Harish-Chandra bimodules for quantized Slodowy slices

The Slodowy slice is an especially nice slice to a given nilpotent conjugacy class in a semisimple Lie algebra. Premet introduced noncommutative quantizaions of the Poisson algebra of polynomial functions on the Slodowy slice. In this paper, we define and study Harish-Chandra bimodules over Premet's algebras. We apply the technique of Harish-Chandra bimodules to prove a conjecture of Premet concerning primitive ideals, and to construct `noncommutative resolutions' of Slodowy slices via translation functors.

1. Geometry of Slodowy slices 1.1. Introduction. Let g be a semisimple Lie algebra, and Ug the universal enveloping algebra of g. For any nilpotent element e ∈ g, Slodowy used the Jacobson-Morozov theorem to construct a slice to the conjugacy class of e inside the nilpotent variety of g. This slice, S, has a natural structure of an affine algebraic Poisson variety.
More recently, Premet [P1] has defined, following earlier works by Kostant [Ko], Kawanaka [Ka], and Moeglin [Moe], for each character c of the center of Ug, a filtered associative algebra A c . The family of algebras A c may be thought of as a family of quantizations of the Slodowy slice S in the sense that, for any c, one has a natural Poisson algebra isomorphism gr A c ∼ = C [S]. For further developments see also [BK1], [BGK], [Lo1], [P2]- [P3].
The algebras A c have quite interesting representation theory which is similar, in a sense, to the representation theory of the Lie algebra g itself. It is well known that category O of Bernstein-Gelfand-Gelfand plays a key role in the representation theory of g. Unfortunately, there seems to be no reasonable analogue of category O for the algebra A c , apart from some special cases, cf. [BK1], [BGK].
In this paper, we propose to remedy the above mentioned difficulty by introducing a category of (weak) Harish-Chandra bimodules over the algebra A c . Our definition of weak Harish-Chandra bimodules actually makes sense for a wide class of algebras, cf. Definition 4.1.1. We show in particular that, in the case of enveloping algebras, a weak Harish-Chandra Ug-bimodule is a Harish-Chandra bimodule in the conventional sense (used in representation theory of semisimple Lie algebras for a long time) if and only if the corresponding D-module on the flag variety has regular singularities, cf. Proposition 6.6.1. Motivated by this result, we use 'micro-local' technique developed in §6 to introduce a notion of 'regular singularities' for A c -bimodules. 1 We then define Harish-Chandra A c -bimodules as weak Harish-Chandra bimodules with regular singularities.
Let I c be the maximal ideal of the center of the algebra Ug corresponding to a regular central character c. There is an associated block O c , of the category O of Bernstein-Gelfand-Gelfand, formed by the objects M ∈ O such that the I c -action on M is nilpotent. One may also consider an analogous category of Harish-Chandra Ug-bimodules. Specifically, one considers Harish-Chandra Ug-bimodules K such that the left I c -action on K is nilpotent and such that one has KI c = 0. It is known that the resulting category is, in fact, equivalent to the category O c , see [BG]. Thus, one may expect the category of Harish-Chandra A c -bimodules to be the right substitute for a category O(A c ) that may or may not exist.
For any algebra A, a basic example of a weak Harish-Chandra A-bimodule is the algebra A itself, viewed as the diagonal bimodule. Sub-bimodules of the diagonal bimodule are nothing but two-sided ideals of A. This shows that the theory of (weak) Harish-Chandra A-bimodules is well suited for studying ideals in A, primitive ideals, in particular.
We construct an analogue of the Whittaker functor from the category of Harish-Chandra Ug-bimodules to the category of Harish-Chandra A c -bimodules. Among our most important results are Theorem 4.1.4 and Theorem 4.2.2 which describe key properties of that functor.
We use the above results to provide an alternative proof of a conjecture of Premet that relates finite dimensional A c -modules to primitive ideals I ⊂ Ug such that the associated variety of I equals Ad G(e), the closure of the conjugacy class of the nilpotent e ∈ g. This conjecture was proved in a special case by Premet [P3], using reduction to positive characteristic, and in full generality by Losev [Lo1], using deformation quantization, and later also by Premet [P4]. Our approach is totally different from the aproaches used by Losev or Premet and is, in a way, more straightforward. Some results closely related to ours were also obtained by Losev [Lo2].
Finally, we introduce translation functors on representations of the algebra A c . In §6, we use those functors to construct 'noncommutative resolutions' of the Slodowy slice by means of a noncommutative Proj-construction. Similar construction has been successfully exploited earlier, in other situations, by Gordon and Stafford [GS], and by Boyarchenko [Bo].
A different approach to 'noncommutative resolutions' of Slodowy slices was also proposed by Losev in an unpublished manuscript.
Remark 1.1.1. We expect that, in the special case of subregular nilpotent elements, our constructuion of noncommutative resolution reduces to that of Boyarchenko. In more detail, Boyarchenko considered noncommutative resolutions of certain noncommutative algebras introduced by Crawley-Boevey and Holland [CBH]. These algebras are quantizations of the coordinate ring of a Kleinian singularity. By a well known result of Brieskorn and Slodowy [Sl], the Slodowy slice to the subregular nilpotent in a simply laced Lie algebra g is isomorphic, as an algebraic variety, to the Kleinian singularity associated with the Dynkin diagram of g. Furthermore, it is expected (although no written proof of this seems to be available, except for type A, see [Hod]) that the algebras A c are, in that case, isomorphic to the algebras constructed by Crawley-Boevey and Holland. Thus, our noncommutative resolutions should correspond, via the isomorphism, to those considered by Boyarchenko.
Boyarchenko for explaining to me some details of his work [Bo], and Iain Gordon for a careful reading of a preliminary draft of this paper. This work was supported in part by the NSF grant DMS-0601050. Remark 1.3.7. We recall that, for any Borel subalgebra b, (each irreducible component of) the set (b ⊥ ∩ O) red is known to be a Lagrangian subvariety in O, see [CG,Theorem 3.3.6].
Thus, Conjecture 1.3.6 says that there exists a Borel subalgebra b such that the two Lagrangian subvarieties b ⊥ ∩ O and Ad M (χ) meet at a single point χ.
Let ̟ : g * → m * be the canonical projection induced by restriction of linear functions from g to m. The map ̟ may be thought of as a moment map associated with the Ad M -action on g * . We put χ m := χ| m = ̟(χ). Observe that χ m ∈ m * is a fixed point of the coadjoint M -action on m * , and we have ̟ −1 (χ m ) = χ + m ⊥ .
We write O Y for the structure sheaf of a scheme Y . Let O χ denote the localization of the polynomial algebra C[m * ] at the point χ m . Corollary 1.3.8. Let Y be a G-scheme, and let f : Y → g * be a G-equivariant morphism such that χ ∈ f (Y ). Then, we have (i) The point χ m is a regular value of the composite map ̟ • f .
Proof. The transversality statement of Proposition 1.3.3(i) may be equivalently reformulated as follows: For any x ∈ g * , the point χ m is a regular value of the composite map Ad G(x) ֒→ g * ̟ → m * . The above statement insures that, for any point y ∈ Y such that ̟(f (y)) = χ m , the differential d̟ • df : T y Y → T χm m * , of the map ̟ • f , is a surjective linear map. This yields part (i) of the corollary. Parts (ii)-(iii) follow from (i) combined with Proposition 1.3.3(i).
1.4. Proof of Proposition 1.3.3. First, we recall a few well known results. Write g x ⊂ g for the centralizer of an element x ∈ g. To prove formula (1.3.2), we compute . By sl 2 -theory, one has that dim g(0) + dim g(1) = dim g e . Hence, we find Observe next that, for the element f of our sl 2 -triple, we have g f ⊂ i≤0 g(i), by sl 2theory. It follows that S ⊂ χ + m ⊥ . Also, it is easy to see that the set χ + m ⊥ is stable under the coadjoint M -action. Moreover, it was proved in [GG,Lemma 2.1] that the M -action in χ + m ⊥ is free, and the action-map induces an M -equivariant isomorphism of algebraic varieties: 4.2) where M acts on M × S via its action on the first factor by left translations.
We may exponentiate the Lie algebra map sl 2 → g to a rational group homomorphism SL 2 → G. Restricting the latter map to the torus C × ⊂ SL 2 , of diagonal matrices, one gets a morphism γ : C × → G, t → γ t . Following Slodowy, one defines a •-action of C × on g by Since Ad γ t (e) = t 2 · e, the •-action fixes e. Dualizing, one gets a •-action of C × on g * that fixes the point χ ∈ g * . It is easy to see that each of the spaces, χ + S and χ + m ⊥ , is •-stable and, moreover, the •-action contracts the space χ + m ⊥ to χ.
