D-Modules on Flag Varieties and Localization of G-Modules

D-modules on ﬂag varieties and localization of g-modules.

Jos´eSimental. October 7, 2013

1 (Twisted) Diﬀerential operators and D-modules. 1.1 Diﬀerential Operators.

Let X be an algebraic variety. In this text, we assume all algebraic varieties are over C. Let R be a sheaf of rings on X. For R-modules F and G, deﬁne the sheaf HomR(F, G) as follows. For any open set U ⊂ X, the sections of HomR(F, G) on U are HomR|U (F|U , G|U ). For any algebraic variety X, let CX be the constant sheaf of complex numbers C.

Now assume that X is a smooth complex algebraic variety. Let OX be its sheaf of regular functions and VX be its sheaf of vector fields, that is, VX := DerC(OX ) := {θ ∈ EndCX (OX ): θ(fg) = fθ(g) + θ(f)g, f, g ∈ OX }. The sheaf DX of differential operators on X, is, by definition, the subsheaf (of associative rings) of EndCX (OX ) generated by OX and VX .

Theorem 1.1 In the setup of the previous paragraph, if dim X = n, then for every point p ∈ X there exists an aﬃne open neighborhood U of p, x1, . . . , xn ∈ OX (U) and ∂1, . . . , ∂n ∈ VX (U) such that [∂i, ∂j] = 0, ∂i(xj) = δij and VX (U) is free over OX (U) with basis ∂1, . . . , ∂n.

Proof. Let mp be the maximal ideal of OX,p. Since X is smooth of dimension n, there exist functions x1, . . . , xn generating mp. Then, dx1, . . . , dxn is a basis of ΩX,p, where ΩX is the cotangent sheaf of X. Take ∂1, . . . , ∂n ∈ VX to be the dual basis of dx1, . . . , dxn.

Deﬁnition 1.2 If x1, . . . , xn, ∂1, . . . , ∂n are as in Theorem 1.1, then we say that {xi} is a local coordinate system. Note that, since dxi are linearly independent, a local coordinate system determines a basis ∂1, . . . , ∂n of VX (U). Thus, sometimes we also refer to {xi, ∂i} as a local coordinate system.

Remark 1.3 A local coordinate system does not give an isomorphism from U to an aﬃne subvariety of Cn. We only have an ´etale morphism U → Cn.

Theorem 1.4 Let U be an aﬃne open subset of X and {xi, ∂i} a local coordinate system on U. Then, any diﬀerential operator of order ≤ k on U can be uniquely written in the form

X k1 kn fk1,...,kn ∂1 ··· ∂n , k1+···+kn≤k where fk1,...,kn ∈ OX (U).

Note that DX can be described by generators and relations as follows: it is generated by OX and VX with relations:

• f1 · f2 = f1f2, • f · ξ = fξ,

• ξ · f = ξ(f) + fξ,

• ξ1 · ξ2 − ξ2 · ξ1 = [ξ1, ξ2], where f, f1, f2 ∈ OX and ξ, ξ1, ξ2 ∈ VX .

1 1.2 A filtration on DX . ≤n Let A be a commutative ring and let M,N be A-modules. Define A-modules DiffA (M,N) ⊆ HomZ(M,N) inductively by:

≤0 1. DiffA (M,N) := HomA(M,N). ≤n+1 ≤n 2. DiffA := {θ ∈ HomZ(M,N):[f, θ] ∈ DiffA (M,N) for everyf ∈ A}. S ≤n Set DiffA(M,N) = n DiffA (M,N). n Remark 1.5 If, moreover, A is a k-algebra, then we add the extra condition that every map inside DiffA(M,N) is k-linear.

Definition 1.6 Let M, N be OX modules. Define DX (M, N ) by gluing DiffOX (M, N ) on affine open subsets, that is, for any affine open subset U ⊆ X,

Γ(U, DX (M, N )) := DiﬀO(U)(M(U), N (U)).

Note that DX = DX (OX , OX ). It follows that DX is a ﬁltered sheaf. In local coordinates (U, {xi, ∂i}), Fj DX (U) = P α1 α2 αn P OU ∂1 ∂2 . . . ∂n , αm ≤ j, that is, Fj DX (U) is given by diﬀerential operators of order ≤ j. Note that if P ∈ Fl DX , Q ∈ Fm DX , then [P,Q] ∈ Fl+m−1 DX . Then, gr DX is a sheaf of commutative algebras. Let us ¯ look closer at gr DX . Take a local coordinate system (U, {xi, ∂i}). Set ξi = ∂i ∈ F1 DX (U)/F0 DX (U). Then, gr DX (U) = OU [ξ1, ξ2, . . . , ξn].

∗ ∗ Let π : T X → X be the projection. Regard ξ1, . . . , ξn as the local coordinate system on the cotangent space Tx X. Then, OU [ξ1, . . . , ξn] is identiﬁed with π∗OT ∗X (U). These identiﬁcations are natural and they can be glued together, so that we have an isomorphism,

∼ gr DX = π∗OT ∗X .

A natural isomorphism can be constructed as follows. First of all, note that F0 DX = OX . This follows easily by the description of DX in local coordinates. Alternatively, we have a monomorphism OX ,→ F0 DX . An inverse map is given by F0 DX 3 P 7→ P (1). Now we see that F1 DX = OX ⊕ VX . Note that, if P ∈ F1 DX , then f 7→ [P, f] is a derivation on OX . Then, we get a morphism F1 DX → OX ⊕ VX , P 7→ (P (1), [P, •]). This is an isomorphism.

So we have isomorphisms OX = gr0 DX , VX = gr1 DX . These give rise to a map π∗OT ∗X → gr DX . By the above considerations in local coordinates, this is an isomorphism.

1.3 Twisted diﬀerential operators.

Deﬁnition 1.7 Let D be a sheaf of rings on X that admits an inclusion ι : OX ,→ D. We say that D is a sheaf ι of twisted diﬀerential operators (TDO) if the embedding OX ,→ D is locally isomorphic to the standard embedding ιX OX ,→ DX = DX (OX , OX ).

