WALL-CROSSING FUNCTORS and D-MODULES Contents 0

REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 3, Pages 1–31 (January 11, 1999) S 1088-4165(99)00063-1

WALL-CROSSING FUNCTORS AND -MODULES D

ALEXANDER BEILINSON AND VICTOR GINZBURG

Abstract. We study Translation functors and Wall-Crossing functors on inﬁ- nite dimensional representations of a complex semisimple Lie algebra using - modules. This functorial machinery is then used to prove the Endomorphism-D theorem and the Structure-theorem; two important results were established earlier by W. Soergel in a totally diﬀerent way. Other applications to the category of Bernstein-Gelfand-Gelfand are given, and some conjectural re- lationshipsO between Koszul duality, Verdier duality and convolution functors are discussed. A geometric interpretation of tilting modules is given.

Contents 0. Introduction 1 1. Translation functors for the extended enveloping algebra 3 2. Translation functors via -modules 6 3. Properties of translationD functors 10 4. Applications to the category 12 5. Convolution of Harish-ChandraO modules 15 6. Projective functors 23 References 30

0. Introduction We will be concerned here with infinite dimensional representations of a complex semisimple Lie algebra g. In more detail, let Ug be the universal enveloping algebra of g, and let Z(g)be the center of Ug. We consider the category of Ug-modules annihilated by a great enough (unspecified) power of a maximal ideal in Z(g). It has been observed by many people during the 1970’s (see e.g., [W], [LW], [Ja], and [Z]) that various results can be usefully transferred between the categories corresponding to two different maximal ideals, using tensor products with finite-dimensional representations. The most relevant for us is the work of Jantzen [Ja], who introduced certain functors between the two categories, called translation functors. Jantzen showed that if both maximal ideals satisfy certain regularity and integrality conditions, then the translation functor establishes an equivalence of the two categories. If one of the two ideals is regular while the other is not, the corresponding translation functor is

Received by the editors March 23, 1998 and, in revised form, September 25, 1998 and December 7, 1998. 1991 Mathematics Subject Classiﬁcation. Primary 05E99, 17B37.

c 1999 American Mathematical Society

1 2 ALEXANDER BEILINSON AND VICTOR GINZBURG no longer an equivalence. The composition of the translation functor that sends the category at a regular maximal ideal to the category at a non-regular maximal ideal with the translation functor acting in the opposite direction is called a wall-crossing functor. The terminology stems from the identification of (integral) maximal ideals of Z(g) with Weyl group orbits in the weight lattice of the maximal torus. Non- regular ideals correspond to the orbits contained in the union of walls of the Weyl chambers. In this paper we will be mainly interested in the “most singular” case, where the non-regular maximal ideal corresponds to the fixed point of the Weyl group, that is the point contained in all the walls. Our study was partly motivated by trying to understand two important results, the “Endomorphism-theorem” and the “Structure-theorem,” proved by W. Soergel [S1] in the course of the proof of the Koszul duality conjecture; see [BG], [BGS]. Soergel’s argument was very clever, but rather technical. A shorter proof of the Endomorphism-theorem was found by J. Bernstein [Be]. An alternative entirely geometric approach to the Structure- theorem, based on perverse sheaves and leaving almost all of the representation theory aside was given in [Gi2]. We propose below new proofs of both theorems in the framework of representation theory, and based on the technique of -modules. An advantage of our approach is that the bulk of the argument goes throughD in a quite general setting, while the proofs in [S1] and [Be] relied heavily on some special features of the category right from the beginning. One of our goals is to convince the reader (or at least ourselves)O that the results in question are not so “hard,” and that they follow quite naturally from the g-module -module correspondence combined with some basic functoriality properties. ←→D It would be very interesting to apply the general results obtained in Section 3 below to the category of Harish-Chandra modules instead of the category .It is also tempting to extend our results to representations of affine Lie algebrasO (cf. [FrMa]). We remark that the -module approach of the present paper provides natural and simple proofs of mostD of the results concerning tilting modules (over the finite-dimensional semisimple Lie algebra g), proved by Soergel in [S2] for tilting modules over an affine Lie algebra. The main problem of extending our results to the affine case is that the category at the regular positive level is known ([KT1]) to be related to -modules on one affine flag manifold (which is the union of cells of finite codimension);D the category at the regular negative level is known ([KT2]) to be related to -modules on another affine flag manifold (which is the union of cells of finite dimension),D and the category at the “most singular” = critical level is expected [FM] to be related to -modules on yet another, so-called periodic flag manifold, (which is the union ofD cells of semi-infinite dimension). It is therefore especially intriguing how to find a geometric construction of translation functors to and from the “most singular” maximal ideal. We now briefly outline the contents of the paper. In Section 1 we introduce a Galois extension of Ug with the Galois group W , the Weyl group. We com- pare modules over Ug with those over the Galois extension. Section 2 provides a self-contained exposition of translation functors from the geometric view-point. Although some of the results of this section are undoubtedly known to experts, we could not find a relevant reference in the literature. Section 3 is the heart of the paper containing our main results about wall-crossing functors. In Section 4 we WALL-CROSSING FUNCTORS AND -MODULES 3 D prove the Endomorphism-theorem and the Structure-theorem of Soergel and give some applications to the category of Bernstein-Gelfand-Gelfand [BGG]. Section 5 is devoted to various convolutionO functors. A convolution functor was first introduced in the geometric setting of perverse sheaves back in 1980, indepen- dently by Beilinson-Bernstein, Brylinski, Lusztig, MacPherson and others, in the course of the proof of the Kazhdan-Lusztig conjecture. An algebraic counterpart of that convolution in terms of Harish-Chandra bi-modules was considered in [Gi1]. It was claimed in [Gi1] (without proof) that the algebraic convolution of Harish- Chandra bi-modules goes into the geometric convolution under the localization functor of [BB1]. This claim turns out to be not quite correct, and in Section 5 we establish a precise relationship between the algebraic and the geometric convolution functors, respectively. We also propose conjectural relationships (see Conjecture 5.18 and Theorem 5.24) between the various convolutions, Koszul duality of [BGS], and the Verdier duality functors. In Section 6 we express projective functors defined in [BeGe] in terms of the convolution functors studied in the previous section. We establish a direct connection between projective functors and projective Harish-Chandra bi-modules (somewhat analogous to a result of [BeGe] that motivated the name “projective functor”). We discuss some applications. In particular, we introduce tilting Harish-Chandra modules, establish their relation to the tilting objects of the categery , and derive their basic properties using convolution functors. O The present paper grew out of an unpublished chapter of one of the (many) preliminary versions of [BGS], written in 1992. That chapter was not directly related to the subject of [BGS], and we decided to publish it separately to keep the size of [BGS] to a minimum.

1. Translation functors for the extended enveloping algebra Throughout the paper we fix g, a complex semisimple Lie algebra. We write h for the corresponding abstract Cartan subalgebra. The reader should be warned that h is not a subalgebra of g; it is defined (see e.g. [CG, p. 137]) as the quotient of a Borel subalgebra modulo its nil-radical, and this quotient is independent of any choices. Recall further (see loc. cit.)thath∗, the dual space, comes equipped with a root system R h∗ which has a preferred choice of simple roots. Thus there ⊂ is a well defined element ρ h∗= half-sum of positive roots.Anelementλ h∗is called ρ-dominant if λ + ρ,∈αˇ 0,for every positive corootα ˇ. ∈ Let W be the Weylh groupi≥ of g, the Coxeter group generated by reflections with respect to roots. The group W acts naturally on h and on h∗.Inthispaperwe will always use the so-called dot-action of W on h∗. The dot-action of w W ∈ is obtained by “twisting” the standard w-action h∗ λ w(λ) as follows: λ w λ := w(λ + ρ) ρ. Thus, the point ( ρ) is the3 unique7→ W -fixed point of the7→ · − − dot-action. Given λ h∗,letWλ W denote the isotropy group, and λ W/Wλ the W -orbit of λ with∈ respect to⊂ the dot-action. | |' The dot-action on h∗ induces a W -action on the polynomial algebra C[h∗]. Let W C[h∗] C[h∗] be the subalgebra of W -invariant polynomials on h∗.Wehave ⊂ W Specm C[h∗] = h∗/W , where Specm stands for the maximal spectrum of a commutative C-algebra. Given λ h∗,welet λ C[h∗] denote the maximal ideal of ∈ J ⊂ all polynomials vanishing at λ. We will often identify C[h∗] with Sh, the symmetric algebra on h. 4 ALEXANDER BEILINSON AND VICTOR GINZBURG

Let Z(g) be the center of the universal enveloping algebra Ug. The Harish- Chandra isomorphism Z(g) ShW gives rise to the following composition of algebra morphisms: ' W χ : Z(g) ∼ S h , S h ∼ C [ h ∗ ] . −→ → −→ W We will identify Z(g) with the image of χ, the subalgebra C[h∗] of dot-invariant polynomials. Thus, the map Specm C[h∗] Specm Z(g) induced by the imbedding → χ can (and will) be identiﬁed with the projection π : h∗ h∗/W .Givenλ h∗, ∈ we will often view the W -orbit λ as a point of the orbi-space h∗/W , and we let 1 | | I λ = π− ( λ) Specm Z(g) denote the corresponding maximal ideal in Z(g). | | J ∈

Deﬁnition. The extended enveloping algebra is deﬁned by U := Ug Z(g) C[h∗]. ⊗ Thus U contains both Ug and C[h∗] as subalgebras ande the subalgebra C[h∗] coincides with the center of U. Furthermore, the W -action on C[h∗] gives rise to a W -actione on U such that U W =Ug.