Proof of Proposition 1.3.3. Observe first the statement of part (i) is clear for the orbit O = Ad G(χ), since the space χ + m ⊥ contains a transverse slice to O. Below, we will use the identification κ : g ∼ → g * , so χ gets identified with e, and we may write χ + m ⊥ ⊂ g. Now, let x ∈ χ + m ⊥ be an arbitrary element and put O := Ad G(x). We are going to reduce the statement (i) for O to the special case of the orbit O using the •-action as follows.
Observe that the tangent space to O at the point x equals T x O = [g, x] = ad x(g), a vector subspace of g. Proving part (i) amounts to showing that the composite pr • κ • ad x : g → g ∼ → g * ։ m * is a surjective linear map, for any x ∈ χ + m ⊥ . It is clear that, for any x ∈ g sufficiently close to e, the map pr • κ • ad x is surjective, by continuity. Since the •-action on χ + m ⊥ is a contraction, we deduce by C × -equivariance that the surjectivity holds for any x ∈ e + m ⊥ . Part (i) is proved.
The isomorphism of part (iii) follows from Proposition 1.2.1, by restricting isomorphism (1.4.2) to N . All the other claims of part (iii) then follow from the isomorphism.
To prove (ii), we observe first, that since χ vanishes on [m, m], the vector space ad m(χ) is an isotropic subspace of the tangent space T χ O. This implies, by M -equivariance, that Ad M (χ) is an isotropic submanifold of O. This Ad M -orbit is closed since M is a unipotent group. Finally, the M -action being free we find dim Ad M (χ) = dim M = 1 2 dim O, by formula (1.3.2). It follows that Ad M (χ) is a Lagrangian submanifold of O.

Springer resulution of Slodowy slices
2.1. The Slodowy variety. Let B be the flag variety, i.e., the variety of all Borel subalgebras in g. Let T * B be the total space of the cotangent bundle on B, equipped with the standard symplectic structure and the natural Hamiltonian G-action. An associated moment map is given by the first projection (2.1.1) The map π, called Springer resolution, is a symplectic resolution of N . This means that, one has N = π(T * B) and, moreover, π is a resolution of singularities of N such that the pull-back morphism π * : O g * → O T * B intertwines the Kirillov-Kostant Poisson bracket on O g * with the Poisson bracket on O T * B coming from the symplectic structure on T * B.
Proposition 2.1.2. (i) The map π : S → S is a symplectic resolution, in particular, S is a smooth and connected symplectic submanifold in T * B of dimension dim S = 2 dim B χ .
(ii) The Springer fiber B χ is a (not necessarily irreducible) Lagrangian subvariety of S.
Our next goal is to find a Hamiltonian reduction construction for the variety S. Specifically, we would like to get an analogue of formula (1.3.5) where the variety S is replaced by S and where the symplectic manifold T * B plays the role of the Poisson variety N . To do so, it is natural to try to replace, in formula (1.3.5), the space χ + m ⊥ by π −1 (χ + m ⊥ ). Thus, we are led to introduce a scheme (2. There is a natural C × -action on T * B along the fibers of the cotangent bundle projection T * B → B. Thus, the group G × C × acts on The Springer resolution (2.1.1) is a G × C × -equivariant morphism. Hence, the map π commutes with the •-action as well. It follows in particular that B χ , S, and Σ, are all •stable subschemes of T * B. The •-action retracts χ + m ⊥ to χ, hence, provides a retraction of Σ = π −1 (χ + m ⊥ ) to B χ = π −1 (χ).
Further, a result of Spaltenstein [Spa] says that the Springer fiber is a connected variety and, moreover, all irreducible components of B χ have the same dimension, cf. also [CG,Corollaries 7.6.16 and 3.3.24], which is equal to Proof of Proposition 2.1.4. Corollary 1.3.8(i) insures that the point χ m ∈ m * is a regular value of the map ̟ • π : T * B → m * . It follows, in particular, that Σ is a (reduced) smooth subscheme of T * B and that dim Σ = dim T * B −dim m. Furthermore, (2.1.5) holds as a scheme theoretic isomorphism and we have dim Σ = dim S + dim m.
Since B χ is connected and the •-action contracts S, resp. Σ, to B χ , we deduce that S, resp. Σ, is a connected manifold. This completes the proof of Proposition 2.1.4(i).
Part (ii) of the proposition is immediate from the isomorphism of Corollary 1.3.3(i). Part (iii) is a general property of the fiber of a moment map over a regular value, see eg. [GuS].
Proof of Proposition 2.1.2. First of all, we observe that the smoothness of Σ, combined with (2.1.5), implies that S is a smooth scheme. Furthermore, from part (iii) of Proposition 2.1.4 and the isomorphism S ∼ = Σ/M we deduce that the symplectic 2-form on T * B restricts to a nondegenerate 2-form on S. This yields Proposition 2.1.2(i).
Further, we know that π : S ։ S is a projective and dominant morphism which is an isomorphism over the open dense subset of S formed by regular nilpotent elements. It follows that the map π : S ։ S is a symplectic resolution. The scheme S being irreducible, we deduce that S is connected (we have already proved this fact differently in the course of the proof of Proposition 2.1.4).
By Proposition 1.2.1, we get dim S = dim S = dim g − rk g − dim O. Hence, using (2.2.1) and the equality 2 dim B + rk g = dim g, we find dim S = (2 dim B + rk g) − rk g − 2(dim B − dim B χ ) = 2 dim B χ . This completes the proof of part (i) of Proposition 2.1.2.
To prove part (ii), let y = (b, χ) ∈ B χ . Recall that any tangent vector to T * B at y can be written in the form ad * a(y) + α, for some a ∈ g and some vertical vector α ∈ b ⊥ (i.e. a vector tangent to the fiber of the cotangent bundle). It is clear that such a vector ad * a(y)+ α is tangent to B χ if and only if one has α + ad * a(χ) = 0. Now, let ad * b(y) + β be a second tangent vector at y which is tangent to B χ at the point y. Thus, we have ad * a(χ) = −α and ad * b(χ) = −β.
Remark 2.2.2. The smoothness statement in Proposition 2.1.2(i) is a special case of the following elementary general result.
Let X be a G-scheme, and let F : X → g * be a G-equivariant morphism such that χ ∈ F (X). Set S := F −1 (S), a scheme theoretic preimage of S, and write F S = F | e S : S → S.
Then, for any 2.3. Given a manifold Y , we write p : T * Y → Y for the cotangent bundle projection.
Let X ⊂ Y be a submanifold. Below, we will use the following Definition 2.3.1. A submanifold Λ ⊂ p −1 (X) is said to be a twisted conormal bundle on X if Λ is a Lagrangian submanifold of T * Y and, moreover, the map p makes the projection Λ → X an affine bundle.
Let σ be the restriction to Σ ⊂ T * B of the projection p : T * B → B. It is clear that σ(Σ) is an M -stable subset of B, and one has the following diagram of M -equivariant maps (ii) For any M -orbit X ⊂ σ(Σ), the map σ makes the projection σ −1 (X) → X a twisted conormal bundle on the submanifold X ⊂ B. Proof. Clearly, b ∈ σ(Σ) if and only if the fiber σ −1 (b) is non-empty. By definition, we have (2.3.5) Note that χ| m∩b = 0 says that χ ∈ (b ∩ m) ⊥ . Thus, the equivalences below yield part (i), To prove (ii), fix a point b ∈ σ(Σ), and let X := M · b ⊂ B be the M -orbit of the point b. Thus, writing B for the Borel subgroup corresponding to b, we have X ∼ = M/M ∩ B. Further, we may use the last displayed formula above and choose λ b ∈ b ⊥ and µ b ∈ m ⊥ such that The above claim implies part (ii) since the fibration φ, of the claim, is well known to be a twisted conormal bundle on the submanifold X ⊂ B = G/B, cf. [CG,Proposition 1.4.14].
To prove Claim 2.3.6, we write Hence, using (2.3.5), we may write Now, we identify the tangent space to B, resp. to the M -orbit X, at the point b with This proves Claim 2.3.6, and part (ii) follows.

Quantization
3.1. The Premet algebra. Given a Lie algebra k, we write Sym k for the Symmetric, resp. Uk for the enveloping, algebra of k. We keep the notation of the previous section, fix χ ∈ N , and define a linear map m → Sym g, resp. m → Ug, by the assignment m → m − χ(m). Let m χ denote the image. Using the canonical isomorphism C[g * ] = Sym g, one can identify m χ with the space of degree ≤ 1 polynomials on g * vanishing on χ + m ⊥ .
