Endomorphisms of Verma modules for rational Cherednik algebras

We study the endomorphism algebra of Verma modules for rational Cherednik algebras at t=0. It is shown that, in many cases, these endomorphism algebras are quotients of the centre of the rational Cherednik algebra. Geometrically, they define Lagrangian subvariaties of the generalized Calogero-Moser space. In the introduction, we motivate our results by describing them in the context of derived intersections of Lagrangians.


Introduction
1.1. The quest for a direct bridge between the geometric world of Fukaya categories and the world of (algebraic or analytic) microlocal sheaves on a symplectic manifold is ongoing, and is currently the subject of intense research. Though considerable progress has recently been made, for instance with Tamarkin's seminal work on quantizations of Lagrangians [31], it seems that concrete, computable examples of some expected correspondence are still desirable to aid one's (or at least the author's) intuition. One abode where such computable, though non-trivial, examples live is that of rational Cherednik algebras, beginning for instance with results of [28]. The corresponding symplectic manifold is the generalized Calegero-Moser space, or often times, more appropriately, a symplectic resolution of this space.
Though we have no idea what the appropriate definition of Fukaya category should be in this case, or its possible relation to microlocal sheaves on the generalized Calogero-Moser space X, we study in this article a shadow of such a hoped for relationship. For arguments sake, we take the objects of the Fukaya category, a 2-category, to be Lagrangian (or more generally coisotropic) subvarieties and the hom spaces given by derived intersections. Then, in the shadow of the Fukaya category, the derived intersections are replaced by the Tor and Ext groups between the structure sheaves of the Lagrangians subvarieties i.e. by the (co)homology of the derived intersections. On the microlocal side, we consider not quantized sheaves on the space X, but those sheaves of O Xmodules that are quantizable i.e. that admit some quantization. One major advantage of working on the generalized Calogero-Moser space, even though it is a singular symplectic variety, is that it is equipped with a canonical quantization by virtue of the fact that it is the centre of the rational Cherednik algebra.
Our original motivation for the current work was quite different, see section 1.5.
1.2. Quantizable modules. Associated to each complex reflection group (h, W ), are the rational Cherednik algebras, a family of algebras depending on a pair of parameter (t, c), where t ∈ C.
When t = 0, the algebra H c is a finite module over its centre Z c . The algebra Z c is the coordinate ring of a symplectic variety X c , and the geometry of this symplectic variety is intimately entwined with the representation theory of H c . When t = 0, the rational Cherednik algebra is strongly non-commutative and provides a canonical quantization of the symplectic variety X c . For fixed c, one can also consider the C[t]-algebra H t,c , which is flat over t.
A H c -module M is said to be quantizable if it can be extended to a H t,c -module, flat over C [t].
In fact, for what follows, the existence of an extension to the 3rd infinitesimal neighborhood of zero in Spec C[t] suffices. Gabber's Theorem implies that the support Supp M ⊂ X c of a quantizable module M is coisotropic i.e. defined by the vanishing of an involutive ideal.
Let M, N be quantizable left H c -modules and M ′ a quantizable right H c -module. Then, by the construction described in section 3.1 of [1], the graded vector spaces Ext q Hc (M, N ) and Tor Hc q (M ′ , N ) carry a canonical differential, the virtual de Rham differential, making each into a complex. The cohomology of Ext q Hc (M, N ), resp. Tor Hc q (M ′ , N ), is commonly called the virtual de Rham cohomology of the pair (M, N ), resp. the virtual de Rham homology of (M ′ , N ). In the context of symplectic geometry, virtual de Rham (co)homology plays a (conjectural) role in counting intersection numbers of Lagrangian intersections, see [2]. With regards to the possible relationship between virtual de Rham (co)homology and microlocal sheaves, see also [29,Remark 7.7].
A pair of (left or right) quantizable modules M and N are said to have smooth intersection if Supp M ∩ Supp N is contained in the smooth locus of X c . In this case, results of [1], together with some basic Mortia theory imply: (2) The virtual de Rham differential on Tor q Hc (M ′ , N ) makes it into a BV(=Batalin-Vilkovisky)algebra so that Ext q Hc (M, N ) becomes a BV-module over Tor q Hc (M ′ , N ), again via the virtual de Rham differential.
The proof of Proposition 1.1 is explained in section 6. The terms used in the statement of the proposition are defined there too. We refer the reader to the appendix for the definition of BV-algebra. (1) The Lagrangian module ∆(p, λ Ω ) is simple.
(2) The canonical map Z c → End Hc (∆(p, λ Ω )) is surjective. When the support of ∆(p, λ Ω ) is contained in the smooth locus of X c , one can show more.
Namely, E a,Ω := End Hc (∆(p, λ Ω )) is a polynomial ring and ∆(p, λ) is a free E a,Ω -module of rank |W | (here a is the image of p in h * /W ). In general the Verma modules ∆(p, λ) are not simple if λ is not the canonical representative in its Calogero-Moser partition, and its endomorphism ring is non-commutative.
As a consequence of Theorem 1.2 and Proposition 1.1, we deduce 1.4. Self-intersections. In particular, one can consider the derived self-intersections of the modules ∆(p, λ Ω ). We assume for the remainder of this section that Ω = {λ Ω }. Then the endomorphism ring E a,Ω is a quotient Z c /I of Z c . Moreover, it is the coordinate ring of a smooth Lagrangian Λ a,Ω ≃ A n . This means that N ∨ a,Ω := I/I 2 is a free E a,Ω -module. It is the module of sections of the conormal bundle of Λ a,Ω in X c . Its dual N a,Ω := (I/I 2 ) ∨ is the module of sections of the normal bundle of Λ a,Ω in X c . The following is an application of the theory developed in [1].
We outline the proof of Corollaries 1.3 and 1.4 in section 6. The appendix contains a summary of the main results of [1] that are required in the article. The reader can also find the definition of Gerstenhaber algebra there. Corollary 1.4 implies that the virtual de Rham cohomology H q vir (∆(p, λ Ω ), ∆(p, λ Ω )) equals the usual de Rham cohomology of the conormal bundle of Λ a,Ω in X c ; this is just C in degree zero since the space is contractible. Similarly, the virtual de Rham homology H vir q ((p, λ * Ω )∆, ∆(p, λ Ω )) is, up to a degree shift, also equal to the de Rham cohomology of the conormal bundle. See Corollary 8.2 for details.
1.5. This paper was motivated by the close relationship, via the limit t → 0 of Suzuki's functor [32], between the rational Cherednik algebra associated to the symmetric group S n and modules for the affine Lie algebra gl m,κ at the critical level κ = −m. If λ is a partition of n with at most m parts then it is easy to see that this functor sends the Weyl module V(λ) to the Verma module ∆(λ). The endomorphism ring End gl (V(λ)) is commutative and can be the identified with the ring of functions on a certain moduli space of GL m -opers on the formal disc. Moreover, the centre of gl m,κ surjects on to End gl (V(λ)), see [19,Theorem 9.6.1]. This is a perfect analogue of our Theorem 1.2. By the result of Frenkel and Teleman [20], this identification, at least when λ = 0, extends to an identification of Ext q (V(0), V(0)) with the space of differential forms on the moduli space; a result completely analogous to our Corollary 1.4. Their (much more sophisticated) arguments also rely crucially on the fact that the vacuum module V(0) can be quantized i.e. it exists for all levels κ.

