative valuations and the enlarged normalizer

For a nondivisible relative root \(a\), use any order of the absolute root coordinates on its ray. If \(\beta|_S=m_\beta a\), then \(m_\beta\) is one or two by GB.1. Define

For a doubled root use \(\varphi_{2a}=2\varphi_a\) on its group.

Theorem. This definition is independent of coordinate order. It equals \(\omega(u)\) on the restriction-of-scalars SL \(_2\) chart and \(\omega(v)/2\) on (GB.1), with \(\varphi_{2a}(0,v)=\omega(v)\). Its level subgroups satisfy

They are compact open in the root group and give its identity neighborhood basis. If \(b\) is not a negative multiple of \(a\), then

Opposite normalizer completions have the negative input value, and all normalizer valuation changes are the affine root changes. The enlarged normalizer acts on \(V(S)=V_D(S)\oplus V_Z(S)\) by \(tn_w:x\mapsto wx-\nu(t)\), with compact open kernel \(T^b\) and full discrete torus-translation lattice. The resulting apartment and its valuation translation class do not depend on the equivariant pinning.

Proof. On an SL \(_2\) ray its conjugate absolute parameters are the embeddings of one element of \(E\); all have the same valuation by AB.0. They commute, so the result and order independence are immediate. On an A \(_2\) component of a unitary ray, the two short parameters are \(u,\bar u\), with signs, and the long parameter is \(v\) in one order and \(-\bar v\) in the other. Indeed exchanging the two short root factors changes the central parameter by \(u\bar u\), and \(v+u\bar u=-\bar v\). Moving the central root factor does nothing. These are all possible orders. BF.7 proves \(2\omega(u)\ge\omega(v)\), so their weighted minimum is exactly \(\omega(v)/2\) in every order. Different components commute and field embeddings preserve valuation. This proves (GB.13)–(GB.14), including the long-root normalization. The subgroup, compactness, openness and neighborhood assertions are the actual matrix and trace calculations of BF.7, or the additive field-ball assertion of BF.1, transported through GB.1's root-group isomorphisms.

For the commutator estimate first take linearly independent \(a,b\). Choose a positive system containing both: a real vector can have positive values on both precisely because they are not opposite multiples. Its invariant absolute positive system contains all their absolute root factors. Collect the commutator in an absolute root order using BF.1's integral commutator identities. Collection terminates by absolute root height. Each resulting coordinate is a polynomial with integer coefficients in the input parameters, homogeneous for torus weights. Give a parameter over \(a\) degree \((m_\beta,0)\), and a parameter over \(b\) degree \((0,m_\gamma)\). Linear independence makes the degree of an output root \(\delta\) uniquely \((p_\delta,q_\delta)\), where \(\delta|_S=p_\delta a+q_\delta b\). Both coordinates are positive integers for a nonzero commutator output: each term vanishes when either input family is zero, and the collection identities create only positive sums. Its valuation is therefore at least \(p_\delta r+q_\delta s\). Integer structure constants have nonnegative valuation in every characteristic.

Regroup these absolute coordinates by nondivisible relative rays. The block order is \(\Gamma\)-stable; uniqueness of the absolute root factorization makes each resulting block rational. If a block has a nonzero short parameter, its root \(c=p_ca+q_cb\) has positive integral coefficients, and its long coordinates, if any, have degree \((2p_c,2q_c)\). Formula (GB.13) puts the block in \(U_{c,p_cr+q_cs}\). If \(c\) has nonintegral coefficients in the basis \(a,b\), no short absolute coordinate of that degree can occur. The block then has only its long-root coordinate, of integral degree \((p_{2c},q_{2c})\), and lies in \(U_{2c,p_{2c}r+q_{2c}s}\). A block with zero short coordinates and integral degree is already covered by the former bound. These cases exhaust every ray and prove (GB.15). For positive multiples on one ray, the only nonzero case is the short-short unitary commutator: BF.7 gives \([U_{a,r},U_{a,s}]\subset U_{2a,r+s}\). Its long-root group is central; SL \(_2\) is additive. This proves all same-ray cases too.

For rank-one completions use exactly BF.1 and BF.7 and transport by the rational relative Weyl representatives of GB.1. They prove existence and uniqueness of the two opposite factors, each of value \(-\varphi_a(u)\), and the reflection formula. In the absolute coordinates, conjugation by \(t\in T(k)\) scales a \(\beta\)-parameter by \(\beta(t)\). AB.10 makes its valuation \(\beta(\nu(t))=m_\beta a(\nu(t))\). Thus (GB.13) changes by \(a(\nu(t))\). The pinned Weyl representatives permute coordinates with signs and preserve their valuations. This proves the stated normalizer covariance and affine signs on every root group; a doubled root has twice the short-root change.

The absolute coroot space and the central space are stable under \(\Gamma\), so taking invariants gives the displayed direct sum. AB.10 gives \(X_*(S)\subset\nu(T(k))\subset e_L^{-1}X_*(S)\), and its compact open kernel. The rational Weyl representatives identify the full normalizer as \(T(k)\{n_w\}\), so its affine kernel is exactly \(T^b\), its image is a lattice with a finite linear Weyl part, and its action is proper with compact orbit space on the enlarged apartment. As in BF.3's elementary Euclidean argument, compact subgroups have fixed points by averaging and their fixed affine spaces are contractible; invariant partitions give the usual affine equivariant maps. Thus this normalizer apartment is its numerable universal proper space, including the central factor.

Finally a change of equivariant pinning multiplies the simple frames by constants \(c_i\) with Galois-equivariant indices. Pinned transport multiplies each other root frame by the corresponding product of the \(c_i\), with its simple-root exponents; this is the torus-weight relation of the actual pinned comparison proof. Galois invariance of the extended valuation makes \(\omega(c_i)\) constant on each orbit. There is a unique invariant derived vector \(h\) with \(\alpha_i(h)=-\omega(c_i)\); the simple roots are a basis on the absolute derived apartment. For a fixed group element with old parameter \(u\), the new parameter is \(u/c_i\); its new valuation is \(\omega(u)-\omega(c_i)=\omega(u)+\alpha_i(h)\). This is exactly the valuation-origin translation with the pullback convention of BF.2. For a ray with \(\beta|_S=m_\beta a\), division by \(m_\beta\) in (GB.13) gives the same translation \(a(h)\). Hence the affine translation class is intrinsic. Central directions are retained as the independent enlargement. \(\square\)

Reference: Bruhat–Tits II, Sections 4.2.2–4.2.11. The coordinate-order, commutator, normalizer and pinning arguments are proved above.

GB.3. Compact face groups and every-order relative Iwahori coordinates

On \(V_D(S)\), remove the hyperplanes \(a(v)+r=0\) for nondivisible relative roots and attained values \(r\). The arrangement is locally finite: by AB.0 and GB.2 every such value belongs to \((2e_L)^{-1}\mathbb Z\). A chamber is a connected component of its complement. Define

The doubled-root generators are already contained in the short-root ones, so this agrees with using all relative roots. For any positive relative system and any orders on its positive and negative nondivisible roots,

is a homeomorphism. All \(P_v\) are compact open; \(I_C\subset P_{v_0}\) has finite index for every \(v_0\in\overline C\). For \(x=(v,z)\) the groups \(K_x=P_vN_x\) are compact open, normalizer covariant, and satisfy \(K_x\cap N=N_x\).

Proof. On a chamber, membership of an element in \(U_{a,-a(v)}\) is constant: its sign \(\varphi_a(u)+a(v)\) can change only at one of the removed hyperplanes. For a nonidentity member it is strictly positive. In absolute coordinates this says \(\omega(c_\beta)+\beta(v)>0\), by GB.2. Expand any finite word in the relative root generators into absolute root factors over \(L\); move its unit-valued torus factors to the left. Apply the exact finite-polynomial big-cell argument proved in BF.6. We spell out why its hypotheses and bounds are unchanged here. The word-map denominator has constant term one and all its other terms have torus weight zero; each such nonconstant monomial has strictly positive valuation from the strict bounds on its variables. Thus the denominator is a valuation unit. Each output root coordinate has its corresponding torus weight, so every output monomial has \(\omega(c_\beta)+\beta(v)>0\); every torus character and its inverse have unit value. The calculation uses integer coefficient polynomials on one finite word and works with the fractional value group of \(L\); it needs neither an absolute alcove containing \(v\) nor a ramified descent theorem.

Choose an absolute positive system restricting to the chosen positive relative system. Group the arbitrary-order absolute root factors into the nondivisible relative rays, with any of the requested orders. GB.1 identifies each ray block as a rational root group. More precisely each block subgroup is Galois stable, and the block order itself is fixed. The unique absolute factorization of a rational product is therefore fixed block by block, even in an A \(_2\) block where coordinate conjugation is nonlinear. The torus factor is rational as well. The output strict weighted bounds, divided by the ray multiplicities, put every block back in the level group of (GB.17), and put the torus in \(T^b\). Conversely all those factors are generators. The absolute big-cell uniqueness gives uniqueness of this product. Its inverse coordinates are regular on that cell, hence continuous; its forward map is multiplication. This proves the homeomorphism and constancy of \(I_C\).

The rational big cell used here is itself an open chart: the stable absolute open immersion descends coefficient by coefficient to \(U^-\times T\times U^+\to G\). Its coordinate-ring kernel and cokernel vanish after the faithful extension to \(L\), hence already over \(k\); its image open is described by the product of the Galois conjugates of an absolute cell denominator. Thus (GB.17), whose factors contain identity neighborhoods in their root groups and torus, contains an open identity neighborhood in \(G(k)\). Consequently \(I_C\) is open.

For compactness, use BF.4's faithful integral representation of the pinned split model over \(L\), with its absolute weight decomposition. On the norm with weight vector \(v+z\), every absolute root factor in a relative generator has \(\omega(c_\beta)+\beta(v)\ge0\). BF.5's weight-polynomial estimate consequently makes that factor and its inverse norm isometries. The torus \(T^b\) has unit values for every absolute character by AB.10, and is an isometry as well. The group generated by these factors lies in the compact norm-isometry group in \(\operatorname{GL}_N(L)\). The closed immersion and the closed subset \(G(k)\subset G(L)\) make its intersection with \(G(k)\) compact. Choose a chamber with \(v\) in its closure. Its \(I_C\) is contained in \(P_v\), since the strict inequalities become nonnegative in the limit. A subgroup containing an open subgroup is open and closed. Thus \(P_v\) is closed in that compact group and is compact. Openness also follows. Its cosets modulo \(I_C\) are compact and discrete, hence finite.

Normalizer covariance follows from GB.2. The normalizer's affine image is a discrete lattice with finite Weyl group, so a point stabilizer consists of finitely many cosets of \(T^b\); it is compact and open in \(N\). It normalizes \(P_v\), giving the compact open product \(K_x\). If a normalizer element is in \(P_v\), its norm action in the faithful representation preserves the norms of all weight vectors. The integral pinned Weyl representatives are invertible on their integral weight lattices, and the torus contribution changes their norms by its character valuations. Exactly the norm-detection calculation in BF.5 therefore forces its affine action to fix \(v+z\). This holds for every central \(z\), since roots vanish on that factor. Hence \(P_v\cap N\subset N_x\), and multiplying by \(N_x\) proves \(K_x\cap N=N_x\). \(\square\)

GB.4. Relative walls, the pivot identity and affine exchange

Lemma. The relative arrangement is reflection invariant and has bounded reduced chambers. Its wall reflections form an affine Coxeter group acting simply transitively on its chambers; closed chambers are fundamental domains, and point stabilizers are generated by their walls through the point.

Proof. For every attained \(r\), the rank-one completion of GB.0 is a rational normalizer element acting as reflection in \(a(v)+r=0\). GB.2's covariance maps every root level and every corresponding wall to another level and wall. Thus the entire arrangement is invariant. For any root \(a\), the integral cocharacter \(a^\vee_{\mathrm{rel}}\) of GB.1 evaluated at \(\pi\) shifts its root values by two. From one attained level we therefore get an unbounded arithmetic sequence in both directions. The values of \(a\) on any chamber lie between consecutive walls of this sequence, so are bounded. The roots span the dual derived apartment, making its chamber bounded. Its closure is a compact polytope by local finiteness.

The affine reflection group is a subgroup of the normalizer's discrete lattice-with-finite-linear-part image. The complete gallery-disc proof of AB.2 now applies verbatim in its proved general form: it only uses a locally finite reflection-invariant hyperplane arrangement, bounded chambers and finite linear point stabilizers. At a codimension-two intersection the finite linear stabilizer makes consecutive sectors dihedral. Backtracking and these local dihedral braids reduce a closed gallery. This proves transitivity, freeness and the Coxeter presentation, with actual finite pair orders and no relation for infinite pair orders. The local-star argument proves the closed fundamental domain and wall-generated point stabilizers. No identification with the split coroot lattice is asserted for this possibly differently spaced arrangement. \(\square\)

We record one additional local matrix identity needed to use the two-cell theorem for exchange. With \(A,D,s,r\) as in GB.0 and \(u\in A\setminus A^+\),

In SL \(_2\) , the lower-right entry of \(n(c_0)x_+(c)\) is \(-c/c_0\), a unit when \(\omega(c)=r\). Its Gaussian factorization with that lower-right pivot has upper parameter of valuation at least \(r\), lower parameter at least \(-r\), and unit diagonal. In SU \(_3\) , the lower-right entry of \(M(v_0)x_+(u,v)\) is \(v/\bar v_0\), again a unit. For any weighted norm isometry \(g\) with unit \(d=g_{33}\), the upper pair in its reverse Gauss factorization is \((g_{23}/d,g_{13}/d)\), its lower pair is \((-\overline{g_{32}/d},g_{31}/d)\), and its torus parameter is \(t=1/\bar d\). The last column and last row unitary equations give both trace relations. Multiplying the upper factor inverse kills its two upper last-column entries; unitary pairings force the remainder lower triangular with diagonal \(D(t)\). The entry bounds (GB.5) place these pairs in \(A,D\), respectively. This proves (GB.18) with every parameter bound, including characteristic two. The ambient unit torus can be used by GB.0's last paragraph.

Proposition. Let \(C\) be a relative chamber and \(I=I_C\). For each of its walls choose \(\alpha=a+r\) positive on \(C\), with \(a\) nondivisible, and the rank-one representative \(s=m(u_0)\) at that level. For every \(n\in N\),

Proof. At this wall the chamber's positive-ray group is \(A=U_{a,r}\) and its negative-ray group is \(B=U_{-a,-r+}\). To check the second equality note that the positive and negative value sets coincide by the rank-one charts and are closed under negation by the unique opposite completion. A chamber neighboring the wall has no intervening attained parallel level. This proves exactly the strict opposite level, even when the gaps of attained SU \(_3\) values are unequal.

Crossing the wall changes only these two ray groups. Choose a positive system containing \(a\) and order its factors with \(a\) last. GB.3 gives \(I=J A\), with \(J\subset I\cap s^{-1}Is\). As in AB.3 it suffices to examine \(s u n\), \(u\in A\). If \(u\in A^+\), its reflection conjugate is in \(B\), giving \(I s n I\). If \(u\) is at the wall level and \(n^{-1}\alpha\) is positive on \(C\), its conjugate under \(n^{-1}\) is in \(I\), giving the same cell. If that affine root is negative, (GB.18) gives \(s u\in A T^b D\), and \(n^{-1}D n\subset I\) because \(n^{-1}(-\alpha)\) is positive. This gives \(InI\). These are all cases because an attained affine root has constant nonzero sign on a chamber. Finally \(sAs^{-1}\) contains a root element at the opposite level \(-r\), which is excluded by the strict level of \(B\). The unique factorization of GB.3 proves it is not in \(I\). Thus conjugation changes the chamber group. \(\square\)

GB.5. Root generation, full-group cells and the relative affine labels

Let \(N_{\mathrm{aff}}\) be generated by \(T^b\) and the chamber-wall representatives of GB.4, and let \(W_{\mathrm{wall}}\) be their actual affine reflection image. Put \(G^0=\langle I,N_{\mathrm{aff}}\rangle\).

Proposition. One has

where \(\Omega_C=\{n\in N:nC=C\}\), allowing central translations. The subgroup \(G^0\) is open and normal, \(I\cap N=T^b\), and

Any two points and any two chambers in the relative gluing lie in a common translated enlarged apartment.

Proof. Denote the root-generated group on the right in (GB.20) by \(R\). All chamber-wall representatives are opposite-root completions, so \(I,N_{\mathrm{aff}}\subset R\). Conversely every reflection in any wall of the arrangement is a conjugate of a base-chamber wall reflection by GB.4. Its rank-one representative differs from that conjugate only by the affine kernel \(T^b\). Thus it is in \(N_{\mathrm{aff}}\). Take a root \(a\) and a nonzero root element of value \(r\); its conjugates by \(a^\vee_{\mathrm{rel}}(\pi)^n\) have values \(r+2n\) by GB.2. Their reflection products yield translations by nonzero multiples of \(a^\vee_{\mathrm{rel}}\), in both directions, inside \(N_{\mathrm{aff}}\). Conjugation by their powers moves the fixed chamber tail for \(U_a\) to arbitrarily low levels. These tails exhaust every element of \(U_a(k)\). Hence \(R=G^0\), including every doubled-root subgroup. It is open because it contains \(I\).

We prove rational full-group generation explicitly. Apply the actual split Bruhat proof Root data, Weyl chambers and the Bruhat decomposition, Theorem 6.2, over \(L\), to the rational element \(g\). Its absolute Bruhat label is fixed by \(\Gamma\), since the Borel is stable and its cells are distinct. GB.1 identifies that label as a relative Weyl label with a rational representative \(n_w\). The absolute cell has unique coordinates in \(U_w\times T\times U^+\) after selecting that representative. Its inversion-root set is Galois stable and is a union of whole relative rays: all absolute roots on a ray have the same sign before and after the relative Weyl element. Group the coordinates by these rays. Uniqueness forces each block, the torus factor and the positive factor to be rational. Thus \(g\) is a product of rational relative root elements, a torus element and \(n_w\).

For each simple relative reflection choose any nonidentity root element and its rational opposite completion. It represents that reflection. The rational representative used above differs from it by an element of \(T(k)\). Since \(T(k)\) normalizes all relative root groups and \(T^b\), the set \(T(k)R\) is a subgroup. It contains these representatives and all the displayed Bruhat factors, proving \(G=T(k)R\). It also proves normality of \(R\): it is normalized by \(T(k)\), and by its own elements. This argument does not assume that a particular integral unitary Weyl representative is itself an attained level-zero completion.

The normalizer preserves the arrangement. Its chamber action and GB.4 give \(N=N_{\mathrm{aff}}\Omega_C\); elements of \(\Omega_C\) preserve the root-level chamber groups and normalize \(I\). Since \(G^0\) is normal and contains \(N_{\mathrm{aff}}\), this proves the last full-group formula in (GB.20). If an element of \(I\cap N\) fixes all points of \(C\times V_Z\), its affine action is the identity and it is in \(T^b\) by GB.2; conversely \(T^b\subset I\). The fixing assertion follows from GB.3's \(P_v\cap N\subset N_{(v,z)}\), for every chamber point and every \(z\).

We now have exactly the hypotheses of the locally proved exchange theorem AB.4 for \((G^0,I,N_{\mathrm{aff}})\): the quotient is the Coxeter group of GB.4, (GB.19) is exchange, the square is in \(T^b\), and conjugation changes \(I\). Its proof gives distinct affine labels and their multiplication rules. The wall group is normal in the full affine normalizer image, since conjugation maps every wall reflection to another. Decompose that image into its chamber stabilizer and the simply transitive wall group. The explicit extended-label induction of AB.4 therefore applies and proves (GB.21), with this actual relative wall group in place of the split coroot semidirect product. No additional extended BN-pair theorem is cited.

For the core equality, use GB.3's unique root-coordinate factorization of an element of \(I\). A split cocharacter scales every absolute coordinate on a relative ray by its nonzero integral relative-root pairing. Powers with the appropriate sign would lower any nonzero root parameter below its fixed chamber bound. Membership in all those torus conjugates of \(I\) thus forces all root coordinates to be zero. The remaining factor is \(T^b\), which is normal in \(N\). This proves the core assertion. Finally for two point or chamber representatives normalize their reduced coordinates into \(\overline C\) and write their relative group element as \(i_1ni_2\), using the full double-coset partition. The translated apartment \(gi_1V(S)\) contains both, exactly as in the explicit argument AB.5. \(\square\)

GB.6. The full quasi-split enlarged local-field building

Theorem. For every connected quasi-split reductive group over the original nonarchimedean local field, the quotient formed from \(G(k)\), its enlarged apartment and the groups \(K_x\) of GB.3 is locally compact, second countable and Hausdorff, with continuous proper cocompact action. It has a compatible proper complete CAT(0) metric, Euclidean on each apartment. It is a numerable universal proper \(G(k)\)-space. This includes all residue characteristics, all ramification of its splitting field and all nonsplit central tori.

Proof. The hypotheses of the set and proper-topology proof AB.1 are now all proved. The normalizer action is the lattice-with-finite-Weyl action of GB.2; compact open \(K_x\), covariance and normalizer intersection are GB.3. The root-level groups have local upper-containment at every point: near a point the only possible level jumps are its finitely many incident walls, and at the wall the non-strict level includes both neighboring groups. Hence nearby \(P_y\subset P_x\). The normalizer fixed-point labels have the same upper-containment by the proper discrete affine action and closed fixed hyperplanes. Thus \(K_y\subset K_x\) nearby, the graph is closed, and their union over a compact apartment set is compact, by AB.1's local finite covering argument.

