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Beilinson-Bernstein localization

Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI. Public domain (CC0).

The projective line is not affine: a line bundle can have higher cohomology or no global sections. Yet a quasi-coherent module carrying an ordinary D-action on it has neither obstruction. Differentiation forces enough global sections to recover the module. This is the simplest localization theorem, and we will prove it for every quasi-coherent D-module, not only for connections or holonomic modules.

We work over an algebraically closed characteristic-zero field $k$, with left D-modules quasi-coherent over $\mathcal O$. The general flag-variety theorem is stated over $\mathbb C$. The background is semisimple Lie algebras, Verma modules and category $\mathcal O$, and line bundles and quasi-coherent cohomology on projective space. Category $\mathcal O$ is the subject of the planned prerequisite Category O and the Kazhdan-Lusztig conjecture in the semisimple Lie algebra course. Twists and adjunctions use Equivariant and twisted D-modules.

Our scalar twist on $\mathbb P^1$ is \[ \mathcal D^{(n)}=\mathcal D_{\mathcal O(1)}^n,\qquad \mathcal D^{(0)}=\mathcal D_{\mathbb P^1}. \] For integer $n$ this acts on $\mathcal O(n)$. The shifted parameter is $\nu=n+1$: regularity means $\nu\ne0$, and integral dominance means $\nu>0$. Thus the singular scalar twist is $n=-1$, not $n=0$.

1. A ring visible in two charts

On the chart with coordinate $x$, put \[ e=\partial_x,\qquad h=-2x\partial_x+n,\qquad f=-x^2\partial_x+nx. \] On the second chart $y=x^{-1}$, the twist transition is $\partial_x=-y^2\partial_y+ny$. The three operators are global and satisfy \[ [h,e]=2e,\qquad[h,f]=-2f,\qquad[e,f]=h. \] Fix the Casimir \[ \mathsf C=h^2+2h+4fe. \] It acts as the scalar $n(n+2)$ in the ring of global operators. The preceding lesson, Formula (3.7), proved for every scalar $n\in k$ that \[ U_n:=\Gamma(\mathbb P^1,\mathcal D^{(n)}) \simeq U(\mathfrak{sl}_2)/(\mathsf C-n(n+2)). \tag{1.1} \] The proof identified the order symbols with the even Veronese ring $\bigoplus_{m\geq0}H^0(\mathcal O(2m))$, and checked that the Casimir generates the entire kernel. We use that proved ring calculation here.

For $n=0$, $e,h,f$ are just vector fields and $\mathsf C=0$. This is the central character of the trivial representation, even though many infinite-dimensional representations also have it.

Write \[ \Delta_n(V)=\mathcal D^{(n)}\otimes_{U_n}V,\qquad \Gamma_n(M)=\Gamma(\mathbb P^1,M). \tag{1.2} \] Tensor-Hom adjunction gives $\Delta_n\dashv\Gamma_n$. We still have to prove that the adjunction maps are isomorphisms. The ring calculation alone does not do this.

2. Jets supply the missing global sections

We first prove a geometric lemma. All tensor products involving $\mathcal D$ on the right below use its right $\mathcal O$-module structure.

Lemma 2.1. For every integer $r\geq0$, the induced left D-module \[ P_r=\mathcal D_{\mathbb P^1}\otimes_{\mathcal O}\mathcal O(-r) \tag{2.1} \] is generated by global sections as an $\mathcal O$-module and has $H^1(P_r)=0$.

Proof. Let $J^m\mathcal O(r)$ denote the bundle of $m$-jets: its fiber at a point consists of local sections modulo those vanishing to order $m+1$. In a coordinate it has basis $1,t,\ldots,t^m$ times a frame. Evaluating a differential operator of order at most $m$ on a jet gives a perfect pairing, with $\partial_t^j(t^j)=j!$. Thus \[ F_m P_r\simeq\bigl(J^m\mathcal O(r)\bigr)^*. \tag{2.2} \] This also follows directly by regarding $P_r$ as differential operators from $\mathcal O(r)$ to $\mathcal O$. It explains why the right $\mathcal O$-tensor structure in (2.1) matters.