Acknowledgments
The author is supported by the EPSRC grant EP-H028153.

2.
Rational Cherednik algebras at t = 0 2.1. Definitions and notation. Let (W, h) be a complex reflection group, where h is the reflection representation for W , and let S(W ) be the set of all complex reflections in W . For each s ∈ S(W ), choose vectors α s ∈ h and α ∨ s ∈ h * that span the one dimensional spaces Im (s−1)| h and Im (s−1)| h * respectively. We normalize α s and α ∨ s so that α ∨ s (α s ) = 2. Let c : S(W ) → C be a W -equivariant function. The rational Cherednik algebra at t = 0, as introduced by Etingof and Ginzburg [18], and denoted H c , is the quotient of the skew group algebra of the tensor algebra T (h ⊕ h * ) ⋊ W by the ideal generated by the relations [x, for all x, x ′ ∈ h * and y, y ′ ∈ h. By the PBW property, there is an isomorphism of vector spaces The trivial idempotent in CW is denoted e. Whenever A is an algebra containing CW , E will denote the functor A-mod → eAe-mod given by multiplication by e. These inclusions define surjective morphisms π : X c → h * /W and ̟ : X c → h/W respectively.
Both π and ̟ are flat of relative dimension dim h. Write for the product morphism Υ := π × ̟. It is a finite, and hence closed, surjective morphism. By putting x ∈ h * in degree one, y ∈ h in degree −1 and each w ∈ W in degree zero, it is clear from the relations (1) that H c is a Z-graded algebra. This implies that Z c is also Z-graded. Thus, there subalgebra of Z c . Proposition 4.15 of [18] implies that Lemma 2.1. Let π −1 (a) denote the scheme-theoretic fiber of π over a ∈ h * /W . Then, The affine scheme π −1 (a) is neither reduced nor irreducible. The generalized Calogero-Moser space X c has a natural Poisson structure, see [18].