Given λ h∗,letModλ(Ue g), resp. Modλ(U), denote the category of finitely- ∈ | | e e n n generated Ug-modules (resp. U-modules) M such that I λ M =0,resp. λ M= e | | · J · 0, for great enough n = n(M) 0. Restricting U-modules to Ug-modules yields e an exact functor Resλ :Modλ(U) Mod λ (Ug). −→ | | W Fix λ h∗. We introduce the intermediate algebrae of Wλ-invariants: C[h∗] W ∈ ⊂ C[h∗] λ C[h∗]. We have e ⊂ Wλ U Ug [h ]W C[h∗] Ug [h ]W C[h∗] [h ]Wλ C[h∗] (1.1) ' ⊗C ∗ ' ⊗C ∗ ⊗C ∗ Wλ U [h ]Wλ C[h∗]. e ' ⊗C ∗ W W Geometrically, introducing the intermediate algebra C[h∗] , C[h∗] λ , C[h∗] e → → corresponds to the factorisation of the projection h∗ h∗/W as the composition Wλ W h∗ h∗/Wλ h∗/W .Let := C[h∗] λ λ be the corresponding max- Jλ ∩J imal ideal. Observe that the projection h∗/W h∗/W is unramified over λ . λ | | Hence, the I λ -adic completion of the algebra Z(g) is isomorphic canonically to | | Wλ Wλ the λ -adic completion of the algebra C[h∗] . This yields a canonical algebra J Wλ isomorphism between the I λ -adic completion of Ug and the λ -adic completion | | W W J W of U λ .Now,letModλ(U λ) be the category of finitely-generated U λ -modules annihilated by a great enough power of the ideal Wλ . For any M Mod (U Wλ ), Jλ ∈ λ theeU Wλ -action on M cane be uniquely extended, by continuity, to ane action of the Wλ e λ -adic completion; similarly, the Ug-actiononanyM Mod λ (Ug)canbeex- J ∈ | | tendede to an action of the I λ -adic completion. The completions being isomorphic, we obtain | |

Lemma 1.2. The Ug-action on any module M Mod λ (Ug) can be extended ∈ | | canonically to a U Wλ -action so that the following restriction functor is an equivalence of categories:

e Wλ Modλ(U ) Mod λ (Ug) . −→ | | W We will often view Ug-modulese as U λ -modules via the lemma. 1 Observe next that for the projection π : h∗ h∗/Wλ we have π− π(λ)=λ. Wλ → ◦ It follows that for any M Mod (Ue ) one has: U W M Mod (U). The ∈ λ ⊗U λ ∈ λ e e e e WALL-CROSSING FUNCTORS AND -MODULES 5 D functor U W ( ) is clearly the left adjoint of the restriction functor Mod (U) ⊗U λ · λ −→ Wλ Modλ(U ). Thus, decomposing the functor Resλ :Modλ(U) Mod λ (Ug)as e −→ | | e Wλ e the composition Modλ(U) Modλ(U ) Mod λ (Ug) and taking adjoints, we get e → → | | e e e Lemma 1.3. The functor M U M is the left adjoint of the functor Res , 7→ ⊗U Wλ λ where M is viewed as a U Wλ -module via Lemma 1.2. e e We will call a weight λ h∗ e ∈ integral if λ, αˇ Z,for any corootα ˇ; • regular if Wh =i∈1 . • λ { } Let E be a finite-dimensional g-module and M Mod λ (Ug), λ h∗. Then, the ∈ | | ∈ Ug-module E C M is annihilated by an ideal in Z(g) of finite codimension [Ko]. Hence there is⊗ a canonical finite direct sum decomposition:

E M = pr µ (E M) , where pr µ (E M) Mod µ (Ug). ⊗ | | ⊗ | | ⊗ ∈ | | µ h /W | |∈M∗ Fix integral ρ-dominant weights λ, µ h∗, and let Eλ µ be an irreducible finite- dimensional g-module with extreme weight∈ λ µ. Following− [Ja], define the translation functor − λ λ θµ :Modµ (Ug) Mod λ (Ug)byθµM=prλ(Eλ µ M). | | −→ | | | | − ⊗ µ λ λ µ The functor θλ is both the left and the right adjoint of θµ. The functors θµ and θλ are both exact. By Lemma 1.2, the functor θλ can be viewed as a functor Mod (U Wµ ) µ µ → W Wµ Wµ Modλ(U λ ). The action of any z C[h∗] on M Modµ(U )isaUg-module ∈ ∈ λ e λ endomorphism, hence induces by functoriality an endomorphism θµ(z):θµM λ → θµM. Onee can describe this endomorphism in terms of the algebrae automorphism Tλ µ : C[h∗] C[h∗] induced by the affine translation: x x +(λ µ),x h∗, as− follows. → 7−→ − ∈ Assume in addition that Wλ Wµ. Then, the automorphisms T (λ µ) preserve ± − Wλ ⊂ Wµ Wλ Wµ the subalgebra C[h∗] . We also have C[h∗] C[h∗] , hence T (λ µ)C[h∗] W ⊂ ± − C[h∗] λ . In the next section we will prove the following result of Soergel ⊂[S1, Thm. 8] using the -module approach. D W Proposition 1.4. Let z C[h∗] µ .Then ∈ Wµ λ λ (i) For any M Modµ(U ) and m θµM we have z m =[Tλ µθµ(z)] m; ∈ ∈ · − · Wλ µ µ (ii) For any M Modλ(U ) and m θλ M we have z m =[θλ(Tλ µz)] m. ∈ e ∈ · − · From now on, fix integral ρ-dominant weights λ, µ h∗, such that Wλ Wµ, and + λ µe ∈ ⊂ set θ := θµ,andθ− := θλ . We are going to extend the functors θ± to U-modules. First, consider the composition of functors

Resλ θ− e Modλ(U) Mod λ (Ug) Mod µ (Ug) . −→ | | −→ | | The action of an element a C[h∗] U on M Modλ(U) induces, by functoriality, e∈ ⊂ ∈ an endomorphism θ−(a):θ−(ResλM) θ−(ResλM). We deﬁne a C[h∗]-action on → θ−(ResλM) by the formula: e e

a m := θ−(Tµ λa) m, m θ−(ResλM). ∗ − · ∈ 6 ALEXANDER BEILINSON AND VICTOR GINZBURG

W When restricted to the subalgebra C[h∗] µ C[h∗], this action coincides, by Propo- Wµ ⊂ Wµ sition 1.4(ii), with the C[h∗] -action on θ−(ResλM) arising from the U -module W structure. Therefore, combining together the C[h∗]- and the U µ -actions we get a Wµ eWµ U [h∗]-action which factors through an action of the algebra U W C C [h ] µ ⊗ e ⊗C ∗ C[h∗]. In view of isomorphism (1.1), that gives a U-action on θ−(ResλM). This e e way we obtain an exact functor θ− :Modλ(U) Modµ(U). → + Resµ e θ Next, consider the composition Modµ(U) Mod µ (Ug) Mod λ (Ug). For | | | | e e−→+ e −→ any N Modµ(U), deﬁne a C[h∗]-action on θ (ResµN)by ∈ + e + a m:= [Tλ µθ (a)] m, m θ (ResµN). e ∗ − · ∈ W Again, when restricted to the subalgebra C[h∗] µ , this action coincides, by Propo- sition 1.4(i), with the action arising from the natural U Wλ -module structure on + W θ (ResµN). This is not necessarily the case, however, for the full algebra C[h∗] λ W ⊃ C[h∗] µ ; the two actions might be diﬀerent! Thus, we pute

+ + Wλ θ N = n θ (Resµ N) a n = a n, a C[h∗] , r { ∈ ∗ · ∀ ∈ } and dually, e + + Wλ + θ N = θ (Resµ N)/ a n a n a C[h∗] ,nθ(Resµ N) . ` { ∗ − · ∈ ∈ } + The Ug-action and the C[h∗]-action on θ (Res µ N) clearly induce similar actions on +e + both θ` N and θr N. Moreover, by construction, these actions ﬁt together, making + + θ` N and θr N into U-modules. This way we obtain two functors e e θ+, θ+ :Mod(U) Mod (U) . e e e ` r µ → λ These functors are not exact in general. Notice however that if both λ and µ are + e +e + e e regular, then one has θr = θ` = θ . Proposition 1.5. The functors θ+ and θ+ are, respectively, the left and the right e e ` r adjoint of the functor θ−. e e Proof. Let M Mod (U)andN Mod (U). Set M := Res M, N := Res N. λe µ λ µ Then we have ∈ ∈ f e e e f e Hom (θ−M,N)= f HomUg(θ−M,N) f(a x)=a f(x), a [h∗] Modµ(U) C { ∈ + ∗ · ∀ ∈ } = g HomUg(M,θ N) g(a m)=a g(m), a C[h∗] . e e f e { ∈ · ∗ ∀ ∈ } By Ug-linearity of the morphism g in the last line above, and Lemma 1.2, we get W W g(a m)=a g(m), a C[h] λ . Hence, a g(m)=a g(m), a C[h∗] λ, · · ∀ ∈ ∗ · ∀ ∈ and Image(g) θ+N.Thus,g Hom (M,θ+N). ⊂ r ∈ Modλ(U) r The second adjunction is proved in a similar way. e e f e e

2. Translation functors via -modules D For a complex variety X let X and X denote the sheaves of regular functions and regular differential operatorsO on X Drespectively. Let G B U be the simply-connected semisimple Lie-group corresponding to g, a Borel⊃ subgroup⊃ of G, and the unipotent radical of B respectively. Let T = B/U be the abstract maximal torus; see [CG, p. 303]. Set = G/U and = G/B,the B B e WALL-CROSSING FUNCTORS AND -MODULES 7 D flag manifold of G. There is a natural T -action on on the right that makes the canonical projection π : G/U G/B a principal G-equivariantB T -bundle. → Let π be the sheaf-theoretic direct image of e to . The right T -action • D D B on inducesB a T -action on the sheaf π by algebraB automorphisms and we let B e • D e π denote the subsheaf of T -invariantB sections of π .Thus, is a sheaf ofD⊂ algebrase •DB on . e • DB D B e The infinitesimale left g-action and the infinitesimal right h-actione on ecommute, giving rise to a homomorphism of the Lie algebra g h into global algebraicB vector × fields on . This Lie algebra homomorphism can be extended to ane associative B algebra homomorphism Ug Uh Γ( , )T (= the algebra of right T -invariant ⊗C → B D global differentiale operators on ). It turnsB out that, for any z Z(g), the differ- B e ∈ ential operator on corresponding undere the homomorphism above to the element z 1 Ug Uh isB equal to thee differential operator corresponding to the element ⊗ ∈ ⊗ W 1 χ(z) Ug Uhe,whereχ:Z(g) ∼ U h is the Harish-Chandra isomorphism. ⊗ ∈ ⊗ −→ It follows that the homomorphism above factors through Ug Uh U (we ⊗Z(g) ' identify Uh with C[h∗]). Furthermore, the associated graded map of “principal symbols” has been shown [BoBr] to be a bijection, so that one obtains an algebrae isomorphism:

(2.1) U ∼ Γ( , )T =Γ( , ). −→ B DB B D Observe that since the toruse T commutese e with itse own Lie algebra action, the image of 1 Uh C[h∗] is contained in the stalk of the sheaf at any point of ⊗ ' D . Notice further that the embedding C[h∗] , thus defined is central. Hence, B → D for any λ h∗, we may define the category Mod ( ) of coherente (sheaves on of) ∈ λ D B -modules M such that n M = 0 for big enoughe n = n(M) 0. Taking global D Jλ · sections of -modules defines (via (2.1)) a functore Γ :Mod ( ) Mod (U). Let D λ λ D → λ ∆e : M M be the localization functor, which is the left adjoint of Γ . λ 7→ D⊗Γ( , ) λ We recall (seee [BB1]B D and also [BB3, Thm. 3.3.1]) the followinge important theorem:e e e Localization theorem 2.2. (i) If λ is ρ-dominant, then the functor Γλ is exact and the canonical adjunction morphism Γλ ∆λ Id is an isomor- · −→ Modλ(U) phism; (ii) If λ is regular, then Γ gives an equivalence of the categoriese Mod ( ) and λ λ D Modλ(U),andthefunctor∆λ is the inverse of Γλ. e Remark 2.3. To M Mod ( ) assign a -module by the formula M.This e ∈ λ D D D ⊗ -module is clearly smooth along the fibresB of the projection π : B D .That DB e e B→Be e gives, for regular λ, a fully faithful imbedding of the category Modλ(U)intothe categorye of T -monodromic -coherent modules. For more details seee [BB3, 2.5, D §§ 3.3], and [Ka]. B e e Recall that a subcategory of an abelian category is called a Serre subcategory if is a full abelian subcategoryA stable under takingC extensions and subquotients inA. Given a Serre subcategory , one can define a quotient-category / . ThisC is an abelian category equippedA⊂C with an exact functor quot : / CsuchA that =Ker(quot), where Ker (quot) stands for the full subcategoryC→C of Aformed by theA objects A such that quot (A)=0. C 8 ALEXANDER BEILINSON AND VICTOR GINZBURG

Conversely, let F : 0 be an exact functor between abelian categories. Then, Ker F is a SerreC→C subcategory. Moreover, there exists a unique functor F : /Ker F 0, such that the functor F factors as the composition C →C quot F / Ker F 0 . C −→ C −→ C Lemma 2.4. Let F : 0 be an exact functor between abelian categories which C→C has a left (resp. right) adjoint functor F † : 0 .Then,F: /Ker F 0 is C →C C →C an equivalence of categories if and only if the canonical morphism Id F F †, C0 → · (resp. F † F Id ) is an isomorphism. · → C From this lemma applied to the functor F =Γλ, and Localization theorem 2.2, we deduce

Corollary 2.5. Let λ be ρ-dominant. Then, the functor Γλ induces an equivalence

Γ : Mod ( )/Ker Γ ∼ Mod (U) . λ λ D λ −→ λ Remark. See [Ka] for a more detailede description of thee category Ker Γλ in the case of a non-regular λ.

Assume further that λ h∗ is integral. Then λ gives rise to a homomorphism ˙ ∈ λ : T C∗.Let (λ) denote the sheaf on formed by all functions f on such that → O B B e (2.6) f(x t)=λ˙(t)f(x) x ,t T. · ∀ ∈B ∈ Note that (0) = , is the structure sheaf of . More generally, (λ)isthesheaf O OB B e O of sections of a line bundle on .Foraρ-dominant λ,wehaveΓ( , (λ)) = Eλ, an irreducible finite-dimensionalBg-module with highest weight λ. BO Let (λ) denote the sheaf of differential operators acting on (λ) (i.e., a sheaf of twistedDB differential operators on ). Observe next that the sheafO (λ)isstable B O under the natural action on functions of T -invariant differential operators on . B This gives an algebra morphism (λ). One can show (see e.g., [BoBr]) that this morphism induces the followingD−→D isomorphisms:B e e (2.7) (λ) / λ and Γ( , (λ)) = Ug/Ug I λ . DB ' D D·J B DB · | | Given λ, µ h∗ such that µ λ is integral, define a geometric translation functor λ ∈ e e − λ Θµ :Modµ( ) Modλ( ) by the formula Θµ : M (λ µ) M.Wehave D → D λ µ 7→ O − ⊗OB λ (ν) ( ν)= , hence Θµ Θλ =IdMod ( ). Thus, the functor Θµ is always O ⊗OBO− OB · λ an equivalencee of categories.e D e Assume now that λ and µ are integral ρ-dominant weights such that Wλ Wµ. Although the following result seems to be well known, we could not find its⊂ proof in the literature. Proposition 2.8. The following diagram commutes (up to canonical equivalence of functors)

Γλ Modλ( ) Modλ(U) Dµ −→ µ Θλ θλ e Γµ e Modµ( ) Modµ(eU) y D −→ y e e WALL-CROSSING FUNCTORS AND -MODULES 9 D Proof (essentially borrowed from [BB1]). Given a finite-dimensional g-module E, let E = E be the trivial sheaf of E-valued regular functions on .Forsuch B ⊗OB B 1 a function f define a function ϕf : G E by the formula ϕf (g)=g− f(g). → G · The assignment f ϕf identifies E with the sheaf IndBE (on )ofgermsof functions: 7−→ B B 1 (2.9) ϕ : G E ϕ(g b)=b− ϕ(g), b B . { → · · ∀ ∈ } The infinitesimal action of g on G by left translation makes IndGE a g-module. B

By Lie’s theorem, one can find a B-stable filtration E0 E1 ... En = E ⊂ ⊂ G⊂ such that dim(Ei/Ei 1)=1,for all i. This gives a filtration on IndBE by the − G g-stable coherent subsheaves IndBEi defined by replacing E by Ei in (2.9). We have G G G IndBEi/IndBEi 1 = IndB(Ei/Ei 1)= (νi) − − O where νi is the character of B corresponding to the 1-dimensional B-module Ei/ Ei 1. − Assume now that E = Eµ λ is an irreducible g-module with extreme weight − G µ λ and M Modλ(D). We endow the sheaf (IndBE) M with the tensor − ∈ ⊗OB product g-module structure. The filtration E0 E1 ... E as above gives a G ⊂ ⊂ ⊂ G g-stable filtration on (Inde BE) M by the subsheaves (IndBEi) M with ⊗OB ⊗OB quotients of the form (ν) M where ν is a weight of Eµ λ. Clearly, for any O ⊗OB − a C[h∗], for the action of a on (ν) M we have the formula: ∈ O ⊗OB (2.10) a (f m)=f (T a) m, f (ν),m M, · ⊗ ⊗ ν · ∈O ∈ where Tν denotes the affine translation by ν on C[h∗] introduced before Proposi- tion 1.4. In particular, Z(g)actson (ν) Mvia the (generalized) infinitesimal O ⊗ character λ + ν h∗/W . We now| use the|∈ following result [BeGe, Lemma 1.5(iii)]: Let ν, ν0 h∗ be such that λ + ν is dominant, λ + ν0 is W -conjugate to λ + ν, and ∈ ν0 ν in some euclidean W -invariant metric on h∗. Then ν0 W ν. k k≤k k k·k ∈ λ · The result above implies that, for any weight ν0 of Eµ λ other than µ λ,the − − point λ+ν0 is not W -conjugate to µ. Hence, the subquotient sheaf (µ λ) M G O − ⊗OB splits off from IndBEµ λ M as a sheaf of Z(g)-modules, and moreover we have − ⊗OB G (2.11) pr µ IndBEµ λ M = (µ λ) M, M Modλ(U) | | − ⊗OB O − ⊗OB ∈ where pr µ stands for the projection to the µ -isotypic component of a Z(g)- | | e module. | | Next, observe that, for any E one has G E Γ(M)=Γ(E M)=Γ(IndBE M). ⊗ B ⊗OB ⊗OB Thus, from (2.11) one obtains

µ θλ Γλ(M)=prµ Eµ λ Γ(M) · | | − ⊗ G =prµΓ IndBEµ λ M | | − OB e ⊗ µ =Γ (µ λ) M =Γµ ΘλM. O − ⊗OB · µ Finally, formula (2.10) shows that the C[h∗]-action on θλ Γλ(M) deﬁned in Section µ · 1 corresponds to the natural C[h∗]-action on ΘλM. 10 ALEXANDER BEILINSON AND VICTOR GINZBURG

3. Properties of translation functors

Throughout this section we ﬁx integral ρ-dominant weights λ, µ h∗ such that + λ µ ∈ + λ Wλ Wµ, and use the notation θ := θµ and θ− := θλ, and similarly, Θ := Θµ ⊂ µ and Θ− := Θλ.

Proof of Proposition 1.4. Let M Mod (U). By Theorem 2.2(i), we can ﬁnd a ∈ λ -module such that M =Γλ( ). Part (ii) of the Proposition now follows from DPropositionM 2.8 and formula (2.10)M for ν = eµ λ. − e To prove part (i) take M Mod (U), N Mod (U)andsetM:= Res M, ∈ µ ∈ λ µ Wµ + N := Resλ N.Anelementa C[h] gives rise to an endomorphism θ (a) + ∈ + + ∈ End (θ M), hence, to an endomorphismf eθ (ea)M of HomeUg(θ M,N), via composi-f tion with θ+e(a). We have a commutative diagram:

+ + Hom(θ M,N)=Hom(M, θ−N) = Hom(M, θ−N) = Hom(M, θ−N) = Hom(θ M, N)

+ θ (a)M aM a θ−(Tµ λa) (Tµ λa) + θ−N − − θ M Hom(θ+M,N)=Hom(M, θ N) = Hom(M, θ N) = Hom( M, θ N) = Hom( θ+M,N). y y − y − y − y + Thus, we deduce that θ (a)=Tµ λa. Here are the most important properties− of the translation functors.

Proposition 3.1. (i) The functor θ− : Mod (U) Mod (U) induces an equiva- λ → µ lence Modλ(U)/Ker θ− ∼ Modµ(U); (ii) The adjunction morphisms−→ inducee isomorphismse of functors:e e e e + + θ− θ =Id = θ− θ ; · ` Modµ(U) · r (iii) We have a naturale isomorphisme of functors:e e eθ+ =Γ Θ+ ∆ . ` λ· · µ Proof. We begin with a general remark. Let be an abelian category, and let C e 2 1 be two Serre subcategories. Then the various quotient categories are Crelated⊂C by⊂C the following canonical transitivity isomorphism (3.1.1) / ( / )/( / ) . C C1 ' C C2 C1 C2

We set := Mod ( ). Since the geometric translation functor Θ− :Mod( ) C λ D λD ∼ Mod ( ) is an equivalence of categories, we may (and will) identify the cate- −→ µ D gory Modµ( ) with .e Further, by Corollary 2.5, we have equivalences e eD C Mod ( )/Ker Γ ∼ Mod (U) , Mod ( )/Ker Γ ∼ Mod (U) . λeD λ −→ λ µ D µ −→ µ Thus, setting := Ker Γ and := Ker Γ we can rewrite the above as follows: eC1 µ C2 e λ e e (3.1.2) / Mod (U), / Mod (U) . C C1 ' µ C C2 ' λ Moreover, Proposition 2.8 insures that and that the natural functor / e C2 ⊂C1 e C C1 → / gets identiﬁed under the equivalences (3.1.2) with the functor θ− :Mod (U) C C2 λ → Modµ(U). Part (i) of Proposition 3.1 now follows from the transitivity equivalence (3.1.1) applied to our categories. With part (i) being established, Lemmae 2.4 yieldse part (ii)e of Proposition 3.1. WALL-CROSSING FUNCTORS AND -MODULES 11 D To prove (iii) we take adjoints in the diagram of Proposition 2.8. This yields the following commutative diagram:

∆λ Modλ( ) Modλ(U) + D ←− + (3.2) Θ θ` e ∆µ e Modxµ( ) Modµx(eU).  D ←−  The diagram implies readily θ+ =Γ Θ+ ∆ ,wherewehaveusedthatevery ` e λ· · µ e object of the category Modµ( ) is isomorphic to an object of the form ∆µ(M), by Theorem 2.2(i). Part (iii) of PropositionDe 2.8 is proved. One may reverse the logic slightlye to derive parts (i) and (ii) of Proposition 3.1 in an alternative way, as follows. First of all one establishes part (iii) of the proposition the same way as above. With the formula of part (iii) at hand we calculate: + + + θ− θ = θ− Γλ Θ ∆µ =Γµ Θ− Θ ∆µ =Γµ ∆µ =Id , · ` · · · · · · · Modµ(U) where the ﬁrst equality holds by Proposition 3.1(iii), the second is Proposition 2.8, e e e e and the last one is due to Theorem 2.2(i). This gives part (ii), and part (i) now follows from the “if” part of Lemma 2.4. Corollary 3.3. For M Mod (U), the following conditions are equivalent: ∈ λ (i) For any N Modλ(U), the following natural morphism is an isomorphism ∈ e Hom (M,N) ∼ Hom (θ−M,θ−N); Modeλ(U) −→ Modµ(U) + (ii) There exists an M 0 eMod (U) such that M =eθ (M 0). ∈ µ `e e Proof. Observe that the canonical adjunction morphism θ+ θ M M is an e e ` − isomorphism if and only if the induced morphism · → + e e Hom(M,N) Hom(θ θ−M,N)=Hom(θ−M,θ−N) → ` · is an isomorphism, for every N. In the latter case we may put M 0 = θ−M.Con- + e e+ + e +e + versely, if M = θ` (M 0), then we have θ` θ−M = θ` (θ− θ` M 0)=θ` M0 =M (by Proposition 3.1(ii)), and the result follows.· · · e e e e e e e e Recall that to any M Mod µ (Ug) we can associate (cf. Lemma 1.3) the U- ∈ | | module M := U U Wµ M. Note that the module M has a natural Wλ-action induced ⊗ e from the Wλ-action on U, which commutes with the Ug-action. By functoriality, f e e f+ this gives a Wλ-actionontheUg-module Resλ θ` (M). We have e · Proposition 3.4. There is a functorial isomorphism: e f W + + λ θ M = Resλ θ` (U Wµ M) ,M Mod µ (U). · ⊗U ∈ | | e Proof. It is clear that theree is ae functor isomorphism θ− Resλ =Resµ θ− : · · Modλ(U) ∼ Mod µ (Ug). Taking adjoints on each side, and using Lemma 1.3 we obtain−→ | | e e + + + + θ]M = U W (θ M )=θ (U W M)=θ (M). ⊗U λ ` ⊗U µ ` The isomorphisms above are compatible with the Wλ-actions. We take Wλ-invari- e e e e e e f ants on each side of the isomorphism. Note that U is a free U Wλ -module isomorphic to [W ] U Wλ as a W -module. Hence taking W -invariants of the leftmost term C λ C λ λ ⊗ e e e 12 ALEXANDER BEILINSON AND VICTOR GINZBURG

+ Wλ + in the above isomorphisms yields U U Wλ (θ M) = θ M. Comparing with the W -invariants of the rightmost term⊗ completes the proof. λ e e Proposition 3.5. There is a functorial isomorphism

+ Wλ θ− θ M Res (U W M). · ' µ ⊗U µ e + + Proof. As in the proof of Proposition 3.4e we have θ]M = θ` (M). The functor

θ− clearly commutes with the functor Res (U U Wµ ) and with the W -actions. Hence, we obtain · ⊗ • e f e + e + Res U W (θ− θ M) = θ− Res θ (M) µ ⊗U λ · · λ · ` + e =Res θ− θ (M)=Res (M). e µ· ·e` f µ Taking Wλ -invariants on each side completes the proof. e e f f 4. Applications to the category O Fix a Cartan and Borel subalgebras h b g. Thus we may identify this Cartan subalgebra with the abstract Cartan⊂ subalgebra⊂ b/[b, b] via the composition h , b b/[b, b] . Given λ h∗, we define the category as the full subcategory → ∈ Oλ of Mod λ (Ug) formed by the Ug-modules M such that U| b| -action on M is locally finite, and • Uh-action on M is diagonalizable. • Given a weight µ h∗, view it as a 1-dimensional b-module, Cµ, via the pro- ∈ µ jection b b/[b, b]=h C . Let Mµ := Ug Ub Cµ be the Verma module with highest weight µ.Wewrite−→ L for its simple⊗ quotient. Then, for any w W , µ ∈ we have Mw λ ,Lwλ λ. The category λ is known ([BGG]) to have enough · · ∈O O projectives, and we let Pw λ denote the indecomposable projective cover of Lw λ in . We refer to [BGG] for· more properties of the category . · Oλ Oλ Clearly ρ is the fixed point of the dot-action. The Verma module M ρ is − − simple and is the unique simple object of the category ρ. Moreover, M ρ is also O− − a projective in ρ,for ρit is a ρ-dominant weight. Thus, ρ is a semisimple O− − O− category and any object of ρ is isomorphic to a direct sum of finitely many copies O− of M ρ. − Now let λ and µ denote integral ρ-dominant weights such that Wλ Wµ.The µ λ ⊂ adjoint functors (θλ,θµ): λ µ are exact, hence, take projectives into projectives. Further, it is knownO and easyO to prove (see e.g. [Ja]) that, for any w W , µ ∈ one has θ Mw λ = Mw µ. Therefore, by the exactness, one gets: λ · ·

µ Lw µ if w Wµ w0 θ Lw λ = · ∈ · λ · 0ifw=Ww. ( 6µ·0 λ It follows, by adjunction, that the exact functor Hom λ (θµPµ, )on λ kills all O • O simple modules but Lλ.Thus,weobtain λ Proposition 4.1. θµPµ = Pλ. From now on let λ be an integral ρ-dominant weight. Proposition 4.1 yields (for µ = ρ): − λ (4.2) Pλ = θ ρM ρ. − − WALL-CROSSING FUNCTORS AND -MODULES 13 D

W Let I be the ideal in C[h∗] generated by all W -invariant polynomials without W constant term. Set C = C[h∗]/I , the coinvariant algebra. The W -action on C[h∗] induces a W -action on C making it a regular representation of W .Foreachλ h∗ ∈ we have the subalgebra CWλ C. The results of the previous⊂ section enable us to give a short proof of the following important theorem that was implicitly conjectured in [BG] and was ﬁrst proved in [S1, Thm. 3] (see also [Be]). Endomorphism-theorem 4.3. There is a canonical algebra isomorphism

Wλ End gP C . U λ ' + λ ρ Proof. Set θ = θ ρ and θ− = θλ− . From (4.2) and Proposition 3.5 we get − + + HomUg(Pλ,Pλ)=HomUg(θ M ρ,θ M ρ) − − + =HomUg(M ρ,θ− θ M ρ) − · − Wλ =HomUg M ρ, Res ρ(U Ug M ρ) − − ⊗ − Wλ Wλ Wλ = U /U ρ C . ·J− ' e To complete the proof we must show that the chain of isomorphisms above gives e eW rise to an algebra map EndUg Pλ C λ . To this end, observe that the functor θ− induces a ring homomomorphism:→ (4.3.1)

+ Wλ EndUg Pλ End W (θ−Pλ)=End W (θ−θ M ρ)=End W (U UgM ρ). → U λ U λ − U λ ⊗ − W W Observe furthere that the action ofe the central subalgebrae Seh λ U λ on Wλ Wλ Wλ⊂ U Ug M ρ gives an algebra isomorphism τ : C ∼ End W (U Ug M ρ), ⊗ − −→ U λ ⊗ − since M ρ is a simple Ug-module. The result now follows by composinge (4.3.1) 1− e withe τ − . e Let B G be the Borel subgroup corresponding to the Borel subalgebra b. Write =⊂ B w B/B for the Bruhat cell in the Flag manifold corresponding Bw · · ⊂B to w W .Letjw: w, denote the embedding, and w∗ := (jw) w ,the ∈ B →B M ∗OB standard -module “of holomorphic distributions,” supported on w.LetDMwλ DB B · denote the dual (in the category ) of the Verma module Mw λ. The following result is known;O see e.g. [BB3]. · Proposition 4.4. Let λ be an integral ρ-dominant weight. Then, for any w W λ and y W , there are natural isomorphisms of (λ)-andUg-modules respectively:∈ ∈ D ∆λ(DMw λ) (λ) ww∗ and Γ( , (λ) yw∗ ) DMy λ . · 'O ⊗OB M 0 B O ⊗M 0 ' · W Following [S1], deﬁne an exact functor V : λ C λ -Mod by the formula O → V : M HomUg(Pλ,M), where the Hom-space is viewed as an EndUgPλ-module, 7−→ hence a CWλ -module, via composition. One may reinterpret the functor V as follows. We have by (4.2), V(M)= ρ W HomUg(M ρ,θ− M). The C λ -module structure on V(M) arises from a natural − λ Wλ ρ C[h∗] -action on θλ− M. The latter action was actually deﬁned in Section 1. Namely, view M as a U Wλ -module, via Lemma 1.2. This gives a C[h]Wλ -action on Wλ ρ M which induces a C[h∗] -action on θλ− M, by functoriality. Furthermore, the e 14 ALEXANDER BEILINSON AND VICTOR GINZBURG

W Wλ ρ subalgebra Z(g) C[h] C[h∗] acts trivially on HomUg(M ρ,θλ− M), for it ' ⊂ − W obviously acts trivially on M ρ. Thus, the action on V(M) factors through C λ . Using the results of the previous− sections we can simplify the proof of the key theorem ([S1, Struktursatz 2]) saying that the functor V is faithful on projectives and injectives; we have

Theorem 4.5. For any injective module I λ and any M λ,thenatural morphism ∈O ∈O

HomUg(I,M) HomCWλ (V(I), V(M)) −→ is an isomorphism.

ρ Remark 4.6. The functor θλ− , hence V, clearly commutes with the standard duality N DN on the category . Furthermore, I is an injective in if and only 7−→ O Oλ if DI is a projective in λ. Thus, dualizing Theorem 4.5 we get that, for any projective P and anyO M , the natural morphism ∈Oλ ∈Oλ HomUg(M,P) HomCW (V(M), V(P )) −→ λ is an isomorphism.