The evaluation map \[ H^0(\mathcal O(r))\otimes\mathcal O \longrightarrow J^r\mathcal O(r) \tag{2.3} \] is an isomorphism. Both bundles have rank $r+1$. At any finite point, polynomials of degree at most $r$ prescribe arbitrary $r$-jets by Taylor expansion. The other coordinate chart gives the same assertion at infinity. Hence (2.3) is an isomorphism on every fiber and therefore on bundles. Dualizing yields \[ F_rP_r\simeq\mathcal O^{\oplus(r+1)}. \]

For $m>r$, the symbol exact sequence is \[ 0\to F_{m-1}P_r\to F_mP_r\to\mathcal O(2m-r)\to0. \tag{2.4} \] On the two standard charts the Čech calculation for $\mathcal O(a)$ gives $H^1(\mathcal O(a))=0$ for $a\geq-1$; for $a\geq0$ it is also generated by its homogeneous polynomial sections. Starting with $F_rP_r$, (2.4) proves inductively that every $F_mP_r$, $m\geq r$, has vanishing $H^1$. Its symbol sections lift globally, and those lifts together with generators of $F_{m-1}P_r$ generate $F_mP_r$ at every point.

Finally $P_r=\varinjlim_m F_mP_r$. Quasi-coherent cohomology commutes with this filtered colimit: use the two affine charts and their affine intersection, so the Čech complex is a finite complex and filtered colimits are exact. This proves both assertions. $\square$

Theorem 2.2 (D-affinity of $\mathbb P^1$). For every quasi-coherent left $\mathcal D_{\mathbb P^1}$-module $M$, \[ H^i(\mathbb P^1,M)=0\quad(i>0), \qquad \mathcal O\otimes_k\Gamma(M)\longrightarrow M \text{ is surjective}. \tag{2.5} \] In particular $\Gamma$ is exact and detects nonzero D-modules.

Proof. Every quasi-coherent sheaf on projective space is generated by maps from negative line bundles. More explicitly, it is the filtered union of coherent subsheaves; after a sufficiently positive twist each such subsheaf is generated by finitely many sections. Thus there is an $\mathcal O$-linear surjection $\bigoplus_\alpha\mathcal O(-r_\alpha)\to M$, with $r_\alpha\geq0$. Induction extends these maps to a D-linear surjection \[ \bigoplus_\alpha P_{r_\alpha}\longrightarrow M. \tag{2.6} \] The image contains the original generating sections, so surjectivity really holds.

Let $K$ be the kernel. It is quasi-coherent over $\mathcal O$. Lemma 2.1 and the same finite Čech complex show that the source of (2.6) has zero $H^1$, even for an infinite sum. Every quasi-coherent sheaf on this two-chart cover has zero $H^2$. The long exact sequence therefore gives $H^1(M)=0$. Higher cohomology already vanishes for this cover.

Every $P_{r_\alpha}$ is $\mathcal O$-generated by its global sections. Their images are global sections of $M$ and generate the image of (2.6), which is all of $M$. This proves the second assertion. The long exact cohomology sequence makes $\Gamma$ exact on D-modules; generation shows that $\Gamma(M)=0$ implies $M=0$. $\square$

The underlying sheaf in (2.5) need not be coherent over $\mathcal O$. The proof applies, for example, to point modules and arbitrary direct sums.

Theorem 2.3 (localization for $\mathbb P^1$). The functors in (1.2) at $n=0$ are inverse exact equivalences \[ \operatorname{Mod}_{\rm qc}(\mathcal D_{\mathbb P^1}) \simeq U_0\text{-}\operatorname{Mod}. \tag{2.7} \] They restrict to coherent D-modules and finitely generated $U_0$-modules.

Proof. First prove the unit $V\to\Gamma\Delta_0(V)$ is an isomorphism for every $U_0$-module. Choose a free presentation $U_0^{(I)}\to U_0^{(J)}\to V\to0$. The functor $\Delta_0$ is right exact, $\Gamma$ is exact by Theorem 2.2, and $\Gamma$ commutes with sums by the finite Čech calculation. For a free module the unit is the identity, since $\Gamma(\mathcal D)=U_0$. Taking cokernels of the two identical free presentations proves the assertion for $V$.