2.3.
H c -modules and fixed points of the C × -action. In this section we define Verma modules and recall some basic facts about the representation theory of rational Cherednik algebras. These can be found for instance in [5]. The stabilizer of p ∈ h, resp. q ∈ h * , under W is denoted W p , resp. W q .
Definition 2.2. The Verma module associated to p and λ ∈ Irr(W p ) is the induced module where the action of C[h * ] on λ is via evaluation at p.
baby Verma module associated to p, λ and b is defined to be The module ∆(0, λ, 0) is the baby Verma module studied in [24]. If L is a simple H c -module then dim L ≤ |W |, with equality if and only if the support of L, a closed point of X c , is contained in the smooth locus. As noted above, the map Υ is C × -equivariant. Since the image of 0 in h * /W ×h/W is the unique C × -fixed point of that space, the finitely many closed point of Υ −1 (0) are the precisely the C × -fixed points in X c . The simple H c -modules supported at each of these fixed points is a graded H c -module. These simple graded modules are naturally parameterized by the set Irr(W ).
Our conventions about graded modules will be that M [n] i = M i−n . The coinvariant ring is a graded copy of the regular representation, considered as a W -module. The fake polynomial associated to λ ∈ Irr(W ) is Define b λ to be the degree of the lowest non-zero monomial. It is called the b-invariant of λ. There is a natural bijection between CM c (W ) and the closed points of Υ −1 (0, 0).
Proof. By Müller's Theorem [12], the blocks of H b c are in bijection with the closed points of Υ c (b, 0). Therefore, by [6,Lemma 4.4.1], there exists at least one simple module L ∈ B i -mod such that . This implies that both L and L ′ occur as composition factors of ⊥ E(L). But This implies that L ≃ L ′ , confirming (1).
Since E is a quotient functor, it sends simple modules to simple modules. Therefore, since eH b c e is a commutative ring, dim eM is at most one-dimensional for any simple module M . This proves (2).
3.1. It has been shown in [11, Theorem 9.6.1] that each Ω ∈ CM c (W ) contains a unique represen- The following is Theorem 9.6.1 of loc. cit.
Thus, it is strongly reminiscent of Soergel's V-functor. Also, by choosing a linear character of the group W other that the trivial representation, the analogue of Theorem 3.2 still holds, but the distinguished element in each block will be different.

Let H +
c -grmod denote the category of finitely generated, Z-graded H + c -modules, and similarly for H 0 c -grmod, where in both cases homomorphisms are homogeneous of degree zero. Let , the inclusion H 0 c -grmod ֒→ H + c -grmod induces a bijection between isomorphism classes of simple graded modules.
H + c -pgr, denote the category of projective H + c -modules, resp. graded, projective H + c -modules. These are exact categories. Then we have the following result, which is presumably standard. (1) The abelian category H + c -grmod is Krull-Schmit. Proof. Since we were unable to find a suitable reference, we sketch the proof. As for the usual proof, e.g. [8,Section 1.4], of the Krull-Schmit property for Artinian rings, it suffices to show that the analogue of Fitting's Lemma holds in this setting. That is, given an indecomposable module M ∈ H + c -grmod and f ∈ End H + c -grmod (M ), we need to show that M = Ker (f n ) ⊕ Im(f n ) for some n ≫ 0. Reading the usual proof of Fitting's Lemma e.g. Lemma 1.4.4 of loc. cit., we just need to show that there is some n ≫ 0 such that Ker (f n+k ) = Ker (f n ) and Im(f n+k ) = Im(f n ) for all k. Let A be the degree zero part of H + c , a finite dimensional algebra. Since M is finitely generated, there exists  (1).
Therefore every idempotent (automatically homogeneous of degree zero) in H 0 c lifts to an idempotent in H + c . Let a ∈ H 0 c be a primitive idempotent, and n(λ) ∈ Z, such that (H 0 c a)[n(λ)] is the projective cover of L(λ) in H 0 c -grmod. We denote by the same letter, the lift of a to H + c . Then the claim is that (H + c a)[n(λ)] is the projective cover of L(λ) in H + c -grmod. It is certainly a graded projective module mapping surjectively onto L(λ). If it is not the projective cover, then it is decomposable. Each summand is free over C[h] W , hence defines, after applying (−) 0 , a non-zero direct summand of (H 0 c a)[n(λ)]. But this contradicts the indecomposability of the latter module. Parts (2) and (3) follow.
Since each P ∈ H + c -pmod is a direct summand of a direct sum of copies of H + c , it is a free C[h] Wmodule. This implies that (−) b is exact. The fact that it maps projectives to projective is just the . This proves part (3). Parts (2) and (3) imply the first part of (4). For the final statement about blocks, we note by part (3) that it suffices to show that Hom H + c (P + (λ), P + (µ)) = 0 if and only if Hom H 0 c (P (λ), P (µ)) = 0.
3.3. We use Lemma 3.4 to lift the statements of Lemma 3.2 to H + c -mod.