A closed reduced chamber is compact by GB.4. The central subspace is rational in the invariant cocharacter lattice, so a full lattice of central cocharacters has a compact parallelepiped. The product of these two sets has locally finite normalizer translates covering the enlarged apartment. Consequently every step of AB.1's proper finite-fiber quotient proof applies: it proves exactly the asserted quotient topology, compact transporters and compact orbit representatives.

For \(v\in\overline C\), \(P_v\subset G^0\). An affine Bruhat representative in \(P_v\) fixes \(v\) by GB.3, so its label is in the finite subgroup generated by the walls through \(v\), by GB.4. Conversely each such wall representative is a product of opposite root factors of value zero there, and lies in \(P_v\). The precise face equality is therefore

This is exactly the face-group input for the chamber realization and retractions, not merely containment in a point stabilizer. Since \(G^0\cap N=N_{\mathrm{aff}}\), the point relation and the closed-chamber fundamental-domain argument give the enlarged homeomorphism \(\mathcal B\simeq\mathcal B^0\times V_Z(S)\), by the explicit set and proper-topology proof AB.5. The wall reflections and their translations have zero central component; the full normalizer supplies the retained central translations. The central projection therefore gives the full group action on that factor.

The chain-metric proof AB.6 applies to these proved affine chamber and face data. It constructs the face-independent chamber retractions, gives their 1-Lipschitz property and the exact distance equality when an endpoint is in the closed center chamber, and proves every apartment is Euclidean and every pair of points has an apartment segment. The proof AB.7 uses just that equality and Euclidean squared-distance identity to prove the strong convexity and full CAT(0) comparison. Its type-fiber proof gives completeness with arbitrary panel multiplicity; here each compact panel group contains the open \(I\) with finite index, so reduced double-coset products give finite panel multiplicities and finitely many chambers meeting each metric ball. The resulting compact-ball argument gives properness and equality of metric and quotient topologies. Product with the entire Euclidean center preserves completeness, properness, curvature and the topology. All the actual statements and inequalities used are proved in AB.6–AB.7; none depends on the split coroot-lattice name.

Finally AB.8's explicit proper CAT(0) argument gives compact-subgroup fixed points and contractible fixed sets, a finite invariant slice partition from the compact metric orbit space and compact open stabilizers, and continuous equivariant squared-distance minimizers for every numerable proper source. Its unique-segment homotopies give uniqueness. This proves universality and numerability for the full group. \(\square\)

GB.0–GB.6 prove the quasi-split relative-root, filtration, Iwahori, rank-one, wall-exchange and metric statements. GB.7–GB.15 give connected models, unramified arithmetic, stable apartments and exact component descent. GB.16–GB.18 prove quasi-splitting over a finite unramified extension, and GB.19–GB.23 prove the rational root and maximal-apartment descent and the full arbitrary-group building theorem.

GB.7. Smooth integral models of every filtered relative root group

This construction supplies actual models, including wildly ramified unitary groups in residue characteristic two. The base in this section may be a local field, or any henselian discretely valued field for which the finite separable root-coordinate extensions have finite free valuation rings. GB.9 proves this latter condition for the maximal unramified extension of a local field. No completeness or compactness is needed for the construction.

Proposition. For every nondivisible relative root \(a\) and every real \(r\), the group \(U_{a,r}\) is the group of integral points of a smooth affine group scheme \(\mathcal U_{a,r}\) with generic fiber \(U_a\). Its underlying scheme is affine space, its special fiber is connected and unipotent, and the inclusion into \(U_a\) uses precisely the coordinates and levels of GB.1--GB.2. The corresponding statement holds for \(U_{2a,r}\) when \(2a\) is a root.

Proof in the reduced case. In the notation of GB.1 the root group is a restriction of scalars of the additive group of a finite separable extension \(E\). The lattice \(I_r=\{c\in E:\omega(c)\geq r\}\) is a principal fractional ideal of its valuation ring. Addition on this finite free base-ring module defines \(\mathcal U_{a,r}\). This is affine space with additive multiplication; its generic parameter is the original \(c\), not a renormalized valuation.

Proof in the unitary case. Work first over \(E\), with the separable quadratic extension \(F/E\), and keep \(\omega\) normalized on the original base field. The upper group has parameters \[ H=\{(u,v)\in F^2:\operatorname {Tr}(v)=-u\bar u\},\qquad (u,v)(w,z)=(u+w,v+z-\bar u w). \tag{GB.23} \] The level \(r\) is exactly \(\omega(v)\geq2r\), by BF.7. Put \[ I=\{v\in F:\omega(v)\geq2r\},\qquad \operatorname {Tr}(I)=\eta\mathcal O_E,\qquad L=\{u\in F:\omega(u\bar u)\geq\omega(\eta)\}. \tag{GB.24} \] The trace is nonzero because the quadratic extension is separable; its image on a nonzero fractional ideal is a nonzero fractional \(\mathcal O_E\)-ideal. Choose \(c\in I\) with \(\operatorname {Tr}(c)=\eta\) and put \(\lambda=c/\eta\). Thus \(\operatorname {Tr}(\lambda)=1\), with no division by two. Conjugation preserves \(\omega\), so \(L\) is a principal fractional \(\mathcal O_F\)-ideal. The trace inequality gives \(\omega(\eta)\geq2r\), and hence every \(u\in L\) satisfies \(\omega(u)\geq r\).

Let \(J=I\cap\ker\operatorname {Tr}\). The trace-zero \(E\)-space has dimension one, and its intersection with the fractional ideal \(I\) is a free rank-one \(\mathcal O_E\)-lattice; write \(J=\delta\mathcal O_E\). Choose an \(\mathcal O_E\)-basis \(b_1,b_2\) of \(L\). Every integral filtered point is uniquely \[ u=b_1s+b_2t,\qquad v=-\lambda\,u\bar u+\delta z,\qquad s,t,z\in\mathcal O_E. \tag{GB.25} \] Indeed the equation forces \(u\bar u\in\eta\mathcal O_E\), so \(u\in L\). Conversely \(-\lambda u\bar u=-c(u\bar u/\eta)\) is in \(I\) and has the required trace. Subtracting it from any permitted \(v\) leaves a unique element of \(J\).

The polynomial \(u\bar u/\eta\), in \(s,t\), has integral coefficients. The two square coefficients do because \(\omega(b_i\bar b_i)\geq\omega(\eta)\). Its mixed coefficient is \(\operatorname {Tr}(b_1\bar b_2)/\eta\), which is integral by the same inequality and the trace inequality. Thus (GB.25) is an actual morphism over \(\mathcal O_E\), with values in the fractional lattice \(I\).

Transporting multiplication through (GB.25) gives \[ (u,z)(w,z')=(u+w,z+z'+C(u,w)/\delta),\quad C(u,w)=\lambda(u\bar w+w\bar u)-\bar u w. \tag{GB.26} \] Here \(\operatorname {Tr}(C)=0\). Also \(u\bar w+w\bar u\in\eta\mathcal O_E\), so its product with \(\lambda\) is in \(I\); and \(\omega(\bar u w)\geq\omega(\eta)\geq2r\). Therefore \(C\in J\). Applying this to each of the four pairs \(b_i,b_j\) proves that the bilinear polynomial \(C/\delta\) has integral coefficients. Equation (GB.26), its evident identity, and its inverse define a group law on \(\mathbf A^3_{\mathcal O_E}\). Associativity holds generically by the matrix product (GB.23), hence everywhere by injection of its torsion-free polynomial coordinate ring into its generic fiber. The central \(z\)-line and the additive \((s,t)\)-quotient give a central extension of two additive groups. Its special fiber is consequently connected and unipotent, and the entire scheme is smooth. Its generic fiber, integral points and valuation are exactly the required \(H\), \(U_{a,r}\) and \(\omega(v)/2\).

For the long root use the additive lattice \(\{v\in\ker\operatorname {Tr}:\omega(v)\geq r\}\); it is a free rank-one \(\mathcal O_E\)-module. The lower root groups use exactly the same construction in the displayed lower matrix chart of BF.7.

Finally restrict scalars from \(\mathcal O_E\) to the original valuation ring \(R\). This operation here requires no general Weil-restriction theorem: choose a finite free \(R\)-basis of \(\mathcal O_E\), expand each coordinate in that basis, and expand the displayed polynomial law using the integral multiplication table of \(\mathcal O_E\). The resulting affine space over \(R\) represents the required functor. Its smoothness and connected special fiber are explicit; the same central additive filtration proves unipotence. Its generic fiber is the restriction of scalars of the original rank-one chart, and its \(R\)-points are exactly the prescribed lattice points. The central covering in GB.1 is an isomorphism on each root group, including inseparable central kernels, so these models apply to the root groups of the original reductive \(G\). \(\square\)

The source of the trace-lattice construction is the freely accessible Bruhat--Tits II, 4.3, together with its explicit rank-one coordinates in 4.1.9--4.1.12. All the ideal choices, integral coefficients and scheme assertions required here have been proved above; a source reference does not replace any step.

GB.8. Smoothening a bounded affine group model, with its components retained

Let \(R\) be a henselian DVR with perfect residue field \(f\), uniformizer \(\pi\), and fraction field \(K\). In the applications to point stabilizers below, \(f\) is algebraically closed. A bounded subset of a finite-dimensional \(K\)-space means one contained in a fractional lattice. Boundedness of a subgroup of a linear algebraic group includes its inverses and is tested in a faithful closed representation.

Lemma (group smoothening). If \(\mathcal H\) is a flat affine group scheme of finite presentation over \(R\) with smooth generic fiber, there is a finite sequence of affine group dilatations \(\mathcal H^{\mathrm{sm}}\longrightarrow\mathcal H\) with smooth final model, unchanged generic fiber and exactly unchanged \(R\)-points.

Proof. Write \(M=e^*\Omega_{\mathcal H/R}\). The generic fiber dimension is \(d\). Translation gives \[ \Omega_{\mathcal H/R}\simeq\mathcal O_{\mathcal H}\otimes_R M. \tag{GB.27} \] One verifies this without a smoothness assumption: a derivation at a point \(g\) is translated to a derivation at the identity by the differential of \(x\mapsto g^{-1}x\); the two translations are inverse on every square-zero extension. They are regular in \(g\), by the Hopf multiplication and antipode. This represents the displayed isomorphism. The finite \(R\)-module \(M\) has generic rank \(d\); let \(\delta\) be the length of its torsion.

Let \(Y\) be the reduced closure in \(\mathcal H_f\) of the reductions of \(\mathcal H(R)\). It is a closed subgroup: products of the dense reduction subset are in that subset, its product with itself is dense in \(Y\times_fY\), and \(Y\times_fY\) is reduced because \(f\) is perfect. Pulling back the ideal of \(Y\) by multiplication and inverse therefore gives zero. A reduced finite-type group over a perfect field is smooth. Here is the needed argument: a reduced finite-type variety over a perfect field has a nonempty smooth open subset on each irreducible component, by the separating-transcendence-basis Jacobian criterion; translating a smooth geometric point to the identity, and then to each other geometric point, proves smoothness of the group everywhere. Thus \(Y\) is smooth.

Dilatate along \(Y\). If \(A=\mathcal O(\mathcal H)\) and \(J\) is its inverse-image ideal, the dilatation has coordinate ring \(A[J/\pi]\subset A[1/\pi]\). It is finite type and flat over the DVR, has unchanged generic fiber, and is universal for maps from flat \(R\)-schemes whose special fibers land in \(Y\). All these assertions are also proved explicitly in the preceding programme lesson Néron models, Proposition 4.1. The universal property gives a group structure because \(Y\) is a subgroup. Every original \(R\)-point reduces into \(Y\), so the new and old \(R\)-points are exactly the same.

We supply the strict defect decrease, rather than invoke a smoothening theorem whose proof has been omitted. If \(\delta>0\), put \(t=\dim_f(M/\pi M)>d\) and \(q=t-d\geq1\). Near the identity embed \(\mathcal H\) in a smooth affine \(R\)-scheme \(Z\) of relative dimension \(t\), with \(\Omega_{Z/R}|_Y\to\Omega_{\mathcal H/R}|_Y\) an isomorphism. To obtain \(Z\), start with any polynomial presentation and eliminate the relations whose Jacobian minors are units at the identity. There are precisely enough of these to leave \(t\) variables. Shrink at the identity; (GB.27) makes the remaining rank \(t\) constant along \(Y\) there. Since \(Y\) is smooth, choose étale coordinates \(y_1,\ldots,y_m,z_1,\ldots,z_{t-m}\) on this ambient neighborhood with the ideal of \(Y\) equal to \(J=(\pi,z_1,\ldots,z_{t-m})\).

For any defining equation \(F\) of \(\mathcal H\) in \(Z\), the differential \(dF|_Y\) is zero, by that cotangent isomorphism. Expressing \(F\in J\) in these coordinates gives \[ F=\pi G+\sum_j z_jG_j,\qquad G_j\in J. \tag{GB.28} \] At every \(R\)-section reducing into this neighborhood of \(Y\), \(F=0\) and \(J\subset\pi R\), whence \(G\) has zero reduction. Those reductions are dense in \(Y\) by its definition, so \(G\in J\) too. Consequently every defining equation is in \(J^2\).

The ambient dilatation has coordinates \(y_i,z'_j=z_j/\pi\). Substitution into \(F\in J^2\) makes \(F/\pi^2\) regular; it vanishes on the dilatation of \(\mathcal H\), since it vanishes generically and its coordinate ring is torsion-free. At its identity section, if the old differential row is \(\sum b_i\,dy_i+\sum c_j\,dz_j\), the new relation row is \[ \sum_i(b_i/\pi^2)\,dy_i+\sum_j(c_j/\pi)\,dz'_j. \tag{GB.29} \] These coefficients are integral by the regular divided equation. The old relation matrix has generic rank \(q\). Over a DVR, diagonalizing a matrix by elementary row and column operations shows that the minimum valuation of its nonzero \(q\)-minors is exactly the torsion length of its cokernel. Choose a minor attaining the old minimum \(\delta\). The corresponding minor of (GB.29) divides it by at least \(\pi^q\), since each of its \(q\) columns has been divided by \(\pi\) or by \(\pi^2\). The new relation module contains these rows. Its generic rank is still \(q\), so its minimum minor valuation is at most \(\delta-q\). Therefore \(\delta_{\mathrm{new}}\leq\delta-1\).

After at most the initial \(\delta\) steps the identity is smooth. For clarity, the criterion used here is the unit-minor Jacobian criterion: torsion length zero lets one cut out a smooth ambient scheme of the generic dimension; the remaining ideal vanishes in its regular local domain because it vanishes generically and the source is flat. This is the complete argument of the actual Néron models, Lemma 4.2. Translation makes all geometric special-fiber points smooth; the generic fiber was already smooth. Thus the final group is smooth over \(R\). This proves the lemma, including termination. \(\square\)

The argument in (GB.28)--(GB.29) is the group version of the locally proved defect calculation in that lesson's Lemma 4.3. The freely accessible Edixhoven--Romagny author paper, Section 5.1--5.6, supplies the dilatation/smoothening framework.

Lemma (integral points detect functions). Assume now that \(f\) is algebraically closed. If \(X=\operatorname {Spec}A\) is smooth affine of finite presentation over \(R\), and \(F\in A[1/\pi]\) has integral value at every \(X(R)\)-point, then \(F\in A\). Hence a generic morphism from \(X_K\) to any affine finite-type \(R\)-scheme extends uniquely if all its coordinate functions have integral values at every \(X(R)\)-point.

Proof. Express \(F=\pi^{-n}g\), with \(g\in A\), choosing minimal \(n\geq0\). If \(n>0\), then \(g\notin\pi A\). The nonzero function \(\bar g\) on the reduced special fiber is nonzero at some \(f\)-point. Smoothness and henselianity lift that point to an \(R\)-point, contradicting integrality of \(F\). If the special fiber is empty, \(\pi\) is a unit in \(A\) and \(n=0\) already. This proves the criterion. Apply it to finitely many affine coordinate generators; their relations hold generically and hence in the torsion-free ring \(A\). Uniqueness follows from that same injection. In particular, two smooth affine models of the same generic group with the same integral-point subgroup are uniquely isomorphic through the generic identity. \(\square\)

Proposition (full and connected bounded torus models). Over this strictly henselian \(R\), any \(K\)-torus \(T\) has a smooth affine model \(\mathcal T^b\) whose integral points are exactly the subgroup \(T(K)^b\) on which every splitting-field character has valuation zero. There is a smooth affine model \(\mathcal T^0\) with the same generic fiber and connected special fiber, and \[ T(K)^0:=\mathcal T^0(R)\ \triangleleft\ T(K)^b,\qquad T(K)^b/T(K)^0\simeq\pi_0(\mathcal T^b_f)(f). \tag{GB.30} \] This is a finite group; it is not silently set equal to zero.

Proof. Every bounded subgroup lies in \(T(K)^b\): powers of an element with a nonzero character valuation are unbounded. Conversely unit character coordinates and their inverses are bounded in a faithful splitting-field diagonal representation, so \(T(K)^b\) is bounded. Choose a faithful \(K\)-representation \(V\). For an initial lattice \(M\), the module \(\Lambda=\sum_{g\in T(K)^b}gM\) contains \(M\) and is contained in a fractional lattice. Projection/induction on rank over the DVR proves that a submodule of a finite free module is finite free: project to one coordinate, lift a generator of its image ideal, split off that rank-one summand, and apply induction to the kernel. Thus \(\Lambda\) is a lattice. Its stabilizer is bounded and contains \(T(K)^b\), so its intersection with \(T(K)\) is exactly \(T(K)^b\). Take the schematic closure of \(T\) in \(\operatorname {GL}(\Lambda)\). Its coordinate ring is the quotient by the kernel of the generic map, and is torsion-free; it is a finite-type flat affine group model with precisely those integral points. Smoothen it by the lemma.

Dilatate the resulting smooth model along its special-fiber identity component. This is the open part consisting of the whole generic fiber and just that special-fiber component: locally on the component the ideal is \((\pi)\), while the other closed components are removed. The affine dilatation description proves that it is affine and of finite presentation. It is smooth, has connected special fiber, and has exactly the integral points reducing into the identity component. Each special-fiber component has an \(f\)-point; smoothness and henselianity lift it. Reduction therefore gives the exact quotient (GB.30). All constructions are unique through their integral points by the preceding lemma. \(\square\)

Corollary (any given maximal bounded subgroup). With the perfect-residue DVR hypotheses of this section, let \(G/K\) be any smooth affine algebraic group and let \(P\subset G(K)\) be maximal among bounded subgroups. Then \(P\) is exactly the integral-point group of a smooth affine finite-type model of \(G\).

Proof. In a faithful generic representation, the module \(\Lambda=\sum_{g\in P}gM\) is a finite free invariant lattice by the projection/induction argument above. The schematic closure of \(G\) in \(\operatorname {GL}(\Lambda)\) is a flat finite-type affine group model. Its integral-point group is bounded and contains \(P\), hence equals \(P\) by the assumed maximality. Group smoothening preserves exactly these points. The proof requires neither connectedness of \(G\) nor an assertion that a maximal bounded subgroup exists in every situation; \(P\)'s maximality is the explicit hypothesis. \(\square\)

GB.9. The maximal unramified field and its finite extensions

Put \(K=k^{\mathrm{ur}}\), the union inside a fixed separable closure of all finite unramified extensions of the original local field \(k\). Its valuation ring is \(R\), its uniformizer is the original \(\pi\), and its residue field is the algebraic closure of the finite residue field of \(k\).

Proposition. \(R\) is a strictly henselian DVR. Every finite separable extension \(L/K\) has a unique extended valuation, with value group \(n^{-1}\mathbf Z\), where \(n=[L:K]\); its valuation ring is finite free of rank \(n\) over \(R\). More precisely \(L=K(\Pi)\), with \(\Pi\) satisfying an Eisenstein polynomial over some finite unramified extension of \(k\), and \(\mathcal O_L=R[\Pi]\). This assertion includes equal-characteristic local fields, for which \(K\) itself is imperfect.

Proof. Every nonzero element of \(R\) lies in a finite unramified valuation ring and is \(\pi^m\) times a unit. Thus \(R\) is a DVR with the stated residue field. A polynomial and a simple residue root involve finitely many coefficients and residue elements. Choose a finite unramified complete subfield containing all of them; Hensel's Newton argument there lifts the root. This proves henselianity of \(R\), and its algebraically closed residue field makes it strictly henselian. The actual unramified construction and Newton lifting used here are proved in AB.9.1.

Choose a primitive element \(\alpha\) for \(L/K\); separability gives one since \(K\) is infinite. Its monic minimal polynomial \(P\) has its finitely many coefficients in some finite unramified \(E/k\). It is irreducible over \(E\), because a factorization there would be one over \(K\). Hence \(M=E(\alpha)\) has degree \(n\) over \(E\), and \(L=MK\). The finite-extension theorem AB.0 applies to \(M/E\), and gives \(n=ef\) for its ramification index and residue degree.