The counit $\Delta_0\Gamma(M)\to M$ is surjective by global generation. Its map on global sections is an isomorphism, by the unit just proved and the adjunction identity. Exactness gives $\Gamma(K)=0$ for its kernel. The detection statement in Theorem 2.2 gives $K=0$. Thus both adjunction maps are isomorphisms.

If $V$ is finitely generated, its localization is a quotient of a finite sum of $\mathcal D$, hence coherent. Conversely a coherent $M$ is locally D-generated by finitely many sections. Global generation and quasi-compactness allow finitely many global sections to replace them. The resulting surjection $\mathcal D^s\to M$ stays surjective under $\Gamma$, so $\Gamma(M)$ is finitely generated over $U_0$. $\square$

3. Positive twists and the singular kernel

The same elementary proof gives more than the untwisted case.

Proposition 3.1. For integer $n\geq0$, $\Gamma_n$ and $\Delta_n$ are inverse equivalences between quasi-coherent $\mathcal D^{(n)}$-modules and $U_n$-modules. For $n=-1$, $\Gamma_{-1}$ is exact, but its kernel is \[ \ker\Gamma_{-1} =\{\mathcal O(-1)\otimes_k V:V\in\operatorname{Vect}_k\}, \tag{3.1} \] with the canonical twisted D-action. In particular it is not an equivalence at the singular twist.

Proof. Integral Morita equivalence writes a twisted module uniquely as $M=\mathcal O(n)\otimes N$, with $N$ an ordinary D-module. By Theorem 2.2 there is an $\mathcal O$-linear surjection $\mathcal O^{(I)}\to N$. Tensoring gives a surjection $\mathcal O(n)^{(I)}\to M$. For $n\geq-1$, $H^1(\mathcal O(n))=0$ and $H^2$ of its kernel is zero. Hence $H^1(M)=0$, proving exactness throughout this range. For $n\geq0$, $\mathcal O(n)$ is generated by sections, so $M$ is too. The free-presentation and counit argument of Theorem 2.3 applies verbatim with $\mathcal D^{(n)}$ and (1.1).

For the kernel at $n=-1$, suppose $\Gamma(N(-1))=0$. Since $N$ is $\mathcal O$-generated by global sections, it is a filtered union of coherent images $E$ of finite sums of $\mathcal O$. Each $E$ is globally generated and $H^0(E(-1))=0$. Its torsion subsheaf must vanish: any nonzero sheaf on finitely many points has a nonzero global section after every line twist, contradicting this equality. Thus $E$ is a vector bundle.

For any point $p$, the exact sequence $0\to E(-p)\to E\to E|_p\to0$ shows that $\Gamma(E)\to E|_p$ is injective. It is also surjective because $E$ is globally generated. Hence the evaluation map $\Gamma(E)\otimes\mathcal O\to E$ is an isomorphism on every fiber, and $E$ is a trivial bundle. Maps between such bundles are constant matrices. Taking the filtered union gives $N\simeq\mathcal O\otimes_k V$.

Its D-action is the ordinary one. Indeed the difference from ordinary differentiation on $1\otimes V$ is a map into global sections of $\Omega^1_{\mathbb P^1}\otimes_k V$, which are zero since $\Omega^1=\mathcal O(-2)$; this remains true for arbitrary $V$. Conversely $\mathcal O(-1)\otimes V$ has no sections. This proves (3.1) for all quasi-coherent modules, not only coherent ones. $\square$

The Casimir at $n=-1$ is $-1$, since $n(n+2)=-1$. The nonzero twisted line $\mathcal O(-1)$ has both $H^0=0$ and $H^1=0$, so even derived global sections kills it. Exactness is therefore insufficient for localization.

The unit $V\to\Gamma_{-1}\Delta_{-1}V$ is still an isomorphism, by the free-presentation proof. Applying $\Gamma_{-1}$ to the counit shows that both its kernel and cokernel lie in (3.1). Thus after quotienting the abelian twisted category by this kernel, the counit becomes an isomorphism and gives an equivalence with $U_{-1}$-modules. This is the precise quotient statement at the singular parameter.