Let M M = U W M be the left adjoint of the functor Res :Mod (U) 7→ ⊗U λ λ λ → Mod λ (Ug); cf. Lemma 1.3. The Wλ-action on U induces a Wλ-action on M and, | | e for any M,Nf Mode λ (Ug), there is a canonical isomorphism e ∈ | | e f Wλ (4.7) Hom g(M,N) ∼ Hom (M,N) . U −→ U We will deduce Theorem 4.5 from the following result. e f e + Lemma 4.8. For any injective I λ, the adjunction morphism θ` θ−(I) I + λ ∈O · → (where θ = θ ρ) is an isomorphism. − e e e e Proof of the Lemma. Let DMλ be the dominant dual Verma module, viewed as a U-module via the projection U U/U λ Ug/Ug λ , and let M ρ be − → ·J ' ·J| | + viewed as a U-module in a similar way. By Proposition 3.1(iii) we have, θ M ρ = ` − +e λ e e e θ` (DM ρ)=Γλ Θ ρ ∆ ρ(DM ρ). Hence, Proposition 4.4 yields − e · − · − − e + θ (M ρ)=DMλ. e ` − + It follows from Corollary 3.3 thate the adjunction morphism θ` θ−(DMλ) DMλ is an isomorphism. · → Now, DMλ is an injective in λ. Furthermore, it was showne e in [BeGe, Thm. 3.3(b)] that any injective in OcanbewrittenintheformI=Φ(DM ) for an Oλ λ appropriate projective functor Φ (see Section 6 below). Hence, I = Φ(DMλ)and the lemma follows from Proposition 6.6 (in Section 6), which is independent of the intervening material. e e

Proof of Theorem 4.5. Let I be an injective in λ. Then, for any M λ, from Lemma 4.8 and Corollary 3.3 we obtain an isomorphismO ∈O

Hom (I,M) ∼ Hom (θ−(I), θ−(M)). U −→ C Taking W -invariants on each side and using (4.7) completes the proof. λ e e f e e e f WALL-CROSSING FUNCTORS AND -MODULES 15 D 5. Convolution of Harish-Chandra modules Throughout this section λ and µ stand for integral ρ-dominant weights. Let Ug and U denote the completions of the algebras Ug and U with respect λ λ | | to the I λ -adic and λ-adic topology, respectively (recall that I λ Z(g)and | | J | | ⊂ λ C[h∗] are maximale ideals). Since Ug is a Noetherian algebra,e its I λ -adic completion,J ⊂ Ug is also Noetherian. Hence, the Artin-Rees lemma implies| | that λ any ﬁnitely-generated| | Ug -module is complete with respect to the I -adic topol- λ λ | | | | ogy. Therefore, ﬁnitely-generated Ug -modules form an abelian category. Similar λ | | considerations apply to the algebra Uλ. By a complete Ug-module, resp. U-module, we will mean an Ug-module M which is complete in the I λ -adic, resp. λe-adic, topology, i.e., such that | | J e M = lim proj (M/In M). λ | | ·

AcompleteUg-module is not necessarily annihilated by some power of the ideal I λ , but it is isomorphic to a limit of the projective system of Ug-bimodules: M/I1 M| | λ· ← M/I2 M .... The algebra Ug is an example of a complete Ug-module.| | λ λ · ← | | Furthermore,| | any complete Ug-module M has a natural structure of a Ug -module, λ and we will often make no distinction between these Ug-modules and Ug| | -module λ structures on M. | | Below, finitely generated left Ug Ugopp-modules will be referred to as finitely λ C λ | | ⊗ | | generated Ug -bimodules, and similar terminology will be used for U U opp- λ λ C λ | | ⊗ modules. Let HC λ ,resp. HCλ, be the category of finitely-generated complete | | e e Ug -bimodules (resp. U -bimodules) M such that the adjoint g-action ad x : λ λ | | g m x m m x on M is locally-finite. Any object of HC λ is finitely-generated both7−→ as a· left− and· as a righte Ug -module. Therefore, the Artin-Rees| | lemma insures, λ | | as explained in the first paragraph of this section, that HC λ is an abelian category. | | Similar remarks apply to the category HCλ. To localize complete Ug -bimodules we recall the imbedding [h ] ,and λ C ∗ | | ⊂ D form the sheaf ,the -adic completiong of the algebra . There is an algebra Dλ Jλ D isomorphism Γ( , )=U . For any sheaf of coherent opp-modulese on B Dλ λ M D D ,thespaceΓ(e , ) has a natural U U opp-modulee structure. However, theB×B assignment eB×BΓ( eM , ) is not an exact⊗ functor,e for Γe is not exact on M7→ B×B M right -modules. Thus, one has to localize Ue-bimodulese in a different way, as we now explain.D Wee begin with a well known observatione that if is a left -module on an M DX algebraic variety X,andΩX is the line bundle of top-degree regular forms on X, then the sheaf ΩX X has a natural right X -module structure. Put another way, there is a canonical⊗O M algebra isomorphism D

opp 1 (5.1) ∼ Ω X X Ω − . DX −→ ⊗ O X D ⊗ O X X Now, let X = be the ﬂag manifold. Then Ω = ( 2ρ); further, it is clear that, for any integralB λ, there is a natural isomorphismB O −

(5.2) λ (λ) ( λ). D 'O ⊗OB DB ⊗OB O− e 16 ALEXANDER BEILINSON AND VICTOR GINZBURG

opp ∼ Formulas (5.1)–(5.2) yield an isomorphism λ λ 2 ρ . Taking global sections we get canonical algebra isomorphisms D −→ D − − opp opp e e (5.3) τ : U Γ( , ) ∼ Γ( , λ 2ρ) U λ 2ρ . λ ' B Dλ −→ B D− − ' − − It was shown in [BB3, 3.2.1] that isomorphisms (5.3) are obtained by specializing e e e e at λ a “universal” isomorphism U opp ∼ U . This way one deduces the following result. −→ Lemma 5.4. (i) There is a principale anti-involutione τ : U U that, for any λ, opp → induces the isomorphism U ∼ U λ 2 ρ of (5.3). λ −→ − − (ii) The anti-involution τ induces anti-involutions τg ande τh eon the subalgebras Ug U and C[h∗] U, respectively.e e The anti-involutions τg and τh are given on ⊂ ⊂ generators x g Ug and h h C[h∗], by the formulas: ∈ ⊂ ∈ ⊂ e e x x, h h 2 ρ(h) . 7→ − 7→ − − · W (iii) The Harish-Chandra isomorphism Z(g) ∼ C [ h ∗ ] intertwines the restric- −→ W tion of τg to Z(g) with the restriction of τh to C[h∗] . Recall next that the Weyl group W acts on U and the action of w W induces an ∈ isomorphism of completed algebras w : Uw µ ∼ U µ . Let w0 be the longest element of W . Observe that, for any λ,wehave·w−→(eλ) 2ρ=w(λ+ρ) ρ=w λ. 0 − 0 − 0· Therefore, composing the automorphisme of U inducede by w0 with isomorphism (5.3) we obtain the following isomorphisms

τ oppe opp w0 opp (5.5) Γ( , ) ∼ U U = U U . w0(λ) w 0 ( λ ) w (λ) 2ρ w0 λ λ B D− −→ − ' 0 − · ' Thus, for any w (λ)-module ,thespaceΓ( , ) acquires, via (5.5) a natural eD− 0 e N e B N e e right Uλ-module structure. Similarly, the assignment Γ( , ) gives a e M7→ B×B M functor from the category of left λ w0(λ)-modules to the category of Uλ- bimodules.e Since both λ and w (λD) are Dρ-dominant− weights, this functor is exact, − 0 by Theorem 2.2. It will be denotede by Γ♦ein the future. e Next, recall that a -module on is said to be G-equivariant if is a G-equivariant sheaf ofD -modulesM andB×B the differential of the G-action coincidesM with the g-action on OB×Barising from the -module structure via the imbedding M D g , λ w0(λ).Let λ be the category of sheaves of complete coherent → D D− HC λ w0(λ)-modules on that are G-equivariant with respect to the diago- D D− B×B nal G-actione e on . For any , the adjoint g-action on the U -bimodule B×B M∈HCλ λ Γ(e e, ) is locally-finite because of the G-equivariance of .Thus,Γ♦( )be- B×B M M M comes an object of HCλ. Moreover, the functor Γ♦ : λ HCλ is ane equivalence of categories provided λ is regular (Theorem 2.2). HC → Given an algebragA, resp. algebras A and B,wewritegA-Mod for the category of left A-modules, resp. Mod-A or A-Mod-B for the category of right A- modules, or left A Bopp-modules (= A B-bimodules). We form the bounded − derived categories Db(U -Mod) ,Db(U-Mod-U ), and Db( -Mod) . Let λ λ µ Dλ Dµ DModλ(U), DHCλ, and D λ be the full triangulated subcategories of the cate- b e HCb e e b e e gories D (U-Mod) , (resp. D (Uλ-Mod-Uλ), and D ( λ w0(λ)-Mod) ) formed e g D D− by the objects whose cohomology belong to the categories Modλ(U), (resp. HCλ and ).e The functor Γ♦ hase a naturale extensione to ae derived functor RΓ♦ : HCλ D λ DHCλ. e g HC → g WALL-CROSSING FUNCTORS AND -MODULES 17 D We now introduce certain convolution functors on our categories. First, define b b b a functor D (Uλ-Mod-Uµ) D (Uµ-Mod-Uν) D ( U λ -Mod-Uν ) by the formula L × −→ M?N=M N. It is easy to verify that the functor so defined restricts to a ⊗eUµ e e e e e functor e ? : DHCλ DHCλ DHCλ . × → Observe next that one may regardg anyg object Ag Modλ(U)asanUλ-module n ∈ on which Uλ acts through a certain quotient Uλ/ λ Uλ. Furthermore, given M J · e e ∈ DHCλ, one verifies that if all the cohomology of an object A DModλ(U) are killed by some powere of the ideal , then a similare vanishinge holds∈ for the cohomology Jλ ofgM?A. This allows us to define a functor DHCλ DModλ(U) eDModλ(U) L × → by the formula M,A M?A=M A. For any M,N,L DHCλ,and 7→ ⊗Uλ g e ∈ e A DModλ(U), there are natural functorial isomorphisms (associativity): ∈ e g (M?N)?L M?(N?L)andM?(N?A) (M?N)?A. e ' ' To define analogous functors on -modules we introduce the projections pij : ,andp : D ,i, j 1,2,3 , along the factors not named.B× B×B → B×B i B×B×B → B ∈{ } Let pij∗ ,pi∗ stand for the corresponding inverse image functors in the category of - b O sheaves. Given a triple of weights (λ, µ, ν), and two objects D ( λ µ-Mod), b M∈b D D and D( µ 2ρ ν-Mod), define an object D ( λ ν-Mod) by the formula:N∈ D− − D M∗N ∈ D eD e e e L e e (5.6) := (Rp13) (p12∗ p23∗ ) , • p M∗N M 2∗Dµ N O where (Rp13) stands for the sheaf-theoretic derivede direct image. It is easy to • verify that (5.6) restricts to a functor : D λ D λ D λ . ∗ HC × HC → HC Using the two projections pri : , one also defines a convolution-functor b b B×B → B b : D ( λ µ-Mod) D ( µ 2ρ-Mod) D ( λ -Mod) by the formula ∗ D D × D− − −→ D L :=(Rpr1) ( pr∗ ) . e e e pr µe 2 M∗A • M ⊗ 2∗D A f This convolution restricts to a functor : D λ DModλ( λ) D Modλ( λ). The following observation will be useful∗ forHC us× in the nextD section.−→ D e e Remark 5.7. Any object M DHCλ is completely determined by the corresponding convolution functor ∈ g M? : DModλ(U) DModλ(U),A M?A. → 7→ To see this, we first have to enlargee the categorye Modλ(U) by adjoining all Uλ- modules which are complete in the λ-adic topology. Write Mod[λ(U) for the latter J e e category, and DMod[λ(U) for the corresponding derived category enlargement of e DModλ(U). The functor M?() is continuous in the λ-adic topology, hence e · J extends uniquely to a well defined functor M? : DMod[λ(U) D Mod[ λ(U). e −→ Observe that Uλ Mod[λ(U), as a left module, and clearly we have ∈ e e n (5.8) M = M?Uλ = lim M?(U/ λ U) , M DHCλ . e e J · ∀ ∈ ←−n e e e g 18 ALEXANDER BEILINSON AND VICTOR GINZBURG