Its residue field extension, of degree \(f\), lifts to an unramified subextension \(E_f/E\) inside \(M\): lift a generator's simple finite-field minimal polynomial by Hensel's lemma in \(M\), using AB.9.1. This unramified field is already in \(K\). If \(f>1\), extension of scalars from \(E\) to \(E_f\subset K\) would lower the degree of \(\alpha\), contradicting irreducibility of \(P\) over \(K\). Therefore \(f=1\) and \(e=n\).

Choose a uniformizer \(\Pi\) of \(M\). Its value is \(1/n\) in the original normalization. Since an extension containing \(\Pi\) has ramification index divisible by \(n\), its degree over \(E\) is at least \(n\); hence \(M=E(\Pi)\). The \(n\) conjugates of \(\Pi\) all have value \(1/n\), by the uniqueness in AB.0. The nonleading coefficients of its minimal polynomial consequently have positive integral valuation, while the constant coefficient has valuation exactly one. Thus this polynomial is Eisenstein.

For every finite unramified \(E'/E\) in \(K\), this same polynomial remains Eisenstein. It defines a complete degree-\(n\), totally ramified extension \(ME'/E'\). Its elements have unique expansions \(\sum_{i=0}^{n-1}c_i\Pi^i\), and the distinct fractional parts of \(\omega(c_i)+i/n\) give \[ \omega\Bigl(\sum_{i=0}^{n-1}c_i\Pi^i\Bigr) =\min_i\{\omega(c_i)+i/n\}. \tag{GB.31} \] In particular an element is integral if and only if every \(c_i\) is integral. The valuations of these finite complete extensions agree on overlaps by AB.0 uniqueness. Their union is \(L\), so they give the unique valuation on \(L\), and (GB.31) gives its value group and its valuation ring \(R[\Pi]\). Any other extension restricts to the unique valuations on all finite complete subextensions, proving uniqueness once more. The argument uses only finite-field perfectness and separable finite extensions, and makes no assumption that \(K\) is perfect. \(\square\)

The same proof works after replacing the original local field by any finite extension of it. In particular every further finite separable root-coordinate extension of \(L\) has a finite free valuation ring. Fractional ideals are finite free over \(R\), with the explicit basis given by (GB.31). Conjugation preserves the valuation. These are exactly the algebraic lattice hypotheses in GB.7--GB.8.

Conditional geometric extension, with its hypothesis explicit. If \(G_K\) is quasi-split, the algebraic constructions GB.0--GB.5 apply over \(K\): replace compactness of a lattice ball by boundedness, and use the finite free rings just proved. The two-cell matrix identities, integral commutators, finite-polynomial Iwahori proof, reflection arrangement, Coxeter labels, face groups and common apartments do not use local compactness. The chamber metric and completeness proofs AB.6--AB.7 allow arbitrary panel multiplicity and therefore construct a complete CAT(0) enlarged building over \(K\). It is not claimed proper, since its residue field is infinite. The hypotheses \(G_K\) quasi-split and the later apartment descent theorem are not inferred from this arithmetic proposition.

GB.10. Actual smooth connected facet models over the unramified field

Assume explicitly in this section that \(G_K\) is quasi-split, where \(K=k^{\mathrm{ur}}\). Use the building constructed in GB.9. For a facet \(F\) in an apartment, choose a rational point \(x\) in its relative interior, including a rational central coordinate. The root levels are constant there. Write \(K_x\) for the full point stabilizer and \(P_F\) for the subgroup generated by \(T(K)^b\) and the root levels at \(x\).

Theorem. There are actual smooth affine finite-presentation \(R\)-models \(\mathcal H_x\) and \(\mathcal G_F\), both with generic fiber \(G_K\), such that \[ \mathcal H_x(R)=K_x,\qquad \mathcal G_F(R)=P_F^0:=\langle T(K)^0,\ U_{a,-a(x)}\rangle. \tag{GB.32} \] The special fiber of \(\mathcal G_F\) is connected. The subgroup \(P_F^0\) is independent of the chosen apartment and interior point, and is normal in the full facet stabilizer. Moreover \[ K_x/P_F^0\simeq \pi_0((\mathcal H_x)_f)(f),\qquad P_F/P_F^0\simeq T(K)^b/T(K)^0. \tag{GB.33} \] The first quotient is finite and includes possible normalizer and torus components. Neither quotient is declared trivial.

Construction of the full model. Choose a finite Galois splitting field \(L/K\), with valuation ring \(B\), and the integral faithful pinned representation of BF.4 over \(B\). View its \(L\)-space as a \(K\)-space. The induced representation of \(G_K\) is a closed immersion: after a separable splitting base change, its restriction-of-scalars target is a product, one projection is the original faithful immersion, and the diagonal source is its closed graph. Its weights are \(w_i=\lambda_i(x)\). The weighted norm is \(\|\sum z_i e_i\|=\max_i\exp(-\omega(z_i)-w_i)\).

For each real \(t\), its ball lattice is \[ M_t=\{\sum z_i e_i:\omega(z_i)+w_i\geq t\}. \tag{GB.34} \] GB.9 makes this a finite free \(R\)-lattice. There are finitely many distinct \(M_t\) for \(0\leq t<1\), and \(M_{t+1}=\pi M_t\). Preserving this finite lattice chain, with inverses, is therefore exactly preserving the norm. Its automorphism scheme is affine of finite presentation: in the product of the finitely many \(\operatorname {GL}(M_t)\), impose that their matrices commute with each inclusion matrix. Its generic fiber is the common \(\operatorname {GL}_K\)-space, since all inclusions are generically invertible. Take the schematic closure of \(G_K\) in it. This is a finite-type flat affine group model.

Its integral points are precisely the norm-isometries in \(G(K)\), which are \(K_x\). Indeed the Iwahori fixes that norm and the full Bruhat partition GB.5 writes every \(g\) as \(i_1ni_2\). Such a \(g\) preserves the norm if and only if \(n\) does. The weight detector BF.5, with the full central lattice retained, says precisely that \(n\) fixes \(x\). Thus \(I N_x I\subset P_xN_x=K_x\), and conversely every defining root level and every \(N_x\)-element preserves the norm. Smoothen this closure by GB.8, obtaining \(\mathcal H_x\) with unchanged integral points.

Dilatate \(\mathcal H_x\) along its special-fiber identity component, as in GB.8. Call this smooth affine model \(\mathcal G_x\). At first its integral points are just the kernel of the special-fiber component map; we now identify that kernel exactly with the subgroup in (GB.32).

The connected subgroup is contained in that kernel. Every root model \(\mathcal U_{a,-a(x)}\) of GB.7 has all its \(R\)-points in \(K_x\). Its generic inclusion therefore extends to \(\mathcal H_x\) by the integral-point criterion GB.8. Its special fiber is connected and its identity maps to the identity, so its special image lands in the identity component and the inclusion factors through \(\mathcal G_x\). The same argument applies to \(\mathcal T^0\). Hence \(P_F^0\subset\mathcal G_x(R)\).

A neighborhood in the connected model has the reverse inclusion. In the split pinned group over \(B\), select a principal open neighborhood \(D(D)\) of the identity inside its integral big cell. Its complement ideal is stable under torus conjugation. Projecting a defining function onto its weight-zero summand and rescaling its identity value gives a function \(D\) of conjugation weight zero with \(D(e)=1\), whose nonvanishing still implies membership in the big cell. This selection may be made over \(B\): some projected identity value is a unit, since their sum is a unit. Each Gaussian root coordinate on this principal open has a numerator of its corresponding torus weight and a power of \(D\) as denominator; each Gaussian torus character and its inverse has weight-zero numerator. Weight projections supply these homogeneous numerators. The integral faithful closed representation supplies polynomial lifts in matrix entries and inverse determinant of those same weights: its coordinate-ring surjection is torus-equivariant, so projecting any lift to the required weight gives a lift. These are the exact integral big-cell and grading inputs proved in BF.4--BF.6, ultimately Roots and reductive groups of rank one Lemma 3.1 and Pinnings and the classification of split reductive groups Theorem 10.1.

Every matrix monomial of weight \(\beta\) has valuation at least \(-\beta(x)\) on \(K_x\), by the entry inequalities of the norm. Weight-zero monomials consequently have integral values. The inverse determinant has valuation zero because norm isometries and their inverses have opposite determinant inequalities. Thus \(D\) is integral on all \(K_x\), and each coordinate numerator has its asserted weighted lower bound. To apply GB.8 although these functions are \(L\)-valued, expand them in the finite free integral basis of \(B/R\) from GB.9; all coefficient functions are \(K\)-regular and integral on \(\mathcal H_x(R)\), hence extend to \(\mathcal H_x\). Let \(V\subset\mathcal H_x\) be the principal open where \(\operatorname {N}_{L/K}(D)\) is a unit. It contains the identity, and every \(V(R)\)-point has \(D\) a unit in \(B\).

Gaussian decomposition of any \(V(R)\)-point therefore has root coordinates satisfying \(\omega(c_\beta)+\beta(x)\geq0\). The torus characters and their inverses have nonnegative valuation, hence valuation zero. Uniqueness of the decomposition in a root order grouped into relative-root blocks makes these factors \(K\)-rational. GB.2 and GB.7 put the root factors in precisely their root models, and the torus factor in \(T(K)^b\). The generic Gaussian maps from \(V_K\) to these affine models extend over \(R\) by GB.8, because they have integral coordinates at every \(V(R)\)-point.

Restrict these maps to the inverse image \(V^0\subset\mathcal G_x\). Its special fiber is a nonempty open subset of a connected smooth group over the algebraically closed \(f\), and is irreducible. For completeness, in a smooth algebraic group the irreducible components are disjoint, since its regular local rings are domains; translation identifies each with a coset of the component through the identity. That component is a subgroup by irreducibility of its product and inverse images. Hence connectedness implies irreducibility. The special Gaussian torus map on \(V^0_f\) lands in a single component of \(\mathcal T^b_f\); at the identity that component is the neutral one. Consequently \[ V^0(R)\subset P_F^0. \tag{GB.35} \]

For any \(h\in\mathcal G_x(R)\), the two nonempty opens \(V^0_f\) and \(h^{-1}V^0_f\) intersect. Choose an \(f\)-point in their intersection and lift it smoothly to \(z\in\mathcal G_x(R)\). Both \(z\) and \(hz\) belong to \(V^0(R)\), so (GB.35) gives \(h=(hz)z^{-1}\in P_F^0\). This proves the equality in (GB.32).

Components, torus intersection and independence. Every component of \((\mathcal H_x)_f\) has an \(f\)-point and that point lifts smoothly. This proves the first exact quotient in (GB.33). Also \(T(K)^b\cap P_F^0=T(K)^0\). To see the potentially delicate reverse containment, take \(t\in T(K)^b\cap\mathcal G_x(R)\), and choose \(z\) with \(z,tz\in V^0(R)\) by the preceding open-intersection argument. Their Gaussian torus factors are both in \(T(K)^0\). Left multiplication by \(t\) conjugates the negative root factor and multiplies the torus factor by \(t\), while preserving its relative-root order. Thus these two factors differ by multiplication by \(t\), and \(t\in T(K)^0\). Since \(P_F\) is generated by \(T(K)^b\) and these same root levels, its quotient by \(P_F^0\) is exactly the second quotient in (GB.33).

The full stabilizer \(K_x\) is intrinsic to the building. Smooth affine models with that same integral-point subgroup are uniquely isomorphic by GB.8, so the full smooth model and its identity-component model are independent of the chosen weighted representation and apartment. Within a given facet the root levels and \(T(K)^0\) are constant, so (GB.32) is independent of the interior rational point too. Rational points exist because the arrangement has rational discrete levels. Conjugation or a valuation-preserving field automorphism carries \(K_x\) to \(K_{gx}\) or \(K_{\sigma x}\); the generic map extends uniquely through these smooth models by their integral points. It therefore carries the neutral special component, and hence \(P_F^0\), to the corresponding neutral component. This proves apartment independence, equivariance and normality under the full facet stabilizer. We may now write \(\mathcal G_F\) for \(\mathcal G_x\). \(\square\)

In particular, a Galois-stable facet has a Galois-stable smooth connected model. Its affine coordinate algebra is an \(R\)-subalgebra of \(K[G]\); every element has a finite Galois orbit because its finitely many coefficients lie in a finite unramified field. Thus the semilinear action is continuous on this algebra, and the explicitly proved affine descent in AB.9 descends \(\mathcal G_F\) to a smooth connected model over the original valuation ring. This proves model existence, rather than assuming it before invoking the descent lemma.

GB.11. Exact Galois descent and the full-stabilizer component cocycle

The result here applies to a Galois-stable facet and its models constructed in GB.10. Galois cohomology uses the usual discrete topology on algebraic points, so continuous cocycles have finite image. It does not assert that arbitrary fixed points already lie in descended apartments.

Proposition. Let \(\Gamma=\operatorname {Gal}(k^{\mathrm{ur}}/k)\), and let \(F\) be a \(\Gamma\)-stable facet. Then \[ H^1_{\mathrm{cts}}(\Gamma,P_F^0)=1. \tag{GB.36} \] For a \(\Gamma\)-fixed point \(x\in F\), put \(D_x=K_x/P_F^0\), with its actual finite \(\Gamma\)-action. The map \[ H^1_{\mathrm{cts}}(\Gamma,K_x)\longrightarrow H^1_{\mathrm{cts}}(\Gamma,D_x) \tag{GB.37} \] is injective. Consequently a fixed coset \(gK_x\) has a representative in \(G(k)\) if and only if the component cocycle of \(g^{-1}\sigma(g)\) is trivial in \(H^1_{\mathrm{cts}}(\Gamma,D_x)\). The image of the fixed-coset orbit map is precisely \[ \ker\{H^1_{\mathrm{cts}}(\Gamma,K_x) \longrightarrow H^1_{\mathrm{cts}}(\Gamma,G(K))\}. \tag{GB.38} \] Thus full bounded stabilizers cannot be substituted for connected facet groups in a Lang argument without this component test.

Proof. GB.10 actually constructs the descended smooth connected facet model. The integral Lang proof in AB.9 therefore applies and gives (GB.36), including degrees divisible by the residue characteristic. For infinite unramified descent a continuous cocycle has finite image and involves finitely many field coefficients; it is trivial on an open subgroup. After enlarging a finite unramified field to contain these coefficients, it is a cocycle over its finite unramified Galois group and is killed by the finite integral Lang proof. This is exactly the direct-limit argument already proved in AB.9.

For injectivity, take two \(K_x\)-cocycles whose images have the same class in \(D_x\). Lift the conjugating \(D_x\)-element to \(K_x\) by the surjection in (GB.33), and change one cocycle by that cochain. They now have the same image pointwise. Write them \(c_\sigma\) and \(c'_\sigma\). The elements \(b_\sigma=c'_\sigma c_\sigma^{-1}\) belong to \(P_F^0\) and form a cocycle for the twisted action \[ \sigma * p=c_\sigma\,\sigma(p)\,c_\sigma^{-1}. \tag{GB.39} \] The full \(K_x\) normalizes \(P_F^0\). Its generic conjugations therefore extend uniquely to the connected model, by the integral-point criterion GB.8. The cocycle identity gives a semilinear descent datum on that model. All its coefficients lie in a finite unramified field, as above. The affine descent proof AB.9 supplies a smooth connected descended twisted model, and its integral Lang proof kills \(b_\sigma\). Thus \(c_\sigma\) and \(c'_\sigma\) are cohomologous in \(K_x\), proving (GB.37). This argument includes possible nontrivial torus and diagram components; it assumes none of them vanish.

If \(gK_x\) is fixed, then \(c_\sigma=g^{-1}\sigma(g)\in K_x\) is a cocycle. Changing the representative multiplies it by a \(K_x\)-coboundary. If its \(D_x\)-class is zero, lift the \(D_x\)-cochain to \(b\in K_x\); the adjusted cocycle \(b^{-1}c_\sigma\sigma(b)\) is \(P_F^0\)-valued and, by (GB.36), equals \(a^{-1}\sigma(a)\) for some \(a\in P_F^0\). Then \(gba^{-1}\) is Galois-fixed and represents the same coset. Conversely a fixed representative gives the zero cocycle. This proves the exact component criterion. Finally any \(K_x\)-cocycle trivial in \(G(K)\) is \(g^{-1}\sigma(g)\) for a \(g\), and therefore gives a fixed coset. Two such cosets are in the same \(G(k)\)-orbit exactly when their \(K_x\)-cocycles are cohomologous, by multiplying their representatives and checking Galois invariance. This proves (GB.38). \(\square\)

The connected-coset equality \[ (G(K)/P_F^0)^\Gamma=G(k)/P_F^{0,\Gamma} \tag{GB.40} \] has no component obstruction, by (GB.36) and the same explicit computation. Equations (GB.37)--(GB.38), rather than (GB.40) with its denominator replaced, are the correct full-stabilizer statement.

GB.12. Torus lifting and transporters for the constructed models

The following proof gives the specific deformation mechanism used here. It does not invoke Artin approximation to turn a formal homomorphism into an actual one.

Lemma (finite free linearization). Every smooth affine model in GB.8 or GB.10 over the strictly henselian \(R\) has a faithful closed representation on a finite free \(R\)-module. The same holds for its unramified descended models.

Proof. Its integral-point subgroup is bounded and Zariski dense in its generic group. Density follows directly from smoothness near the identity: an étale coordinate chart gives a valuation-open polydisc of integral points. Such a polydisc is Zariski dense. To verify the latter assertion, pass to the completion, where the étale inverse is a convergent local power series. A nonzero regular function has a nonzero lowest homogeneous term and cannot vanish on an entire polydisc over an infinite valued field; choose values at which that term is nonzero and then scale them sufficiently far toward zero. The original field is dense in its completion, and henselianity supplies the inverses there already in the original field, so the same conclusion holds before completion. This also applies to the torus models.

Let \(A\) be the model's coordinate algebra. Take a finite-dimensional generic right-regular comodule \(V\subset K[G]\) containing a finite generating set of \(A\); existence is the actual finite-comodule construction in the earlier Group schemes, actions and Hopf algebras, Lemma 5.1. Put \(M=V\cap A\). The integral-point criterion GB.8 says exactly that \(M\) consists of the functions in \(V\) integral on the integral-point subgroup. Choose \(\dim V\) integral group points whose evaluation functionals are linearly independent; density supplies them by induction. Their inverse evaluation matrix bounds every coefficient of an element of \(M\). On the other hand clearing denominators of a basis of \(V\) gives a full lattice contained in \(M\). Projection/induction over the DVR therefore makes \(M\) a finite free full lattice.

Right translation by every integral group point preserves \(M\). Relative to its free basis, the generic representation matrices consequently have integral values on all integral group points. GB.8 extends their coefficient functions to \(A\), including their inverse-determinant coordinate. The Hopf identities hold generically and hence over \(R\). This is an actual representation of the model on \(M\). It is a closed immersion: evaluating its comultiplication at the identity expresses every chosen generator of \(A\) as an \(R\)-linear combination of its matrix coefficients, so the matrix coordinate algebra surjects onto \(A\).

For an unramified descended model, first base change to the strictly henselian \(R\), so that the algebraically closed residue hypothesis of GB.8 applies. Choose the initial generators and generic comodule there together with their finitely many Galois translates. The lattice \(M=V\cap A\) is then Galois-stable. The explicit finite-free-module descent in AB.9 descends it and the representation, and faithful flatness descends the coordinate surjection. This proves the descended assertion too. \(\square\)

Lemma (lifting a fiber torus). Let \(\mathcal H\) be one of these smooth affine models, over \(R\) or over the original henselian valuation ring. An embedded torus in its special fiber lifts to an embedded torus in \(\mathcal H\), with the same residue torus. Transporters and centralizers between embedded tori are smooth. A prescribed subtorus may be kept fixed when lifting in its smooth centralizer.

Proof of the finite parameter space. First split the source torus by a finite unramified cover. In the finite free faithful representation just constructed, its special-fiber action has a finite set \(S\) of weights, with positive multiplicities \(r_m\). Homomorphisms with these weights are represented by finitely many endomorphism variables \(P_m\) satisfying \[ P_mP_n=\delta_{m,n}P_m,\qquad \sum_{m\in S}P_m=1, \tag{GB.41} \] and the open-and-closed conditions \(\operatorname {rank}P_m=r_m\). Their homomorphism is \(t\mapsto\sum_m t^mP_m\). To require its image to lie in \(\mathcal H\), substitute this matrix into the finitely many equations of the closed faithful representation; each gives a finite Laurent polynomial, so vanishing of its coefficients gives finitely many additional equations. Thus this is a finite-presentation parameter scheme near the original embedding. The weights generate the character lattice, since the original torus embedding is faithful. Near its given point, each positive-rank projector has a unit matrix entry, and matrix coefficients of \(P_m\rho(t)\) give \(t^m\); coefficients of \(\rho(t)^{-1}P_m\) give \(t^{-m}\). Hence pullback on the torus coordinate algebra is surjective, and every homomorphism in this neighborhood is an embedding. The source torus over the unsplit original ring is obtained by descending its finite-field character lattice along the finite unramified cover, using AB.9. That same descent, with the Galois permutation of \(S\), descends the finite parameter scheme and its equations.