At $n=-2$ even exactness fails: the twisted module $\mathcal O(-2)$ has $H^1=k$. The two parameters $0$ and $-2$ have the same Casimir scalar, but their global-section functors behave differently. A central character alone does not specify a favorable localization parameter.

4. The general flag theorem and its normalization

Let $G$ be simply connected semisimple over $\mathbb C$, $X=G/B$, and $\rho$ half the sum of the positive roots. To state the twist unambiguously, use the homogeneous line bundles $\mathcal L(\eta)$ of Miličić's Localization and Representation Theory of Reductive Lie Groups, Chapter 2, where they are written $\mathcal O(\eta)$: for $\mathrm{SL}_2$, $\mathcal L(m\rho)=\mathcal O(-m)$. Our geometrically positive parameter $\lambda$ specifies operators on $\mathcal L(-\lambda)$ in integral cases. In Miličić's notation this is \[ \mathcal D^{(\lambda)}=\mathcal D^{\rm Mil}_{-\lambda-\rho}. \tag{4.1} \] For arbitrary complex $\lambda$ the TDO is defined by the analogous Cartan-torsor scalar reduction. The Harish-Chandra parameter of its central character, in Miličić's conventions, is $-\lambda-\rho$, considered modulo the Weyl group. Denote this character by $\chi_\lambda^{\rm geom}$. At $\lambda=0$ it is the character of the trivial representation.

Theorem 4.1 (Beilinson-Bernstein; statement). For every such parameter, \[ \Gamma(X,\mathcal D^{(\lambda)}) \simeq U(\mathfrak g)/ U(\mathfrak g)\ker\chi_\lambda^{\rm geom}. \tag{4.2} \] If for every positive root $\alpha$, \[ \langle\lambda+\rho,\alpha^\vee\rangle\notin\mathbb Z_{<0}, \tag{4.3} \] global sections is exact on quasi-coherent $\mathcal D^{(\lambda)}$-modules. If also none of these pairings is zero, it is an equivalence with the modules over (4.2), inverse to localization. It restricts to coherent modules and finitely generated representations.

For integral dominant $\lambda$, all pairings in (4.3) are positive integers and the equivalence applies. In particular the ordinary flag variety is D-affine. The usual expression “dominant regular” refers here to the shifted parameter $\lambda+\rho$ and these conditions. For $\mathrm{SL}_2$, it reduces to the calculation $\nu=n+1$ already fixed.

Miličić's Localization and Representation Theory of Reductive Lie Groups proves this theorem for arbitrary complex parameters: Chapter 2 computes the global sections of the twisted operator rings and proves exactness of global sections, and Chapter 3, §1, proves the equivalence and its restriction to coherent modules and finitely generated representations. Its antidominance condition on $-\lambda-\rho$, that no pairing with a positive coroot is a positive integer, is (4.3), and its regular antidominant case is the equivalence. The theorem is due to Beilinson and Bernstein, Localisation de $\mathfrak g$-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), 15–18. This lesson proves Theorems 2.2–2.3 and Proposition 3.1 on $\mathbb P^1$; it does not prove the general theorem.

5. Two orbits and two highest-weight simples

Choose the Borel whose Lie algebra is spanned by $e,h$. Its action on the $x$ chart has open orbit $\mathbb A^1$ and closed orbit $\{\infty\}$. In the action convention of the preceding lesson, the group is chosen so that its induced action on functions differentiates to the displayed $e,h$.

Localization and global sections respect this strong equivariance. One direction follows from the balanced tensor construction of the preceding lesson. For the other, global sections inherit the rational Borel action, and its derivative agrees with the $\mathfrak b$-action. Thus (2.7) restricts to finitely generated $U_0$-modules whose $\mathfrak b$-action integrates rationally.

For the principal integral parameters, these are the modules in category $\mathcal O$ on which the Casimir acts exactly as zero. Rational Borel action implies integral semisimple $h$ and locally nilpotent $e$; conversely these properties integrate on each finite $\mathfrak b$-span, compatibly, to the Borel. Finite generation gives the usual category $\mathcal O$ conditions. This is the exact-central-character part of the principal block. A block defined using generalized central character can contain modules with nonzero nilpotent action of $\mathsf C$ and is larger. Such modules require a thickened central-character or monodromic twisting formulation, rather than the single ring $U_0$.