L For any line bundle L on X, DX = DX (L, L) is a TDO, as L is locally isomorphic to OX .

1 S One can also get a sheaf of TDO from a closed 1-cocycle α ∈ Ωcl as follows. Consider an open cover X = Ui and 1 a 1-cocycle α = (αij) ∈ Ωcl. Then, D(Ui) := DX (Ui), and the transition function from Uj to Ui maps a vector ﬁeld ξ to ξ + hξ, αiji. In fact, by this procedure we can get all TDO.

1 Proposition 1.8 Let ϕ : DX → DX be an endomorphism such that ϕ|OX = id. Then, there exists ω ∈ Ωcl such that ϕ(θ) = θ − ω(θ) for any vector ﬁeld θ ∈ VX . Moreover, ϕ is completely determined by ω and it is an automorphism of DX .

Proof. Let f ∈ OX , θ ∈ VX . Then, [ϕ(θ), f] = ϕ([θ, f]) = ϕ(θ(f)) = θ(f). Then,[ϕ(θ), f](1) = θ(f)(1), so ϕ(θ)(f) = θ(f) + fϕ(θ)(1). Set ω(θ) = −ϕ(θ)(1). This is a 1-form. Note that ω([θ, η]) = −ϕ([θ, η])(1) = −[ϕ(θ), ϕ(η)](1) = ϕ(θ)(ω(η)) − ϕ(η)(ω(θ)) = θ(ω(η)) − η(ω(θ)). It follows that ω is closed. The last statement of the Proposition is clear.

The following is then an exercise in Cechˇ cohomology.

2 1 1 Proposition 1.9 Twisted differential operators on a smooth variety X are classified by the first cohomology HZar(X, Ωcl).

To prove Proposition 1.9, one uses a covering Ui of X by open aﬃne subsets such that the sheaf D is locally isomorphic to DX in each Ui, and the transition morphisms in each intersection Ui ∩ Uj.

L Remark 1.10 If L is a line bundle, then we know that DX is a TDO. The Picard group is naturally isomorphic 1 ∗ ∗ ∗ 1 to H (X, OX ), where OX is the subsheaf of invertible elements in OX . There exists a homomorphism OX → Ωcl −1 p p ∗ given by taking the logarithmic derivative, f 7→ dlog(f) = f df, which induces morphisms H (dlog): H (X, OX ) → p 1 L 1 H (X, Ωcl). It is an exercise to show that the 1-cocycle corresponding to DX is H (dlog)(L).

Note that it follows that any TDO D is ﬁltered, and moreover, gr D = π∗OT ∗X .

1.4 Homogeneous TDO. In this subsection, we assume that X is a homogeneous G-variety, where G is a semisimple algebraic group. Later, we will specialize the results of this subsection to the case X = G/B, where B is a Borel subgroup of G. Let g := Lie(G). Recall that the universal enveloping algebra of g is U(g) = T g /(u ⊗ v − v ⊗ u − [u, v], u, v ∈ g).

Diﬀerentiating the action of G on OX , we get a G-equivariant Lie algebra homomorphism τ : g → Γ(X, VX ), that can be extented to a G-equivariant algebra homomorphism τ : U(g) → Γ(X, DX ). More generally, we make the following deﬁnition.

Deﬁnition 1.11 let D be a TDO on X with an algebraic action γ of G on D and a G-equivariant algebra homomorphism α : U(g) → Γ(X, D) satisfying the following conditions: 1. The multiplication in D is G-equivariant.

2. For ξ ∈ g, we have an equality ξX = [α(ξ), •], where ξX is the the derivation induced by diﬀerentiating the G-action on D. In this case, we say that α is a quantum comoment map and we call (D, γ, α) a homogeneous sheaf of twisted diﬀerential operators (HTDO).

For example, (DX , γX , τ) is an HTDO, where γX is the natural action of G on DX . Our next goal is to classify HTDO. To do this, we study (DX , γX , τ) more closely.

◦ ◦ Consider the trivial bundle g ×X X, and its sheaf of sections g := OX ⊗C g. We can deﬁne a bracket on g by

[f ⊗ ξ, g ⊗ η] := fτ(ξ)g ⊗ η − gτ(η)f ⊗ ξ + fg ⊗ [ξ, η]. ◦ This makes g a sheaf of Lie algebras. It is also an OX -bimodule. Note, however, that the Lie bracket is not ◦ ◦ ◦ OX -bilinear. We can extend τ : g → Γ(X, VX ) to τ : g → VX by τ (f ⊗ ξ) = fτ(ξ). Note that this satisﬁes the following identity:

[f ⊗ ξ, h(g ⊗ η)] = τ ◦(f ⊗ ξ)(h)(g ⊗ η) + h[f ⊗ ξ, g ⊗ η], (1) where f, g, h ∈ OX , ξ, η ∈ g.

◦ ◦ Let U be the sheaf of algebras generated by g and OX subject to the following relations:

(a) ι(fg) = ι(f) · ι(g). (b) ι([a, b]) = ι(a) · ι(b) − ι(b) · ι(a). (c) ι(fa) = ι(f) · ι(a). (d) ι(a) · ι(f) − ι(f) · ι(a) = ι(τ ◦(a)(f)).

◦ Where f, g ∈ OX , a, b ∈ g .

◦ ◦ ◦ Remark 1.12 In the formalism of Lie algebroids, g is a Lie algebroid with τ : g → VX the anchor map (this follows from Equation (1)), and U ◦ is the universal enveloping algebra of g◦.

3 ◦ ◦ ◦ ◦ Note that τ induces a map τ : U := U(g ) → DX . This map is an epimorphism. This follows from the fact that DX is generated by VX and OX , and the following Proposition.

◦ ◦ Proposition 1.13 The morphism τ : g → VX is an epimorphism.

◦ ◦ Proof. Since g and VX are locally free, it suﬃces to show that the induced map on the geometric ﬁbers of g and VX is surjective. But this is clear.

◦ ◦ ◦ So DX = U / Ker(τ ). We now give a description of Ker(τ ). Of course, ﬁrst we need an explicit characterization of U ◦.