Proof of smoothness. For a square-zero extension, smoothness of the affine target first lifts the source's scheme map. Its multiplication defect is a regular two-cocycle in the Lie algebra tensored with the square-zero ideal, with action through the original homomorphism. For a split torus integration means taking the constant Laurent coefficient. It is invariant under translation in the integrated variable, over any base ring. If \(c(t,u)\) is a regular two-cocycle and \(b(t)=\int c(t,v)\,dv\), integration of its cocycle equation gives \[ c(t,u)=t\cdot b(u)-b(tu)+b(t). \tag{GB.42} \] Thus correcting the map by this one-cochain removes the defect. For a one-cocycle \(a(t)\), integration of \(a(tu)=a(t)+t\cdot a(u)\) gives \(a(t)=A-t\cdot A\), where \(A=\int a(u)\,du\). Hence two lifts are infinitesimally conjugate. The same assertions descend from a split cover by exactness of the weight-zero projection. A finite free torus representation acquires no additional weights over a square-zero extension: its weight projectors reducing to zero are zero by Nakayama. Thus the corrected map remains in the finite parameter space (GB.41). The infinitesimal lifting criterion proves that space smooth at the original embedding.

Henselian lifting in a smooth finite-presentation scheme now gives an actual homomorphism over the original ring, in the embedding neighborhood. This follows from an étale coordinate chart and the simple-root Hensel argument of AB.9.3.

For a transporter point, lift its ambient group element smoothly. The transported embedding and the required embedding have the same reduction; the one-cocycle calculation corrects that lift by infinitesimal conjugation. The transporter equations are finite-presentation closed equations: expand equality of the two conjugated torus homomorphisms in their finitely many matrix Laurent coefficients. If only equality of subgroups is required, a fixed character-lattice isomorphism gives one such chart, and these charts are disjoint open pieces indexed by the integral lattice automorphisms. The same argument gives smoothness of the centralizer, where the two embeddings are identical. An automorphism of the source torus has no infinitesimal change, as is immediate from its character lattice. These arguments establish precisely the transporter and centralizer assertions. Working in the centralizer of the prescribed subtorus keeps it fixed. \(\square\)

The primary free source for this deformation problem is Conrad's author manuscript Reductive group schemes, Proposition 2.1.3 and Remark 2.1.4 (PDF pp. 58--60), together with Bruhat--Tits II 5.1.10. The actual preceding programme proof Tori, maximal tori and their conjugacy Lemma 3.1, supplies the weight-cohomology mechanism. The finite projector parameter space and henselian lifting above replace the formal-completion/Artin-approximation passage of that earlier presentation for the models used here.

GB.13. Galois action, connected residue tori, and stable apartments

Here \(G\) is defined over the original \(k\), and the explicit hypothesis \(G_K\) quasi-split is retained. The statement below supplies stable apartments; it deliberately does not call every fixed affine subspace a maximal \(k\)-apartment.

Lemma (the semilinear action). The valuation-preserving action of \(\Gamma\) on \(G(K)\) induces an isometric action on the enlarged building of GB.9. On every apartment an open subgroup of \(\Gamma\) acts trivially. Facets and connected models are carried to their corresponding conjugates.

Proof. A rational Borel subgroup of a quasi-split group is conjugate to the chosen one by \(G(K)\): apply the absolute Bruhat charts to its point in the Borel variety. The rational point's absolute cell label is Galois-fixed, its normalizer representative is rational by GB.1, and uniqueness of the ordered cell coordinates makes the remaining unipotent coordinates rational. The same descent argument was used for actual group elements in GB.5. Within a fixed rational Borel, two rational maximal tori are conjugate by its unipotent radical. Indeed projection to its torus quotient identifies each maximal torus with that quotient, since its multiplicative-type kernel cannot lie in a unipotent group. Filter the unipotent radical by absolute root height, grouping Galois-stable relative-root coordinates. Successive quotients are vector groups; the degree-one torus calculation of GB.12 kills the difference of the two sections in each quotient, and induction conjugates them. The assertion about the kernel follows from that same weight projection: a group of multiplicative type has no nontrivial morphism into a vector group. For a diagonalizable source this is the degree-one integration calculation with trivial coefficient action, using its group-algebra character basis; it works for finite non-smooth sources such as \(\mu_p\) too. Descent and the filtration give none into the unipotent radical.

Thus an image of the pinned Borel--torus pair under \(\sigma\) can be compared with the original pair by an actual \(G(K)\)-conjugation. The pinning comparison in GB.2 gives precisely the resulting affine change of root coordinates: the valuation change on each root is its pairing with the comparison translation. On the central factor use the canonical character valuation \(\nu_Z(g)(\chi)=-\omega(\chi(g))\), with the same sign convention as the apartment action; compensate the comparison conjugation by its central translation. These comparisons carry every defining root level and normalizer stabilizer to its transformed one. They therefore preserve exactly the quotient equivalence relation defining the building.

Independence of the comparison follows by composing two comparisons: their conjugating elements differ by a torus element, whose root-coordinate valuation change and entire affine translation are exactly the torus covariance formula GB.2. The compensating translation cancels both its root and central contributions. Composition gives the comparison for \(\sigma\tau\) for the same reason. Hence the quotient maps form an actual semilinear action. They are Euclidean isometries on apartments, after averaging the root inner product over the finite based-root-datum action; they preserve chain lengths, so AB.6's chain metric makes them global isometries. The coordinates of the chosen pair, pinning, conjugation and splitting field are finite field data. For an open Galois subgroup fixing those data the comparison is the identity and its character action is trivial. Thus it fixes that apartment pointwise. GB.10's intrinsic integral-point characterization gives the asserted action on connected facet models. \(\square\)

Lemma (the torus rank of a connected facet fiber). If \(S\) is the maximal \(K\)-split torus for an apartment through \(F\), its split integral torus \(S_R\) embeds in \(\mathcal G_F\), and its special image is a geometrically maximal torus of \((\mathcal G_F)_f\).

Proof. First embed \(S_R\) in \(\mathcal T^0\). Its integral points are in \(T(K)^b\), so GB.8 extends its generic inclusion to \(\mathcal T^b\); connectedness of its special fiber factors the map through \(\mathcal T^0\). This extension is an embedding, including on the special fiber. For every absolute character \(\chi\) of \(T_L\), \(\chi\) and \(\chi^{-1}\) have integral values on \(T(K)^b\). Expand them in the finite free basis of \(B/R\) and apply GB.8; they become actual characters on \(\mathcal T^b_B\). Their special pullbacks to \(S_f\) give all its characters, because \(S_L\subset T_L\) is a subtorus and restriction on character lattices is surjective. This proves the special embedding. Over \(R\), grade the target coordinate algebra by translation by \(S_R\). The coefficient ideal of each character in the pullback to \(R[X^*(S)]\) has full special fiber and hence is the unit ideal in this local ring. Thus pullback is surjective in every weight, proving the actual closed immersion. This is also the explicit graded closed-immersion criterion in Tori, maximal tori and their conjugacy.

Any special-fiber torus of \(\mathcal T^0\) lifts by GB.12 to an embedded torus over \(R\). Strict henselianity splits it, so its generic rank is at most \(\operatorname {rank}S\) by maximality of \(S\) in \(T\). Together with the preceding embedding this proves \[ \operatorname {rank}(\mathcal T^0_f)=\operatorname {rank}S. \tag{GB.43} \]

The Gaussian maps in GB.10 and their product map are mutually inverse on an open neighborhood of the identity, also on the special fiber: both compositions are the identity generically and hence on their flat smooth domains. Consequently the special tangent space decomposes into the torus-model tangent space and the positive and negative root-model tangent spaces. Conjugation by \(S_R\) on the latter spaces has the nonzero characters \(a\) or \(2a\), in the explicit coordinates of GB.7. These are nonzero group characters even in characteristic two. The weight-zero tangent space is therefore exactly that of \(\mathcal T^0_f\).

The centralizer of this split torus in the smooth facet fiber is smooth, by the transporter proof GB.12. Its identity component consequently has dimension \(\dim T\). The special torus-model map is injective: any kernel element maps into the Gaussian neighborhood of the identity, whose inverse Gaussian torus map recovers it. Its connected commutative image has dimension \(\dim T\), by that same local inverse. Its map to the smooth centralizer identity component has invertible differential everywhere by translation, and trivial kernel; it is an étale monomorphism and hence an open immersion. Its image is an open subgroup of a connected group, so it is the entire identity component. A maximal torus containing the image of \(S_f\) lies in this identity component, whose torus rank is (GB.43); it cannot properly enlarge \(S_f\). Finally all maximal tori of a smooth connected affine group over an algebraically closed field are conjugate, by the preceding programme Tori, maximal tori and their conjugacy Theorem 2.2. This proves the assertion. \(\square\)

Theorem (stable apartments through stable facets). Every \(\Gamma\)-stable facet \(F\) is contained in an apartment associated to a maximal \(K\)-split torus defined over \(k\). In particular every Galois-fixed building point lies in such a stable apartment.

Proof. The descended connected facet model exists by GB.10. Its special fiber over the finite residue field has a geometrically maximal torus \(Q\) defined over that field: this is the complete finite-field Lang proof in actual Tori, maximal tori and their conjugacy Theorem 4.2, also supplied in AB.9. Lift \(Q\) to a torus \(\mathcal Q\) inside the descended smooth facet model using GB.12. After base change to \(R\), it is split. Its generic torus \(S'\) is therefore \(K\)-split, is defined over \(k\), and has rank \(\operatorname {rank}S\) by the preceding lemma. Hence it is maximal \(K\)-split.

Over the algebraically closed residue field, the special tori \(S_f\) and \(Q_f\) are conjugate by a special facet-model point. Identify their split character lattices by that conjugation and extend this identification to their split \(R\)-tori. Their transporter is smooth by GB.12; henselian lifting gives a conjugating element \(g\in\mathcal G_F(R)=P_F^0\). On generic fibers \(gSg^{-1}=S'\). Since \(P_F^0\) fixes every point of \(F\), the apartment \(gA\) contains \(F\). The torus \(S'\) is defined over \(k\), so the semilinear action preserves its apartment. A fixed point's unique open facet is stable, giving the final assertion. \(\square\)

On a stable apartment the Galois action factors through a finite affine isometry group, so its fixed affine subspace is nonempty: average the orbit of any point. Its direction is \((X_*(S')\otimes\mathbf R)^\Gamma=X_*(S'_k{}_{\mathrm{split}})\otimes\mathbf R\). The enlarged central fixed factor is likewise the full invariant character/cocharacter subspace, with its Euclidean metric; it is not discarded.

These fixed affine subspaces need not all have the maximal \(k\)-split rank. For example, in a split rank-one group an unramified nonsplit maximal torus gives a stable apartment with zero-dimensional reduced fixed subspace. A proof that the fixed set is the \(G(k)\)-union of apartments for maximal \(k\)-split tori, with the precise descended facet arrangement and locally finite metric, must therefore still establish that stronger apartment theorem. Stable apartments alone do not prove it.

GB.14. The arithmetic dimension-one input and finite unramified definition

Proposition. Every finite algebraic extension \(L/k^{\mathrm{ur}}\) has zero Brauer group. In particular the precise field condition “all finite-extension Brauer groups vanish” is proved here, in both mixed and equal characteristic.

Proof. First let \(L/K\) be finite separable. In GB.9 we obtained \(L=MK\), with \(M/E\) a finite totally ramified extension of a finite unramified local field \(E/k\). This field \(MK\) is \(M^{\mathrm{ur}}\): the compositum of \(M\) with each unramified degree-\(d\) extension of \(E\) is unramified of degree \(d\) over \(M\), as follows either from its Eisenstein basis in GB.9 or from the identical residue extensions and ramification indices. Conversely the unramified extension of any given degree is unique by the finite-field/Hensel construction AB.9.1. Taking the union proves the equality.

A central simple algebra over \(M^{\mathrm{ur}}\) descends to some finite unramified local field \(M_d/M\): choose its finite basis and finitely many multiplication and unit constants there. Associativity holds at that level. A proper ideal or an extra central element there would persist after faithful field extension, so the descended algebra is central simple. The preceding programme Brauer groups of local and global fields, Theorem 24.2, proves \(\operatorname {Br}(M_d)=\mathbf Q/\mathbf Z\), with restriction multiplying its invariant by the extension degree. Its class is consequently killed by a further finite unramified extension whose degree is divisible by the invariant's denominator. That field is still in \(M^{\mathrm{ur}}\), so the original algebra splits. This proves the separable case, including every characteristic and every ramification of \(M/E\). The complete earlier proof of the local theorem, rather than a citation to a local-class-field textbook, is the provider.

In characteristic \(p>0\), each complete equal-characteristic local field \(M\) is \(\mathbf F_{q_M}((\Pi))\). The coefficient field is obtained by lifting the simple roots of \(X^{q_M}-X\); successive subtraction of the unique coefficient times a power of \(\Pi\), followed by completeness, gives the unique Laurent series expansion. Its unramified extensions change only this coefficient field. Grouping powers of \(\Pi\) by their residues modulo \(p\), and taking the unique \(p\)-roots of finite-field coefficients, proves \[ M^{\mathrm{ur}}=\bigoplus_{i=0}^{p-1} \Pi^i(M^{\mathrm{ur}})^p. \tag{GB.44} \] Independence follows from the distinct valuation residues modulo \(p\). Thus this field has \(p\)-degree exactly \(p\). Its finite purely inseparable extensions are exactly \((M^{\mathrm{ur}})^{1/p^r}\): if the exponent of such an extension is \(r\), an element of exponent \(r\) generates degree \(p^r\), which is already the degree of the entire \(p^r\)-root field by (GB.44). Frobenius identifies that extension abstractly with \(M^{\mathrm{ur}}\), so it too has zero Brauer group. Every finite algebraic extension of \(K\) is purely inseparable over a finite separable one: take sufficiently high \(p\)-powers of a finite generating set to obtain separable generators. Apply the preceding separable case and then this Frobenius argument. In characteristic zero there is no further case. \(\square\)

Proposition (finite definition of quasi-splitting). If \(G_{k^{\mathrm{ur}}}\) is quasi-split, then \(G\) is quasi-split over some finite unramified extension of \(k\).

Proof. Its Borel subgroup has a finitely presented coordinate algebra and finitely many defining equations inside \(G_K\). Descend those equations, its group identities and its identity/inverse maps to a finite unramified field \(E/k\). Faithful extension from \(E\) to \(K\) detects smoothness, geometric connectedness, solvability and the geometric maximal dimension of that solvable subgroup. Therefore the descended subgroup is a Borel of \(G_E\). A chosen Borel torus or pinning likewise descends after enlarging \(E\), since it consists of finitely many equations and maps. The converse implication is immediate by field extension. This proves the finite-definition assertion; it does not assume that \(K\) is perfect. \(\square\)

Over a field whose finite algebraic extensions have zero Brauer group, GB.16 proves the required torus-cocycle vanishing, GB.17 gives the rational regular-class representative and GB.18 proves inner-form descent to a rational Borel. Together with the arithmetic proposition and finite-definition proposition above, these prove quasi-splitting of every connected reductive group over a finite unramified extension. Steinberg, Regular elements of semi-simple algebraic groups, and Borel–Springer, Rationality properties of linear algebraic groups II, Section 8.6, give the historical context, including imperfect fields.

GB.15. Fixed points and the descended CAT(0) metric

For the next metric argument assume \(G_K\) quasi-split; GB.18 proves this hypothesis for every connected reductive group. The semilinear action of GB.13 has finite orbit on every point: an apartment is fixed pointwise by an open subgroup, and a group translate uses finitely many field coefficients. Since the building of GB.9 is complete CAT(0), the entire Galois group has a fixed point. Here is the needed existence proof without assuming properness.

For a finite orbit \(O\), put \(r(z)=\max_{o\in O}d(z,o)\) and \(r_0=\inf_zr(z)\). Choose \(z_n\) with \(r(z_n)^2\leq r_0^2+1/n\). Strong convexity AB.7 at the midpoint gives \[ r((z_n+z_m)/2)^2 \leq\frac{r(z_n)^2+r(z_m)^2}{2} -\frac{d(z_n,z_m)^2}{4}. \tag{GB.45} \] Its left side is at least \(r_0^2\), so \(d(z_n,z_m)^2\leq2/n+2/m\). Completeness gives a limit attaining the infimum. The same inequality proves uniqueness, and invariance of \(O\) makes this circumcenter Galois-fixed.

The fixed set \(X=\mathcal B_K^\Gamma\) is closed and convex: an isometry fixing both endpoints fixes their unique geodesic. Thus its induced metric is complete CAT(0), with the same geodesic distances and comparison inequalities. By GB.13 every point of \(X\) lies in a stable apartment, and the induced metric on that apartment's fixed affine subspace is exactly its Euclidean metric. The central product is \[ X=(\mathcal B_K^0)^\Gamma\times V_Z^\Gamma. \tag{GB.46} \] The central factor is the entire invariant cocharacter/character space, hence exactly the split central directions defined over \(k\), with its retained Euclidean metric. The semilinear central action is linear by the character-valuation construction in GB.13, so no central translation is hidden in this equality.

The maximal rational apartment and the rational stabilizers are established in GB.19–GB.23. In particular GB.22 proves \[ X=G(k)\,A_k,\qquad A_k=A^\Gamma, \tag{GB.47} \] where the apartment is attached to a maximal \(K\)-split torus whose maximal \(k\)-split subtorus is maximal in \(G\). GB.19 proves the unique reduced fixed point and bounded rational subgroup for its root-zero centralizer, retaining the full split centre. GB.20 gives the weighted rational root filtration and its exact half-apartment bounds; GB.21 identifies facet cofaces with parabolics of the connected residue reductive quotient; GB.22 proves opposite completion and descended wall exchange. The full-stabilizer component test is GB.11, and GB.23 proves local compactness, properness, cocompactness and equality of the descended quotient and inherited metrics. Thus the metric conclusions above combine with these exact rational apartment and face-group statements.

GB.16. Vanishing for every torus over the strict unramified field

Let \(K=k^{\mathrm{ur}}\), or any finite extension of it. GB.14 proves \(\operatorname{Br}(E)=0\) for every finite extension \(E/K\), including the imperfect equal-characteristic case. We prove \[ H^1(K,T)=1 \quad\hbox{for every \(K\)-torus \(T\)}. \tag{GB16.1} \] All cohomology below is discrete. The complete bar resolution and its Tate extension are proved in the actual programme lesson Brauer groups of local and global fields, §3: induced modules are Tate-acyclic, short exact sequences give long exact sequences, restriction followed by corestriction is multiplication by the subgroup index, and all Tate groups are killed by the group order. We supply the extra tensor argument; vanishing for \(\mathbf G_m\) does not by itself settle a twisted torus.

Finite-group algebra. Let \(D\) be finite and \(A\) a \(D\)-module. If \(H^1(H,A)=H^2(H,A)=0\) for every subgroup \(H\subset D\), then there is a resolution \[ 0\longrightarrow Q\longrightarrow P\longrightarrow A \longrightarrow0 \tag{GB16.2} \] with \(P,Q\) projective over \(\mathbf Z[D]\). Consequently \(A\otimes_{\mathbf Z}\Lambda\), with the diagonal action, is Tate-acyclic for every subgroup if \(\Lambda\) is a finite free abelian group with a \(D\)-action. Here are full algebraic details, including torsion in \(A\).

For a finite \(p\)-group \(D\), the augmentation ideal \(J\subset\mathbf F_p[D]\) is nilpotent. Induct on \(|D|\): the class equation gives a nontrivial centre, hence a central element \(z\) of order \(p\). The central ideal \((z-1)\) has \(p\)-th power zero, and the quotient is \(\mathbf F_p[D/\langle z\rangle]\). Induction gives \(J^m\subset(z-1)\), hence \(J^{mp}=0\). Thus a module with zero coinvariants is zero, even if infinitely generated.

If a module \(V\) killed by \(p\) has \(H_1(D,V)=0\), lift a basis of \(V/JV\). The resulting map from a free \(\mathbf F_p[D]\)-module \(F\) is surjective, since its cokernel has zero coinvariants. For its kernel \(W\), the homology exact sequence gives \[ 0\longrightarrow W/JW\longrightarrow F/JF \longrightarrow V/JV\longrightarrow0 . \] The last map is an isomorphism, so \(W=0\) and \(V\) is free. If any single Tate group of \(V\) vanishes, use the usual two exact sequences with induced modules to shift that degree to \(-2\), keeping all modules killed by \(p\). The shifted module has \(H_1=0\) and is free. All its Tate groups vanish; shifting back proves the same for \(V\), and the argument with \(H_1\) makes \(V\) itself free.

If \(B\) has no \(p\)-torsion and two consecutive Tate groups vanish, the exact sequence \[ 0\longrightarrow B\xrightarrow{p}B\longrightarrow B/pB \longrightarrow0 \] makes one Tate group of \(B/pB\) zero. Thus \(B/pB\) is free over \(\mathbf F_p[D]\). Multiplication by \(p\) is now an isomorphism on every Tate group of \(B\) and of every subgroup. These groups are killed by powers of \(p\), so all are zero.