Further, notice that multiplication by u U on the right gives an endomorphism ∈ of the left module U/ n U, hence induces by functoriality a right U -action on Jλ · λ M?(U/ n U). This way, formula (5.8) recoverse M from the functor M?(), as Jλ · · a Uλ-bimodule. e e e e e eLet (T ) be the algebra of globally-defined regular differential operators on the D torus T . View Sh C[h∗] as the subalgebra of (T ) consisting of T -invariant differential operators.' For each integer n =1,2,...D , define the (T )-module n = n D n E (T )/ (T ) 0 where 0 denotes the augmentation ideal in C[h∗]. Then, gives Drise toD a sheaf·J of regularJ holonomic -modules on T with unipotent monodromy,E for T is an affine variety. These D-modules form a natural projective system: D 1 2 ... . E E 1 Now, let O be the unique open G-orbit in ,O:= π− (O) the inverse image B×B of O in ,andj:O, the imbedding. The group G acts freely on O B×B →B×B on the left and the space of left cosets, G O, is canonicallye isomorphic to T , due to the Bruhate e decomposition.e e Thus,e we have\ a diagram: e e O (5.9.1) µ π .& T=GOOe \ For each n =1,2,... , define a -module of “multivalued functions” on eD n n B×B n O by = j (µ∗ ). The sheaf π has a natural -module ⊂ B×B O∗ ∗ E e e • O∗ D D structure. For any weight µ h∗,weset ∈ e e e e e e n (5.9.2) µ := lim ( µ w 0 ( µ ) ) π . R D D − • O ∗ ←− n O e e D D We have . The module has the followinge e local description. Locally on Rµ ∈HCµ Rµ O, one can trivialize the projection π : . Using such a trivialization, one can write B×B→B×B e e = T T and µ w0(µ) = C[h∗]µ C[h∗] w0(µ), B×B∼B×B× × D D− ∼ DB×B ⊗ ⊗ − where C[h∗]µ denotes the µ-adic completion of C[h∗]. Accordingly, locally on O, e e J e e e one has µ = C[h∗]µ. An element 1 a1 a2 C[h∗] C[h∗], R OO ⊗ ⊗ ⊗ ∈ DB×B ⊗ ⊗ viewed as an element of µ w0(µ) via the local factorization above, acts on D D− µ = C[h∗]µ as multiplication by 1 a1 w0(τ(a2)); seee (5.3) for the meaning R OO ⊗ ⊗ · of τ(a2). e e We now define convolution functors D λ D λ D λ and D λ HC × HC → HC HC × DModλ(D) DModλ(D) by the formula: → (5.10) ? = w (λ) 2ρ , e e M N M∗R 0 − ∗N where D λ and is an object of either D λ or DModλ(D). Notice that unlikeM∈ the ?-convolutionHC N in (5.10), the -convolution,HC , without the middle ∗ M∗N factor w0(λ) 2ρ is undefined, for the corresponding parameters λ,e µ, ν in (5.6) do not matchR in− the right way. It turns out that the convolution (5.10) on -modules corresponds to the alge- D braic ?-convolution on Uλ-bimodules, i.e., we have the following:

e WALL-CROSSING FUNCTORS AND -MODULES 19 D

Proposition 5.11. For any D λ and D λ,resp. DModλ(U), there is a natural isomorphismM∈ HC N∈ HC N∈ e RΓ♦( ? ) RΓ♦( ) ?RΓ♦( ), resp. RΓ( ? ) RΓ( ) ?RΓ( ) . M N ' M N M N ' M N Proof. We have

RΓ( ? )=RΓ( w (λ) 2ρ ) M N M∗R 0 − ∗N L

(5.12) =RΓ w0(λ) 2ρ RΓ( ) , M∗R − Uλ N O where in the second isomorphism we have that Γ( , )=e U and that the isomor- B Dλ λ phism clearly holds if w0(λ) 2ρ and are replaced by free -modules. M∗R − N e e D To compute the factor RΓ( w (λ) 2ρ ) in (5.12) we apply an extension M∗R 0 − ∗N e of the localization theorem which holds for the derived category of µ-modules even if µ is not dominant; see [BB2, 12]. Recall ﬁrst that we haveD constructed opp § an algebra isomorphism Uλ U w0(λ); see (5.3). It gives rise toe an exact ' − functor opp : U w (λ)-Mod Mod-Uλ that can be extended to derived categories. − 0 e → e b On the other hand, the functor w (λ) 2ρ takes D ( w (λ)-Mod) to 0 − − 0 b e M7→M∗Re Dopp D ( λ 2ρ-Mod). Using the canonical isomorphism λ 2ρ we can view − − − − λ D b D 'b De the latter functor as a functor : D ( w0(λ)-Mod) D (Mod- λ). Thus, Lemmae 5.4 combined with [BB2,∗R theoremD in− 12] yielde → e D § For any ρ-dominant λ the following diagrame commutes: e

b b D ( w (λ)-Mod) ∗R D (Mod- λ) D− 0 −→ D RΓ RΓ Db(Ue -Mod) opp D b (Mod-Ue ) . w0(λ)  λ −y −→ y Applying to (5.12) a bimodule version of this theorem we obtain e e RΓ( , w (λ) 2ρ)=opp(RΓ( , )). B×B M∗R 0 − B×B M L Hence, RΓ( ? ) RΓ( ) RΓ( ), and the proposition follows. M N ' M ⊗Uλ N It should be mentioned, perhaps, that the -convolution given by formula (5.6) e ∗ is a -analogue of a more familiar convolution of holonomic -modules. This latterD convolution is deﬁned by DB×B e L L (5.13) := (p13) (p12∗ p23∗ ) = p12∗ p23∗ , M∗N ∗ M ⊗ N M N p13 OB×B×B Z O where we are using the same notation for the projections pij : , as in (5.6), and stands for a direct image of -modules.B×B×B→B×B Notice that (5.13) p13 D goes under the Riemann-Hilbert correspondence to the convolution on Db( ) R deﬁned by the expression in the middle of (5.13). B×B To proceed further we have to recall some generalities. Recall that there are Verdier duality functors D both on the category of holonomic -modules and on the category of perverse sheaves, and that these two functorsD go to each other under the Riemann-Hilbert correspondence. It is well known that convolution (5.13) commutes with Verdier duality. This is no longer the case for the convolution (5.6) of -modules, due to the fact that is not compact. In more detail, let us restrict toD the special case λ = 0, the only caseB we need. Verdier duality takes an object of e e 20 ALEXANDER BEILINSON AND VICTOR GINZBURG

D 0 into (in general) a direct limit of objects of D 0,anind-object of D 0. OneHC can easily find the commutation relation betweenHC convolution (5.6) and VerdierHC duality on D 0 using the fact that all objects of this category are smooth along HC the fibers of the projection .We have the following result whose proof is left to the reader. B×B→B×B e e Lemma 5.14. For any , D 0 there is a functorial isomorphism: M N∈ HC (D ) (D ) D( )[dim T ] , M ∗ N ' M∗N C where [dimC T ] stands for the shift in the derived category by the dimension of the fiber of the projection . B→B Next, let t be the functor on induced by the flip of the two e D 0 factors of theM7→M manifold . This functor clearlyHC commutes with Verdier duality, and we let Dt denote theB×B composition functor D( t)=(D )t.With this understood, one has the following standard result.M7→ M M

Lemma 5.15. For any , , D 0 there is a functorial isomorphism: M N L∈ HC Hom , Hom , (Dt ) . M∗N L ' M L∗ N In the remainder of this section we are going to formulate a conjecture relating the various convolution functors introduced above to Koszul duality. More precisely, we will be dealing with an extension of the Koszul duality considered in [BGS] to the Harish-Chandra setup; see [S3]. b Write DG( )fortheG-equivariant derived category on , as deﬁned in B×Bmix B×B [BeLu]. Let DG ( )beitsmixed version (see [BGS, 4.3] where it is referred B×B mix § to as a graded version). On the other hand, let denote the mixed version of HC0 the category 0. The following result, conjectured in [BG], is an extension of the main theoremHC of [BGS] to an equivariant setup. Theorem 5.16. There is a contravariant equivalence of triangulated categories

mix mix K : D ( ) ∼ D , G B×B −→ HC0 mix that takes pure perverse sheaves in DG ( )to indecomposable projectives in mix B×B . HC0 Unfortunately, so far no complete proof of this theorem has been written down (although, see [S3]). The remarks below should hopefully help the reader to better understand the result.