◦ Proposition 1.14 As a sheaf, U = OX ⊗C U(g). The product is given by

(f ⊗ ξ)(g ⊗ η) = fτ(ξ)g ⊗ η + fg ⊗ ξη.

Where f, g ∈ OX , ξ ∈ g, η ∈ U(g).

◦ Proof. Follows from the relations deﬁning U .

◦ ◦ Note that it also follows that U is a filtered algebra, by setting FpU := OX ⊗C FpU(g), where FpU(g) is the ◦ ◦ ◦ ◦ ◦ standard filtration on U(g). Note that τ is a filtered morphism, that F0U = OX , F1U = OX ⊕ g and that U is ◦ generated by F1U as a sheaf of algebras.

◦ ◦ ◦ ◦ P P Let b = Ker(τ : g → VX ), so that b consists of those sections fi ⊗ ξi such that fiτ(ξi) = 0 ∈ VX . ◦ ◦ ◦ P ◦ Note that J0 := b U is a sheaf of two-sided ideals in U : it is easy to see that, if fi ⊗ ξi ∈ b and η ∈ g, then P ◦ [1 ⊗ η, fi ⊗ ξi] ∈ b . Similarly, if g ∈ OX , then hX i X fi ⊗ ξi, g ⊗ 1 = fiτ(ξi)g ⊗ 1 = 0.

Alternatively, one can see that J0 is a sheaf of two-sided ideals from the following Proposition.

◦ ◦ Proposition 1.15 Ker(τ : U → DX ) = J0.

◦ Proof. It is clear that J0 is contained in Ker(τ). Moreover, for any x ∈ X, the geometric fiber Tx(J0) = bx U(g) is ◦ the kernel of the induced map from the geometric fiber Tx(U ) = U(g) to the geometric fiber of DX at x. The result follows.

◦ Then, we have that DX = U /J0.

Now we show that any HTDO admits a similar description. Let (D, γ, α) be an HTDO. Then, for every ξ ∈ g and f ∈ OX,[α(ξ), f] = τ(ξ)f. It follows that α : g → F1 D. Note that we can also extend α to an algebra homomorphism ◦ ◦ ◦ ◦ ◦ ◦ ◦ α : U → D, f ⊗ ξ 7→ fα(ξ). Moreover, α is ﬁltered and gr α = gr τ , so α (b ) ⊆ F0 D = OX. Then, an HTDO ◦ determines a G-equivariant morphism from the G-homogeneous OX-module b to OX.

◦ ∗ ∗ Fix a point x0 ∈ X. Let B0 = StabX (x0), and let b0 be the ﬁber of b at x0. Then, B0 acts on b0. Let I(b0) ∗ be the space of B0-invariants. There is a natural linear isomorphism between I(b0) and the space of G-equivariant ◦ morphisms σ of b to OX. This follows from the correspondence between G-equivariant vector bundles on X and ∗ representations of B0, see e.g. [3, Section 9.11]. Then, for each λ ∈ I(b0), let σλ be the associated G-equivariant ◦ ◦ ◦ morphism σλ : b → OX. Let ϕλ : b → U be given by ϕλ(s) = s − σλ(s). Let Jλ be the sheaf of two-sided ideals of ◦ ◦ U generated by the image of ϕλ. Finally, set DX,λ := U /Jλ.

∗ Theorem 1.16 DX,λ is an HTDO. Moreover, the map λ 7→ DX,λ is an isomorphism between I(b0) and the set of isoclasses of HTDO on X.

For a proof of Theorem 1.16 see, for example, [5, Section 1.2].

4 1.5 DX -modules.

By DX -module, we mean a left module over the sheaf DX of diﬀerential operators. Clearly, every DX -module is also an OX -module. On the other hand, given an OX -module M, giving a DX -module structure to M is equivalent to giving a C-linear morphism ∇ : VX → EndC(M), θ 7→ ∇θ satisfying the following conditions:

(1) ∇fθ(s) = f∇θ(s).

(2) ∇θ(fs) = θ(f)s + f∇θ(s).

(3) ∇[θ1,θ2](s) = [∇θ1 , ∇θ2 ](s).

For f ∈ OX , s ∈ M, θ, θ1, θ2 ∈ VX .

Note that if V is a vector bundle, then ∇ deﬁnes a connection on V, and condition (3) amounts to saying that this connection is ﬂat.

A DX -module is called quasi-coherent if it is quasi-coherent as an OX -module. Denote by Modqc(DX ) the category of quasi-coherent DX -modules.

An algebraic variety X is said to be DX -affine if the global sections functor Γ : Modqc(DX ) → Mod(Γ(X, DX )), M 7→ Γ(X, M) is exact and if every module in Modqc(DX ) is generated by its global sections. Clearly, every affine algebraic variety is DX -affine. We’ll see later that, for a semisimple algebraic group G and a Borel subgroup B, the corresponding flag variety G/B is DG/B-affine.

2 D-modules on G/B. 2.1 Universal enveloping algebras and the Harish-Chandra isomorphism. Let g be a semisimple Lie algebra, and let h ⊆ g be a Cartan subalgebra. Let Φ be the root system of g relative to h, and + + ∨ Φ a choice of positive roots. Also, let {α1, . . . , αl} ⊆ Φ be a choice of simple roots. For each i = 1, . . . , l, let αi ∈ h ∗ ∨ ∨ be the coroot of αi, and let πi ∈ h be the fundamental weight dual to the coroot αi , that is hπi, αj i = δij. Denote by + P 1 P Pl Q = ZΦ the root lattice, Q its positive part, and P = Zπi the weight lattice. Finally, let ρ = 2 α∈Φ+ α = i=1 πi be the Weyl vector.