Endomorphism Algebras
(3) The graded character of E Ω is where d 1 , . . . , d n are the degrees of (W, h).
Proof. All the claims of the theorem follow more or less directly from Proposition 3.5. Let B be the block of H + c corresponding to Ω and P + (λ Ω ) the projective cover of L(λ Ω ) in H + c -grmod. Applying Hom H + c (P + (λ Ω ), −) to the exact sequence P + (λ Ω ) → ∆(λ Ω ) → 0 gives a surjection of eBe-modules, eBe → e∆(λ Ω ) → 0. Thus, e∆(λ Ω ) is a cyclic eBe-module. Since the action of Z c on e∆(λ Ω ) factors through eBe, it is also a cyclic Z c -module.
Recall that b Ω := b λ * Ω . As in the proof of [24,Theorem 5.6], an easy calculation shows that where H λ (q) is the hook polynomial of the partition λ.

4.2.
When Ω = {λ Ω } consists of a single element, one can say a great deal more about the endomorphism ring E Ω .  Let's show dim T = n. It must have dimension at least n. Since Spec(E Ω ) is a closed subvariety of X c , T is a subspace of T x Ω X c . The fact that x Ω is a C × -fixed point implies that the space T x Ω X c is a C × -module and T is a submodule. We decompose T x Ω X c = T − ⊕ T 0 ⊕ T + , where T − consists of all weight spaces of strictly negative weights etc. We have T ⊂ T + . Since x Ω is an isolated fixed point, it follows from [10, Corollary 2.2] that T 0 = 0. By assumption, x Ω is contained in the smooth locus of X c . Therefore, T x Ω X c is a symplectic vector space. The symplectic form on T x Ω X c is C × -invariant. This implies that dim T + = dim T − = n. Thus, T ⊂ T + implies that dim T ≤ n as required.
Proof. Since we have shown in Corollary 4.4 that E Ω is a polynomial ring, and ∆(λ Ω ) is a finitely generated E Ω -module, it suffices to show that ∆(λ Ω ) is projective. Let m ⊂ E Ω be a maximal ideal and consider the quotient L = ∆(λ Ω )/m · ∆(λ Ω ). This space is non-zero since any endomorphism φ ∈ E Ω whose image in E Ω /m · E Ω is non-zero induces a non-zero endomorphism of L. On the other hand, since e∆(λ Ω ) is a cyclic E Ω -module, the map is an isomorphism. In particular, dim eL = 1. By Lemma 4.3, the support of ∆(λ Ω ), and hence of L too, is contained in the smooth locus of X c . Therefore, L is simple and has dimension |W |.

We have shown in Theorem 4.1 that Z c surjects on to End Hc (∆(λ)).
Proposition 4.6. Assume that X c is smooth. Then, multiplication defines a graded isomorphism Proof. The supports of the modules ∆(λ) are also disjoint because they are precisely the attracting sets for the C × -action c.f. Lemma 4.3. Therefore End Hc λ∈Irr(W ) ∆(λ) ≃ λ∈Irr(W ) End Hc (∆(λ)).