Remark 5.17. (a) Let erG ( ) denote the abelian category of G-equivariant P B×B b perverse sheaves on . This is an abelian subcategory of DG( ), yet the b B×B b B×B natural functor D erG ( ) DG( ) is not an equivalence. (b) The categoryP er B×B( )→ is knownB×B to be equivalent to the (proper) sub- P G B×B category of the category 0 (= category λ with λ = 0) formed by the ﬁnitely generated Ug-modules M Osuch that O

The augmentation ideal Z+(g)=Z(g) Ug+ annihilates M; • Ub-action on M is locally finite; ∩ • Uh-action on M is diagonalizable. • WALL-CROSSING FUNCTORS AND -MODULES 21 D (c) The category , in its turn, is known to be equivalent to the (proper) O0 subcategory of the category formed by the -modules with the trivial mon- HC0 D odromy along the fiber of the projection , the first factor of the map B→B . e B×(d)B→B×BIn [BGS] we used instead of the contravariante duality K, a covariant duality mix thate e sends pure perverse sheaves in DG ( ) to indecomposable injectives in mix B×B 0 . The duality K used here is obtained from that of [BGS] via composition HCwith Verdier duality D. Our definitions of the -convolution, resp. ?-convolution, extend verbatim to the mix ∗ mix category D ( ), resp. D , and we have G B×B HC0 Conjecture 5.18. The Koszul duality functor K intertwines the -convolution mix mix ∗ (5.13) on D ( )with the ?-convolution on D 0 , that is, G B×B HC mix K( )=K( )?K( ), , D ( ). M∗N M N ∀M N∈ G B×B Note that Decomposition theorem ([BBD]) implies that -convolution of pure perverse sheaves is pure. On the other hand, we will see in the∗ next section that ?- convolution of two projectives in 0 is again a projective. This gives a supporting evidence for our conjecture. HC We need some more notation. Let i :∆, denote the diagonal, and let ∆ →B×B be its inverse image under the projection . For any point (x ,x ) ∆ B×B→B×B 1 2 ∈ thereisauniqueelementt=t(x,x ) T such that the point (x t, x e) ⊂B×B 1 2 ∈ 1 · 2 belongs to the diagonal of . The assignmente e (x ,x ) t(x ,x )givesamape B×B 1 2 7→ 1 2 ν:e∆ eT, which is analogous to the map µ in (5.9.1). Let i : ∆ , denote → e e →mixB×B the imbedding. We define the following objects of the categories DG ( )and mix B×B e , respectively: e e e e HC0 n (5.19) ∆ := i ∆, ∆ := lim ( 0 0 ) π ( i ν ∗ ) . L ∗O L D D • ∗ E ←−n DO D e e e These objects are the units with respect to -convolutions,e e that is, for any mix mix ∗ M∈ D ( )and D ,wehave G B×B M∈ HC0 (5.20) = = , = = . M∗L∆ M L∆ ∗M M∗L∆ M L∆ ∗M Further, there is the unique open G-orbit j : O , . We define the following mix →B×B objects of D ( ): G B×B ! := j! , := j . R OO R∗ ∗OO These objects are -inverse to each other, i.e., it is easy to check that ∗ (5.21) ! = ! = ∆ . R ∗ R∗ R∗ ∗ R L Next, recall diagram (5.9.1), write j! for the direct image with compact support functor corresponding to the open imbedding j : O , . We define the mixe → B×B following object of the category λ HC e e e e n (5.22) ! := lim ( λ w 0 ( λ ) ) π j ! ( µ ∗ ) . R D D − • E ←−n DO D e e e e e 22 ALEXANDER BEILINSON AND VICTOR GINZBURG

Then, one can show that the sheaf ! w0(λ) 2ρ is supported on the diagonal in and, moreover, the correspondingR ∗R sheaf on− the diagonal is the indecomposable B×B projective local system with the monodromy representation (viewed as C[h]-module) isomorphic to C[h] λ 2ρ. It follows that the object ! is the -inverse of w0(λ) 2ρ, i.e., one has − − R ∗ R −

(5.23) ! w (λ) 2ρ = w (λ) 2ρ ! = ∆. R ∗R 0 − R 0 − ∗R L

If λ = 0, as we will assume below, we will write instead of w0(λ) 2ρ,and mix R∗ R − view it as an object of D , using a twist by (2ρ). HC0 O

Remark. An important motivation for Theorem 5.16 comes from the equation K( ∆)= ! (see (5.25) below) in view of the following. The object ∆ is a simpleL G-equivariantR perverse sheaf supported on the diagonal of ,andoneL has: B×B

ExtD• b ( )( ∆, ∆) HG• ( ) Sh∗ . G B×B L L ' B '

This corresponds, on the other side of Koszul duality, to the fact that ! is a mix R projective object of the category , and moreover one has an isomorphism HC0 End mix ( !) Sh. Similarly, let C be the constant sheaf, viewed as a simple 0 R ' B×B HC b object of DG( ), and let ∆ be the indecomposable projective cover of ∆ in mix B×B P L . Then one has graded algebra isomorphisms: HC0

ExtD• b ( )(C , C ) HG• ( ) G B×B B×B B×B ' B×B Sh W Sh End mix ( , ) . ∗ S(h∗) ∗ ∆ ∆ ' ⊗ ' HC0 P P

Assuming Conjecture 5.18 holds true, we establish the following commutation relation between the Koszul duality functor K and the Verdier duality functor D.

mix Theorem 5.24. If Conjecture 5.18 holds, then, for any DG ( ), one has a functorial isomorphism M∈ B×B

t t mix K(D ) D K( ! in D 0 . M ' R∗ ∗ M∗R HC Proof. We begin with the following two easy isomorphisms, which we leave for the reader to prove:

(5.25) K( )= , K( )= . L∆ R! R! L∆ WALL-CROSSING FUNCTORS AND -MODULES 23 D

mix Now, for any , D ( )wehave M N∈ G B×B

Hom( , ) = by (5.20) N M Hom( , ) = by Lemma 5.15 N L∆ ∗M Hom (Dt ), = by Theorem 5.16 N∗ M L∆ Hom (K ) , K (Dt ) = by Conjecture 5.18 L∆ N∗ M t Hom K ∆ , K K(D ) = by Lemma 5.15 L N∗R∗∗ M t t Hom (K ∆) (D K D ) , (K ) = by (5.20) L ∗ · · M N ∗R∗ t t Hom (K ∆) (D K D ) , (K ) ∆ = by Conjecture 5.18 L ∗ · · M N ∗R∗ ∗L 1 1 1 t t Hom (K− K ) (K− ) , K− (K (D K D ) = N ∗ L∆ L∆ ∗ · · M 1 1 t t Hom (K− ),K− K( ) (D K D ) = by (5.25) N∗ L∆ L∆ ∗ · · M 1 t t Hom !,K− K( ∆) (D K D ) = convolve with , use (5.21) N∗R L ∗ · · M R∗ 1 t t Hom , K− K ∆ (D K D ) . N L ∗ · · M ∗ R∗ Since these isomorphisms hold for any , we deduce that N 1 t t = K− K ∆ (D K D ) . M L ∗ · · M ∗ R∗ Hence, convolving each side with , using (5.21) and applying K we get R! K( )=(K ) (Dt K Dt ). M∗R! L∆ ∗ · · M

Thus, using (5.23), (5.25) we calculate (K ∆) = ! = ∆. Hence, convolving each side with ,weobtain L ∗R∗ R ∗R∗ L R∗ t t D( K( !)) = K(D ) . R∗∗ M∗R M The theorem is proved

6. Projective functors

Fix a ρ-dominant integral weight λ and write pr λ for the projection of a locally ﬁnite Z(g)-module to its λ -isotypic component (see| | after Lemma 1.3). | | Following [BeGe, 3.1], we say that a functor Φ : Mod λ (Ug) Mod λ (Ug)isa § | | → | | projective functor if Φ is a direct summand of the functor M pr λ (E C M), for a certain ﬁnite-dimensional g-module E. Any such functor can7→ be| written| ⊗ in terms of ?-convolution as follows. Given E as above, make E C Ug intoaUg-bimodule by the formulas ⊗ x (e u)=x e u+e x u · ⊗ · ⊗ ⊗ · and (e u) x = e u x, x g,e E, u Ug. ⊗ · ⊗ · ∈ ∈ ∈ 24 ALEXANDER BEILINSON AND VICTOR GINZBURG

Similar formulas make E C Ug λ into a Ug-Ug λ -bimodule, and for any M ⊗ | | | | ∈ Modλ(Ug), we clearly have

Φ(M)=prλ(E CM) | | ⊗ =pr (E Ug) Ug M = pr (E Ug ) Ug M. λ C λ C λ λ | | ⊗ ⊗ | | ⊗ | | ⊗ | | We see that our functor Φ is a direct summand of the ?-convolution with the bimodule pr (E Ug ) HC . Notice that no (higher) derived tensor product λ C λ λ | | ⊗ | | ∈ | | is required above, because the bimodule pr (E Ug ) is a projective right Ug - λ C λ λ | | ⊗ | | | | module, as a direct summand of the free right Ug -module E Ug . λ C λ | | ⊗ | | It is known that the category HC λ has enough projectives; cf. [BGG]. Fur- thermore, it was shown in [BeGe, 2.2,| | 4.1] that any Ug -bimodule of the form λ §§ | | pr (E Ug ) is a projective in the category HC , and conversely, any pro- λ C λ λ | | ⊗ | | | | jective in HC is a direct summand of pr (E Ug ), for an appropriate ﬁnite λ λ C λ | | | | ⊗ | | dimensional E. Convolution with such a direct summand is by deﬁnition a projective functor. Moreover, we have the following: Proposition 6.1. (i) For any projective functor Φ there exists a uniquely determined projective object PΦ HC λ such that ∈ | | n Φ(M)=PΦ?M where PΦ = lim Φ U g /I λ Ug ; | | · ←−n The assignment Φ P sets up a (1-1)-correspondence between projective functors 7→ Φ and projectives in HC λ . (ii) Any projective| functor| is exact. (iii) Composition of projective functors is a projective functor.

Proof. Part (i) follows from an analogue of Remark 5.7 (for Ug instead of U). Part

(ii) follows from the exactness of the functor pr λ (E C ). Part (iii) is clear from | | ⊗ • deﬁnition. e

For any projective functor Φ, there is a projective functor Φ† whichisboththe left and the right adjoint of Φ; see [BeGe, Lemma 3.2(v)].