For a concrete example of that difference, take basis vectors $w_m,z_m$, $m\geq0$, put $w_{-1}=z_{-1}=0$, and define \[ \begin{aligned} &fw_m=w_{m+1},\quad hw_m=-2m w_m,\quad ew_m=m(1-m)w_{m-1},\\ &fz_m=z_{m+1},\quad hz_m=-2(m+1)z_m,\quad ez_m=w_m-m(m+1)z_{m-1}. \end{aligned} \] The three Lie relations hold directly; for example $(ef-fe)z_m=-2(m+1)z_m$. This module is generated by $w_0,z_0$, its weight spaces are finite dimensional, and $e$ is locally nilpotent. It belongs to the generalized principal block, but $\mathsf Cw_m=0$ and $\mathsf Cz_m=4w_{m+1}\ne0$. Thus $\mathsf C^2=0$ without $\mathsf C=0$, and it is not a module over $U_0$.

The two simple strongly equivariant coherent D-modules are \[ \mathcal O_{\mathbb P^1},\qquad \delta_\infty. \tag{5.1} \] To verify completeness on this curve, choose a finite set of D-generators and its finite-dimensional Borel orbit span. A good filtration generated from that span makes the orbit vector fields preserve filtration degree, so their principal symbols annihilate the associated graded module. Its characteristic variety lies in the union of the two orbit conormals, of dimension one; it is holonomic. Its simple factors are minimal extensions of irreducible connections on an orbit, by Preservation of holonomicity and minimal extensions. The open orbit is homogeneous with connected stabilizer; equivariant descent there reduces its irreducible connection to the trivial rank-one one. At the closed orbit, connected stabilizer likewise permits only the trivial irreducible strong representation. Minimal extension of the former is $\mathcal O_{\mathbb P^1}$, while Kashiwara's equivalence gives $\delta_\infty$ for the latter.

Their sections are $L(0)=k$ and $L(-2)=M(-2)$. Indeed on the $y$ chart $\delta_\infty$ has basis $\partial_y^m\delta$, $m\geq0$, with $y\delta=0$. The global operators give \[ h\,\partial_y^m\delta=-2(m+1)\partial_y^m\delta,\quad f\,\partial_y^m\delta=\partial_y^{m+1}\delta,\quad e\,\partial_y^m\delta=-m(m+1)\partial_y^{m-1}\delta. \tag{5.2} \] The last expression is zero at $m=0$. These are the Verma formulas for highest weight $-2$, which is irreducible. Thus the two orbit simples match the two highest-weight simples of the principal block.

Standard and costandard modules

Let $j:\mathbb A^1\hookrightarrow\mathbb P^1$. Direct images and duality give nonsplit sequences \[ 0\to\mathcal O_{\mathbb P^1}\to j_*\mathcal O_{\mathbb A^1} \to\delta_\infty\to0, \qquad 0\to\delta_\infty\to j_!\mathcal O_{\mathbb A^1} \to\mathcal O_{\mathbb P^1}\to0. \tag{5.3} \] Their local forms are the Laurent star and shriek extensions calculated in Holonomic D-modules and duality. The second is the holonomic dual of the first; the first nonsplitting follows because a section killed by $y$ cannot be a nonzero Laurent function. These are strong Borel-equivariant modules.

For the star extension, global sections are $k[x]$, with \[ e x^m=m x^{m-1},\quad h x^m=-2m x^m,\quad f x^m=-m x^{m+1}. \tag{5.4} \] This is the restricted dual Verma module $M(0)^\vee$, where the duality uses the anti-involution $e\leftrightarrow f$, $h\mapsto h$. To check rather than merely name it, let $v_m=f^m v_0$ be the Verma basis. Its actions are $e v_m=m(1-m)v_{m-1}$ and $h v_m=-2m v_m$. On the dual basis $v_m^*$, $e v_m^*=v_{m-1}^*$ and $f v_m^*=-m(m+1)v_{m+1}^*$. The map $v_m^*\mapsto x^m/m!$ identifies these with (5.4). The constants form its submodule $L(0)$ and the quotient is $L(-2)$.