A free abelian, cohomologically trivial \(D\)-module \(B\) is \(\mathbf Z[D]\)-projective. Map \(\operatorname{Ind}(B)=\mathbf Z[D]\otimes_{\mathbf Z}B\) onto \(B\) by \(g\otimes b\mapsto gb\), and call the kernel \(B_*\). It is free abelian and cohomologically trivial. A basis of \(B\) gives an underlying abelian-group section, so \[ 0\to\operatorname{Hom}_{\mathbf Z}(B,B_*)\to \operatorname{Hom}_{\mathbf Z}(B,\operatorname{Ind}(B))\to \operatorname{Hom}_{\mathbf Z}(B,B)\to0 \tag{GB16.3} \] is exact. Restrict to a Sylow \(p\)-subgroup \(D_p\). We just proved that \(B/pB\) is \(\mathbf F_p[D_p]\)-free. Since \(B\) is free abelian and \(B_*\) has no \(p\)-torsion, \[ \operatorname{Hom}_{\mathbf Z}(B,B_*)/ p\operatorname{Hom}_{\mathbf Z}(B,B_*) =\operatorname{Hom}_{\mathbf F_p}(B/pB,B_*/pB_*). \] The right module is coinduced: on every free group-algebra summand evaluate at its group basis and untwist the diagonal action by applying the inverse group element to its value. Products over the basis index still have the form \(\operatorname{Maps}(D_p,C)\), because \(D_p\) is finite. Thus its Tate groups vanish. The integral Hom module has no \(p\)-torsion, and the preceding \(p\)-multiplication argument makes it Tate-acyclic. Restriction/corestriction for each Sylow group proves its cohomological triviality for \(D\) and every subgroup. Taking invariants in (GB16.3) lifts the identity of \(B\) to an equivariant section of \(\operatorname{Ind}(B)\to B\). Hence \(B\) is projective.

Subgroups of the possibly infinite free abelian modules here are free: well-order an ambient basis, intersect the subgroup with successive initial spans, and at a successor stage project to the new coordinate. Its image is \(d\mathbf Z\). If nonzero, lift \(d\) and split off its free cyclic span; the preceding intersection is the kernel. At limit stages take the unions, since vectors have finite support. The chosen lifts form a basis. No finite generator list was presupposed.

Finally choose a free \(\mathbf Z[D]\)-module \(P\) surjecting onto \(A\), with kernel \(Q\). For every \(H\subset D\), the Tate sequence gives \(H^2(H,Q)=H^1(H,A)=0\) and \(H^3(H,Q)=H^2(H,A)=0\). The module \(Q\) is free abelian. The no-\(p\)-torsion argument, followed by Sylow restriction/corestriction, makes it cohomologically trivial. The projectivity criterion proves (GB16.2). Tensor it with \(\Lambda\); exactness remains because \(\Lambda\) is free abelian. Projectives remain projective after this tensor: for a free summand use \[ \mathbf Z[D]\otimes_{\mathbf Z}\Lambda \simeq\mathbf Z[D]\otimes_{\mathbf Z}\Lambda_{\rm underlying}, \qquad g\otimes v\longmapsto g\otimes g^{-1}v , \] with the action on the first factor on the right. Projectives restrict to projectives and are summands of induced modules. Their Tate groups vanish, and the exact sequence proves the tensor assertion. This includes all roots-of-unity torsion in \(A\).

Apply the lemma. Choose a finite Galois splitting field \(L/K\) of \(T\), with group \(D\). Hilbert90 gives \(H^1(H,L^\times)=0\) for every \(H\subset D\); its actual programme averaging proof is Hilbert's Theorem90 and Kummer theory, Theorem3.1. The crossed-product identification of \(H^2(H,L^\times)\) with the relative Brauer kernel is the actual provider bound in GB.14 and Brauer groups of local and global fields, §1. Since \(\operatorname{Br}(L^H)=0\), degree two also vanishes. With the cocharacter lattice \(\Lambda=X_*(T)\), the lemma gives \[ T(L)=\Lambda\otimes_{\mathbf Z}L^\times,\qquad H^1(D,T(L))=0 . \] Over \(L\), the torus is a product of multiplicative groups, so Hilbert90 kills its first cohomology. The degree-one inflation/restriction sequence, proved by the double-bar complex in Brauer groups of local and global fields, §1, identifies \(H^1(K,T)\) with \(H^1(D,T(L))\). One can apply that same calculation over a larger finite Galois field containing all values of any given continuous cocycle. This proves (GB16.1).

The integral lattice argument is the Tate-cohomology method discussed in R. Sharifi, Group and Galois Cohomology, Section 1.11, Lemmas 1.11.4–1.11.9 and Theorem 1.11.11. This result holds in mixed characteristic, in residue characteristic two, and for the imperfect equal-characteristic strict unramified field. It also proves norm surjectivity for every finite cyclic \(L/K\): cyclic Tate periodicity identifies \(K^\times/N_{L/K}L^\times=H^2(D,L^\times)\), whose relative Brauer group has just vanished.

GB.17. A rational representative of a strongly regular class

We need the following specific part of Steinberg's conjugacy argument. Over \(K=k^{\mathrm{ur}}\), if \(H\) is quasi-split, semisimple and simply connected, then every geometrically strongly regular semisimple conjugacy class invariant under \(\operatorname{Gal}(K^{\mathrm{sep}}/K)\) has a representative in \(H(K)\). Invariance here is an assertion about the geometric class, not a prior assertion that its conjugating element is rational. The proof is given below. It uses the freely available original Steinberg paper, §§6–7 and10–11, for the character/section mechanism; the unitary case is proved directly by a trace form and GB.16. No perfectness of \(K\), smoothness of a centre, or division by a residue characteristic is assumed.

Integral representations used in the argument. Let \(H_{\mathbf Z}\) be the pinned simply connected split group of a given root system. The actual programme construction is Pinnings and the classification of split reductive groups, Theorems5.1 and10.1, and its §6 supplies the rational characteristic-zero modules \(V_i=L(\lambda_i)\) of fundamental highest weights. We explain the needed integral extension of those particular modules.

Choose a highest vector \(v_i\) and expand \(u^-v_i\), for the universal ordered negative-root product, as a polynomial in its root coordinates. Let \(M_i\) be the \(\mathbf Z\)-span of its finitely many coefficients in \(V_i\). It is a finite free lattice: it is finitely generated and torsion-free, and it spans \(V_i\) over \(\mathbf Q\), since the highest vector generates the characteristic-zero module under negative root operators. Its weight decomposition is integral, because each coefficient has the weight \(\lambda_i\) minus the corresponding sum of positive roots. The weight-\(\lambda_i\) part is exactly \(\mathbf Z v_i\).

The ordered integral multiplication in \(U^-\), Pinnings and the classification of split reductive groups §7, proves stability under every negative root group. For a positive root group, factor \(x_\alpha(t)u^-\) formally at \((t,u^-)=(0,1)\) in the integral big cell \(U^-TU^+\). The integral cell and the formal inverse at its identity are provided by the integral group of Pinnings and the classification of split reductive groups; the root and torus coordinates of that factorization are power series with integer coefficients. For the inverse, solve its coordinates degree by degree: its linear coefficient matrix at the identity is the integral identity, so each new coefficient is obtained by subtracting an already integral polynomial. Acting on the highest vector kills \(U^+\). The torus factor acts by the integral character \(\lambda_i\); expansion of a positive or negative integral power of a series with constant term one has integral binomial coefficients. The coefficient of every monomial in \(t\) and the negative-root parameters consequently lies in \(M_i\). On the characteristic-zero module the result is already polynomial; comparing coefficients proves that every coefficient of \(x_\alpha(t)\) preserves \(M_i\). The torus preserves its weight lattice, and the simple normalizers, being products of positive and negative root elements with parameters \(1,-1\), preserve \(M_i\) as well.

These actions extend to \(H_{\mathbf Z}\). On its open cell their ordered product gives a morphism to \(\operatorname{GL}(M_i)\). Its determinant and the determinant of the inverse are integral units, since each root factor is unipotent and the torus factor is diagonal with integral characters. The homomorphism identity on the open subset where \(x,y,xy\) all lie in the cell holds over \(\mathbf Q\), hence integrally: that subset is flat over \(\mathbf Z\), and its regular functions inject into their rational scalar extension. Pinnings and the classification of split reductive groups Lemma4.1, whose complete translation proof is, extends the cell map to the entire group. Base change gives a module in every characteristic with highest weight \(\lambda_i\), a one-dimensional highest line, and all weights \(\lambda_i-\sum n_\alpha\alpha\), \(n_\alpha\geq0\). Nothing requires its reduction to be irreducible. Write \(\chi_i\) for its trace.

The rank-one action on the highest line in the \(i\)-th module is the two-dimensional standard \(\operatorname{SL}_2\)-action: the pairing with \(\alpha_i^\vee\) is one; for \(j\ne i\) that pairing is zero. Its usual matrices over \(\mathbf Q\) preserve the lattice just constructed, with the two primitive highest-string vectors and coefficients \(1\). Thus these statements hold after reduction in every characteristic. Transporting a pinning transports this construction and the traces: a pinned diagram automorphism permutes \(\lambda_i,M_i,\chi_i\). One can also see trace uniqueness directly from their characteristic-zero weight multiplicities and the integral lattice decomposition.

Characters separate semisimple classes. Work first over an algebraically closed field. On a maximal torus, every \(\chi_i\) is a Weyl-invariant Laurent polynomial with highest dominant weight \(\lambda_i\), whose coefficient is one; all other dominant weights are strictly lower in the positive-root order. Every dominant weight is \(\lambda=\sum n_i\lambda_i\), \(n_i\geq0\). The product \(\prod\chi_i^{n_i}\) has highest weight \(\lambda\) with coefficient one and otherwise lower dominant weights. The sums of the distinct Weyl translates of a dominant weight form a basis of the Weyl-invariant Laurent polynomials. Subtracting this highest term and inducting expresses every such orbit sum as a polynomial in the \(\chi_i\). The induction terminates: for weights between zero and a fixed dominant highest weight, pair with \(\rho^\vee\), where \(\alpha_i(\rho^\vee)=1\); every positive-root subtraction strictly lowers a bounded discrete nonnegative value on dominant weights. There are only finitely many lattice points with bounded value in the dominant cone.

The invariant Laurent polynomials separate finite Weyl orbits in all characteristics. Given two disjoint finite orbits, interpolate a Laurent polynomial equal to zero on one and one on the other; the maximal ideals of distinct torus points are comaximal. Its product over all Weyl translates is invariant and still has values zero and one. This uses no division by the Weyl-group order. Finally, semisimple elements are conjugate into tori by the actual proof Regular elements and centralizers Theorem3.1. If two elements of a torus are conjugate, their torus centralizers can first be conjugated within the centralizer of the common element using Tori, maximal tori and their conjugacy Theorem2.2; the resulting conjugator normalizes the torus. Thus semisimple classes are precisely Weyl orbits and are separated by the \(\chi_i\).

Trace on any module has the same value on an element and its semisimple Jordan factor: the commuting unipotent factor acts unipotently on each eigenspace, so its trace is the eigenspace dimension. A fibre of the tuple \(\chi=(\chi_i)\) whose semisimple class is strongly regular therefore consists only of that class. Indeed its semisimple factor is conjugate to the given torus element, and the commuting unipotent factor lies in its torus centralizer, hence is one. Jordan factors belong to the group, as in the actual calculation in Regular elements and centralizers.

A section when simple-root orbits are orthogonal. Use the pinned representatives \[ n_i=x_{\alpha_i}(1)x_{-\alpha_i}(-1)x_{\alpha_i}(1), \qquad N(c)=\prod_i x_{\alpha_i}(c_i)n_i . \tag{GB17.1} \] Choose an order putting each Galois orbit of simple roots together. Suppose roots in each orbit are mutually orthogonal. Its factors commute, so the morphism (GB17.1) is Galois-equivariant for the permutation action on its coordinates. These coordinates form the vector space descended from the permutation root set.

The morphism \(\chi N\) is a polynomial isomorphism. Here is the trace computation, valid on the integral modules just constructed. On a weight \(\mu\), the factor \(x_{\alpha_j}(c_j)n_j\) first sends the weight to \(s_j\mu\) and then adds nonnegative multiples of \(\alpha_j\). A contribution to a diagonal matrix block of the complete product must leave each simple-root coordinate unchanged: each \(\alpha_j\) occurs in just that one factor and the simple roots are independent. Thus each factor in that contribution returns to the same weight \(\mu\). Such a contribution is zero unless every \(\langle\mu,\alpha_j^\vee\rangle\geq0\), so only dominant weights contribute. For a dominant \(\mu\), its diagonal block depends only on those \(c_j\) for which \(\langle\mu,\alpha_j^\vee\rangle>0\), because the positive root action adds exactly this many root steps. On the highest line of \(M_i\), all factors except the \(i\)-th fix the line; the standard rank-one matrix gives the diagonal coefficient \(c_i\).

For any other dominant weight \(\mu\) in \(M_i\), write \(\mu=\sum q_j\lambda_j\). If \(q_j>0\), then \[ \langle\lambda_j,\rho^\vee\rangle \leq\langle\mu,\rho^\vee\rangle <\langle\lambda_i,\rho^\vee\rangle . \] Consequently \[ \chi_i(N(c))=c_i+F_i\bigl(c_j: \langle\lambda_j,\rho^\vee\rangle <\langle\lambda_i,\rho^\vee\rangle\bigr), \tag{GB17.2} \] with integral coefficients. Solve recursively in increasing \(\rho^\vee\)-value. This yields an integral polynomial inverse. Since \(\chi N\) is equivariant, its inverse is equivariant; it descends over the original field. An invariant tuple of character values thus produces a rational \(N(c)\). For a strongly regular semisimple class the preceding fibre argument proves that this is the desired representative. No descent through an inseparable singleton has been assumed: the inverse is an actual polynomial morphism.

The remaining diagram orbits. An orbit of adjacent simple roots in an irreducible finite Dynkin diagram occurs only in type \(A_{2m}\). Here is a proof requiring no unchecked classification list. Normalize simple vectors to squared length two. Their Gram matrix has diagonal two and an edge entry \(-\sqrt{d}\), where \(d\in\{1,2,3\}\) is the product of the two Cartan integers. Positive definiteness first makes the graph a tree: a cycle with coefficient one at all its vertices would have squared norm at most zero. A vertex of degree four is impossible, by assigning coefficient two to that vertex and one to four neighbours. Two degree-three vertices are impossible too: choose the path between them, put coefficient two on its vertices and one on two external neighbours at either end. The resulting norm is at most zero. Thus there is at most one branch vertex. There is also at most one multiple edge on a path. For the segment between two nearest such edges, replace their weights by \(\sqrt2\) and all intermediate weights by one. Its Gram matrix has a positive null vector: use coefficient one at the two ends and \(\sqrt2\) at every intervening vertex. Increasing either edge weight makes its quadratic value nonpositive, contradicting positive definiteness.

A tree automorphism preserves its centre, obtained by repeatedly removing leaves. Two adjacent vertices in the same automorphism orbit must be the two vertices of the central edge: away from it adjacent vertices have different distances from the centre. Some automorphism therefore swaps that edge and its two sides. There can be no unique branch vertex, because this swap fixes no vertex. The diagram is consequently a path. There can be no multiple edge off the centre, since it would have a distinct mirrored companion; the central edge is not multiple either, since its swapped endpoints have equal root lengths. Thus the path has only simple edges and an even number \(2m\) of vertices, which is exactly \(A_{2m}\). The pinned matrix realization of that path as \(\operatorname{SL}_{2m+1}\) follows from its displayed standard root datum and Pinnings and the classification of split reductive groups Theorem5.1, rather than an algebraic-group classification citation. For reducible data, first descend a transitive set of simple components to its stabilizer field \(E\); the original group is its restriction of scalars. Its rational points and invariant geometric classes are identified with those over \(E\) by projecting to one component. GB.14 and GB.16 apply to that finite extension too. Thus it remains to handle the quasi-split simply connected unitary group of dimension \(2m+1\) over a separable quadratic \(F/E\). Its identification with the pinned outer \(A_{2m}\)-form is Automorphisms, forms and parabolic subgroups Proposition5.1 together with the explicit transpose-inverse pinned automorphism and the pinned isomorphism theorem, already bound in GB.1.

In fact the following argument handles every unitary dimension. Let the strongly regular class have characteristic polynomial \(P(t)\in F[t]\), of degree \(n\). It is separable, its constant term is nonzero, the product of its roots is one, and its multiset of roots is invariant under \(z\mapsto\bar z^{-1}\). These are exactly the conditions forced by the special-unitary equation on its geometric class and its invariance. The coefficient descent assertion is literal: after the group is identified over \(F\) with \(\operatorname{SL}_n\), the subgroup fixing \(F\) fixes every characteristic coefficient, and the remaining involution sends the polynomial to its conjugate reciprocal. Hence \(P\) is defined over \(F\), with that reciprocal identity, even when \(E\) is imperfect.

Put \(A=F[t]/P(t)\). It is a finite separable \(F\)-algebra, and the reciprocal identity defines a semilinear involution \(*\) with \(t^*=t^{-1}\). The form \[ h(x,y)=\operatorname{Tr}_{A/F}(x^*y) \tag{GB17.3} \] is hermitian and nondegenerate: after a separable splitting field its trace pairing is the sum of coordinate products, and \(*\) is an invertible semilinear map. Multiplication by \(t\) preserves \(h\), since \(t^*t=1\), and its determinant is one.

Every nondegenerate hermitian form for a separable quadratic \(F/E\) admits an orthogonal basis with diagonal entries in \(E^\times\), also in characteristic two. There is a vector \(v\) with \(h(v,v)\ne0\). Otherwise \(h(x,x)=h(y,y)=0\) and the equality for \(x+cy\), as \(c\) ranges over \(F\), would give \(\bar c\,h(y,x)+c\,h(x,y)=0\) for every \(c\). Taking \(c=1\), then a \(c\) not fixed by the involution, forces \(h(x,y)=0\), contrary to nondegeneracy. Split off this line, whose orthogonal complement is nondegenerate, and induct. Norm surjectivity \(F^\times\to E^\times\), proved in GB.16 from the vanished relative Brauer group, rescales every diagonal entry to one. Hence all nondegenerate hermitian forms of dimension \(n\) are isometric. In particular (GB17.3) is isometric to the standard quasi-split form: the latter is itself a nondegenerate form of that dimension. This statement does not silently divide by two; the orthogonalization argument works in characteristic two.

In such an isometry, multiplication by \(t\) becomes a matrix in the standard \(\operatorname{SU}_n(E)\) with characteristic polynomial \(P\). Over an algebraic closure its distinct eigenvalues identify its \(\operatorname{GL}_n\)-class. The same class is a single \(\operatorname{SL}_n\)-class: adjust the determinant of a conjugator by a diagonal centralizer element in an eigenbasis. Thus it is a representative of the specified geometric special-unitary class. This proves the missing type \(A_{2m}\) case without invoking the perfect-field semisimple-part construction of a cross-section.

Products and restrictions of scalars now prove the stated rational-representative result for every quasi-split semisimple simply connected \(H/K\). The construction used exact integral coordinates and separable trace forms. In particular it includes wildly ramified quadratic extensions in characteristic two and groups with nonsmooth finite centre.

GB.18. Quasi-splitting over \(k^{\mathrm{ur}}\), including nonsmooth centres

For every connected reductive \(G/k\), \[ G_{k^{\mathrm{ur}}}\text{ is quasi-split}. \tag{GB18.1} \] Its Borel is defined over some finite unramified extension. We prove the assertion without citing the dimension-one theorem as a proof.

Put \(K=k^{\mathrm{ur}}\). First let \(H/K\) be quasi-split, semisimple and adjoint, and let \(c_\sigma\in H(K^{\mathrm{sep}})\) be a continuous cocycle. Its simply connected covering \(\widetilde H\) is quasi-split too: the inverse image of its Borel is a Borel, as is checked in the central-quotient root coordinates of Root data, Weyl chambers and the Bruhat decomposition Theorem7.1. The adjoint group acts algebraically on \(\widetilde H\) by conjugation. The central kernel acts trivially, so conjugation descends faithfully flat through the central quotient; this is also the group-scheme automorphism decomposition of Automorphisms, forms and parabolic subgroups Theorem3.1. Thus \(c\) twists \(\widetilde H\) to a reductive \(K\)-group \({}_c\widetilde H\), even if that kernel is nonsmooth. We do not lift the individual cocycle values through it.

A smooth connected affine group over \(K\) has Zariski-dense rational points, by the local henselian argument proved in GB.12: take an étale coordinate chart at the identity, spread its finitely many equations over a finite local-field subextension, and Hensel-lift a sufficiently small valuation polydisc. In the completion, the étale inverse expands a nonzero regular function into a nonzero convergent power series: the local completion is faithfully flat and its étale coordinates are formal parameters. Choose parameter values at which its first nonzero homogeneous polynomial is nonzero, possible by the one-variable root bound and induction over the infinite field. Scaling those values sufficiently toward zero makes that term dominate the later terms. Henselianity places the inverse points in the original field, dense in its completion. Thus no nonzero regular function vanishes throughout the polydisc. This proves density in the chart and hence the group. It applies to the twisted group and does not assume that \(K\) is perfect.