Let Zg denote the center of Ug and write Specm Zg for the set of maximal ideals of the algebra Zg. Given c ∈ Specm Zg, let I c ⊂ Zg denote the corresponding maximal ideal, and put U c := Ug/Ug · I c . Let U c · m χ denote the left ideal of the algebra U c generated by the image of the composite m χ ֒→ Ug ։ U c .
Similarly, let I o denote the augmentation ideal of the algebra (Sym g) Ad G , and put S o := Sym g/ Sym g · The left ideal U c · m χ ⊂ U c , resp. the ideal S o ·m χ ⊂ S o , is ad m-stable. It is clear that the scheme isomorphism (1.3.5) translates into the following algebra isomorphism C[S] ∼ = (S o /S o ·m χ ) ad m ; by analogy, we put A c := (U c /U c ·m χ ) ad m . (3.1.2) It is easy to see that multiplication in Ug gives rise to a well defined (not necessarily commutative) associative algebra structure on A c . Furthermore, the above formulas show that the algebra C[S] is obtained from C[N ] by a classical Hamiltonian reduction, resp. the algebra A c is obtained from U c by a quantum Hamiltonian reduction. In particular, C[S] has a natural Poisson algebra structure. The algebra A The family of algebras {A c , c ∈ Specm Zg} may be viewed as 'quantizations' of the Poisson algebra C[S].
Remark 3.1.3. Each of the algebras A c is a quotient of a single algebra A := (Ug/Ug ·m χ ) ad m . This algebra A, that has been introduced and studied by Premet in [P1], is a Hamiltonian reduction of the algebra Ug. The natural imbedding Zg ֒→ Ug descends to a well-defined algebra map  : Zg → A with central image. It is easy to see that, for any central character c, one has A/A · (I c ) = A c .
It is immediate to check that, for any right U c -module N , the assignment u : n → nu induces a well defined right A c -action on the coinvariant space N/N · m χ . This yields, in particular, a right A c -action on Q c , cf. (3.1.1), that commutes with the natural left U caction. In this way, Q c becomes an (U c , A c )-bimodule. Moreover, the right action of A c gives an algebra isomorphism A op

Kazhdan filtrations.
Given an algebra A with an ascending Z-filtration F qA, one puts gr F A := n∈Z F n A/F n−1 A, an associated graded algebra, resp. Rees F A := n∈Z F n A, the Rees algebra. From now on, we assume that gr F A is a finitely generated commutative algebra.
Let V be a left A module equipped with an ascending Z-filtration F qV which is compatible with the one on A. Then, Rees F V := n∈Z F n V acquires the structure of a left Rees F Amodule. The filtration on V is called good if that module is a finitely generated. In such a case, gr F V := n∈Z F n V /F n−1 V is a finitely generated gr F A-module, and the support of gr F V is a closed subset Supp(gr F V ) ⊂ Spec(gr F A) (equipped with reduced scheme structure).
The following standard result is well known, cf. [ABO], Theorem 1.8 and Proposition 2.6.
Lemma 3.2.1. Let F qA be a Z-filtration on an algebra A such that Rees F A is both left and right noetherian, and gr F A is commutative. Then, for any left A-module V , we have (i) All good filtrations on V are equivalent to each other and the set Var V := Supp(gr F V ) is independent of the choice of such a filtration.
(ii) For any good filtration F qV and any A-submodule N ⊂ V , the induced filtration Given a vector space V equipped with an ascending Z-filtration F qV and with a direct sum decomposition V = a∈C V a , one defines an associated Kazhdan filtration, a new ascending Z-filtration on V , as follows (3.2.2) Let Ug = i∈Z Ug i be the Z-grading induced by some Lie algebra grading g = i∈Z g i . We may take V := Ug and let F j V := U ≤j g, j = 0, 1, . . . , be the canonical ascending PBW filtration on the enveloping algebra. Then, formula (3.2.2) gives an associated Kazhdan filtration K qU g, on Ug. This filtration is multiplicative, i.e., one has K i Ug · K j Ug ⊂ K i+j Ug, ∀i, j.
For any i ∈ Z, we have g i ∈ K i+2 Ug. We see as in [GG,§4.2] that the identity map g → g extends to a Poisson algebra isomorphism gr K Ug ∼ = Sym g. Further, it follows easily that the algebra Rees K Ug is isomorphic to a quotient of T g ⊗ C[t] by the two-sided ideal generated by the elements x⊗y −y ⊗x−[x, y]⊗t 2 , x, y ∈ g. Thus, Rees K Ug, is a finitely generated algebra. Moreover, Rees K Ug, viewed as an algebra without grading, is independent of the grading on g used in (3.2.2) (for V = Ug). For the trivial grading g = g 0 , the filtration K qU g is clearly non-negative, and the corresponding algebra Rees K Ug is easily seen to be both left and right noetherian. It follows that Rees K Ug is both left and right noetherian for any Lie algebra Z-grading on g. In particular, Lemma 3.2.1 applies.
From now on, we let K qU g be the Kazhdan filtration associated with the grading (1.3.1) by formula (3.2.2). Given a left Ug-module V , one may consider ascending filtrations F qV which are compatible with the PBW filtration on Ug, in the sense that U ≤i g · F j V ⊂ F i+j V holds for any i, j ∈ Z. One may also consider filtrations K qV which are compatible with the above defined Kazhdan filtration on Ug, to be referred to as Kazhdan filtrations on V .
Assume, in addition, that the h-action on V is locally finite. Then there is a direct sum decomposition V = a∈C V a where, for any a ∈ C, one defines a generalized a-eigenspace by the formula V a : For any h-stable ascending filtration F qV which is compatible with the PBW filtration on Ug, formula (3.2.2) gives an associated Kazhdan filtration K qV , on V . Clearly, one has F j V = a∈C F j V ∩ V a . Using this, one obtains by an appropriate 're-grading procedure' a canonical isomorphism of Sym g-modules, resp. Rees K Ug-modules, (that does not necessarily respect the natural gradings): Corollary 3.2.4. Let V be a finitely generated Ug-module such that the h-action on V is locally finite. Then, one has (i) For any good h-stable filtration F qV, on V , formula (3.2.2) gives a good and separated Kazhdan filtration on V .
(ii) Any good Kazhdan filtration K ′ qV , on V , is equivalent to (3.2.2). Hence, it is a separated filtration and, in g * , one has Var K ′ V = Var F V (set theoretic equality).
Proof. Let F qV be a good h-stable filtration. Clearly, it is bounded from below, hence, separated. It follows that Rees F V is a finitely generated Rees F Ug-module and, moreover, we have ∩ j∈Z t j · Rees F V = 0. Thus, from (3.2.3) we deduce that Rees K V is a finitely generated Rees K Ug-module and, moreover, we have ∩ j∈Z t j · Rees K V = 0. This yields (i). Part (ii) is now a consequence of Lemma 3.2.1.
From now on, in the setting of Corollary 3.2.4, we will use simplified notation Var V for Associated with any left Ug-module, resp. U c -module, V , is its Whittaker subspace: The right A c -action on Q c gives the space Wh m V a structure of left A c -module. Similarly, for any right U c -module N , the right A c -action on Q c gives the coinvariant space N/N m χ = N ⊗ Uc Q c a structure of right A c -module.
Below, we are also going to use a version of the Whittaker functor for bimodules. Given an (U c ′ , U c )-bimodule K, the subspace Km χ ⊂ K is stable under the adjoint m-action on K. Hence, the latter action descends to a well defined ad m-action on K/Km χ . It is clear that, for any x ∈ K/Km χ and m ∈ m, one has mx = ad m(x) + χ(m) · x. We see in particular that the adjoint m-action and the above defined right A c -action on K/Km χ commute.
Finally, we put Definition 3.3.3. Let (U c , m χ )-mod be the abelian category of finitely generated U c -modules V satisfying the following condition: for any v ∈ V there exists an integer n = n(v) ≫ 0 such that, we have (m 1 · m 2 · . . . · m n )v = 0, ∀m 1 , . . . , m n ∈ m χ .
Objects of the category (U c , m χ )-mod are called Whittaker modules. It is clear that Q c is a Whittaker module. We let K qU c , resp. K qQ c , be the quotient filtration induced by the Kazhdan filtration on Ug. The Kazhdan filtration on Q c induces, by restriction, an algebra filtration K qA c , on A c ⊂ Q c . Note that, unlike the case of K qU c , the filtration K qQ c , hence also K qA c , is non-negative.