For the shriek extension, a generator $u$ on the $y$ chart satisfies $y\partial_y u=0$ and restricts to $1$ on the $x$ chart. It is therefore a global section $v_0$. Its global descendants satisfy \[ v_m=f^m v_0,\quad h v_m=-2m v_m,\quad e v_m=m(1-m)v_{m-1}. \tag{5.5} \] Here $f v_0=\partial_yu$ is a nonzero point-supported section. For $m\geq1$ all descendants lie in $\delta_\infty$ and span it. Exactness of $\Gamma$ applied to (5.3) shows that these are all global sections, one in each displayed weight. Thus \[ \Gamma(j_!\mathcal O_{\mathbb A^1})=M(0),\qquad \Delta_0(M(0))=j_!\mathcal O_{\mathbb A^1}. \tag{5.6} \] This fixes the standard/costandard convention by actual operators.

The general Kazhdan-Lusztig character theorem is a further statement. With Weyl group $W$ and its Kazhdan-Lusztig polynomials, \[ [L(-w\rho-\rho)] =\sum_{y\leq w}(-1)^{\ell(w)-\ell(y)}P_{y,w}(1) [M(-y\rho-\rho)]. \tag{5.7} \] Kazhdan and Lusztig formulated it in this normalization in Representations of Coxeter groups and Hecke algebras, Conjecture 1.5, and Beilinson–Bernstein and Brylinski–Kashiwara proved it. Localization turns the simples into intersection-cohomology objects; Riemann-Hilbert and the intersection-cohomology calculation on Schubert varieties yield the coefficients. That last calculation, due to Kazhdan and Lusztig, is an additional geometric input, not a formal consequence of localization alone; Springer's Bourbaki exposé Quelques applications de la cohomologie d'intersection explains it together with the proof of the multiplicity formula. For $\mathfrak{sl}_2$, $P_{e,s}=1$, so (5.7) says $[L(0)]=[M(0)]-[M(-2)]$, exactly (5.3)–(5.6).

6. What survives in the affine setting

There are two different comparisons to keep apart.

Beilinson–Drinfeld, Theorem 7.15.6, treats the affine flag ind-scheme $G((t))/I$ and a twist $\chi$. In its conventions $\chi$ is antidominant when its affine Verma module is irreducible; it is regular when $\chi+\rho$ has trivial affine Weyl stabilizer. Under these conditions global sections is an equivalence with discrete right modules over the topological global TDO ring. Theorem 7.15.8 gives higher-cohomology vanishing and nonzero-section detection. The definition in §7.15.5 explicitly excludes its critical central value $c=-1/2$. Moreover §7.15.7 distinguishes this global-ring category from the category of all affine Lie algebra modules. Thus this theorem cannot be read as critical-level localization onto all Kac-Moody modules.

At critical level, use instead Frenkel–Gaitsgory, Localization of $\widehat{\mathfrak g}$-modules on the affine Grassmannian. The Grassmannian is $G((t))/G[[t]]$. Its critical D-modules carry convolution by $\operatorname{Rep}(\check G)$, and global sections have the regular quotient of the Feigin-Frenkel center. Ordinary global sections is not full. The paper's subsection The statement of the main result, parts (1)–(2), proves an exact equivalence after imposing both the regular-center Hecke eigen-object structure and strong $I^0$-equivariance, where $I^0$ is the pro-unipotent radical of the Iwahori. The target is critical modules with the regular-center action and algebraic $I^0$-action. The unrestricted eigen-object equivalence is presented there as a conjecture; it is not what that theorem proves. These are statements with precise source hypotheses, not extensions of our finite-dimensional proof.

The global construction used in Proof of the geometric Langlands conjecture II: Kac-Moody localization and the FLE is another local-to-global functor. Section 10 constructs localization from the factorization family of Kac-Moody categories to twisted D-modules on $\mathrm{Bun}_G$. Section 10.3 obtains adjunctions on quasi-compact open substacks and assembles them by a limit. This is not a literal single adjunction between the two displayed global categories: the auxiliary global-section functor is defined on the paper's “co” category. At critical level its twist is the half-twist discussed in the preceding lesson. The introduction explains its Hecke eigen-property. We cite this construction and its compatibility, without asserting that all critical modules and all D-modules on $\mathrm{Bun}_G$ are equivalent.