In this subsection, we want to study the center z = Z(U(g)) of the universal enveloping algebra of g. Since, in particular, z commutes with h, any element z ∈ z acts as a scalar on any highest weight module M = U(g)vλ with ∗ highest weight λ ∈ h . Since every such module is a quotient of the Verma module ∆λ, this scalar only depends on λ. ∗ Then, for any z ∈ z, we get a function Ξz : h → C. L We show that Ξz is polynomial. Indeed, it follows by the PBW theorem that U(g) = U(h) (n− U(g) + U(g) n+). Since h is abelian, U(h) = S h, the symmetric algebra. Consider then the projection pr : U(g) → U(h). It follows that, for any highest weight module M = U(g)vλ, and any u ∈ U(g), uvλ = pr(u)(λ)vλ+ terms of lower weight. Then, ∗ Ξz(λ) = pr(z)(λ), so Ξz is indeed polynomial. We get a map Ξ : z → U(h) = C[h ], which is clearly an algebra morphism.

Remark 2.1 Even though the restriction of pr : U(g) → U(h) to z is an algebra homomorphism, the map pr is not.

∗ Now consider the automorphism t−ρ of C[h ], t−ρ : f(λ) 7→ f(λ−ρ), and let HC : z → S(h) be HC = t−ρΞ, so that any z ∈ z acts on any highest weight representation with highest weight λ by HC(z)(λ + ρ). The reason for twisting the morphism Ξ is that HC has its image in S(h)W . Indeed, it is known that, for any integral dominant weight λ and any w ∈ W , there exists a nonzero morphism ∆w∗λ → ∆λ, where w ∗ λ = w(λ + ρ) − ρ is called the ρ-shifted action of W on h. It follows that Ξz(λ) = Ξz(w ∗ λ), or, equivalently, that HC(z)(λ) = HC(z)(wλ). Since the lattice of integral dominant weights is dense in Zariski topology, it follows that HC(z) ∈ S(h)W .

Theorem 2.2 The morphism HC : z → S(h)W is an isomorphism. It is called the Harish-Chandra isomorphism.

A strategy to prove Theorem 2.2 is to compare HC with the Chevalley isomorphism. Recall that this is an isomorphism res : S(g)G → S(h)W that is given by restriction of a polynomial map in g∗ to h∗. On the other hand, we have an isomorphism of g-modules sym : S(g) → U(g), given by

5 1 X sym : x . . . x 7→ x x . . . x . 1 n n! s(1) s(2) s(n) s∈Sn This map induces an isomorphism of vector spaces sym : S(g)G → z whenever G is connected.

We have the following diagram,

S(g)G I sym | II res || II || II || II | HC $ z ~| / S(h)W . This diagram is not commutative. However, it is commutative ‘up to lower degree terms’, as follows. Recall that each of the vector spaces z,S(g)G and S(h)W are ﬁltered. Then, we have that, for any p ∈ (Sn(g))G, HC(sym(p)) ≡ res(p) mod Sn−1(h)W . Note that it suﬃces to show that pr(sym(p)) ≡ res(p) mod Sn−1(h)W . In fact, a more general statement holds.

Lemma 2.3 For any p ∈ Sn(g), pr(sym(p)) ≡ res(p) mod Sn−1(h).

Q kα Q ni Q mα n Proof. Let p = ( α∈Φ+ fα )( i hi )( α∈Φ+ eα ) ∈ S(g) . Then, ( Q hni , if m , k = 0, pr(sym(p)) ≡ i i α α mod Sn−1(h). 0 otherwise

The same is true for res(p). Then, the result follows.

∗ An algebra homomorphism z → C is called a central character. For any λ ∈ h , deﬁne a central character χλ : z → C by deﬁning χλ(z) = HC(z)(λ). Theorem 2.2 has the following easy consequence.

∗ Corollary 2.4 Any central character is of the form χλ for some λ ∈ h . Moreover, χλ = χµ if and only if λ and µ lie in the same W -orbit.

2.2 Equivariant vector bundles on the flag variety. Let T ⊆ B be a maximal torus with Lie(T ) = h, and N be the unipotent radical of B so that B = NT . Note that the fiber at B ∈ B of any G-equivariant vector bundle is a representation of B. Conversely, given a representation U of B, consider the trivial vector bundle U ×G on G. This descends to a G-equivariant vector bundle on B whose fiber at B is precisely U. In other words, G-equivariant vector bundles of B are in one-to-one correspondence with representations of B. In particular, G-equivariant line bundles on B are in one-to-one correspondence with 1-dimensional representations of B. These, in turn, are in one-to-one correspondence with characters of T , as N acts trivially on any 1-dimensional ∗ representation of B. Then, for any character λ ∈ P = Hom(T, C×) ⊆ h , we get a G-equivariant line bundle L(λ). Note that, since L(λ) is G-equivariant, the sheaf of differential operators DL(λ) is an HTDO. OB Theorem 2.5 (Borel-Weil-Bott) Let λ ∈ P . Then, we have,

(1) If λ is antidominant, then the line bundle L(λ) is generated by its global sections. (2) The line bundle L(λ) is ample if and only if λ is antidominant and regular. (3) If λ − ρ is not regular, then Hi(B, L(λ)) = 0 for i ≥ 0. (4) If λ − ρ is regular, then there exists w ∈ W such that w · λ := w(λ − ρ) + ρ is antidominant, and

( L−(w · λ) if i = l(w), Hi(X, L(λ)) = 0 otherwise.

Where L−(w · λ) is the irreducible module with lowest weight w · λ.

Note that it follows that if λ is antidominant, then Γ(X, L(λ)) = L−(λ), and Hi(X, L(λ)) = 0 for i > 0.

6 2.3 HTDO on the ﬂag variety. Let G be a semisimple algebraic group with Lie algebra g, and let B be a Borel subgroup, with B = G/B the corresponding ﬂag variety. Then, B is an homogeneous G-variety, so we can apply the results of Subsection 1.4 to B.

◦ Recall the sheaf g = OB ⊗C g, which is the sheaf of sections of the trivial bundle B × g. Inside this bundle, we have the homogeneous bundle of Borel subalgebras, F, whose ﬁber at a point x ∈ B is the Borel subalgebra bx corresponding to x. Let b◦ be the sheaf of sections of this bundle. Also, we have the homogeneous bundle whose ﬁber at each point ◦ ◦ ◦ ◦ x is [bx, bx]. Let n be the sheaf of sections of this bundle. Clearly, we have n ,→ b ,→ g .