The proof of the following result repeats the proof of [GG,Proposition 5.2].
Proposition 3.3.4. For any c ∈ Specm Zg, one has a graded Poisson algebra isomorphism From this proposition, using Proposition 1.2.1 and a result of Bjork [Bj], we deduce Corollary 3.3.5. The algebra A c is Cohen-Macaulay and Auslander-Gorenstein.
Let K qV be a good Kazhdan filtration on an object V ∈ (U c , m χ )-mod. Using that the Kazhdan filtration on Q c is nonnegative, one proves that the filtration K qV is bounded from below. If, in addition, the filtration K qV is m-stable then gr One has a graded algebra isomorphism that results from the scheme isomorphism of Proposition 1.3.3(iii). Here, the grading on C[M ] is the weight grading with respect to the adjoint action of the 1-parameter subgroup t → γ t −1 . Hence, we get gr For any noetherian algebra B, let mod B denote the category of finitely generated left B-modules. Also, in part (ii) of the proposition below, given a filtration on a A c -module N , we equip Q c ⊗ Ac N with the tensor product filtration using the Kazhdan filtration on Q c .
The equivalences in parts (i)-(ii) of the above proposition are due to Skryabin, [Sk], and the graded isomorphisms in parts (ii)-(iii) are immediate consequences of the results of [GG].

Weak Harish-Chandra bimodules
4.1. Let B and B ′ be an arbitrary pair of nonnegatively filtered algebras such that gr B and gr B ′ , the corresponding associated graded algebras, are finitely generated commutative algebras isomorphic to each other. Thus, there is a well defined subset ∆ ⊂ Spec(gr B) × Spec(gr B ′ ), the diagonal.
Associated with any finitely generated (B, It is straightforward to show the following Given a closed subset Z ⊂ Spec(gr B) = Spec(gr B ′ ), let mod Z B, resp. mod Z B ′ be the category of finitely generated left B-modules, resp. B ′ -modules, K such that Var K ⊂ Z. One similarly defines W H C Z (B, B ′ ) to be the Serre subcategory of W H C (B, B ′ ) formed by the wHC bimodules K such that Var K ⊂ Z.
Tensor product over B ′ gives a bi-functor (4.1.3) We have, in particular, the category W H C (Ug, Ug), where the enveloping algebra Ug is equipped with the PBW filtration, not with Kazhdan filtration. Similarly, for any c ′ , c ∈ Specm Zg, one has the category W H C A finitely generated (Ug, Ug)-bimodule, resp. (U c ′ , U c )-bimodule, K such that the adjoint g-action ad a : v → av − va, on K, is locally finite is called a Harish-Chandra bimodule. It is clear that any Harish-Chandra bimodule is a weak Harish-Chandra bimodule. However, the converse is not true, in general, cf. section 6.6. We write H C (U c ′ , U c ) for the abelian category of Harish-Chandra (U c ′ , U c )-bimodules.
Let K be a Harish-Chandra (U c ′ , U c )-bimodule and let F qK be a good ad g-stable filtration compatible with the tensor product of PBW filtrations on U c ′ and on U op c . The ad h-action on K being locally finite, one can use formula (3.2.2) to define an associated Kazhdan filtration K qK, on K. The latter induces a quotient filtration on K/Km χ which gives, by restriction, a filtration K q(Wh m m K), on Wh m m K. It is easy to see that this filtration is compatible with the algebra filtration on The first main result of the paper is the following Then, the following holds: The functor K ⊗ Uc (−) takes Whittaker modules to Whittaker modules, and there is a natural isomorphism of functors that makes the following diagram commute The proof of Theorem 4.1.4 occupies subsections § §4.3-4.4. From part (i) of the theorem, one immediately obtains The following direct consequence of Theorem 4.1.4(ii) says that Wh m m is a monoidal functor.
the subspace of I(N ) formed by ad g-locally finite elements. Part (i) of the theorem below will be proved later, in section 6.5. It is stated here for reference purposes.
We begin the proof of (ii) by showing the adjunction property. The latter says that, To prove this, using the definition of I(N ), we compute To complete the proof of part (ii) we must show that I(N ) is a finitely generated (U c ′ , Ug)bimodule. To this end, we need to enlarge the category H C (U c ′ , U c ) as follows. Recall first that I c ⊂ Zg denotes the maximal ideal in the center of the enveloping algebra Ug. Let K be a finitely generated (U c ′ , Ug)-bimodule. We say that K is a Harish-Chandra (U c ′ , U c )-bimodule if the adjoint g-action on K is locally finite and, moreover, there exists an large enough integer ℓ = ℓ(K) ≫ 0 such that K is annihilated by the right action of the ideal (I c ) ℓ , that is, we have K · (I c ) ℓ = 0. Let H C (U c ′ , U c ) be the full subcategory of (U c ′ , Ug)-bimod whose objects are Harish-Chandra (U c ′ , U c )-bimodules. The structure of the category H C (U c ′ , U c ) has been analyzed by Bernstein and Gelfand [BGe]. It turns out that any Harish-Chandra (U c ′ , U c )-bimodule has finite length. Furthermore, it was shown in loc cit that the category H C (U c ′ , U c ) has enough projectives and there are only finitely many nonisomorphic indecomposable projectives P j , j = 1, . . . , m, say.
Next, we observe that the algebra A, introduced in Remark 3.1.3, comes equipped with a natural ascending filtration such that one has gr A = C[S]. A finitely generated (A c ′ , A)bimodule N will be called a weak Harish-Chandra (A c ′ , A)-bimodule if one has Var N ⊂ S ⊂ S × S, where S ֒→ S × S is the diagonal imbedding. We let W H C (A c ′ , A c ) denote the full subcategory of the category (A c ′ , A)-bimod whose objects are weak Harish-Chandra (A c ′ , A)bimodules N such that N · (I c ) ℓ = 0 holds for a large enough integer ℓ = ℓ(N ) ≫ 0. As we will see later, the arguments of §6.5 can be used to show that any object of the category W H C (A c ′ , A c ) has finite length.
Clearly, the category H C (U c ′ , U c ) may be viewed as a full subcategory in H C (U c ′ , U c ), resp. the category W H C (A c ′ , A c ) may be viewed as a full subcategory in W H C (A c ′ , A c ). It is straightforward to extend our earlier definitions and introduce a functor Wh m m : One also defines a functor I in the opposite direction such that an analogue of formula (4.2.3) holds.
We are now ready to complete the proof of Theorem 4.2.2(ii) by showing that I(N ) is a finitely generated (U c ′ , Ug)-bimodule, for any N ∈ W H C (A c ′ , A c ). It suffices to show, in view of the results of [BGe] cited above, that, for each j = 1, . . . , m, the vector space Uc)-bimod (P j , I(N )) has finite dimension. To see this, we use the analogue of formula (4.2.3), which yields But, we know that Wh m m P j ∈ W H C (A c ′ , A c ) and that the object N ∈ W H C (A c ′ , A c ) has finite length. It follows that the dimension on the right of formula (4.2.6) is finite, and we are done.

Homology vanishing.
Recall the Lie subalgebra m χ ⊂ Ug. The Kazhdan filtration on Ug restricts to a filtration on m χ . The latter induces an ascending filtration K j (∧ q m χ ), j ≥ 0, on the exterior algebra of m χ .
Given a right Ug-module V equipped with a Kazhdan filtration K qV , we form a tensor product C := V ⊗ ∧ q m χ , and let K n C = n=i+j K i V ⊗ K j (∧ q m χ ) be the tensor product filtration.
We view V as a m χ -module, and write H q(m χ , V ) for the corresponding Lie algebra homology with coefficients in V . The latter may be computed by means of the complex (C, ∂), where ∂ : C → C is the standard Chevalley-Eilenberg differential, of degree (−1). It is immediate to check that the differential respects the filtration K qC, making C a filtered complex.
Write B ⊂ Z ⊂ C for the subspaces of boundaries and cycles of the complex C, respectively. Thus, we have H q(m χ , V ) = H q(C) = Z/B. The filtration on C induces, by restriction, a filtration K p Z := Z ∩ K p C, on the space of cycles. The latter filtration induces a quotient filtration on homology. Explicitly, the induced filtration on homology is given by There is an associated standard spectral sequence with 0-th term, cf. [CE], Ch.15, Recall the local algebra O χ introduced above Corollary 1.3.8. The lemma below is a slight generalization of a result due to Holland, cf. [Ho], §2.4. Lemma 4.3.3. Let V be a right Ug-module equipped with a good Kazhdan filtration K qV . Assume that the localization of gr K V at χ + m ⊥ is a flat ̟ q O χ -module and, moreover, the induced filtration on C, the Chevalley-Eilenberg complex, is convergent in the sense of [CE], p. 321.