The geometrically strongly regular semisimple locus of \(\widetilde H\) is nonempty and open. In a torus avoid \(\alpha(t)=1\) for every root. The Regular elements and centralizers §3 root-weight proof makes the centralizer smooth with identity component that torus. Every other component normalizes it because the identity component is characteristic. Avoid also the finitely many proper loci \(w(t)=t\), \(w\ne1\), in the faithful Weyl action. No other component remains. The differential of conjugation is surjective on this locus, by Regular elements and centralizers, so its image is open. It is invariant under automorphisms and therefore descends to the twist. Density supplies a strongly regular \(y\in{}_c\widetilde H(K)\). On identifying the two groups over \(K^{\mathrm{sep}}\), its twisted rationality is \[ c_\sigma\sigma(y)c_\sigma^{-1}=y, \tag{GB18.2} \] where the formula means the algebraic adjoint action. The geometric conjugacy class in the original quasi-split \(\widetilde H\) is invariant. GB.17 gives a representative \(y_0\in\widetilde H(K)\) of that class.

There is a conjugator \(a\in H(K^{\mathrm{sep}})\) with \(y=ay_0a^{-1}\). Here geometric conjugacy alone would not justify the field of definition. Its transporter in the adjoint group is a torsor under \(T=C_H(y_0)\). This is a \(K\)-torus: over an algebraic closure the root factors \(\alpha(y_0)-1\) are units and absence of a Weyl stabilizer excludes additional components, while the diagonalizable fixed-point argument makes the centralizer scheme smooth. The geometric transporter is consequently nonempty and smooth over \(K^{\mathrm{sep}}\). A nonempty smooth finite-type scheme over a separably closed field has a point: take an étale chart to affine space, choose a rational point of its nonempty open image, and a point in its étale fibre has a finite separable residue field, hence the same field. This proves the choice of \(a\) without a purely inseparable adjunction.

Substitution in (GB18.2) shows that \(a^{-1}c_\sigma\sigma(a)\) is a cocycle in \(T(K^{\mathrm{sep}})\). GB.16 kills this class, so \(c\) is trivial. We have proved \(H^1(K,H)=1\) for every quasi-split semisimple adjoint \(H\), with nonsmooth covering centres treated explicitly.

For arbitrary connected reductive \(G/K\), choose a split pinned form \(G_0\) of its root datum. The Automorphisms, forms and parabolic subgroups Theorem3.1 proves, as group schemes, \[ \operatorname{Aut}(G_0)= G_0^{\mathrm{ad}}\rtimes\operatorname{Aut}(\text{based datum}). \] The form is split over a finite separable extension: a geometrically maximal torus spreads through its smooth transporter, the torus character lattice has a finite separable splitting field, and the split root lines can then be pinned using the Roots and reductive groups of rank one root construction. Its descent cocycle has an outer datum part \(e_\sigma\) and an inner part \(c_\sigma\). The datum part has finite image even when the full integral-lattice automorphism group is infinite, since the descent is defined over that finite splitting field. Twisting by \(e\) gives a quasi-split \(G_{\rm qs}\): its pinned Borel is preserved and descends, as proved in Automorphisms, forms and parabolic subgroups Proposition5.1. The remaining \(c\) lies in the quasi-split semisimple adjoint group \(G_{\rm qs}^{\rm ad}\). Its first cohomology has just been proved trivial. Correcting the descent isomorphism by the coboundary identifies \(G\) with \(G_{\rm qs}\) and transports its rational Borel. This proves (GB18.1).

The Borel and its inclusion use finitely many coefficients in \(K\), so are already defined over a finite unramified extension \(E/k\). Smoothness, connectedness, solvability and the geometric Borel dimension descend faithfully flat from \(K\) to \(E\). Equivalently its point in the projective Borel scheme uses finitely many such coordinates. This proves the finite-unramified assertion with all the original arithmetic hypotheses. In particular imperfect equal characteristic, arbitrary residue characteristic, the full reductive centre and nonsmooth finite centres have been retained.

GB.19. The centralizer of a maximal \(k\)-split torus

This section closes the root-zero and boundedness part of the descent problem. Let \(S\subset G\) be a maximal \(k\)-split torus and put \(M=Z_G(S)\). Then \(M\) is connected reductive by the actual torus-centralizer proof Regular elements and centralizers Theorems2.1–2.2. The torus \(S\) is central in \(M\); there is no larger \(k\)-split torus in \(M\), because a larger one in \(M\) would also be one in \(G\). More explicitly any noncentral \(k\)-split torus \(S_1\subset M\) commutes with \(S\), and their product is a split torus of dimension greater than \(\dim S\). Thus all \(k\)-split directions of \(M\) are exactly those of its central \(S\).

By GB.18, \(M_K\), \(K=k^{\mathrm{ur}}\), is quasi-split. GB.0–GB.6 and AB.6–AB.7 give its complete enlarged building \(\mathcal B(M,K)\), its root coordinates, and its product \(\mathcal B(M,K)^0\times V_{Z(M),K}\). GB.13 supplies its natural Galois isometric action and GB.15 supplies a fixed point. We prove \[ \mathcal B(M,K)^\Gamma =\{z_M\}\times V_{Z(M),K}^\Gamma, \qquad \dim V_{Z(M),K}^\Gamma=\dim S . \tag{GB19.1} \] The point \(z_M\) is unique in the reduced building. The second equality is the torus-lattice calculation AB.10, applied to the full centre; anisotropic central directions have no invariant cocharacters.

Let \(x\) be a fixed point, and \(F\) its open facet. The connected model \(\mathcal M_F\) constructed in GB.10 descends to \(\mathcal O_k\), by GB.13 and AB.9. The central split torus \(S_{\mathcal O_k}\) embeds in it: in a \(K\)-apartment every central torus character and its inverse have value zero on the bounded central torus, so the character-extension and faithful weight argument of GB.13 embeds its integral model in the bounded torus model and then the facet model. It is central there, since it is central generically and the models and source are flat.

Every geometric maximal torus of the special fibre contains this central \(S_{\mathbf F_q}\): the product with that central torus is a torus, so maximality forces inclusion. Choose a maximal torus \(Q\) defined over \(\mathbf F_q\). Its existence is the actual finite-field Lang proof in Tori, maximal tori and their conjugacy §4, already bound in GB.13. In using that proof one may start with a torus containing the central \(S\), and all conjugations retain \(S\). Lift \(Q\) to an unramified torus \(\mathcal Q\subset\mathcal M_F\) by GB.12, prescribing the already embedded subtorus \(S_{\mathcal O_k}\). The prescribed-subtorus part follows from the same finite-projector and degree-one transporter calculation there; centrality ensures that its centralizer is the whole group. Hence this is an actual lift containing \(S\), not merely a formal lift with an unspecified embedding of that subtorus.

The maximal split part of an unramified torus has the same rank generically and on its finite residue field: both are the invariant cocharacters of its finite étale character lattice. Consequently the generic maximal \(k\)-split part of \(\mathcal Q_k\) contains \(S\). Maximality of \(S\) in \(M\) makes it exactly \(S\). GB.13's special-fibre torus-conjugacy and smooth transporter argument identifies \(\mathcal Q_K\) with a maximal \(K\)-split torus of \(M_K\), and gives a \(\Gamma\)-stable apartment \(A_x\) containing the whole facet \(F\). In this paragraph \(\mathcal Q_K\) means the maximal \(K\)-split part of the generic unramified torus; its generic rank equals the geometric torus rank of the facet model, as proved in GB.13. That apartment has \[ A_x^\Gamma=x+V_S. \tag{GB19.2} \] Indeed its invariant cocharacters are exactly the just identified split part \(S\), and its affine Galois action has the fixed point \(x\). Central directions in the enlarged building are global, so this is a translate of the same central vector space at every \(x\).

To deduce uniqueness of the reduced fixed point, take two fixed points \(x,y\). They have a common \(K\)-apartment by GB.5 and AB.4, and their segment is fixed by convexity. That segment meets finitely many facets: in that apartment the affine root hyperplanes form the locally finite arrangement proved in GB.4, and only finitely many meet a bounded segment. Every facet whose open part contains a fixed segment point is \(\Gamma\)-stable, by uniqueness of the open facet of a point. For each such \(F'\), repeat the preceding construction. Its stable apartment contains the whole \(F'\), and (GB19.2) shows that \(F'^\Gamma\) has only central directions. Partition the segment into its finitely many facet pieces; on each piece its reduced-building projection is constant. Continuity at their endpoints makes this projection constant on the entire segment. Thus \(x,y\) have the same reduced projection. Conversely one fixed point plus all invariant central directions is fixed, by the product action in GB.13–GB.15. This proves (GB19.1). This centralizer argument uses only the stable-apartment and centralizer statements already established; the general residue/coface and maximal-apartment results are proved separately in GB.21–GB.22.

The rational group \(M(k)\) preserves its unique reduced fixed point. On the remaining \(S\)-flat it acts by translations. Its central character valuation is \[ \langle\chi,\nu_M(m)\rangle=-\omega(\chi(m)), \qquad \chi\in X_k^*(M), \tag{GB19.3} \] with the same sign as BF.2, GB.2 and AB.10. These characters span the dual of the split central factor: the root datum of \(M/M^{\rm der}\) identifies them with a finite-index sublattice of the centre's rational character space. An isometry commuting with all central \(S(k)\)-translations has identity linear part on this flat, so (GB19.3) gives its full action. The image is discrete because each displayed valuation is integral; it contains a full lattice from the cocharacters \(S(k)\), by the nonsingular character/cocharacter pairing. Thus it is itself a full lattice.

The kernel of (GB19.3) is compact. Its elements fix an enlarged point, hence belong to the full norm stabilizer constructed in GB.10. In the faithful integral representation over a finite splitting extension, the displayed weighted norm bounds every matrix coordinate and inverse coordinate. To transfer those bounds to any faithful \(k\)-representation, express its matrix entries and inverse entries as regular functions in the first closed matrix immersion. Each is a polynomial in finitely many matrix and inverse coordinates with fixed coefficients in that splitting field, so has a uniform valuation bound. Its \(k\)-values consequently lie in closed bounded subsets of the original locally compact field. The same holds for its inverse matrices. The subgroup is closed, and its image in the affine closed immersion over \(k\) is a closed subset of a compact product of valuation balls. Hence it is compact. This uses compactness of the original \(k\), not of \(K\). In particular an anisotropic connected reductive \(k\)-group has compact rational points and a unique fixed point in the enlarged \(K\)-building. With a split centre present, that entire centre has instead been retained in (GB19.1).

For completeness the centralizer building embeds in that of \(G_K\) with the same apartment metric. Choose a maximal \(K\)-split torus of \(M_K\) containing \(S\). It is maximal in \(G_K\): its centralizer in \(G_K\) is contained in \(M_K\), since it contains \(S\), and is the torus centralizer already computed for quasi-split \(M_K\). Its apartment is the same full cocharacter space. Restrict the integral root and weight representation of \(G_K\) to \(M_K\). The \(M\)-root coordinates are exactly the roots vanishing on \(S\), as follows from the actual centralizer weight calculation and big-cell root product. The full norm stabilizer at a point therefore satisfies \[ K_x(G,K)\cap M(K)=K_x(M,K). \tag{GB19.4} \] One may use the restricted faithful representation in GB.10: its integral root coefficients and weight coordinates are still those already used, and it detects a normalizer's movement of every apartment coordinate. Thus the quotient map \(M(K)\times A\to G(K)\times A\) is well defined and injective after the point-stabilizer relations. Two points of the \(M\)-building have a common \(M\)-apartment, on which this map is the identity Euclidean map; hence it is isometric. Its image is the union of apartments whose maximal \(K\)-split tori contain \(S\), since such tori lie in \(M\) and are conjugate there by the quasi-split root/Borel argument of GB.13. The image is \(\Gamma\)-invariant and its fixed set is the single full \(S\)-flat (GB19.1). This gives an actual maximal-split flat in the general fixed-point space, together with its exact root-zero compact kernel.

GB.20. Rational filtered root parameters and exact half-apartment bounds

Let \(S\subset G\) be maximal \(k\)-split, and choose the \(\Gamma\)-stable \(K\)-apartment \(A\) through the centralizer flat of GB.19. Its maximal \(K\)-split torus is defined over \(k\) and contains \(S\); choose \(x_0\in A^\Gamma\). For a primitive character \(b\in X^*(S)\) on a nonzero root-weight ray, let \(\Psi_b\) be the set of nondivisible \(K\)-roots \(a\) with \[ a|_S=m_a b,\qquad m_a>0 . \] The integers \(m_a\) are the actual restrictions of the characters. Internal \(2a\) coordinates in a nonreduced \(K\)-root group have weight \(2m_ab\). Define \(U[b]_K\) as the product of these \(K\)-root groups, in any order refining positive \(S\)-weight. It is a smooth connected unipotent subgroup and descends to \(k\). To verify descent without an assumed nonsplit root classification, choose a cocharacter of \(S\) positive on \(b\); the ambient positive group has ordered root coordinates. Conjugation by \(S\) assigns each coordinate its character. A semilinear automorphism fixing \(S\) sends an output coordinate to a polynomial of the same \(S\)-weight. On the ray subgroup all its inputs have weights \(m b\), \(m>0\), so every output of a weight off that ray is zero. Hence the subgroup is \(\Gamma\)-stable. Positive closure of the ray and the integral root collection formula give its group law. This construction allows every actual positive integer multiple; it does not assume an unproved claim that all original nonsplit rank-one groups are \(\operatorname{SL}_2\) or \(\operatorname{SU}_3\).

Normalize the \(K\)-root valuations at \(x_0\): \[ \phi_{a,x_0}(u)=\phi_a(u)+a(x_0),\qquad U[b]_r(K)= \prod_{a\in\Psi_b}\{u\in U_a(K): \phi_{a,x_0}(u)\geq m_a r\}. \tag{GB20.1} \] The same definition includes the \(2a\) coordinates at twice that bound. The coordinate product is a subgroup by the weighted commutator calculation GB.2. Root order changes preserve it, because every new collected coordinate has the sum of the input weights and at least the sum of their valuation bounds. The \(x_0\)-normalization makes it \(\Gamma\)-stable: GB.13's affine covariance, evaluated at the fixed \(x_0\), removes the translation part. Define \(U[b]_{r+}\) by strict inequalities on the intrinsic root valuations, equivalently \(U[b]_{r+\epsilon}\) for a sufficiently small positive \(\epsilon\). There is such a common \(\epsilon\), since only finitely many coordinate weights occur and all their valuation sets are discrete.

The exact integral models of these groups are products of the finite-free additive and unitary parameter spaces of GB.7, with the scaled bounds (GB20.1). Their group law and inverse are the integral collected polynomials, so they are smooth affine models with connected fibres. They are canonically \(\Gamma\)-stable. Indeed the generic semilinear maps preserve their entire \(R\)-point groups. The integral-point criterion of GB.8 extends these maps and their inverses, since \(R=\mathcal O_K\) is henselian with algebraically closed residue field; the group and action identities extend by flatness. Infinite continuous descent is effective by AB.9, after putting the finitely many equations and character weights over a finite unramified subextension.

Here is the cohomology needed for rational lifting: \[ H^1_{\rm cts}(\Gamma,U[b]_r(R))=1, \qquad H^1_{\rm cts}(\Gamma,U[b]_{r+}(R))=1. \tag{GB20.2} \] Here \(U[b]_r(R)\) denotes the point group of its integral model, namely the bounded group in (GB20.1); no separate integral-point assumption on a \(K\)-coordinate is intended. Filter this group by increasing positive \(S\)-character weight. In coordinates the subgroup with all weights less than \(j b\) zero is closed and normal, and its quotient by the next such subgroup is a vector group on a finite free \(R\)-module. The commutator of weights \(i b,j b\) has weight \((i+j)b\), so these quotients are central at the relevant stage. A semilinear \(S\)-equivariant automorphism on a quotient of the one weight \(j b\) is linear: every monomial of degree different from one has a different algebraic character, even when the integer \(j\) is divisible by the residue characteristic. Differentiating the character is neither necessary nor sufficient for this assertion.

The positive cohomology of every such semilinear \(R\)-module is zero. AB.9 descends it to a finite free \(\mathcal O_k\)-module. A given continuous cochain and the finite descent equations lie over a finite unramified \(E/k\). The ring \(\mathcal O_E\) has a normal integral basis: choose a normal basis in its separable residue extension, Hensel-lift its generator, and the determinant of its conjugates is a unit by residue independence. Thus \(\mathcal O_E\otimes_{\mathcal O_k}N\), under \(\operatorname{Gal}(E/k)\), is an induced module for every coefficient module \(N\). The bar insertion contraction proves vanishing in all positive degrees, without dividing by the extension degree. Increasing \(E\) to contain all values of a continuous cocycle proves the same for \(\Gamma\). This proof works for finite-length modules too: the same normal basis identifies their scalar extensions with induced modules, regardless of \(\mathcal O_k\)-torsion.

Now kill a group cocycle in successive central vector quotients. After making its first vector image a coboundary, correct by a lift of that vector; the remaining cocycle lies in the next group. Each vector correction lifts, because the actual coordinates give surjective maps on \(R\)-points. The finitely many weights make the induction terminate. This proves (GB20.2). The same proof applies to the strict model and, if required, to an inner twist of these actions by a cocycle: inner conjugation changes a parameter only by terms of larger weight, so its action on each vector quotient is unchanged.

The boundary map on rational points is consequently surjective: \[ U[b]_r(k)\longrightarrow \bigl(U[b]_r(K)/U[b]_{r+}(K)\bigr)^\Gamma . \tag{GB20.3} \] Indeed lift a fixed boundary class to \(u\in U[b]_r(K)\). Its defect \(u^{-1}\sigma(u)\) is a cocycle in the strict group. Write it \(a^{-1}\sigma(a)\) using (GB20.2). Then \(ua^{-1}\) is fixed, remains in the same bounded group, and has the chosen boundary class. The same argument applies to quotients by a positive-weight tail, such as the \(2b\)-tail when that is the only higher root multiple. Thus there is no unproved averaging of nonlinear unitary coordinates and no failure at residue characteristic two.

The boundary group itself has a finite central filtration with successive finite-length \(R\)-modules. To see this, filter by its coordinate character weights as above. At a fixed weight the change from non-strict to strict bounds is inclusion of fractional lattices; its quotient is a finite-length \(R\)-module. Changes of unitary coordinates may add quadratic terms, but those have twice the character weight and disappear in the preceding vector quotient. Its semilinear action descends to the corresponding finite residue modules by AB.9 and the normal-basis calculation. In particular a nonzero vector boundary piece has nonzero rational points after descent to \(\mathbf F_q\), including \(q=2\). Apply (GB20.3) to lift them.

Finally these numerical bounds really give the half-apartment fixed by each parameter. For \(u=\prod u_a\in U[b](k)\), put \[ \phi_b(u)= \min_{u_a\ne1} \frac{\phi_{a,x_0}(u_a)}{m_a}. \tag{GB20.4} \] This minimum is independent of the ordered coordinates: reordering adds terms with summed weights and no smaller weighted valuation. Applying the inverse reordering gives equality of the two minima. In the unitary coordinate group use the exact BF.7 formula \(\phi_a(u,v)=\omega(v)/2\) and \(\omega(u)\geq\omega(v)/2\), with no lost \(2a\) parameter. The filtered model and its bounds, rather than the chosen trace-one coordinate change, define (GB20.4).

The full point-stabilizer calculation GB.10 gives \[ u\text{ fixes }x\in A^\Gamma \quad\Longleftrightarrow\quad b(x-x_0)+\phi_b(u)\geq0 . \tag{GB20.5} \] For this equivalence, the root Gauss coordinates of \(u\) have torus component one and denominator one. Membership in the full weighted norm stabilizer is therefore exactly the root-coordinate inequalities \(\phi_{a,x_0}(u_a)+a(x-x_0)\geq0\), with the doubled coordinate bounds in the unitary group. Substituting \(a|_{A^\Gamma}=m_a b\) proves (GB20.5). The sign is the same as the split formula; replacing it by its negative would interchange the fixed and moving half-apartments. These levels, rational boundary lifting and all same-sign weighted commutator bounds are now proved for the original arbitrary connected reductive group.

The ray construction does not identify an original nonsplit rank-one group with a classical matrix group. GB.21 proves the exact residue wall data, and GB.22 proves the opposite rational completion and wall exchange at every attained wall using (GB20.3). All actual restricted character weights and higher tails remain in the ordered ray groups; that proof route needs no additional standalone classification of the original nonsplit forms.

GB.21. The residue group and its coface parabolics

Let \(F\) be a \(K\)-facet, let \(\mathcal H_F\) be the connected model of GB.10, and write \(H_F\) for its special fibre over \(\bar f\), the algebraically closed residue field. Its maximal torus is the special torus \(S_K\) of GB.13. Its centralizer is the connected bounded torus special fibre \(T_F\), with reductive torus quotient \(S_K\). We prove that reduction identifies the cofaces of \(F\), in reverse order, with the parabolics of \[ \overline H_F=H_F/R_u(H_F). \tag{GB21.1} \] It identifies cofaces in a given apartment with parabolics containing that apartment's special maximal torus. This is the concrete input for maximal-apartment descent. The primary free comparison is BTII4.6.8–4.6.14 and4.6.33–4.6.35; the root and point-group arguments needed here follow.

Fix \(x\in F\). A root coordinate is called strict if its weighted value at \(x\) is positive. These strict subgroups do not depend on \(x\in F\). Non-strict boundary coordinates occur precisely at the affine root walls containing \(F\); the opposite coordinate has a boundary too, by the exact normalizer completion and valuation covariance GB.0–GB.2. For a reduced field root, its boundary quotient is one additive \(\bar f\)-line: consecutive fractional ideals in the totally ramified root field have the same algebraically closed residue field by GB.9. We give the corresponding calculation for every ramified unitary wall, rather than assume its residue quotient.