◦ ◦ ◦ Recall the epimorphism τ : g → VB. Its kernel is precisely b . Pick a basepoint x0 = B ∈ B. Then, we have ∗ ◦ ◦ b0 = b. An element λ ∈ h determines a B-invariant function b → C, and therefore a G-equivariant map λ : b → OB. ◦ ◦ ◦ ◦ Let Jλ be the ideal in U generated by elements of the form ξ − (λ + ρ) (ξ) for ξ ∈ b , and let Dλ := U /Jλ. The following is then a consequence of Theorem 1.16.

Proposition 2.6 Dλ is an HTDO.

We remark that D−ρ = DB, and that for an integral weight λ, Dλ is the sheaf of diﬀerential operators on the line bundle L(λ + ρ).

By deﬁnition of an HTDO, we have a G-equivariant algebra homomorphism Ψλ : U(g) → Γ(B, Dλ). Recall the central character χλ : z → C. Let Jλ be the ideal of U(g) generated by Ker(χλ) ⊆ z. Let Uλ := U(g)/Jλ. ∗ Lemma 2.7 For any λ ∈ h , the morphism Ψλ : U(g) → Γ(B, Dλ) factors through Uλ.

Proof. Since the map Ψλ : U(g) → Γ(B, Dλ) is G-equivariant, it is enough to show that the induced map on ﬁbers P maps Ker(χλ) to 0. Note that the ﬁbers of Dλ have the form U(g)/ x∈b(x − hλ + ρ, xi)U(g). Then, for z ∈ z, by the PBW theorem, z ∈ nU(g) + f for a unique f ∈ U(h), so that χλ(z) = f(λ + ρ). It follows that if f(λ + ρ) = 0, then P z ∈ x∈b(x − hλ + ρ, xi)U(g).

Abusing notation, we denote by Ψλ the induced morphism Ψλ : Uλ → Γ(B, Dλ).

∗ Theorem 2.8 (Beilinson-Bernstein) For any λ ∈ h , the morphism Ψλ : Uλ → Γ(B, Dλ) is an isomorphism.

The key step to prove Theorem 2.8 is to prove its quasiclassical version, that is, to prove that its associated graded morphism is an isomorphism. In fact, we are going to relate this isomorphism to the Springer resolution.

By the PBW theorem, we know that gr U(g) = S(g). We canonically identify g =∼ g∗ via the Killing form, so gr U(g) =∼ C[g]. The next Lemma characterizes the ideal of the nilpotent cone N in S(g). G G T L Lemma 2.9 The ideal in S(g) deﬁning N is generated by S(g)+ = S(g) ( p>0 S(g)p).

G W G W Proof. Recall the Chevalley isomorphism, res : S(g) → S(h) . This maps the ideal S(g)+ to S(h)+ , which is the G ∗ ideal of 0 in h /W . Then, the ideal generated by S(g)+ deﬁnes the elements x ∈ g such that Gx ∩ h = 0, that is, elements whose Jordan decomposition doesn’t have a semisimple part. This is precisely N . By results of Kostant, N is a normal variety and a complete intersection in g. The result now follows.

G Note that, by lifting elements of S(g)+ to U(g), it follows that we have an epimorphism C[N ] gr Uλ. We then have the following commutative diagram.

S(g) / gr Γ(B, Dλ) F 9 FF rr FF rrr FF rr F# rrr gr Uλ O

C[N ] ∗ ∗ ∗ Recall the Springer resolution γ : T B → N , and its pullback γ : C[N ] → Γ(T B, OT*B). Since γ is a resolution of singularities and N is normal, γ∗ is actually an isomorphism. Moreover, the following diagram commutes.

7 S(g) / gr Γ(B, Dλ) F 8 FF qqq FF qq FF qqq F" qqq gr Uλ Γ(B, gr Dλ) O

∗ γ ∗ C[N ] / Γ(T B, OT*B)

Since gr Γ(B, Dλ) → Γ(B, gr Dλ) is injective, it follows that C[N ] gr Uλ, gr Uλ → gr Γ(B, Dλ) and gr Γ(B, Dλ) → Γ(B, gr Dλ) are all isomorphisms. Then, gr Ψλ is an isomorphism. Since all the algebras we’re working with are posi- tively graded, it follows that Ψλ is an isomorphism.

We have two induced functors. The ﬁrst functor is the global sections functor:

Modqc(Dλ) → Mod -U(g) M 7→ Γ(B, M).

Note that Γ(B, M) = HomOB (OB, M). Also, note that HomOB (OB, M) = HomDλ (Dλ, M): if M has a Dλ-module structure, then any OB-homomorphism OB → M admits a unique extension to a Dλ-linear homomorphism Dλ → M. The next functor is the localization functor:

Mod -U(g) → Modqc(Dλ) M 7→ Dλ ⊗U(g)M.

Note that, even if U(g) is an algebra and Dλ a sheaf, the localization functor makes sense. Indeed, it follows by Theorem 2.8 that, for every open set U, Dλ(U) is an U(g) module. Note that the global sections functor is right adjoint to the localization functor. Our next goal is to study these functors.

2.4 Abelian Beilinson-Bernstein theorem. The goal of this subsection is to state and prove two fundamental theorems of Beilinson-Bernstein on the cohomology of OB-coherent Dλ-modules. The ﬁrst one, Theorem 2.11, concerns the vanishing of the higher cohomology of modules. The second Theorem 2.15, tells us when every OB-coherent Dλ-module is generated by its global sections. A strategy to do this is to realize every OB-coherent submodule of such a module as a direct summand in a sheaf without higher cohomology. To do this, we will use the Borel-Weil-Bott theorem, which tells us that the sheaf L(λ) is ample whenever λ ∈ P is antidominant and regular.

Assume that µ ∈ P is antidominant. By the Borel-Weil-Bott theorem, Theorem 2.5 (1), L(µ) is generated by its global sections. We know that the global sections of L(µ) are L−(µ), the simple module with lowest weight − µ. Then, we have pµ : OB ⊗CL (µ) L(µ). Taking the dual of pµ, we get a morphism HomOB (L(µ), OB) → − + OB ⊗C HomC(L (µ), C). This is injective. Rewriting, we have an injective morphism L(−µ) → OB ⊗CL (−µ). If we tensor with the locally free module L(µ), we get an injective morphism,

+ iµ : OB → L(µ) ⊗C L (−µ).