Then, for any j > 0, we have H j (m χ , gr K V ) = 0 and H j (m χ , V ) = 0. Moreover, the natural projection yields a canonical graded C[g * ]-module isomorphism Proof. We recall various standard objects associated with the spectral sequence of a filtered complex. We follow [CE], Ch. 15, §1-2 closely, except that we use homological, rather than cohomological notation.
First of all, for each p ∈ Z, one defines a pair of vector spaces Z ∞ p and B ∞ p , in such a way that one has (4.3.4) Further, one defines a chain of vector spaces, cf. [CE], p.317, . . . Precise definitions of these objects are not important for our purposes, they are given e.g. in [CE], Ch. 15, §1. What is important for us is that the definitions imply B ∞ p = ∪ r≥0 B r p . The assumption of the lemma that the filtration K qC be convergent means that, in addition, one has, see [CE], ch. 15, §2, Now, to prove the lemma, we observe that the zeroth page (E 0 , d 0 ) of the spectral sequence (4.3.2) may be identified with the Koszul complex associated with the subscheme ̟ −1 (χ m ) ⊂ g * . Thus, the assumption of the lemma that the localization of gr K V at χ + m ⊥ be a flat ̟ q O χ -module forces the Koszul complex be acyclic in positive degrees. Therefore, the spectral sequence degenerates at E 1 .
The degeneration implies that, for any p ∈ Z and r ≥ 1, one has Z r p = Z 1 p and B r p = B 1 p . Therefore, we have B 1 p = B ∞ p , and using the first equation in (4.3.5), we get Z 1 , by (4.3.4). We conclude that one has gr K H 0 (m χ , V ) = (gr k V )/(gr k V )m χ and, moreover, gr K H j (m χ , V ) = 0 for any j > 0. Finally, thanks to the second equation in (4.3.5), we have gr K H j (m χ , V ) = 0 ⇒ H j (m χ , V ) = 0, and the lemma is proved.
It is important to observe that the filtration K qC, on C, gives rise to two natural filtrations on the subspace B = ∂C, of the boundaries. These two filtrations are defined as follows It is clear that on has K qB ⊂ K ′ qB, but this inclusion need not be an equality, in general. We have the following result Lemma 4.3.7. Assume the above filtrations K qB and K ′ qB are equivalent, i.e. there exists an integer ℓ ≥ 1 such that, for all p ∈ Z, one has K ′ p−ℓ B ⊂ K p−1 B. Then, we have Z ∞ p = ∩ r≥0 Z r p , ∀p ∈ Z, i.e. the first equation in (4.3.5), holds. Proof. For any ℓ ≥ 1, we have K q −ℓ C ⊂ K q −1 C, hence, one gets an obvious imbedding B ∩ K q −ℓ C ֒→ Z ∩ K q −1 C. Using the definitions of various filtrations introduced above, this imbedding may be rewritten as follows K ′ q −ℓ B ֒→ K q −1 Z. We may further compose the imbedding with a projection to homology to obtain the following composite Next, we fix p ∈ Z and consider the complex K p C/K p−ℓ C. By definition, we have The differential ∂ clearly annihilates the space K p−ℓ Z + ∂(K p C), in the denominator of the rightmost term. Therefore, we see that applying the differential ∂ to the numerator of that term yields a well defined map . Thus, we have constructed the following diagram Now, let ℓ be such that the assumption of the lemma holds, so that we have K ′ q −ℓ B ⊂ K q −1 B. Then the corresponding map δ, in (4.3.8), clearly vanishes. It follows that the composite map δ • ∂ vanishes as well, and we have The last equation insures that we are in a position to apply a criterion given in [CE,ch. 15,Proposition 2.1]. Applying that criterion yields the statement of the lemma.

Proof of Theorem 4.1.4. Throughout this subsection, we fix
Choose a finite dimensional ad g-stable subspace K 0 ⊂ K that generates K as a bimodule. For any ℓ ≥ 0, let In this way, one may define a good ad g-stable filtration on K. Now, let F qK be an arbitrary good ad g-stable filtration on K. Let K qK be the Kazhdan filtration associated with the filtration F qK via formula (3.2.2).
We first view K as a right m χ -module. A key step in the proof of Theorem 4.1.4 is played by the following Lemma 4.4.1. For all j > 0, we have H j (m χ , gr K K) = 0 and H j (m χ , K) = 0.
Furthermore, the canonical projection gr K K/(gr K K)m χ Proof. The result is clearly a consequence of Lemma 4.3.3, provided we show that all the assumptions of that lemma hold in our present setup.
First, we verify the assumption that, for the above defined Kazhdan filtration on K, the To this end, we note that the construction of the filtration F qK insures that gr F K is an Ad G-equivariant finitely generated C[N ]-module, where N ⊂ N × N is the diagonal copy of the nilpotent variety. By Corollary 1.3.8(ii), we conclude that the localization of gr F K at χ + m ⊥ is a flat ̟ q O χ -module. Further, by (3.2.3), we have a C[N ]-module isomorphism gr F K ∼ = gr K K. Moreover, it is immediate from definitions that this isomorphism is compatible with ad m-actions on each side. It follows, in particular, that ̟ * O χ C[N ] gr K K is a flat ̟ q O χ -module, and we have an Ad M -equivariant C[N ]-module isomorphism gr K K/(gr K K)m χ ∼ = gr F K/(gr F K)m χ .
To complete the proof, we show that our filtration K qK is convergent, i.e., both equations in (4.3.5) hold in the situation at hand.
To see this, we observe that the filtration on K being ad g-stable, it is good as a filtration on K viewed as a left U c ′ -module. Further, the differential in the Chevalley-Eilenberg complex C := ∧ q m χ ⊗ K, involved in Lemma 4.3.3, is clearly a morphism of left U c ′ -modules. Thus, the subspace B = ∂C ⊂ C, of the boundaries of the Chevalley-Eilenberg complex, as well as each homology group H q(m χ , K), acquires a natural structure of left U c ′ -module. Being a subquotient of C, any of these U c ′ -modules is finitely generated. Hence, each of the two Kazhdan filtrations on B defined in (4.3.6), as well as the Kazhdan filtration on H q(m χ , K) defined in (4.3.1), is a good filtration on the corresponding left U c ′ -module. It follows, in particular, that the two filtrations on B are equivalent, cf. Lemma 3.2.1. Thus, the first equation in (4.3.5) holds by Lemma 4.3.7.
It remains to prove that the Kazhdan filtration on H q(m χ , K) defined in (4.3.1) is separated. Recall that any good Kazhdan filtration on an object of the category (U c , m χ )-mod is bounded below, hence, separated. Thus, it suffices to show that, for any j ≥ 0, H j (m χ , K) viewed as a left U c -module, is an object of (U c , m χ )-mod.
To this end, we recall that the Chevalley-Eilenberg complex of a right module over a Lie algebra has a natural action of that Lie algebra, by the 'Lie derivative'. It is well known that the Lie derivative action on the Chevalley-Eilenberg complex induces the zero action on each homology group. Applying this to our Harish-Chandra Ug-bimodule K, we see that the complex C = K ⊗ ∧ q m χ has a left g-action, defined as a tensor product of the left g-action on K and the zero g-action on ∧ q m χ . There is also an m χ -action, by the 'Lie derivative'. The left g-action and the m χ -action on C commute, and the difference of the left m χ -action and the Lie derivative m χ -action gives a a well defined m χ -action on C, to be called the adjoint action. The adjoint g-action on K being locally finite and the Lie algebra m χ being nilpotent, it follows easily that the adjoint m χ -action on C is locally nilpotent. We conclude that the left m χ -action on H j (m χ , K), induced by the left m χ -action on C, may be written as a sum of a locally nilpotent adjoint action and of the Lie derivative action, the latter being known to be the zero action. Thus, we have shown that H j (m χ , K) ∈ (U c , m χ )-mod. This completes the proof.