Normalize the valuation on its intermediate field \(E\) to integers. A quadratic \(F_1/E\) has an Eisenstein uniformizer \(\theta\), with \[ \theta^2-t\theta+n=0,\qquad \omega_E(n)=1,\quad \omega_E(t)\geq1,\quad \mathcal O_{F_1}=\mathcal O_E\oplus\mathcal O_E\theta . \tag{GB21.2} \] Here \(\omega_{F_1}(\theta)=1/2\); in characteristic two separability says \(t\ne0\). This is the finite-extension theorem AB.0/GB.9. For \(s=2r\in\frac12\mathbf Z\), the lattice \(I_s\) of the \(v\)-coordinate is \[ \begin{array}{ll} s=j:& I_s=\pi^j\mathcal O_E\oplus\pi^j\mathcal O_E\theta,\\ s=j+\frac12:& I_s=\pi^{j+1}\mathcal O_E\oplus\pi^j\mathcal O_E\theta . \end{array} \] Its trace ideal is generated by the displayed basis traces \(2\pi^j,t\pi^j\), or \(2\pi^{j+1},t\pi^j\), respectively. Put \(h_2=\omega_E(2)\), with \(h_2=\infty\) in characteristic two, and \(h_t=\omega_E(t)\), with \(h_t=\infty\) if \(t=0\). Choose a basis element \(c\) whose trace generates this ideal; if both traces have equal valuation, choose the element with the larger coordinate valuation. Then \(\lambda=c/\operatorname{Tr}(c)\) has trace one. The formula of GB.7 is \[ u\in\{u:\omega_E(Nu)\geq\tau\},\qquad v=-\lambda Nu+\delta z,\qquad \tau=\omega_E(\operatorname{Tr}I_s), \tag{GB21.3} \] where \(\delta\mathcal O_E=I_s\cap\ker\operatorname{Tr}\). In characteristic two this trace-zero line is \(E\). Otherwise use \(\theta-t/2\); its valuation is \(1/2\) if \(h_t\geq h_2+1\), and the integer \(h_t-h_2\) if \(h_t\leq h_2\). These alternatives follow from unequal valuations of its two terms, and include \(t=0\).

Exactly one scalar parameter survives modulo the strict group. At \(s=j\), it is the leading \(u\)-coordinate when \(h_t\geq h_2+1\), and the leading trace-zero coordinate when \(h_t\leq h_2\). At \(s=j+1/2\), the choices are reversed. To check this list, the minimum valuation of \(-\lambda Nu\) is \(\omega(c)\), because \(\omega(Nu)=\tau\) is attained and \(u\) ranges over the principal ideal of valuation \(\tau/2\). The minimum trace-zero valuation has the integer or half-integer part just computed. In each case one minimum equals \(s\) and the other is strictly larger. The surviving principal-ideal quotient has dimension one over \(\bar f\). In characteristic two \(h_2=\infty\): at integral \(s\) the trace-zero parameter survives, and at half-integral \(s\) the \(u\)-parameter survives. Thus no missing division by two occurs at these walls.

The surviving coordinate is additive. If it is \(u\), this follows from the first coordinate of the exact group law; if it is trace-zero \(v\), the product term \(-\bar u u'\) has strictly larger valuation and disappears. The strict model maps onto the kernel of that one coordinate in the special root model; (GB21.3) and the explicit polynomial group law prove the assertion on schemes, not just their field points. The torus character of the surviving line is respectively \(a\) or \(2a\). The opposite line has its negative character, since the boundary normalizer conjugates the two models and fixes the wall. Rescaling \(\omega_E\) back to the original \(k\)-normalization divides all these valuations by the same ramification index and preserves the list. This proves the boundary-line assertion for every quadratic ramification and every residue characteristic.

Let \(R_F\) be the reduced connected closed subgroup generated by all special strict root subgroups and \(R_u(T_F)\). It is normal in \(H_F\). Same-sign normalization is the integral commutator collection: one strict input makes the resulting weighted value positive. For opposite coordinates, one strict input makes the rank-one Gauss denominator a unit congruent to one in its effective boundary coordinate; BF.7/GB.3's Gauss identities put its root corrections in the strict groups and its torus correction in \(R_u(T_F)\). These identities use the original unitary formula, so the list above includes both effective \(a\) and \(2a\) walls. The bounded torus normalizes these subgroups. The wall normalizers preserve strict values and normalize \(R_u(T_F)\). Finally reduction of the exact group \(P_F^0\) is surjective, by henselian smooth lifting, and \(P_F^0\) is generated by the root groups and \(T^0\), by GB.10. Thus the normalization checks on these generators prove normality in the entire special group.

This normal subgroup is unipotent. Use the faithful lattice-chain representation from GB.10 and its associated graded representation. Every strict root matrix raises the weighted filtration and is trivial on its associated graded. The special maximal torus acts faithfully there: the weights of a closed faithful torus immersion generate its character lattice, as in GB.13. The reduced identity kernel of this graded representation has no torus. Indeed every torus of \(H_F\) is conjugate into its maximal torus, and normality of the kernel would put a conjugate of any kernel torus in that faithful torus. The no-torus argument Regular elements and centralizers §1 makes its identity kernel unipotent. The image of \(R_u(T_F)\) is central in the graded image, since its commutators with boundary root lines are strict: a unipotent group has no nontrivial homomorphism to the scalar character group of a boundary line. Thus the connected subgroup generated by the strict kernel and this torus-centralizer radical is an extension of unipotent groups, hence unipotent (triangularize its normal kernel and then its quotient). This proves \(R_F\subset R_u(H_F)\).

Pass to the smooth affine quotient \(Q=H_F/R_F\). Here quotient existence has a short local proof. Choose a finite-dimensional translation comodule in \(\bar f[H_F]\) containing generators of the ideal of \(R_F\), using the actual finite-comodule argument AG-GS Group schemes, actions and Hopf algebras, Lemma5.1. The top exterior power of its intersection with that ideal gives a line whose scheme stabilizer is \(R_F\): stability of the subspace is exactly stability of its generated ideal, and translation preserves \(R_F\) exactly for its own points. The normal unipotent \(R_F\) acts trivially on that line, since it has no characters, and normality makes it fix the span of all its translates. The resulting representation has kernel exactly \(R_F\). Its reduced image is a closed affine group: a constructible subgroup contains an open of its closure, and translations make it the whole closure. The homomorphism onto it is flat by generic flatness and translations; its fibres are translates of the smooth \(R_F\), so it is a smooth surjection and an \(R_F\)-torsor. This represents the quotient and makes \(Q\) smooth connected affine.

The special Gauss cell shows that \(R_F\) has precisely the strict-coordinate and \(R_u(T_F)\) directions. Its ordered product has that dimension. Conversely every word in these generators, whenever in the Gauss cell, has zero effective boundary coordinates, by the strict commutator and opposite-Gauss calculations above; the same equations hold on their reduced closure. Hence its intersection with that cell has no additional directions. Its scheme intersection with each boundary additive line is trivial: that line's effective coordinate is one of the displayed linear \(u\)- or trace-zero coordinates, while this coordinate vanishes on \(R_F\). It follows that \(Q\) has a torus \(S_K\), one-dimensional boundary root lines, and no other moving tangent directions. Its Gauss cell is the product of those boundary lines and that torus.

There is no further unipotent radical in \(Q\). If its smooth normal radical had a nonzero boundary weight, centralize the codimension-one kernel of that weight and take its positive limit group. The actual smooth cocharacter construction Roots and reductive groups of rank one Lemma3.1, isolates a smooth connected one-dimensional subgroup with that tangent line; in \(Q\) the only positive weight on this ray is its one boundary line. The equal dimensions and closed inclusion force that whole additive line into the radical. Normality and the boundary normalizer put the opposite line into it too. But the exact GB.0 normalizer formula expresses the wall representative as a product of those two lines modulo strict coordinates. It would then be in the radical while acting by a nontrivial reflection on the surviving torus. This is impossible: a normal unipotent radical has trivial scheme intersection with a torus, by diagonalizing multiplicative-type representations and triangularizing unipotent ones, so that torus embeds in the radical quotient, where a radical element acts trivially. This also excludes an infinitesimal boundary intersection: a nontrivial torus-stable finite subgroup of an additive line has nonzero tangent, giving the same positive-limit contradiction. Weight zero in a putative radical lies in the centralizer of the torus, which in \(Q\) is the torus itself, by the Gauss coordinates. It therefore vanishes too. A smooth connected positive-dimensional group has nonzero Lie algebra, so the radical is trivial. Thus \(Q\) is reductive and \(R_F=R_u(H_F)\). The quotient (GB21.1) is therefore reductive. Its roots are precisely the surviving boundary characters, one of \(a,2a\) for each wall direction, with the negative character on the opposite line. Its maximal torus is the image of \(S_K\), and its Weyl group is the finite wall group \(W_F\): every root reflection has the reduced boundary representative, and these generate its Weyl group by the Root data, Weyl chambers and the Bruhat decomposition root/Weyl proof. Their actions are exactly the linear actions of the walls through \(F\). This identifies the local spherical root data without replacing ramified unitary roots by a reduced-field formula.

Let \(F'\) be a coface of \(F\) in the base apartment and \(d\) a small rational displacement from \(F\) into \(F'\). A boundary root at \(F\) keeps its full level if its character is nonnegative on \(d\), and becomes strict if negative. Previously strict roots stay strict: choose the displacement smaller than every positive gap in the finitely many discrete root levels. The maps \(\mathcal H_{F'}\to\mathcal H_F\) exist by the integral-point extension criterion GB.8, because \(P_{F'}^0\subset P_F^0\). Their special image contains \(R_F\) and the torus, and its reductive image is generated by the boundary lines nonnegative on \(d\). It is exactly the cocharacter parabolic \(P_{\overline H_F}(d)\), by the actual classification proof Automorphisms, forms and parabolic subgroups Theorem1.1. All parabolics containing this torus occur by the same finite root-chamber classification. Different cofaces give different sign/zero patterns.

The exact inverse-image assertion is \[ P_{F'}^0= \{g\in P_F^0: \operatorname{red}(g)\in \operatorname{preimage}(P_{\overline H_F}(d))(\bar f)\}. \tag{GB21.4} \] The reduction kernel is contained in every such \(P_{F'}^0\): in the Gauss coordinates of an element reducing to one, each root parameter is strict, and the torus parameter is in \(T^0\). The desired parabolic's points are generated by its torus, its permitted boundary root groups and the radical. Their parameters lift in the displayed root models, and their torus parameters lift by smooth henselian lifting. They consequently lift to \(P_{F'}^0\). This proves both inclusions in (GB21.4).

Finally all cofaces can be moved into the base apartment by \(P_F^0\). Choose a chamber of their common \(K\)-apartment having \(F\) in its closure. The finite-wall Bruhat decomposition of GB.5/AB.4 makes \(P_F\) transitive on the chambers containing \(F\). Apartments containing a chamber are its Iwahori translates, by the explicit apartment argument AB.6. The factors \(T_b\) normalize the base apartment and can be absorbed, leaving \(P_F^0\); GB.10 gives \(P_F=T_bP_F^0\). Thus the coface orbits are \(P_F^0/P_{F'}^0\). By (GB21.4) these are exactly the residue flag orbits. The latter exhaust the parabolics by Automorphisms, forms and parabolic subgroups Theorem1.1, and equal parabolics have equal stabilizers. This proves the claimed bijection, its order, and the apartment assertion.

When \(F\) is \(\Gamma\)-stable all maps and radicals are canonical and descend. A stable coface therefore corresponds exactly to an \(\mathbf F_q\)-parabolic of the descended reductive quotient. The parabolic scheme and its faithful-flat descent are proved in Automorphisms, forms and parabolic subgroups Theorem2.1. Its finite set of \(\mathbf F_q\)-points gives finitely many stable cofaces. Smooth henselian lifting gives \(P_F^0(k)\twoheadrightarrow H_F(\mathbf F_q)\). Lang for the connected special group, and for its connected parabolic preimages, makes this connected rational point group transitive on the cofaces of any rational parabolic type. This is a statement about the connected model. No assertion that \(H^1\) of the full disconnected bounded stabilizer vanishes has been used.

GB.22. Maximal rational apartments and descended wall exchange

Let \(X=\mathcal B(G,K)^\Gamma\), with its closed convex metric from GB.15. We prove that its apartments are the maximal-split flats of GB.19 and that one such flat \(A_k\) satisfies \[ X=G(k)A_k . \tag{GB22.1} \] We use the coface theorem just proved, not an unproved equality of fixed cosets for a disconnected stabilizer.

We first record the finite-field torus fact needed in this argument. In a connected reductive group over \(\mathbf F_q\), every maximal split torus has torus centralizer and all maximal split tori are rationally conjugate. Such a group is quasi-split: apply the pinned outer/inner decomposition Automorphisms, forms and parabolic subgroups, and kill its adjoint inner cocycle by the actual finite-field Lang proof Tori, maximal tori and their conjugacy §4. If the centralizer of a maximal split torus had a nontrivial semisimple derived factor, that factor would be quasi-split by the same argument and would have a nontrivial split cocharacter: sum the coroots in a diagram orbit as in GB.1. Multiplying by the original central split torus would enlarge it. Its centralizer is therefore a torus. A generic cocharacter in the split torus now determines a rational Borel, since no absolute root is zero on it. Rational Borels are conjugate: their smooth connected stabilizer has trivial \(H^1\) by Lang, so the rational points of the Borel variety are one rational orbit. Within a rational Borel, the root-height vector filtration and additive Hilbert90 conjugate the maximal tori by its unipotent radical, as in the Automorphisms, forms and parabolic subgroups Proposition5.1 proof. This proves the split-torus conjugacy assertion. For a smooth connected nonreductive group over \(\mathbf F_q\), lift through its split unipotent radical. Tori have trivial intersection with it, their transporters are smooth by GB.12, and the vector-filtration/Lang calculation makes the same conjugacy rational. These arguments also show that a maximal split torus belongs to every rational parabolic after a rational conjugation.

At a fixed point \(x\), with stable open facet \(F\), choose a maximal \(\mathbf F_q\)-split torus \(Q_0\) of \(H_F\), then a rational geometric maximal torus \(Q\) containing it. GB.12 lifts them, with the inclusion prescribed, to actual unramified tori in the descended \(\mathcal H_F\). GB.13 gives a stable \(K\)-apartment containing \(F\), whose generic maximal \(k\)-split part is \(S_0\), the lift of \(Q_0\). We claim that \(S_0\) is maximal split in \(G\).

The centralizer-model comparison used in this claim is exact. For any integral split subtorus \(\mathcal S_0\subset\mathcal H_F\), the smooth centralizer \(C_{\mathcal H_F}(\mathcal S_0)\) has connected fibres, by the Regular elements and centralizers Lemma1.1 and its weight proof. Its generic fibre is \(M_0=Z_G(S_0)\). Its Gauss coordinates are the bounded torus and precisely the root coordinates of zero \(S_0\)-weight. Thus its integral points are \(P_F^0\cap M_0(K)\), which is the connected facet point group of \(M_0\): the torus model is the same \(T^0\), and root coordinates of nonzero \(S_0\)-weight vanish in the centralizer. GB.10 and the integral uniqueness criterion GB.8 identify it with the actual \(M_0\)-facet model. Its special reductive quotient is the centralizer of the image of \(Q_0\) in \(\overline H_F\). Indeed the same zero-weight Gauss coordinates remain after killing the strict radical of GB.21. This is a torus by the finite-field fact above. This proves the comparison on models and point groups; it does not assume that centralizers commute with arbitrary nonsmooth reduction.

The \(M_0\)-building embeds in the \(G_K\)-building by the root-zero restriction argument at the end of GB.19; that embedding argument applies to any split subtorus, irrespective of maximality. The point \(x\) belongs to it. Since the residue reductive group of its \(x\)-facet is a torus, GB.21 gives no proper stable coface there. A sufficiently small ball about \(x\) in the \(M_0\)-building meets only cofaces of this facet, by the explicit local-star radius of AB.7. Its fixed part is therefore locally contained in the facet's fixed affine space. That space has precisely the directions of \(S_0\): choose a special maximal torus containing \(Q_0\) and apply GB.13; every additional invariant cocharacter would enlarge the maximal residue split torus. Since \(S_0\) is central in \(M_0\), these directions are a global central metric factor. After removing it, the fixed set has an isolated point \(x\). The fixed set is convex and connected; a geodesic from an isolated point cannot reach a second point. Hence its entire fixed set is this one \(S_0\)-flat.

If \(S_0\) were contained in a larger split torus, enlarge to one maximal split torus \(S_1\) in \(M_0\). GB.19 applied to \(Z_{M_0}(S_1)\), and its isometric embedding, gives an \(S_1\)-flat in the same fixed set of the \(M_0\)-building. Its dimension is larger, a contradiction. This proves the claim without importing rational conjugacy of maximal split tori as a premise. In particular every fixed point belongs to a stable apartment whose fixed affine space is a maximal-split flat.

The fixed cells are the nonempty intersections \(F^\Gamma\) of stable \(K\)-facets with the fixed set. GB.13 puts each in a stable apartment, so it is a convex polyhedral cell. Its cofaces are exactly the rational parabolics of GB.21. They are finite in number. Maximal cofaces correspond to rational Borels, because the finite residue quotient is quasi-split, as just proved. Their fixed dimensions agree locally: all those Borels contain conjugate maximal residue split tori, and the lift has the same generic split rank. These locally equal dimensions agree throughout \(X\). To check this last assertion without a purity assumption, take a segment between any two points in a common \(K\)-apartment. Its finitely many facet pieces have overlapping local stars at their endpoints; the maximal-coface dimensions therefore agree successively along it. Write the resulting constant dimension as \(r\), including the invariant centre. Every maximal \(k\)-split flat has dimension \(r\): its local cells have that dimension, and every residue-maximal lift is a maximal split torus, while a split torus of largest dimension contributes its own flat by GB.19. Thus no hidden lower-rank apartment has been substituted.

The maximal cells are gallery connected, after dividing out the full central Euclidean factor. A segment meets finitely many cells. In each local star the rational parabolics form the finite spherical building of the residue group, by GB.21 and the actual finite Bruhat theorem; its maximal chambers are gallery connected (write any finite Weyl label as a word in simple reflections). Replace the finitely many star crossings of the segment by these galleries. This gives a gallery between any two maximal cells, also when the segment originally crossed a face of larger codimension. In rank one it is the succession of vertex stars; in rank zero the reduced fixed space is a point.

The group \(P_F^0(k)\) is transitive on the maximal cofaces at every face \(F^\Gamma\), by the final paragraph of GB.21. Moving across a gallery thus makes \(G(k)\) transitive on maximal fixed cells. Choose one of them \(C_k\), in a maximal-split flat \(A_k\). Every point lies in the closure of a maximal cell, so it lies in some \(G(k)\)-translate of \(\overline C_k\), proving (GB22.1).

We also need apartment transitivity with a fixed cell. Let two maximal-split flats contain a fixed point \(x\). Their integral split tori embed in \(\mathcal H_F\): their units fix \(F^\Gamma\), and the weighted root-character and integral extension argument of GB.13, over \(R\), gives their actual split integral embeddings; on \(F\), take the minimal \(K\)-facet containing \(x\), whose norm inequalities those units also satisfy. Their special split tori are maximal in \(H_F\), by the claim and its residue-rank argument. Conjugate them by \(H_F(\mathbf F_q)\) using the finite-field fact. The transporter of the two prescribed split torus embeddings is smooth by GB.12, so that conjugator lifts to \(P_F^0(k)\). Its generic conjugation carries the two split tori, hence carries their unique centralizer flats by GB.19. It fixes \(x\). When both flats contain a maximal cell, use its model and the same transporter; the lifted element fixes that cell pointwise. Thus the action is strongly transitive on a flat together with a maximal cell in it.

The walls in \(A_k\) are exactly the nonzero restrictions of the \(K\)-walls that meet it. They are locally finite, by the finite discrete root-value sets of GB.4. The cell containing a given open fixed point is the intersection with its unique \(K\)-facet, so this restricted arrangement is the actual fixed-cell arrangement. Its noncentral normal directions span: a cocharacter annihilating all restricted roots is central in \(G\), by the absolute big-cell centralizer computation. Translation by \(S(k)\) supplies a full lattice. Its maximal cells are bounded modulo the \(k\)-split centre of \(G\): each root direction has walls at an unbounded arithmetic sequence in both signs, so their finite independent directions bound a cell.

Every wall reflection lifts to \(N_G(S)(k)\). At a relative panel the residue group has relative semisimple rank one. Its relative normalizer, computed from the finite-field orbit classification GB.1, contains its reflection. Lift that residue normalizer point preserving the split torus. First correct its lift in the special group's unipotent radical so that it normalizes the chosen split torus (the split-torus transporter and Lang argument above); then use the smooth normalizer of GB.12 and henselian lifting. The resulting \(n\in P_F^0(k)\cap N_G(S)(k)\) fixes the panel pointwise and acts on \(A_k\) by its nontrivial rank-one linear reflection. An affine isometry with those properties is precisely the wall reflection, with no translation along the panel. It preserves the full arrangement because it acts on \(X\). AB.2's proved gallery-disc argument now gives the affine Coxeter system for this arrangement and the alcove \(C_k\). This proof does not identify its translation lattice with a coroot lattice that may be inappropriate for the original group.