Tensoring with an OB-coherent Dλ module M we get,

+ iµ,M : M → M ⊗OB L(µ) ⊗C L (−µ).

Note that iµ,M is always injective because iµ locally splits. We want to show that, if λ is antidominant, iµ,M splits as morphism of sheaves of vector spaces. Note that this splitting will be constructed using diﬀerential operators, so it 1 is not a splitting of OB-modules. For example, take g = sl2, so B = P , and µ = −1. To produce a splitting of iµ,M as sheaf of vector spaces, we will realize the image of iµ,M as a generalized eigensheaf for the action of the center z of + U(g) on M ⊗OB L(µ) ⊗C L (−µ).

An essential ingredient will be the following construction. Let F be a ﬁnite dimensional g-module. Recall that F has a ﬁltration by b-submodules 0 = F0 ⊂ F1 ⊂ · · · ⊂ Fm = F , where dimFi = i, n Fi ⊆ Fi−1 and h acts on the 1- dimensional quotient Fi/Fi−1 by an integral character νi. The νi’s are just the weights of F . Consider the trivial vector

8 bundle B ×F B. Its sheaf of sections is F := OB ⊗CF . Note that B ×F has a ﬁltration 0 = U0 ⊂ U1 ⊂ · · · ⊂ Um, where

Ui := {(gB, v) ∈ B ×F : v ∈ g(Fi)}.

This deﬁnes a ﬁltration on F, 0 = F 0 ⊂ F 1 ⊂ · · · ⊂ F m = F. These are G-equivariant coherent sheaves on B. Recall that we have an equivalence between the category of G-equivariant coherent sheaves on B and the category of representations of the Borel subgroup B. Under this equivalence, F i corresponds to Fi. It then follows that F i / F i−1 = L(νi).

It follows that, more generally, for any quasi-coherent OB-module M, M ⊗C F has a ﬁltration with succesive ◦ quotients being M ⊗OB L(νi). Now assume that M is a Dλ-module. Then, M is a g = OB ⊗C g-module such that ◦ ◦ ◦ ◦ the subbundle of Borel subalgebras b acts with character (λ + ρ) . Similarly, b acts with character νi on L(νi). It ◦ ◦ ◦ follows that M ⊗OB L(νi) is a g -module and the b acts on it with character (λ + νi + ρ) . In other words, the action ◦ of U on M ⊗OB L(νi) factors through the quotient Dλ+νi . By Theorem 2.8, the center z of U(g) acts on M ⊗OB L(νi) with character χλ+νi . It follows that Y (z − χλ+νi (z)) i annihilates M ⊗CF for every z ∈ z. Then, the action of z on M ⊗CF is locally ﬁnite, and M ⊗CF decomposes into the direct sum of its generalized z-eigensheaves.

◦ ∗ For a U -module M and λ ∈ h , denote by M[λ] the generalized z-eigensheaf of M with eigencharacter χλ. Note that M[λ] = M[µ] whenever λ, µ belong to the same W -orbit.

∗ Lemma 2.10 Let λ ∈ h be antidominant. Then, for every OB-quasi-coherent Dλ-module M, and every antidominant ∼ + integral weight µ, iµ,M splits. In particular, M = [M ⊗OB L(µ) ⊗C L (−µ)][λ].

+ Proof. We know that the eigencharacters of M ⊗OB L(µ) ⊗C L (−µ) are of the form χλ+µ+νi where νi is a weight of + + L (−µ). Assume that χλ+µ+νi = χλ for some weight νi of L (−µ). Then, for some w ∈ W , w(λ) = λ + µ + νi, so (−µ−νi)+w(λ)−λ = 0. But λ is antidominant, so w(λ)−λ is positive, that is, it is a non-negative linear combination + of simple roots. Since L (−µ) is the irreducible module with highest weight −µ, −µ − νi is also positive. It follows that w(λ) = λ and µ = −νi. Then, the generalized eigensheaf with eigencharacter χλ is M ⊗OB L(µ)⊗OB L(−µ) = M. Theorem 2.11 (Beilinson-Bernstein) Let λ ∈ h∗ be antidominant. Then, Hi(B, M) = 0 for every quasi-coherent Dλ-module M and i > 0. In particular, the global sections functor Γ(B, •) : Modqc(Dλ) → Mod -Uλ is exact.

1 Proof. Let W be an OB-coherent submodule of M. By the Borel-Weil-Bott Theorem (Theorem 2.5 (2)) we can i i + ﬁnd an antidominant weight µ such that H (B, W ⊗OB L(µ)) = 0 for i > 0. Then, H (B, W ⊗OB L(µ)⊗C L (−µ)) = 0. Now consider the following commutative diagram:

Hi(B, W) / Hi(B, M) .

i + i + 0 = H (B, W ⊗OB L(µ) ⊗C L (−µ)) / H (B, M ⊗OB L(µ) ⊗C L (−µ)) i i + Since the diagram commutes and, by the previous lemma, H (B, M) → H (B, M ⊗OB L(µ)⊗C L (−µ)) is injective, i i we get that H (B, W) → H (B, M) is the zero map. Since M is the direct limit of its OB-coherent submodules and i cohomology commutes with direct limits, we get that H (B, M) = 0.

∗ Corollary 2.12 Let λ ∈ h be antidominant. Then, for every Uλ-module V , the natural map ϕV from V to Γ(B, OB ⊗U(g)V ) is an isomorphism of g-modules.

Proof. By the previous theorem, the global sections functor Γ is exact. Then, the functor Γ(B, OB ⊗U(g)•) is right ⊕I ⊕J exact. Now let V ∈ Mod -Uλ. There exists an exact sequence (Uλ) → (Uλ) → V → 0. Then, we get a commutative diagram,

1 Note that the argument I gave on the October 18 talk is incorrect: the morphism iµ,W does not necessarily split.

9 ⊕I ⊕J (Uλ) / (Uλ) / V / 0

⊕I ⊕J Γ(B, Dλ) / Γ(B, Dλ) / Γ(B, OB ⊗U(g)V ) / 0.