Proof of Theorem 4.1.4. It follows from the preceding paragraph that we have K/Km χ = H 0 (m χ , K) ∈ (U c , m χ )-mod, cf. (3.3.2). Thus, we get a functor Wh m : H C (U c ′ , U c ) → (U c , m χ )-mod, K → K/Km χ . The homology vanishing of Lemma 4.4.1 implies that this functor is exact. The functor Wh m : (U c , m χ )-mod → A c -mod being an equivalence, cf. Proposition 3.3.6(i), we deduce the exactness of the composite functor Wh m • Wh m . The exactness statement of part (iii) of the theorem now follows by writing Wh m m = Wh m • Wh m . Next, fix an ad g-stable good filtration on K, and write Wh m m K = Wh m (K/Km χ ). It follows, in particular, that the induced filtration on K/Km χ is m-stable. Further, by Lemma 4.4.1, we get (below, (−)| S stands for a restriction of a C[N ]-module to the subvariety S ⊂ g * ) gr Wh m m K = gr Wh m (K/Km χ ) = [gr(K/Km χ )] S = [gr K K/(gr K K)m χ ] S = (gr K K) S . where the second equality is due to Proposition 3.3.6(ii) applied to the object V = K/Km χ ∈ (U c , m χ )-mod. This proves part (i) of the theorem.
Observe that proving commutativity of the diagram of part (ii) is equivalent, thanks to Skryabin's equivalences, cf. Proposition 3.3.6(i)-(ii), to showing commutativity of the following diagram To prove (4.4.2), write Wh m m K = Wh m • Wh m K. Thus, one has a canonical map (4.4.3) Since Wh m K ∈ (U c ′ , m χ )-mod, the map Φ is actually an isomorphism, by Skryabin's equivalence. Hence, tensoring diagram (4.4.3) with a left A c -module N , we get a chain of isomorphisms The composite isomorphism above provides the isomorphism of functors that makes diagram (4.4.2) commute, and Theorem 4.1.4(ii) follows.
We now complete the proof of part (iii) of the theorem. To this end, pick a good ad g stable filtration on our wHC bimodule K. We know by part (i) that gr Wh m m K = (gr K K)| S . This is clearly a finitely generated C[S × S]-module supported on the diagonal in S × S. It follows that Wh m m K is itself finitely generated and we have Wh m It remains to show that the resulting functor Wh m m is faithful. To prove this, observe that We conclude that Wh m m (U c /I) = 0, by Proposition 3.3.6(i). Now, (4.5.1) follows from the first equivalence of Corollary 4.1.6 applied to K = U c /I.
As a second application of our technique, we provide a new proof of a result (Theorem 4.5.2 below) conjectured by Premet [P2,Conjecture 3.2]. In the special case of rational central characters, the conjecture was first proved (using characteristic p methods) by Premet in [P3], and shortly afterwards by Losev [Lo1] in full generality. An alternate proof of the general case was later obtained in [P4]. Our approach is totally different from those used by Losev and Premet. Since, I ⊂ J, we conclude that dim Var(U c /I) = dim Var(U c /J). Now, N being a simple finite dimensional left A c -module, we deduce using Skryabin's equivalence that Q c ⊗ Ac N is a simple U c -module. Thus, J is a primitive ideal in U c . The equation dim Var(U c /I) = dim Var(U c /J) combined with the inclusion I ⊂ J forces I = J, due to a result by Borho-Kraft [BoK], Korollar 3.6.
Remark 4.5.3. Set n := dim S. The assignment N → Ext n Ac-mod (N, A c ) is an exact functor on the category of finite dimensional left A c -modules, thanks to Corollary 3.3.5. That functor gives a contravariant duality

Whittaker D-modules.
Let h denote the abstract Cartan algebra for the Lie aldgebra g, and let X + ⊂ h * be the semigroup of dominant integral weights. For any integral weight λ ∈ h * one has a G-equivariant line bundle O(λ) on B. For any ν ∈ h * , let D ν denote the sheaf of ν-twisted algebraic differential operators on B, see [BB1]. In the case where ν is integral, we have D ν = D(O(ν)), is the sheaf of differential operators acting in the sections of the line bundle O(ν). There is a canonical algebra isomorphism U ν ∼ = Γ(B, D ν ), see [BB1].
Let V be a coherent D ν -module on B that has, viewed as a quasi-coherent O B -module, an additional M -equivariant structure. Given an element x ∈ m, we write x D for the action on V of the vector field corresponding to x via the D ν -module structure, and x M for the action on V obtained by differentiating the M -action arising from the equivariant structure.
We say that an M -equivariant D ν -module V is an (m, χ)-Whittaker D ν -module if, for any x ∈ m and v ∈ V, we have (x D − x M )v = χ(x) · v. Write (D ν , m χ )-mod for the abelian category of (m, χ)-Whittaker coherent D ν -modules on B.
Remark 5.1.1. The natural D-module projection D ν ։ Q ν induces an U ν -module map Q ν = U ν /U ν m χ → Γ(B, Q ν ). The latter map turns out to be an isomorphism, according to a special case (λ = 0) of Corollary 5.4.2(ii). If ν is a dominant weight then the same isomorphism follows from the Beilinson-Bernstein theorem [BB1]. In any case, one concludes that the canonical algebra map A op ν → Hom Dν (Q ν , Q ν ) is an isomorphism as well.
For any regular and dominant ν ∈ h * , the localization theorem of Beilinson and Bernstein [BB1] yields a category equivalence Γ(B, −) : (D ν , m χ )-mod ∼ → (U ν , m χ )-mod. Combining the Beilinson-Bernstein and the Skryabin equivalences, one obtains, see [GG,Proposition 6.2], the following result where H q(n, −) denotes the Lie algebra homology functor, and the subscript 'o' stands for a certain particular weight space of the natural Cartan subalgebra action on homology.
Assume now that V ∈ (D ν , m χ )-mod. Then, V ∈ (U c , m χ )-mod. Therefore, for any element x ∈ m ∩ n, the natural action of x on H q(n, V ) is such that the operator x − χ(x) is nilpotent. On the other hand, the action of any Lie algebra on its homology is trivial, hence, the element x induces the zero operator on H q(n, V ). Thus, we must have χ(x) = 0. Proposition 2.3.3(i) completes the proof. The space A ν+λ ν comes equipped with a natural (A λ+ν , A ν )-bimodule structure. The standard filtration on D ν by the order of differential operator gives rise to a tensor product filtration on O(λ) ⊗ O B D ν , where the factor O(λ) is assigned filtration degree 0. This induces a natural ascending filtration on the vector space Q ν+λ ν , resp. A ν+λ ν . Further, the •-action on B gives a Z-grading on the above vector spaces. Hence, formula (3.2.2) provides the vector space Q ν+λ ν , resp. A ν+λ ν , with a natural Kazhdan filtration F K Q ν+λ ν , resp. F K A ν+λ ν . Main propeties of the bimodules A ν+λ ν are summarized in the following result.

Translation bimodules. Let
Proposition 5.2.1. Let ν be dominant regular weights. Then, for any λ, µ ∈ X + , we have (i) A ν+λ ν is finitely generated and projective as a left A ν+λ -module, as well as a right A ν -module; (ii) The natural pairing A ν+λ+µ which is compatible with the pairings in (ii).
(iv) The following translation functor is an equivalence Remark 5.2.3. Applying gr(−) to the pairing of part (ii) of the Proposition, one obtaines a pairing gr K A ν+λ+µ The last statement of Proposition 5.2.1(iii) means that the latter pairing gets identified, via the the isomorphisms in (5.2.2), with the natural pairing . We begin the proof of Proposition 5.2.1 with the following result that relates 'geometric' and 'algebraic' translation functors.
Lemma 5.2.4. For any dominant regular weight ν and any λ ∈ X + , the following diagram commutes To prove this, we are going to use Corollary 5.4.2 from section 5.4 below, which is independent of the intervening material. We have where the last equality holds thanks to part (ii) of Corollary 5.4.2. The rightmost term in the above chain of equalities is nothing but Wh m m Γ(B, D ν+λ ν ), and formula (5.2.5) is proved. given by the vertical arrow on the right of the diagram is an equivalence as well, and (iv) is proved.
Note that the equivalence of part (iv) yields, in particular, an algebra isomorphism Thus, the (A ν+λ , A ν )-bimodule A ν+λ ν fits the standard Morita context. Hence, the general Morita theory implies all the statements from part (i) of the proposition.
We now prove (ii). To this end, we observe that the composite functor T ν+λ+µ ν+λ • T ν+λ ν is clearly given by tensoring with A ν+λ+µ ν+λ ⊗ A ν+λ A ν+λ ν , an (A ν+λ+µ , A ν )-bimodule. Therefore, the canonical pairing in (ii) induces a morphism of functors (5.2.8) We may transport the latter morphism via the equivalences provided by Proposition 5.1.2. In this way, we get a morphism O(µ) It is clear from commutativity of diagram (4.1.5) that the latter morphism of functors is the one induced by the canonical morphism of sheaves ψ : Now, the morphism of sheaves ψ is clearly an isomorphism. It follows that the associated morphism of functors is an isomorphism as well. Thus, we conclude that the morphism in (5.2.8) is an isomorphism. For the corresponding bimodules, this implies that the pairing in (ii) must be an isomorphism.