Generalized Verma modules
In this section we extend Theorem 4.1 to Verma modules for which p = 0. This is done by showing that the endomorphism ring of ∆(p, λ) is isomorphic to the endomorphism ring of the H c ′ (W p )module ∆(λ), where c ′ denotes the restriction of c to the reflections in W p and H c ′ (W p ) denotes the rational Cherednik algebra associated to (h, W p , c ′ ). First, we recall a certain completion of H c (W ), as defined by Bezrukavnikov and Etingof [9]. Our presentation will be slightly different from loc.
cit., since it will be useful for the applications in [4] that it agrees with Wilson's factorization of the Calogero-Moser space when W is the symmetric group.
Let p ∈ h * and let a be the image of p in h * /W . We denote by m a for the maximal ideal of a local ring. For all k ≥ 0, the PBW property for rational Cherednik algebras implies that Hence, we have embeddings and taking the inductive limit, C[h * ] a ֒→ H c (W ) a , where we have used the fact that the functor of inverse limit is left exact. Therefore, we have e i ∈ H c (W ) a for all i and Let We write C[h * ] p i for the completion of C[h * ] with respect to the maximal ideal n p i in order to distinguish it from C[h * ] p i . Write also W i,j for the subset of W consisting of all elements w such that w·e i = e j . Before proving the main result of the section we require some preparatory lemmata.
Lemma 5.1. Multiplication defines a vector space isomorphism preserving map it follows that it is also an isomorphism.
The second claim now follows from the fact that C[h * ] p j e j w = w C[h * ] p k e k for all w ∈ W j,k and Proof. Let s ∈ W be a reflection. Recall the vectors α s ∈ h and α ∨ s ∈ h * as defined in (2.1). It is shown in section 3.5 of [24] that the functional α ∨ s can be extend to a C-linear operator on C[h * ] by setting This operator satisfies It is also shown that α ∨ s (f ) = 0 for all s ∈ S and f ∈ C[h * ] W . Therefore α ∨ s extends to an operator on C[h * ] p such that relation (4) holds for f ∈ C[h * ] p . Applying α ∨ s to e i and using the fact that e i is an idempotent gives Multiplying by e i and using the fact that C[h * ] p e i ≃ C[h * ] p i , which is a domain, we must have (1) we have an isomorphism of completed algebras we have an isomorphism of commutative algebras Proof. By Lemma 5.1 we can define a map and Corollary 5.4. Let p ∈ h * and a ∈ h * /W as above. Let a i be the image of p i in h * /W p i . Then, we have an isomorphism of schemes π −1 W (a) ≃ π −1 Wp i (a i ).
Proof. Since the isomorphism θ i of Proposition 5.3 maps the space n p i onto e i m a e i , the map φ i of It is proved in [3, Lemma 3.9] that Z( H c (W ) a ) is the completion of Z(H c (W )) with respect to the ideal generated by m a and, similarly, that Z( with respect to the ideal generated by n p i . Therefore, φ i induces an isomorphism of commutative algebras .
, then the algebra E a,Ω is a polynomial ring of dimension dim h.
Let w ∈ W , then ∆(p, λ) ≃ ∆(w(p), w(λ)), where w(λ) is the representation of W w(p) corresponding to λ under the isomorphism w : W p ∼ −→ W w(p) of conjugation. Therefore, if a is the image of p in h * /W , then we denote by Λ a,Ω the support of the Z c -module ∆(p, λ), thought of as a subscheme of X c . If Ω = {λ Ω }, then Λ a,Ω is a smooth Lagrangian subvariety of X c , isomorphic to A n , as a closed subscheme of X c . The varieties Λ a,Ω play a key role in [4].

5.
3. An equivalence of categories. Let H c -mod a,Ω denote the category of finitely generated H cmodules scheme-theoretically supported on Λ a,Ω i.e. those modules M such that I · M = 0, where I is the ideal defining Λ a,Ω . The category of coherent O Λ a,Ω -modules is denoted Coh(Λ a,Ω ). In this section we prove the following theorem: (1) The functor e is an equivalence if and only if X c is smooth.
Part (2): by Lemma 4.3, the assumption of part (2) implies that Λ a,Ω ⊆ X sm c . Since Λ a,Ω ∩X sing c = ∅, Hilbert's Nullstellensatz implies that I(Λ a,Ω ) + I(X sing c ) = Z c and we can find a characteristic function f ∈ Z c taking the value 1 at all points of Λ a,Ω and vanishing on X sing c . Replacing H c by its localization at f and Z c by its localization, we may assume that Z c is a regular affine algebra and H c an Azumaya algebra over Z c . We remark that part (1) still holds after localization. Recall that the centre Z(A) of an abelian category A is defined to be the ring of endomorphisms End A (id A ) of the identity functor. For a Noetherian k-algebra A, the centre of A-mod is canonically isomorphic to the centre of A. The equivalence e induces an isomorphism Z(H c -mod) is just the identity map. Let I = I(Λ a,Ω ). We can identify H c -mod a,