Here is the group exchange, including the actual opposite-parameter input. Let \(I_k\) be the full pointwise stabilizer of \(C_k\), let \(\alpha=b+r\geq0\) be one of its walls, and let \(s\) be the reflection lift. Use \(L=Z_G((\ker b)^0)\), with its split central tangential torus removed; its rational split semisimple rank is one. Its positive and negative groups are the ray groups \(U[b],U[-b]\) of GB.20. The exact centralizer-model comparison above applies to its panel model. Its special reductive quotient has finite-field relative rank one. GB.0–GB.1, with the zero valuation and the finite root fields, prove its two Bruhat cells, including the unitary finite-field cell.

Put \(A=U[b]_r(k)\) and \(A^+=U[b]_{r+}(k)\). The boundary lifting GB.20 makes \(A\) surject onto the positive residue root group. The coface inverse-image formula (GB21.4) consequently gives \[ I_k=J A,\qquad J=I_k\cap s^{-1}I_ks . \] The reduction kernel and the residue torus are in \(J\), and the rank-one Borel is its torus times its positive root group. For the full \(I_k\), first use transitivity of the connected point group on flats containing \(C_k\): an element fixing \(C_k\) can be adjusted by its connected facet group to normalize \(A_k\), where it belongs to the pointwise kernel \(M(k)^0\). Thus \(I_k\) is its connected coface group times \(M(k)^0\). This compact kernel is normal under \(s\) and lies in \(J\). Hence the same factorization holds for the full \(I_k\), including its component classes; they have not been discarded.

For a nontrivial residue class \(u\in A\setminus A^+\), the residue element \(su\) is in the opposite Gauss cell. This cell has an actual integral inverse. Apply Roots and reductive groups of rank one Lemma3.1 to the panel model of \(L\) and the split cocharacter normal to the panel: its positive, zero and negative weight models are precisely the filtered positive ray, the \(M=Z_G(S)\) model and the filtered negative ray. Their multiplication is an open immersion. Its special image is exactly the inverse image of the residue opposite Gauss cell. Here is the radical check in this assertion. The smooth connected radical over the perfect residue field has a torus-stable central filtration with additive-line quotients: triangularize its semidirect product with the acting torus and take the reduced identity components of the successive upper-entry kernels, as in the full solvable-structure proof Tori, maximal tori and their conjugacy §2. Each quotient is an additive line with one algebraic torus character. On it, multiplication of its negative-, zero- and positive-weight limit subgroups is an isomorphism, since exactly one of those three groups is that line. Induct up the central filtration: a factorization in the quotient lifts, and its remaining central coordinate is put in its uniquely prescribed sign factor; uniqueness follows by projecting and then using the central line. This proves that the corresponding three factors exhaust the radical, with polynomial inverses. Their formation is the actual cocharacter limit construction and commutes with reduction by Roots and reductive groups of rank one Lemma3.1. Thus killing the radical sends the integral cell to the residue Gauss cell and the sign-factor argument lifts its entire inverse image. The strict-coordinate calculations of GB.21 identify these factors with the displayed positive, zero and negative filtered models, including the unitary higher-coordinate tails. Thus \(su\), whose reduction is in that cell, lies in the integral open cell, and its inverse coordinates give \[ su=v\,m\,w,\qquad v\in U[b]_r(k),\quad m\in M(k)^0,\quad w\in U[-b]_{-r}(k). \tag{GB22.2} \] Here \(M(k)^0=\ker\nu_M\): the middle factor fixes the panel point, hence its centralizer-flat translation is zero by GB.19. The coordinates are rational because the open-cell inverse is defined over \(\mathcal O_k\); an external Bruhat assertion has not replaced this factorization.

Equation (GB22.2) is also the normalizer completion. Rearranging gives \[ u=(s^{-1}vs)(s^{-1}m)w ; \] the outside factors are in the negative ray, while \(s^{-1}m\) normalizes \(S\) and acts by the wall reflection. Thus every nonzero boundary parameter has its exact opposite rational completion. Apply this same panel-model argument at each attainable wall to obtain the assertion at every level; it does not require different wall levels to be conjugate under \(S(k)\). All levels and half-apartment signs are those of (GB20.4)–(GB20.5). Higher central parameters in a nonreduced ray remain in these groups throughout.

For \(n\) in the apartment normalizer, \(J\) can be moved through \(s\). A strict \(u\in A^+\) conjugates through \(s\) into \(I_k\). For a boundary \(u\), if \(n^{-1}\alpha\) is positive on \(C_k\), (GB20.5) gives \(n^{-1}un\in I_k\). If it is negative, use (GB22.2): \(v,m\in I_k\), and \(n^{-1}wn\in I_k\) by that same exact half-apartment inequality. These two cases prove \[ sI_kn\subset I_ksnI_k\ \cup\ I_knI_k . \tag{GB22.3} \] The group nontriviality required by AB.4 is also exact. The positive boundary group at this panel is nonzero and has a nonzero \(\mathbf F_q\)-point, by GB.20 and its finite-field rank-one identification. Lift it to \(u\in U[b]_r(k)\setminus U[b]_{r+}(k)\). It belongs to \(I_k\), by (GB21.4), and its exact inequality (GB20.5) fixes the base side of the wall and fails on the interior of \(sC_k\). Thus \(u\notin sI_ks^{-1}\), and these two subgroups differ. The intersection of \(I_k\) with the apartment normalizer is its pointwise kernel, because \(C_k\) has an open subset of the apartment.

Finally galleries and panel transitivity generate \(G(k)\) by \(I_k\) and the normalizer. Each adjacent chamber is an \(I_k\)-translate of \(sC_k\) at the base panel. Move this argument along a gallery. A residual element stabilizing \(C_k\) can be adjusted by \(I_k\) to normalize \(A_k\), using the just proved transitivity on flats containing that cell. Thus no extra rational-group generation premise is needed. Apply AB.4 to the affine Coxeter subgroup, as follows. Let \(W_{\rm aff}\) be the reflection subgroup proved above, let \(N_{\rm aff}\subset N_G(S)(k)\) be its inverse image, and put \(G_{\rm aff}=\langle I_k,N_{\rm aff}\rangle\). Its quotient \(N_{\rm aff}/(I_k\cap N_{\rm aff})\) is precisely \(W_{\rm aff}\): the intersection is the compact pointwise kernel \(M(k)^0\), proved directly in GB.23 below. The reflection group is normal in the full affine normalizer, since conjugation sends a wall reflection to the reflection in its image wall. Every full normalizer element can be followed by an element of \(N_{\rm aff}\) to preserve the base alcove; the remainder normalizes \(I_k\). Consequently \(G_{\rm aff}\) is normal in the generated full group and the full group is \(G_{\rm aff}\) times these residual normalizer elements. Gallery induction itself makes \(G_{\rm aff}\) transitive on maximal cells. It is also transitive on apartments containing the base cell: the connected transporter used above lies in its subgroup \(I_k\). Thus \(X=G_{\rm aff}A_k\), and the Coxeter arrangement, generation within \(G_{\rm aff}\) and (GB22.3) are exactly AB.4's hypotheses. Its complete double-coset proof supplies common maximal rational apartments for any two cells and the exact finite-face parahorics. This use does not misidentify the full extended affine normalizer, which can contain central translations and residual alcove symmetries, with a Coxeter group. Thus the descended construction has the common maximal apartments and wall exchange required for the enlarged building.

GB.23. Full stabilizers, topology, metric and properness over \(k\)

The group \(G(k)\) is locally compact and second countable: a faithful affine \(k\)-matrix immersion realizes it as a closed subgroup of \(\operatorname{GL}_n(k)\), and the original nonarchimedean local field has compact valuation balls and a countable valuation/residue basis. The action on \(X\) is continuous. Indeed each point stabilizer is open: the faithful weighted norm description GB.10 is a finite set of closed matrix inequalities with discrete coefficient valuations, and a sufficiently small matrix neighbourhood of the identity satisfies them. Joint continuity then follows from the isometry inequality \(d(gx,gy)=d(x,y)\), using a fixed point's open stabilizer.

Every full point stabilizer \(K_x(k)=K_x(G,K)\cap G(k)\) is compact. The valuation-bound transfer to a faithful \(k\)-matrix immersion is exactly the polynomial coordinate argument of GB.19: the weighted norm bounds matrices and inverses in a finite splitting field, hence bounds each regular \(k\)-matrix coordinate. The stabilizer is closed, so is a closed subset of a compact product of original \(k\)-valuation balls. No compactness of \(K=k^{\mathrm{ur}}\) is being asserted.

We first prove the pointwise and setwise stabilizers of the entire maximal-split flat; neither assertion requires the double-coset argument. If \(g\in G(k)\) fixes every point of \(A_k\), then \(sgs^{-1}\) fixes \(x_0\) for every \(s\in S(k)\). Those matrices lie in the single uniformly bounded full norm stabilizer just proved. Decompose the faithful representation over a finite splitting field into its algebraic \(S\)-weight spaces. Its matrix block from weight \(\eta\) to weight \(\xi\) is multiplied under this conjugation by \((\xi-\eta)(s)\). If a block of nonzero difference weight were nonzero, choose a \(k\)-cocharacter \(\lambda\) with nonzero pairing and put \(s=\lambda(\pi)^j\), letting \(j\) tend to either sign of infinity. That block would violate the uniform valuation bound. Thus all nonzero-difference blocks vanish. Faithfulness makes \(g\) centralize \(S\), so \(g\in M(k)\). Its translation on the centralizer flat is zero, hence \(g\in M(k)^0=\ker\nu_M\). Conversely GB.19 shows that this kernel fixes the whole flat.

The subgroup \(M(k)^0\) is Zariski dense in \(M\). It contains an ordinary open neighbourhood of the identity in \(M(k)\), since the finitely many character valuations in (GB19.3) vanish near one. A \(k\)-analytic open neighbourhood of a smooth rational point is Zariski dense in the smooth connected group: take an étale coordinate chart there; the finite polynomial Jacobian equations, after scaling to a sufficiently small valuation ball, have unit Jacobian and their Newton corrections converge, giving a branch over a full \(k\)-ball. A nonzero regular function vanishing on that branch would have a nonzero field norm to the coordinate function field (the chart is generically finite and separable) vanishing on that ball wherever its denominators are defined. A nonzero polynomial cannot vanish on a full product of infinite \(k\)-balls, by induction on the number of variables and the one-variable root bound. This is a contradiction. One may restrict the chart and ball to remove all those denominators; the coordinate norm remains nonzero. This proves the density also in imperfect equal characteristic.

An element carrying \(A_k\) to itself conjugates its pointwise kernel to itself. By this density it therefore normalizes the algebraic group \(M\). The torus \(S\) is the unique maximal \(k\)-split subtorus of \(Z(M)^0_{\rm red}\): it is central and split, and any additional central split direction would enlarge it in \(G\). Equivalently this unique torus is defined by the invariant cocharacter sublattice of the centre, so every \(k\)-automorphism of \(M\) preserves it. Hence the element normalizes \(S\). Conversely \(N_G(S)(k)\) preserves the centralizer building and its unique fixed flat by GB.19. We have proved that the setwise stabilizer of \(A_k\) is exactly \(N=N_G(S)(k)\), and its pointwise kernel is exactly the compact \(M(k)^0\).

Let this \(N\) act on \(A_k\). Its linear image is finite: it permutes the finite nonzero root restrictions on \(S\), acts trivially on the rational characters of \(G\), and these together span the apartment's dual. For each linear label its translation vectors form a coset of the discrete lattice \(\nu_M(M(k))\), by (GB19.3). Thus the full affine image is discrete and its point stabilizers are finite; \(N/M(k)^0\) acts properly on the apartment. Wall reflections are the ones proved in GB.22, so these facts supply the finite-stabilizer hypothesis of AB.2 used there. Wall-pair translations span the semisimple directions: every wall direction occurs in an unbounded arithmetic family; parallel reflection products give nonzero translations in its coroot direction. The full centre has the lattice from \(S(k)\) in GB.19. Thus the affine Coxeter alcove has compact closure modulo that entire central factor, and the extended normalizer has only a finite residual alcove action in the semisimple directions.

For \(x\in A_k\), the full stabilizer formula is \[ K_x(k)=P_F^0(k)\,N_x,\qquad N_x=\{n\in N:nx=x\}, \tag{GB23.1} \] where \(F\) is the stable minimal \(K\)-facet of \(x\). If \(g\) fixes \(x\), its translated maximal-split flat \(gA_k\) also contains \(x\). The connected-model torus transporter of GB.22 supplies \(p\in P_F^0(k)\) fixing \(x\) and carrying that flat back to \(A_k\). Then \(pg\in N_x\). Conversely both factors fix \(x\). The setwise-normalizer assertion here is precisely the uniform-conjugation and Zariski-density proof above; thus the adjustment gives an actual element of \(N_G(S)(k)\).

Connected point groups are exactly those of the actual descended models GB.10–GB.13. Their full bounded models can have nontrivial finite component groups. Formula (GB23.1) retains \(N_x\), including its compact \(M(k)^0\) kernel. If a fixed full-stabilizer coset is used, the exact component-cocycle test is GB.11: its class must vanish in the finite component group, after which the connected Lang calculation supplies the correction. This argument used connected coface and torus transporters, so it never assumed vanishing of \(H^1(\Gamma,K_x(G,K))\). In particular it has not equated the full bounded torus with its connected integral model \(T^0\).

There is an exact quotient description \[ X\simeq (G(k)\times A_k)/\!\sim,\qquad (g,x)\sim(h,y)\ \Longleftrightarrow\ \exists n\in N:\ y=nx,\quad g^{-1}hn\in K_x(k), \tag{GB23.2} \] with the normalizer convention chosen consistently with AB.1. Surjectivity is (GB22.1). For injectivity, if the two represented points agree, the connected point transporter in GB.22 carries their two maximal flats to each other while fixing that point; after it is removed the remaining comparison is a normalizer element, giving the displayed witness. The reverse implication follows from the point stabilizer identity. Thus the quotient uses the actual full stabilizers.

All hypotheses of the topological theorem AB.1 now hold over the original locally compact field. Compactness and closedness of \(K_x\) were just proved. The local upper-containment \(K_y\subset K_x\) near \(x\), and closed incidence, follow directly from the finite weighted norm inequalities: round their finitely many real weight differences to the discrete coefficient valuation lattices. At a boundary a neighbouring inequality may become stronger, while it cannot become weaker than the inequality at that boundary; away from a boundary it is constant. Matrices and inverses give the same conclusion for norm equality. Normalizer covariance and properness were proved above.

A chamber's pointwise stabilizer \(I_k\) is compact open. A face has only finitely many adjacent rational chambers, by the finite set of residue parabolics GB.21. For a point of a face in \(\overline C_k\), its compact full point stabilizer contains the open \(I_k\) and therefore has finite index over \(I_k\): its disjoint open cosets have a finite subcover by compactness. The same applies to the pointwise face stabilizer. A setwise chamber stabilizer has only a finite residual action on its vertices modulo the central factor; its possible central translations are retained in the normalizer, while a point stabilizer has zero central translation. Root wall exchange (GB22.3) gives precisely the finite face groups required by AB.4. Finally the compact set consisting of the closed semisimple alcove times a central lattice parallelepiped covers all orbits, by (GB22.1) and the retained lattice translations. AB.1 therefore proves that (GB23.2) has a locally compact Hausdorff second-countable topology and a proper continuous \(G(k)\)-action with compact orbit space.

The quotient metric and the metric inherited from \(\mathcal B(G,K)\) agree. AB.4 supplies a common maximal rational apartment for any two cells. Its straight segment is a geodesic in the \(K\)-building, with the same Euclidean metric as \(A_k\). Conversely every quotient chamber chain has at least that endpoint distance, since each chamber metric is the restricted \(K\)-metric and the triangle inequality holds there. Thus both distances equal the apartment distance. AB.6's retractions and AB.7's local-star comparison show that this metric gives exactly the quotient topology; finiteness of the residue panel sets makes bounded closed balls compact. Completeness and the CAT(0) inequality already hold on the closed convex fixed set by GB.15, and are thereby also established for the descended quotient. The central factor is exactly \[ V_{Z(G),K}^\Gamma , \] with its full Euclidean metric and full lattice of rational central translations. No reduced-building properness assertion has silently removed it.

Lastly this enlarged space is a numerable universal proper \(G(k)\)-space. Compact subgroups have bounded orbits in its proper metric, and the circumcentre argument GB.15 makes their fixed sets nonempty; they are convex and contractible. Point stabilizers are compact. Properness, second countability and the compact quotient give the invariant finite partition of unity on slice neighbourhoods constructed in AB.8. Its partition-of-unity and squared-distance-minimization proof then gives the universal property for numerable proper spaces. Theorem 3.2 also gives the mapping property directly for every locally compact proper source. This proves the arbitrary connected reductive local-field building and properness scope, in every residue characteristic, with the exact finite-unramified arithmetic, connected models, component correction and metric comparisons bound above.

The argument used generating root rays and their exact integral filtration, opposite completion and wall exchange. A separate classification of the original nonsplit \(k\)-rank-one groups as classical matrix groups is not needed or claimed: their residue rank-one group is quasi-split over the finite field and has the proved orbit classification GB.1. Internal \(2a\) unitary parameters and every actual restricted character weight have remained in (GB20.1)–(GB22.2). Consequently no unproved standalone list of nonsplit classical forms is a prerequisite of this construction.

What this lesson does not prove

The equivariant Kasparov-cycle, product and homotopy comparison results used in Proposition 4.4 are proved in Equivariant KK-theory and the Green–Julg theorem, Theorems 2.3, 4.3 and Proposition 6.1. The topological and Haar prerequisites for Theorems 1.3 and 2.1 are proved in Noncompact foundations for equivariant induction, NCF.1 and NCF.3–NCF.5. The exact earlier algebraic-group and local-field cohomology proofs used by the building construction are named in BF.0, AB.9 and GB.14–GB.18. They are prerequisites, not replacements by external literature citations.

The numerable recognition theorem, the almost-connected homogeneous-space theorem, the Rips-complex theorem and the nonequivariant geometric/analytic comparison are stated and unused in Section 5. Their precise statements and freely accessible sources are given there. The assembly homomorphism, its conjectural isomorphism and its consequences belong to the following lesson.

References

• Wolfgang Lück, Survey on Classifying Spaces for Families of Subgroups, free author preprint, revised 2004. Definitions 1.8 and 2.1, Theorem 2.5, and Theorems 4.3, 4.5–4.7, 4.13 and 4.16 discuss the mapping category and the models named here.
• Maria Paula Gómez Aparicio, Pierre Julg and Alain Valette, The Baum–Connes conjecture: an extended survey, 2019, Section 4.2, Definition 4.2, and Section 4.3, Remark 4.3. This gives the universal-space and equivariant K-homology formulation due to Baum, Connes and Higson, including cutoff normalization.
• Paul Baum, Nigel Higson and Thomas Schick, On the Equivalence of Geometric and Analytic K-Homology, free author preprint, revised 2009. Definitions 5.1–5.7 and Theorem 6.2 give the finite CW-pair comparison stated in Section 5.
• François Bruhat and Jacques Tits, Groupes réductifs sur un corps local I. Données radicielles valuées, 1972. Sections 2.3–2.5 and 3.2 provide the apartment, retraction and metric constructions; Definition 6.2.1 and Examples 6.2.3 give the root-valuation conventions. The proofs used here are in BF, AB and GB.
• François Bruhat and Jacques Tits, Groupes réductifs sur un corps local II. Schémas en groupes. Existence d’une donnée radicielle valuée, 1984. Sections 4.1–4.2 and 5.1 treat nonreduced roots and unramified descent.
• Michel Demazure, SGA 3, Exposé XXIII, freely accessible revised text, 2024. Sections 1.1–1.2 and Corollaries 5.2–5.9 concern integral pinned root coordinates.
• Brian Conrad, Reductive group schemes, free author text. Lemma 6.2.8 treats integral root constants. His Lang’s theorem and unirationality, Section 2, gives the finite-field context of the local proof in AB.9.
• James S. Milne, Algebraic Number Theory, free author notes, Proposition 2.29 and Theorem 7.38. The finite-extension valuation and DVR proof needed here is given in AB.0.
• Todd Trimble, BN-pairs and Boolean Hecke algebras, 2005. AB.4 proves the exchange consequences needed here.
• Bas Edixhoven and Matthieu Romagny, Group schemes out of birational group laws, Néron models, revised 2013. The integral model arguments are supplied in GB.7–GB.13.
• Armand Borel and Tonny A. Springer, Rationality properties of linear algebraic groups II, 1968, Section 8.6. The arithmetic-dimension-one argument, including imperfect fields, is proved in GB.14 and GB.16–GB.18.
• Robert Steinberg, Regular elements of semi-simple algebraic groups, 1965. The regular-class and cocycle arguments needed for quasi-splitting are proved in GB.17–GB.18.
• Romyar Sharifi, Group and Galois Cohomology, free author notes, Section 1.11, Lemmas 1.11.4–1.11.9 and Theorem 1.11.11. GB.16 proves the integral lattice criterion and torus vanishing used here.