The proof of part (iii) of the proposition will be given in §5.4.

Characteristic varieties.
We are going to define a Kazhdan filtration on D ν . To this end, view B as a C × -variety via the action C × ∋ t : b → Ad γ t (b). Any C × -orbit in an arbitrary quasi-projective C × -variety is known to be contained in an affine Zariski-open C × -stable subset. Thus, we may view D ν as a sheaf in the topology formed by Zariski-open C × -stable subsets of B.
For any Zariski-open subset U ⊂ B, the order filtration on differential operators gives a filtration on the vector space Γ(U, D ν ). If, in addition, U is C × -stable, then the C × -action gives a weight decomposition Γ(U, D ν ) = i∈Z Γ(U, D ν ) i . We are therefore in a position to define an associated Kazhdan filtration on Γ(U, D ν ) by formula (3.2.2). For the associated graded sheaf, one has a canonical isomorphism gr K D ν = p qO T * B .
Let K qV be a good M -stable Kazhdan filtration on a D-module V ∈ (D ν , m χ )-mod. Write gr K V for the M -equivariant coherent sheaf on T * B such that p q gr K V = gr K V. Lemma 5.3.1. In the above setting, for all sufficiently dominant λ ∈ X + , there is a canonical isomorphism Proof. Since V ∈ (D ν , m χ )-mod, any good Kazhdan filtration K qV on V is bounded below and we have Supp gr K V ⊂ Σ. For any integral λ, put We apply the functor Wh m Γ(B, −) to V(λ), a filtered sheaf. Thus, there is a standard convergent spectral sequence involving R q Wh m Γ(B, −), the right derived functors of the composite functor Wh m Γ(B, −). The spectral sequence reads where R q Wh m Γ(B, −) stand for the derived functors of Wh m Γ(B, −), a left exact functor. We claim that, for all λ dominant enough, one has R i Wh m Γ(B, gr K V(λ)) for any i > 0. That would yield the collapse of the above spectral sequence which would give, in turn, the required canonical isomorphism Wh m Γ(B, gr K V(λ)) ∼ → gr K Wh m Γ(B, V(λ)).
The M -action on Σ being free, one can repeat the argument in the proof of [GG], formula (6.1) and Proposition 5.2, to show that the functor Inv M q q is exact (and is isomorphic to the functor of restriction to the closed submanifold S ⊂ Σ). Further, using the Springer resolution π : S → S, we can write Γ( S, −) = Γ(S, π q(−)). Therefore, for any i ≥ 0, we have isomorphisms of derived functors where in the last isomorphism we have used that Γ(S, −) is an exact functor since S is affine.
Recall next that, for a dominant regular weight λ the sheaf O Σ (λ) is relatively ample with respect to the Springer resolution π. Therefore, for λ dominant enough, one has R i π q(Inv M q q gr K V(λ)) = 0, for any i > 0. Hence, for such λ, and i > 0, we obtain R i Wh m Γ(B, gr K V(λ)) = 0, and we are done.
5.4. Harish-Chandra D-modules. For any pair µ, ν ∈ h * , we put D µ,ν := D µ ⊠ D ν , a sheaf of twisted differential operators on B × B. Let G sc denote a simply-connected cover of the semisimple group G. The group G sc acts diagonally on B × B. We may consider the restriction of this action to the 1-parameter subgroup C × and corresponding Kazhdan filtrations on the sheaf D µ,ν as well as on D µ,ν -modules.
Recall that a D µ,ν -module is called G-monodromic if it is G sc -equivariant (with respect to the diagonal action on B × B). Let U ⊂ B × B a Zariski open subset stable under the C × -diagonal action. Then, for any G-monodromic D µ,ν -module V, the induced C × -action on Γ(U, V) is locally finite. Hence, any G-stable filtration on V gives an associated Kazhdan filtration (3.2.2).
We define a set Z : where π is the Springer resolution (2.1.1). We will view Z as a closed reduced subscheme in T * B × T * B = T * (B × B), called Steinberg variety. The assignment u × v → π(u) gives a proper morphism π : Z → N . For any G-monodromic D µ,ν -module V, one has Var V ⊂ Z.
Let V be a coherent D µ,ν -module and let pr : B × B → B denote the first projection. We abuse the notation and write V/Vm χ := pr q[V/(1 ⊠ m χ )V] for a sheaf-theoretic direct image of the sheaf of coinvariants with respect to the action of the Lie algebra 1 ⊠ m χ ⊂ D µ ⊠ D ν . A filtration on V induces one on V/Vm χ .
Proposition 5.4.1. Let V be a G-monodromic D µ,ν -module equipped with a G-stable good filtration. Then, the induced filtration on the D µ -module V/Vm χ is good and we have V/Vm χ ∈ (D ν , m χ )-mod. Furthermore, the corresponding Kazhdan filtration on V/Vm χ is good as well.
In addition, we have H j (m χ , gr K V) = 0 and H j (m χ , V) = 0, for any j > 0; moreover, the canonical map gr K V/(gr K V)m χ → gr K (V/Vm χ ) is an isomorphism.
Proof. Given a G-stable good filtration F qV, write gr F V for the coherent O T * B×T * B -module such that p q gr F V = gr F V. Then, we have that gr F V is a G sc -equivariant coherent sheaf on Z. The composite pr : Z ֒→ T * B × T * B → T * B, of the closed imbedding and the first projection, is a proper morphism. Hence, the push-forward pr q( gr F V) is a coherent sheaf on T * B. It follows that the induced filtration on V/Vm χ , viewed as a left D µ -module, is good and that V/Vm χ is a coherent D µ -module. Now, applying Corollary 1.3.8(ii) to the morphism π : Z → N we deduce that the stalk of the sheaf gr F V at any point of π −1 (χ + m ⊥ ) is a flat (̟ • π) q O χ -module. At this point, the proof of Lemma 4.4.1 (cf. the proof of Theorem 4.1.4 given in §4.4) goes through verbatim in our present situation.
The Kazhdan filtration on D ν induces a good Kazhdan filtration on each of the objects D ν+λ ν , Q ν , and also on Q ν+λ ν . One may consider D ν as a D ν -bimodule, resp. D ν+λ ν as a (D ν+λ , D ν )-bimodule. Such a bimodule may be viewed as a G-monodromic left D ν,µ -module, resp. D ν+λ,µ -module, supported on the diagonal B ⊂ B × B. Here, the weight µ is given by the formula µ = −w 0 (λ) + 2ρ, where w 0 stands for the longest element in the Weyl group.
Recall that the group C × acts on S via the •-action.
Corollary 5.4.2. For an ν, λ ∈ h * , we have Recall that the Cohen-Macaulay property claimed in part (i) of the corollay says that the sheaves E xt q D ν+λ (Q ν+λ ν , D ν+λ ) vanish in all degrees but one.
We see that the complex on the right of (5.4.3) may be identified, up to degree shift, with the Chevalley-Eilenberg complex D µ−λ µ ⊗ ∧ q m χ . But the latter complex is a resolution of the D µ−λ -module Q µ−λ , by the first paragraph of the proof. Therefore, the complex (5.4.3) has a single nonvanishing cohomology group which is isomorphic to Q µ−λ . This completes the proof of part (i).
To prove part (ii) we recall that, for λ ∈ X + and any j > 0, one has the cohomology vanishing H j (T * B, O T * B (λ)) = 0, cf. [Br]. Using that grD ν+λ ν = O T * B (λ), by a standard spectral sequence argument, we deduce H j (B, D ν+λ ν ) = 0 for all j > 0. Thus, the Chevalley-Eilenberg complex D ν+λ ν ⊗ ∧ q m χ yields a Γ-acyclic resolution of Q ν+λ ν , the latter being viewed as a sheaf on B. We conclude that the sheaf cohomology groups H j (B, Q ν+λ ν ) may be computed as the cohomology groups of the complex Proof of Proposition 5.2.1(iii). For any weights ν, λ, we have (5.4.5) We apply Lemma 5.3.1 to the D ν -module V = Wh m D ν := D ν /D ν m χ . The lemma says that, for all sufficiently dominant λ ∈ X + , one has  Thus,using (5.4.5)-(5.4.6), we get ). The verification of compatibilities with the pairings is left for the reader.