Lagrangian subvarieties.
It is shown in [25,Proposition 4.5] that X c is a symplectic variety, see [21] for the definition and properties of symplectic varieties. This implies that the smooth locus X sm c is a symplectic leaf in X c and hence its compliment has codimension at least two in X c .
The goal of this subsection is to prove the following proposition. It is a consequence of Gabber's Integrability Theorem [22]. Proof. It is clear from the definition of the Poisson bracket on Z c that the ideal generated by m a in Z c is involutive. However, it seems that this does not in general imply that J is involutive.
Therefore, we need to work a bit harder. Since we have not assumed any smoothness condition on X c , we are also unable to use results from previous sections. Let p 1 , . . . , p ℓ ∈ h * be the elements in the orbit a. Set Let Y = (Supp M ) red . Then, I claim that Y = V (J). Since m a · M = 0, we have Y ⊂ V (J). Let x ∈ V (J) and choose some simple H c -module L supported on x. As a C[h * ]-module, L = ⊕ ℓ i=1 L p i , where L p i is supported at p i . Without loss of generality, we may assume that L p 1 = 0. Let λ ⊂ L p 1 be an irreducible W p 1 -module in the socle of L p 1 . Then, there is a non-zero homomorphism ∆(p 1 , λ) → L. This implies that Hom Hc (M, L) = 0 and hence x ∈ Y . Now, let C[ǫ] be functions on the 3rd infinitesimal neighborhood of 0 in C, so that ǫ 3 = 0. It will be easier to work, via the Satake isomorphism, with the spherical subalgebra eH c e. The usual rational Cherednik algebra H t,c has an additional parameter t, which we have assume throughout is set to zero. Specializing instead to t = ǫ, we have a C[ǫ]-algebra eH ǫ,c e such that eH ǫ,c e/ǫeH ǫ,c e ≃ eH c e.
Then the Poisson structure on eH c e is constructed as in [22]. We can define ∆ ǫ (p, λ) in the obvious way. It is a H ǫ,c -module, free over C [ǫ]. This gives us a eH ǫ,c e-module eM ǫ , free over C [ǫ]. and, similarly, Ext and eN | X sm c are simple, they are quotients of Z c | X sm c . Therefore, they are naturally commutative algebras. Now the claims of Proposition 1.1 are consequences of the theory developed in [1]; in particular, Corollary 1.1.3. The assumptions of loc. cit. that eM | X sm c and eN | X sm c define smooth, closed subvarities of X sm c is unnecessary in our case since that assumption is only required in order to guarantee the existence of a third order quantization of the modules. But, in our case, the existence of the quantization is built into the definition of quantizable modules.
Fix p ∈ h * , Ω ∈ CM c ′ (W p ) and let a be the image of p in h * /W . Let I be the ideal defining the closed subscheme Λ a,Ω in X c . We assume Ω = {λ Ω }, so that Λ a,Ω is smooth. Then N ∨ a,Ω := I/I 2 is a free Z c /I-module. As noted in the introduction, it is the module of sections of the conormal bundle of Λ a,Ω in X c . Its dual N a,Ω := (I/I 2 ) ∨ is the module of sections of the normal bundle of Λ a,Ω in X c .
Proof of Corollary 1.4. Let λ * = Hom C (λ, C), an irreducible right W p -module. We denote by (p, λ * )∆ the right H c -module induced from the right C[h * ] ⋊ W p -module λ * . Standard arguments using the Kozsul resolution show that Ext n Hc (∆(p, λ), The results of Theorem 1.2 apply equal well to the right H c -module (p, λ * )∆. In order to be able to apply Proposition 1.1, the only thing left to check is that e∆(p, λ) and (p, λ * )∆e define the same smooth, closed subvariety of X c i.e. that they are isomorphic as Z c -modules. By completing H c at b ∈ h * /W , we may assume that p = 0. Then, Ext n Hc (∆(p, λ), H c ) ≃ (p, λ * )∆ implies that the support of (p, λ * )∆ is contained in the support of ∆(p, λ). In particular, it is contained in the smooth locus of X c . Thus, the "right block" of H 0 c containing λ * consists of a single element and Corollary 4.4 implies that the support of (p, λ * )∆ is isomorphic to A n . Thus, e∆(p, λ) ≃ (p, λ * )∆e as Z c -modules. The statement of the corollary now follows from Theorem 8.1 of the appendix.
Recall from Theorem 4.1 that the graded character of . . , d n are the degrees of (W, h). Since C[Λ 0,Ω ] is a polynomial ring, this implies that there exists 1 integers 0 < e 1 ≤ · · · ≤ e n such that Corollary 6.1. When Ω = {λ Ω }, we have, as bigraded vector spaces, and Proof. Let x be the C × -fixed point in X c over which L(λ Ω ) lives. As in the proof of Corollary 4.4, the integers e i are defined so that the graded character of T x Λ 0,Ω is given by n i=1 q −e i . We have T x X c = T x Λ 0,Ω ⊕ (T x Λ 0,Ω ) ⊥ with respect to the symplectic form on X c , and we may identify Then formula (7) follows from the fact that as graded vector spaces, together with the fact that if V is a graded vector space with character q m 1 + · · · + q m k then the bigraded character of ∧ q V is given by (1 + tq m 1 ) · · · (1 + tq m k ). The proof of formula (6) is similar.
We end the section by noting one other situation were one can easily calculate virtual deRham (co)homology groups. Let q ∈ h and µ ∈ Irr(W q ). We let the dual Verma module associated to Proof. Our assumptions on Ω imply that there is a regular sequence of homogeneous elements z = z 1 , . . . , z n such that Λ 0,Ω = V (z). Let Π Ω denote the support of ∇(λ Ω ). The analogue of Theorem 1.2 in this context implies that Π Ω is a smooth Lagrangian subvariety of X c , isomorphic to A n .
,Ω , O Π Ω , so it suffices to calculate the latter groups. It is explained in chapter 17 of [17] how one does this.
We use the notation of loc. cit. Using the fact that both Λ 0,Ω and Π Ω are smooth subvariety of X c intersecting transversely in a single point, we have We note that if Y is the connected component labeled by Ω, then one can study the generic fibers of π| : Y → h/W by considering the representation theory of H b c . Namely, statements (2) and (3) above would imply the following: (4) Assume that b is generic. Then the head of P + (λ Ω ) b equals ∆(0, λ Ω , b), a semi-simple module, with pairwise non-isomorphic simple summands.