7. Exercises and solutions

Exercise 16.1 (easy). Verify the $\mathfrak{sl}_2$ relations for $\partial_x,-2x\partial_x,-x^2\partial_x$.

Solution. For vector fields $a\partial,b\partial$, the bracket is $(ab'-ba')\partial$. The three coefficient pairs give $2$, $2x^2$ and $-2x$, respectively. These are $2e,-2f,h$, so all three relations hold. For first-order operators the formula is $[a\partial+b,c\partial+d]=(ac'-ca')\partial+ad'-cb'$. Substitution of the actual coefficients in Section 1 also verifies every twisted relation.

Exercise 16.2 (easy). Compute the Casimir in the global operator ring and on a highest-weight vector.

Solution. Substitute $h=-2x\partial+n$, $f=-x^2\partial+nx$, $e=\partial$ and use $\partial x=x\partial+1$. The order-two and order-one terms in $h^2+2h+4fe$ cancel, leaving $n(n+2)$. On a vector $v$ with $ev=0$, $hv=\mu v$, the same Casimir acts as $\mu(\mu+2)v$. Thus the untwisted scalar zero allows highest weights $0,-2$, and the singular scalar $-1$ has the repeated highest-weight solution $\mu=-1$.

Exercise 16.3 (medium). Find the D-module whose global sections are the Verma module $M(0)$.

Solution. It is $j_!\mathcal O_{\mathbb A^1}$ for the open Borel orbit. The global generator and all three operator actions are computed in (5.5). The nonzero vector $fv_0$ generates the point submodule $M(-2)$, and quotienting leaves the trivial representation. Every weight space has dimension one, so the resulting surjective map from $M(0)$ is injective as well. The localization equivalence proves uniqueness. The star extension gives the dual Verma, as checked in (5.4), and is a different answer.

Exercise 16.4 (medium). Prove $H^1(\mathbb P^1,M)=0$ for every quasi-coherent ordinary D-module, without assuming finite generation.

Solution. The jet evaluation map (2.3) trivializes $F_rP_r$. Positive symbol quotients in (2.4) then show $H^1(P_r)=0$. Resolve $M$ surjectively by any sum of these induced modules, as in (2.6). Cohomology commutes with the sum on the finite affine Čech cover, so the source has zero $H^1$. The kernel has zero $H^2$ because the cover has two opens. The long exact sequence gives $H^1(M)=0$. This is the full argument for arbitrary quasi-coherent modules; reducing to finite-rank connections would not prove the assertion.

Exercise 16.5 (hard). Locate the singular twist, show localization is not an equivalence there, and identify the entire kernel of global sections.

Solution. The shifted parameter is $n+1$, so the singular twist is $\mathcal D^{(-1)}$. It kills the nonzero module $\mathcal O(-1)$, which has no cohomology. Proposition 3.1 proves the full kernel is exactly all $\mathcal O(-1)\otimes V$, by untwisting, using globally generated coherent images and their vanishing sections after a negative twist, and then ruling out a nontrivial connection by $H^0(\Omega^1)=0$. Coherent objects in the kernel have finite-dimensional $V$. Global sections is nevertheless exact and gives the quotient category described after (3.1). The ordinary untwisted parameter $n=0$ is regular and its localization equivalence remains valid.

What this lesson does not prove

We use quasi-coherent generation by negative line bundles, filtered unions of coherent subsheaves on a noetherian scheme, and the two-chart computation of line-bundle cohomology as algebraic geometry prerequisites. Basic Verma-module and restricted-dual constructions belong to the category $\mathcal O$ prerequisite.

The general flag-variety theorem is stated here and proved in Miličić's Localization and Representation Theory of Reductive Lie Groups, Chapters 2–3. The general Kazhdan-Lusztig theorem and its intersection-cohomology input are stated with the sources named in Section 5. We do not prove the infinite-dimensional constructions, the affine flag theorem, the critical Grassmannian equivalence, or the localization and Hecke constructions of the geometric Langlands series. Their exact scopes and locators are given in Section 6. The complete projective-line D-affinity, equivalence, Verma dictionary and singular kernel are proved here.

References

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