A directed algebra is a vector space B := µ ν B µν , graded by pairs µ, ν ∈ Λ such that µ ν, and equipped, for each triple µ ν λ, with a bilinear multiplication pairing B µν ⊗ B νλ → B µλ . These pairings are required to satisfy, for each quadruple µ ν λ ̺, a natural associativity condition, cf. [Mu]. In the special case where the group Λ = Z is equipped with the usual order, our definition reduces to the notion of Z-algebra used in [GS], and [Bo].
In general, for any directed algebra B, the multiplication pairings give each of the spaces B µµ an associative algebra structure. Similarly, for each pair µ ν, the space B µν acquires a (B µµ , B νν )-bimodule structure. Note that, for any µ ν λ, the mutiplication pairing descends to a well defined map B µν ⊗ Bνν B νλ → B µλ . Example 6.1.1. Let B = λ∈Λ B λ be an ordinary Λ-graded associative algebra. For any pair µ ν, put B µν := B µ−ν . Then, the bigraded space ♯ B = µ ν B µν has a natural structure of directed algebra. It is called the directed algebra associated with the graded algebra B.
We say that a directed algebra B is filtered provided, for each pair µ ≻ ν, one has an ascending Z-filtration F qB µ,ν , on the corresponding component B µ,ν , and multiplication pairings B µ,ν ⊗ B ν,λ → B µ,λ respect the filtrations. There is an associated graded directed algebra gr B, with components (gr B) µν := i∈Z F i B µν /F i−1 B µν .
Given a directed algebra B = µ ν B µν , one has the notion of an Λ-graded B-module. Such a module is, by definition, an Λ-graded vector space M = ν∈Λ M ν equipped, for each pair µ ν, with an 'action map' act µ,ν : B µν ⊗ M ν → M µ , that satisfies a natural associativity condition for each triple µ ν λ. Such a module M is said to be finitely generated if there exists a finite collection of elements m 1 ∈ M ν 1 , . . . , m p ∈ M νp such that, one has We let grmod(B) denote the category of finitely generated Λ-graded left B-modules. Given a Λ-graded B-module M , we put Spec M := {ν ∈ Λ | M ν = 0}. It is clear that, for any M ∈ grmod(B), there exists a finite subset S ⊂ Λ such that we have Spec M ⊂ S + Λ + . We say that M is negligible if there exists ν ∈ Λ such that (ν + Λ + ) ∩ Spec M = ∅. Let tails(B) be the full subcategory of grmod(B) whose objects are negligible B-modules.
An directed algebra B is said to be noetherian if, for each λ ∈ Λ, the algebra B λ is left noetherian and, moreover, B λν is a finitely generated left B λ -module, for any λ ν. It is known, see Boyarchenko [Bo,Theorem 4.4(1)], that, for a noetherian directed algebra B, the category grmod(B) is an abelian category, and tails(B) is its Serre subcategory. Thus, one can define Qgrmod(B) := grmod(B)/tails(B), a Serre quotient category.
Fix α ∈ Λ. Given a directed algebra B = µ ν B µν , resp. a Λ-graded B-module M = µ M µ , put B α = µ ν α B µν , resp. M α = µ α M µ . Thus, B α may be viewed as a directed subalgebra of B which has zero homogeneous components (B α ) µν unless µ ν α. Similarly, M α may be viewed as a B-submodule in M . If B is noetherian then B α is clearly noetherian as well.
One easily proves the following result Proposition 6.1.3. Assume that the pair (Λ, Λ + ) satisfies the following two conditions: • The semi-group Λ + is finitely generated.
6.2. Geometric example. Let X be a quasi-projective algebraic variety, and write Coh X for the abelian category of coherent sheaves on X. Given an ample line bundle L on X, one defines B(X, L ) := n≥0 Γ(X, L ⊗n ), a homogeneous coordinate ring of X. For the corresponding Proj-scheme, we have Proj B(X, L ) ∼ = X.
Following Example 6.1.1 in the special case where Λ = Z and Λ + = Z ≥0 , we may form the directed algebra ♯ B(X, L ) associated with B(X, L ), the latter being viewed as a Λ-graded algebra. Then, one can construct a natural equivalence of categories Coh X ∼ = Qgrmod( ♯ B(X, L )). (6.2.1) We return to the setting of §1.1. Let h be the Cartan subalgebra for the semisimple Lie algebra g, and fix X + ⊂ h * , the subsemigroup of integral dominant weights. For each λ ∈ X + , we have the line bundle O e S (λ) on the Slodowy variety S. The direct sum A (e) = λ∈X + Γ S, O e S (λ) has a natural structure of X + -graded algebra. For the corresponding multi-homogeneous Proj-scheme, one has S ∼ = Proj A (e). Now, in the setting of §6.1, we put Λ := h * , resp Λ + := X + , and let ♯ A (e) be the directed algebra associated to the X + -graded algebra A (e). One can show that ♯ A (e) is a noetherian directed algebra and the following multi-homogeneous analogue of the equivalence (6.2.1) holds Coh S ∼ = Qgrmod( ♯ A (e)). (6.2.2) 6.3. We are going to produce a family of quantizations of the directed algebra ♯ A (e). To this end, we exploit translation bimodules A ν+λ ν introduced in §5.2. Given ν ∈ h * , we associate to our nilpotent element e ∈ g a directed algebra A(e, ν) := µ,λ∈X + A µ+λ+ν λ+ν , i.e., using directed algebra notation, for any α β 0, we put A αβ := A α+ν β+ν . The resulting directed algebra A(e, ν) has the following properties: (i) Each homogeneous component A µ+ν λ+ν , of A(e, ν), comes equipped with a Kazhdan filtration K qA µ+ν λ+ν , by nonnegative integers, such that the multiplication pairings respect the filtrations.
(ii) For all λ ∈ X + , one has A λλ = A λ+ν is the Premet algebra associated to the central character that corresponds to the weight λ + ν via the Harish-Chandra isomorphism.
The equivalence of the theorem may be thought of as some sort of the Beilinson-Bernstein localization theorem for Slodowy slices, cf. Introduction to [GS] for more discussions concerning this analogy.
A different approach to a result closely related to Theorem 6.3.2 was also proposed by Losev in an unpublished manuscript.
that sheaf a C × -equivariant structure. Thus, Var M := Supp(qgr M ) is a •-stable closed algebraic subset in S. Now, let ν be an arbitrary, not necessarily dominant and regular, weight. Then, we find a sufficiently dominant integral weight µ such that ν + µ is a dominant and regular weight. Given a Λ-graded A(e, ν)-module M , we may view M ν+µ as a Λ-graded A(e, ν + µ)-module. A good filtration on M induces one on M ν+µ , and we have gr(M ν+µ ) = (gr M ) ν+µ . Thus, one may apply all the above constructions to the A(e, ν + µ)-module M ν+µ . This way, one defines the set Var M in the general case where ν is an arbitrary weight.
One has the following standard result Sketch of Proof. The coisotropicness statement in part (i) is a special case of Gabber's "integrability of characteristics" result [Ga]. Part (ii) is an analogous to a well known result of Borho-Brylinski (cf. [BoBr], proof of Proposition 4.3). A key point is that the map π : S → S is proper. To prove part (iii), let λ ∈ X + . Lemma 5. In general, let F be an arbitrary coherent sheaf on S. By definition of the equivalence (6.2.2), in Coh S, one has an isomorphism Φ λ∈X + Γ( S, O e S (λ) ⊗ O e S F) = F. Applying this observation to the sheaf F := gr K V| e S , and using isomorphisms (6.4.2)-(6.4.3), we deduce part (iii) of Proposition 6.4.1.
To proceed further, we introduce the following terminology. A coherent sheaf F, on a reduced scheme X, is said to be reduced if the annihilator of F is a radical ideal in O X . Definition 6.4.4. An object V ∈ Qgrmod(A(e, ν)) is said to have regular singularities if there exists a representative M ∈ grmod(A(e, ν)), of V , and a good filtration on M such that the corresponding sheaf qgr M ∈ Coh S is reduced and, moreover, Supp(qgr M ) is a Lagrangian subvariety in S.
Next, recall the Steinberg variety Z and observe that Z ∩( S × S) is a Lagrangian subvariety in S × S. An analogue of Proposition 6.4.1 for bimodules yields the following result. In part (ii) above, we have abused the notation Wh m m and, given a left (U β ⊗ U γ )-module N , write Wh m m N for (m χ ⊗ 1)-invariants in N/(1 ⊗ m χ )N. Transition from left D β,γ -modules to (Ug, Ug)-bimodules can be carried out as explained eg. in [BG, §5], esp. Lemma 5.4 and formula (5.5).