Appendix: Batalin-Vilkoviski structures
In this appendix we summarize the main results of [1], as required in this article. We follow the presentation of loc. cit., which the reader is encouraged to consult for further details. All undecorated tensor products will mean tensor product over C.
Let D = i≥0 D i be a graded commutative algebra. If D is equipped with a differential δ : Let X be a smooth affine variety equipped with a symplectic two-form ω. Let Y be a smooth, coisotropic subvariety of X. Let N X/Y denote the sheaf of sections of the normal bundle of Y in X. It is a sheaf of O X -modules supported on Y . Its dual N ∨ X/Y is the sheaf of sections of the conormal bundle. Since the two-form ω is non-degenerate, it induces an isomorphism Γ(X, Ω 2 X ) ≃ Γ(X, ∧ 2 Θ X ). We let P be the image of ω under this isomorphism; it is a Poisson bivector. The graded commutative algebras ∧ • N ∨ X/Y and ∧ • N X/Y have a natural structure of BV algebra. Namely, the differential on ∧ • N ∨ X/Y is given by the formula δ = i P • d DR + d DR • i P , where d DR is the deRham differential of degree one and i P denotes contraction with P (and thus δ has degree −1). The differential on ∧ • N X/Y is given by the Schouten bracket [P, −]. That these operations are indeed well-defined follows from the fact that the ideal defining Y is involutive. Combining Corollary 1.
Moreover, the sheaf Ext admits a canonical structure of Gerstenhaber module over the Gerstenhaber algebra as Gerstenhaber modules, compatible in the obvious sense with the identification (8).
We denote by H vir