The ﬁrst two vertical maps are isomorphisms. Then, the third vertical map is also an isomorphism.

Denote by Qmodqc(Dλ) the quotient category of Modqc(Dλ) modulo the full subcategory formed by quasi-coherent Dλ-modules without global sections.

∗ Corollary 2.13 Let λ ∈ h be antidominant. Then, the localization functor induces an equivalence from Mod -Uλ to Qmodqc(Dλ).

Proof. Let M ∈ Qmodqc(Dλ). By adjointness, we have a natural morphism ψM : Dλ ⊗Uλ Γ(B, M) → M. Let K0 and K00 be the kernel and cokernel of this morphism, respectively. Then, we get an exact sequence 0 → K0 → 00 0 00 Dλ ⊗Uλ Γ(B, M) → M → K → 0. Applying the global sections functor we ﬁnd that Γ(B, K ) = 0, Γ(B, K ) = 0. The result follows.

Now we show another result due to Beilinson-Bernstein, that says that when λ ∈ h∗ is antidominant and regular, every quasi-coherent Dλ-module M is generated by its global sections. The strategy is similar to that of the proof of Theorem 2.11 but somewhat easier. Recall that for any integral antidominant weight µ we have a surjective morphism − pµ : OB ⊗CL (µ) L(µ). Note that this morphism locally splits. Then, for every quasi-coherent module M we get − an epimorphism pµ,M : M ⊗CL (µ) → M ⊗OB L(µ). The following is an analog of Lemma 2.10.

Lemma 2.14 Assume that λ is antidominant and regular. Then, for every quasi-coherent Dλ-module M, and every − antidominant integral weight µ, the epimorphism pµ,M splits. In fact, the generalized χλ+µ-eigensheaf of M ⊗CL (µ) is M ⊗OB L(µ).

− Proof. We argue similarly to Lemma 2.10. Assume there exists a weight νi of L (µ) and w ∈ W such that w(λ + νi) = λ + µ. Then, (w(λ) − λ) + (w(νi) − µ) = 0. Similarly to Lemma 2.10, it follows that νi = µ. The result follows. ∗ Theorem 2.15 (Beilinson-Bernstein) Let λ ∈ h be antidominant and regular. Then, for any quasi-coherent Dλ- module M, the morphism Dλ ⊗U(g)Γ(B, M) → M is surjective. In other words, every quasi-coherent Dλ-module is generated by its global sections.

Proof. Since λ is antidominant, Γ(B, •) is exact. Hence, it suﬃces to show that Γ(B, M) 6= 0 for M= 6 0. We can assume that M is coherent. By the Borel-Weil-Bott Theorem, we can ﬁnd a regular antidominant weight ν such that − Γ(B, M ⊗OB L(ν)) 6= 0. Since ν is regular, Lemma 2.14 implies that L (ν) ⊗ Γ(B, M) 6= 0. We’re done. Corollary 2.16 Let λ ∈ h∗ be antidominant and regular. Then, the global sections functor is an equivalence of categories Modqc(Dλ) → Mod -Uλ. Its inverse is the localization functor.

Proof. Follows from Corollary 2.13 and Theorem 2.15.

As an application of Theorems 2.11, 2.15, we show that the homological dimension of Uλ is ﬁnite whenever λ is regular. It is known that if this is not the case then the homological dimension of Uλ is inﬁnite.

0 ∗ Proposition 2.17 Let λ ∈ h be regular. Then the homological dimension of Uλ0 is ﬁnite.

0 0 Proof. Since Uλ0 = Uwλ for any w ∈ W , we can replace λ by an antidominant weight λ ∈ W λ . Note that, by hypothesis, λ is regular. Since Dλ is a TDO, the homological dimension of each stalk Dλ,x is finite, as this is a filtered algebra whose associated graded algebra has finite homological dimension. Moreover, the homological dimension hd Dλ,x ≤ dim B, so that these homological dimensions are uniformly bounded. It is known that, for any x ∈ B, i ∈ , we have Exti (M, W) = Exti (M , W ), for an O -coherent D -module M and a quasi-coherent D - Z>0 Dλ x Dλ,x x x B λ λ module W. Then, Exti (M, W) = 0 for i > dim B. Dλ

On the other hand, we have the Grothendieck spectral sequence Hp(B, Extq (M, W)) ⇒ Extp+q(M, W). It follows Dλ Dλ that Exti (M, W) = 0 for i > 2 dim B, where M is a coherent D -module and W is a quasi-coherent D -module. Dλ λ λ

10 Since we’re assuming λ is antidominant and regular, Exti (M,W ) = 0 for any ﬁnitely generated U -module M, any Uλ λ U -module W and any i > 2 dim B. Taking direct limits, it follows that Exti (M,W ) = 0 for any two U -modules λ Uλ λ M,W and any i > 2 dim B. Then, hd Uλ ≤ 2 dim B. ∗ Remark 2.18 If λ ∈ h is an integral regular weight, then actually hd Uλ = 2 dim B. For example, if λ = −ρ, then U = D(B), the algebra of (global) diﬀerential operators on B. By Corollary 2.16, Ext• ( , ) = Ext• (O, O) = λ U−ρ C C D(B) H•(B, C), where the last equality follows because B is smooth. Since B is an smooth, irreducible, projective variety, H2dimB(B, C) = C.

2.5 Derived Beilinson-Bernstein Theorem. ∗ Assume that λ ∈ h is regular. Then, Uλ has ﬁnite homological dimension, so the localization functor has a left derived L b b functor Dλ ⊗Uλ • : D (Mod -Uλ) → D (Modqc(Dλ)). Note that the global sections functor admits a right derived b b functor RΓ: D (Modqc(Dλ)) → D (Mod -Uλ).

∗ L Theorem 2.19 Let λ ∈ h be a regular integral weight. Then, Dλ ⊗Uλ • and RΓ are quasi-inverse equivalences of triangulated categories.

Remark 2.20 We remark that Theorem 2.19 is valid in a greater generality for λ ∈ h∗ regular but not necessarily integral.

The rest of this section is devoted to the proof of Theorem 2.19. Let P be a projective Uλ-module. Recall that ⊕|I| this means that P is a direct summand of a free module Uλ , for some set I so, in particular, P is ﬂat. Note that, by Theorem 2.8, the adjunction morphism P → Γ(B, Dλ ⊗Uλ P ) is an isomorphism. Also, it is clear that Dλ ⊗Uλ P is ⊕|I| ∗ a direct summand of Dλ . Note that OT ∗B has no higher cohomology: since T B is symplectic, the canonical bundle on T ∗B is trivial, and then the vanishing of cohomology is a consequence of the Grauert-Riemenschneider Vanishing ∗ ∗ Theorem applied to T B → N . Since gr Dλ = OT B, it follows that Dλ is Γ-acyclic. Hence, Dλ ⊗Uλ P is Γ-acyclic.

· b ∗ · Now, let V be a complex in D (Mod -Uλ), with λ ∈ h a regular weight. By Proposition 2.17, V is quasi- L L · · · · isomorphic to a complex P of projective Uλ-modules, and Dλ ⊗Uλ V = Dλ ⊗Uλ P . It follows that RΓ(Dλ ⊗Uλ V ) = · ∼ · ∼ · Γ(Dλ ⊗Uλ P ) = P = V . We have proved the following.

L ∗ b b Lemma 2.21 Let λ ∈ h be a regular weight. Then, RΓ(Dλ ⊗Uλ •): D (Mod -Uλ) → D (Mod -Uλ) is isomorphic to the identity functor.

Note that it follows that RΓ is a quotient functor of triangulated categories. Then, to prove Theorem 2.19, it suﬃces to show that RΓ(M·) = 0 only when M· = 0. We will follow a strategy that appears in [2, Section 3]. We will need the following result, due to Kontsevich (see e.g. [2, Theorem 3.5.1]):

Lemma 2.22 Let X ⊆ n be a smooth closed subscheme. Then, O (i), −n ≤ i ≤ 0 generate Db(coh X) under shifts, PC X cones, and direct summands.

Corollary 2.23 There exists a ﬁnite set of dominant weights S such that L(µ), µ ∈ S, generate Db(coh B) under shifts, cones, and direct summands.

We will also need a derived version of the splitting method used in the proof of Theorems 2.11, 2.15. Recall that, if M is a (sheaf of) module(s) on which the center z of U(g) acts locally finitely, then by [M][λ] we denote the generalized eigenspace (resp. generalized eigensheaf) with generalized eigencharacter χλ. For integral weights λ, µ with µ − λ µ µ + + dominant, define the translation functor Tλ : Mod -Uλ → Mod -Uµ by Tλ (M) = [L (µ − λ) ⊗ M][µ], where L (µ − λ) is the simple finite dimensional module with highest weight µ − λ.

· b + · Now, let M be in D (Modqc(Dλ)). Then, using the notation of the previous paragraph, L (µ − λ) ⊗ M is a complex of U ◦-modules. Moreover, by the construction before Lemma 2.10, z acts locally ﬁnitely on L+(µ − λ) ⊗ M·. µ · So one can deﬁne Tλ [M ] similarly as above. We have that translation functors commute with RΓ, that is,

µ · ∼ µ · Tλ [RΓλ M ] = RΓµ([Tλ M ]). (2) We’ll use the following Lemma, that is parallel to Lemmas 2.10, 2.14. The proof is also similar. To get the desired combinatorial relations between the weights, it uses [4, Lemma 7.7].

11 · b µ · Lemma 2.24 Assume that λ, µ are in the same chamber. Then, for M ∈ D (Modqc(Dλ)), Tλ (M ) = L(µ − λ) ⊗OB M·.

Finally, Theorem 2.19 follows from the next result.

· b · Lemma 2.25 Let λ be an integral and regular weight, and let M ∈ D (Modqc(Dλ)) be such that RΓ(M ) = 0. Then, M· = 0.

Proof. Let µ be a dominant weight such that λ, λ + µ are in the same chamber. It then follows from Equation λ+µ · · (2) that 0 = Tλ [RΓλ M ] = RΓλ+µ(L(µ) ⊗OB M ). By Corollary 2.23, it follows that for λ deep in its chamber, RΓ(F · ⊗ M·) = 0 for all F · ∈ Db(coh B). Then, M· = 0.

The case where λ is any integral regular weight follows again from (2) but, to pass from an integral regular weight λ to another (integral and regular) weight deep into the chamber of λ, we need to extend the definition of translation µ functors to allow the case when the difference µ − λ is not dominant. Here, define Tλ (M) := [L(µ − λ) ⊗ M]µ, where L(µ − λ) is a finite dimensional g-module with extremal weight µ − λ. Again, we can extend this functor to b D (Modqc(Dλ)), and Equation (2) is valid. Finally, Lemma 2.24 is also valid in this more general setting, with the · b µ · same proof. It follows that, for λ, µ in the same chamber and M ∈ D (Modqc(Dλ)), we have Tλ (M ) = 0 only when · M = 0.

References

[1] A. Beilinson, J. Bernstein, Localisation de g-modules. C. R. Acad. Sci. Paris Ser. I Math. 292 (1981) no.1, 15-18 [2] R. Bezrukavnikov, I. Mirkovi´c,D. Rumynin, Localization of modules for a semisimple Lie algebra in prime char- acteristic. Ann. Math. 167 (2008), 945-991.

[3] R. Hotta, K. Takeuchi, T. Tanisaki, D-modules, Perverse Sheaves and Representation Theory. Birkh¨auserProgress in Mathematics 236. [4] J. Jantzen, Representations of algebraic groups. Pure and Applied Math. 131, Academic Press.

[5] D. Miliˇci´c, Localization and Representation Theory of Reductive Lie Groups. Available at http://www.math. utah.edu/~milicic/Eprints/book.pdf.