Next we extend projective functors to the category Modλ(U) as follows. To any

Ug λ -bimodule M HC λ associate the Uλ-bimodule | | ∈ | | e M := U W M W U HC ; λ U λ e U λ λ λ ⊗ λ ⊗ λ ∈ e e see Lemma 1.3. If M isf a projectivee in HC λ ,thene Mgis a projective in HCλ (by | | adjunction of Resλ and Uλ ( )). Given a projective functor Φ on Mod λ (Ug), ⊗ · f g| | deﬁne a functor ΦonModλ(U)by e e e ^ Φ(M):=P Res M = P M. Φ Ug λ λ Φ U ⊗ | | ⊗ λ Clearly, Φ:Mod(U) Modz(U) is}| an exact{ functore and, moreover, Φ(M)= λ e → λ e PΦ?M. The followinge resulte describese the relation between projective functorse and - D modules.e WALL-CROSSING FUNCTORS AND -MODULES 25 D Proposition 6.2. (i) For any projective the assignment ? P∈HCλ M7→P M gives an exact functor Mod ( ) Mod ( ) and there is a projective functor Φ= λ D → λ D Φ on Modλ(U) such that the following diagram commutes: P e e e Mod ( ) Γ Mod (U) e e λ D −→ λ ? ( ) Φ P · e Γ e Modλ( ) Modλe(U) y D −→ y (ii) For any projective functor Φ on Mod (U) there exists a projective e λ e P∈HCλ such that Φ=Φ , i.e., such that the above diagram commutes. P e e Proof. By Proposition 5.11 we have Γ( ? )=Γ♦( )?Γ(M). If λ is regular, e e P M P then Γ♦ gives an equivalence of the categories λ and HCλ. This proves the proposition if λ is regular. HC To complete the proof of part (i) in the general case chooseg a regular ρ-dominant µ weight µ. Using the geometric translation functor Θλ we obtain the following equivalences of categories:

λ µ HCµ HC µ , Modλ( ) Modµ( ) Modµ(U) Mod µ (Ug). HC 'HC ' ' | | D ' D ' ' | | The proposition being already known for regular µ’s, it follows that the functor g e e e ? ( )isanexactfunctoronMod ( ) and that there exists a projective † P · λ D P ∈HCλ such that the functor † ? ( ) is both the left and the right adjoint of ? ( ). Now, P · P · the category Mod (U) may be viewede as a quotient of the category Mod ( ). By λ λ D [S1, Lemma 6, p. 432], the pair of adjoint functors ( ? ( ), † ? ( )) descends to the P · P · pair (Γ♦( ) ? ( ), Γ♦e( †) ? ( )) of adjoint (exact) functors on Modλ(U). Wee have in particularP · P · e Hom Uλ, Γ♦( †) ?M =Hom Γ♦( ) ? Uλ,M (6.3) Modλ(U) P Modλ(U) P e =Home Γ♦( ),M . e Modλ(U) Pe One checks from the construction that isomorphisms (6.3) stille hold if M is taken to be a bimodule from HCλ and Hom’s are taken in HCλ. But then the functor

HomMod (U) Uλ, Γ♦( †) ? ( ) on the left-hand side of (6.3) is an exact functor. λ P g • g Hence, the functor Hom (Γ♦( ), ) on the right-hand side is an exact functor e e Modλ(U) P • again. Thus, Γ♦( ) is a projective in HC and part (i) follows. P e λ To prove part (ii) it suﬃce to show, by part (i), that any projective P HC ∈ λ can be written in the form P =Γ♦( g), where is a projective in .Wemay P P HCλ put := ∆λP where ∆λ is the localization functor. Then is projective,g by adjunction,P and the result follows from Theorem 2.2(i). P As a simple application of our analysis we get

Corollary 6.4. There is a natural isomorphism Γ♦( ) U . R! ' λ Proof. Observe that formula (5.23) yields e

! w (λ) 2ρ = , for any Modλ( ). R ∗R 0 − ∗M M M∈ D This equation shows that ?-convolution with gives the identity functor on . R! e HCλ On the other hand, view the bimodule Uλ as an object of HCλ. We know that the f e g 26 ALEXANDER BEILINSON AND VICTOR GINZBURG functor Uλ ? ( ) is the identity functor on HCλ, cf. (5.8). Thus, Proposition 5.11 yields the result.· e g Now let λ, µ be integral ρ-dominant weights such that W W and let θ− : λ ⊂ µ Modλ(U) Modµ(U) be the translation functor. Repeating the deﬁnitions, one → e gets a similar functor θ− : HCλ HCµ on bimodules. e e → Corollary 6.5. Let Pebe ag projectiveg in HCλ.Thenθ−Pis a projective in HCµ and the following diagram of functors commutes: P?(g) e g Mod (U) · Mod (U) λ −→ λ θ− θ− e (θ−P )?( ) e Mode µ(U) · Modµe(U). y e −→ y Proof. Let Θ− : be the geometric translation functor. By the proof HCλ −→ H C µ e e of part (ii) of Proposition 6.2, there is a projective such that P =Γ♦( ). P∈HCλ P Then, by Proposition (2.8), we have θ−P =Γ♦(Θ− ). The proof of part (i) of P Proposition 6.2 now shows that Γ♦(Θ− ) is a projective in HCµ. Furthermore, one checks easily that the following -moduleP counterpart of the above diagram D commutes: g ?( ) Mod ( ) P · Mod ( ) λ D −→ λ D Θ− Θ− e (Θ− )?( ) e Mode ( ) P · Mod e ( ). µyD −→ yµ D The result now follows from Propositione 6.2. e e + Recall the functor θ :Mod (U) Mod (U), the left adjoint of θ−.Wehave ` µ → λ Proposition 6.6. Assumee that fore some M eModλ(U) the adjunctione morphism + ∈ θ` θ−M M is an isomorphism. Then, for any projective functor Φ on Modλ(U), · → + e the adjunction morphism θ` θ−(ΦM) ΦM is an isomorphism again. e e · → e e Proof. Let Φ† be the adjointe ofe ae projectivee functor Φ. The pair (Φ, Φ†) descends, by Corollary 6.5 and [S1, Lemma 6, p. 432] to an adjoint pair of exact functors

(Ψ, Ψ†)onMode (U). For any M,N Mod (U) we have,e by adjunction,e e µ ∈ λ + Hom(θ θ−ΦM, N)=Hom(θ−ΦM, θ−N) e e e ` · e =Hom(Ψθ−M, θ−N) e e e e e e =Hom(θ−M, Ψ†θ−N) ee e =Hom(θ−M, θ−Φ†N) e e e + =Hom(θ θ−M, Φ†N). e` · e e + If θ θ−M = M, then the last Hom can be rewritten as ` · e e e Hom(M,Φ N)=Hom(ΦM,N), e e † and the proposition follows. Next let λ =0.Since0isaρe-dominant regulare weight, we may identify the categories 0 and HC0. There is a standard duality functor on the category HC0. Any projectiveHC functor is known to commute with that duality, so does tensoring g g WALL-CROSSING FUNCTORS AND -MODULES 27 D with a finite dimensional representation; see [BeGe]. Moreover, it is known that the standard duality on HC0 goes into the Verdier duality on 0 (this is false for the Harish-Chandra category over the general real reductive group:HC the two dualities may, in general, actg differently already on the simple objects). The two dualities coincide, however, in the special case of a complex reductive group, the case we are interested in. Then the claim boils down to a similar result for the category .In the category case the result is known, although we could not find any writtenO account of it.O Here is a sketch of proof. First, it is immediate to verify that the two dualities agree on every simple object of . Second, each duality is an exact functor, hence, agreement on simple objects impliesO that any object M is sent by both dualities ∈O to isomorphic objects, call it M †. The remaining (most delicate) part of the proof is to verify that, for any M,N , the two dualities induce the same maps: ∈O Hom (M,N) Hom (N †,M†). This is equivalent, by a standard homological algebra,O to the→ claim that,O for any simple objects M,N and any i 0, the i ∈Oi ≥ two dualities induce the same maps: Ext (M,N) Ext (N†,M†). The latter claim is obvious for i = 0, and can be verifiedO by hand→ forO i =1,usingthatfor i= 1 one can take M to be a Verma module instead of a simple module. The general case i 1 now follows from the main theorem of [BGS] which implies that the Ext-algebra≥ of all the simple objects is Koszul, in particular, is generated by Ext0’s and Ext1’s. We now return to our convolutions and note that for λ = 0 we may assume, twisting by the canonical bundle if necessary, that both -convolution, and the ∗ ?-convolution take the category D 0 into itself and, moreover, that the object involved in formula (5.10) is alsoHC an object of . Recall further that the R HC0 Verdier duality functor, D, is a contravariant exact functor on 0. Hence, it takes projective objects into injective objects. HC

Proposition 6.7. For any projective 0, in the notation of (5.9.1)–(5.9.2) we have P∈HC n D = ? ∨ where ∨ := lim ( 0 0 ) π j µ ∗ ( D ) . P P R R D D • ∗ E −→ n O e e D D Proof. Let Φ : M ?M be the projective functore e corresponding to .It follows from theP discussion7→ P of two dualities above the proposition that the functorP Φ on 0 commutes with Verdier duality. Therefore, setting r =dimT and using LemmaP HC 5.14, for any , we ﬁnd M∈HC0 (D )= ?(D ) P∗R∗ M P M =Φ (D ) since Φ commutes with Verdier duality P M = D Φ ( )=D( ? ) · P M P M =D( ) by Lemma 5.14 P∗R∗M = D( ) D( )[r] . P∗R ∗ M The above isomorphisms imply, by Remark 5.7, that (6.8) =D( )[r]=(D ) (D ), P∗R P∗R P ∗ R where in the last equality we have used Lemma 5.14 once more. Now, it is easy to show that the object ∨ is -inverse to D . Therefore, convolving both sides of R ∗ R 28 ALEXANDER BEILINSON AND VICTOR GINZBURG

(6.8) with ∨ yields R D = ∨ = ? ∨, P P∗R∗R P R and the claim follows.

Tilting -modules. We continue to assume that λ = 0, and use the notation of Section 4.D Let b be the fixed Borel subalgebra, and B G the corresponding Borel ⊂ group. Since 0 is regular we may (and will) identify Ug-modules with -modules via D the equivalences: Mod 0 (Ug) Mod0(U) Mod0( ). Further, there is a natural equivalence (see e.g. [BeGe,| | '5]) between' the categoryD of Ub-locallye finite Ug - 0 § | | e e modules, resp. B-monodromic -modules on (see [BB3]) and the category 0. ↔ D B HC We will write ∆(M ) for the object of 0 corresponding to M Mod 0 (Ug). As HC ∈ | | we mentioned in Remark 5.17(c),e the category goes under the above equivalence O0 to the subcategory of formed by the -modules with the trivial monodromy HC0 D along the fiber of the projection , the first factor of the map B→B ↔ ↔B×B→ . In particular, for each w W , theree are objects ∆(M w 0 ) , ∆(Pw 0 ) B×B ∈ · · ∈ 0 corresponding to the Verma modulee and its indecomposable projectivee cover,e HCrespectively. Given µ h∗,letSh denote the completion of the Symmetric algebra at the ∈ µ corresponding point. We introduce the “universal” Verma module M := Ug/Ug n, where n := [b, b] is the nilradical of b. This module has a natural right Sh-action,· and we set := Sh , the completion of at the point µ. In particular, Mµ M Sh µ M for each w W , there is an Ug-module Mw 0, which is not an object of the category · ∈ ↔N 0.Welet w:= ∆(Mw 0 ) denote the corresponding object in 0, and let w O M · HC P be its indecomposable projective cover in 0. Wenotethat,ifw=w0,then = is the Harish-Chandra module definedHC in (5.22). In general, has a Mw0 R! Mw pro-unipotent monodromy (i.e., free Shw 0-action) along the fiber of the projection · ↔ (the first factor of the map ), and ∆(M w 0 ) is obtained B→Bfrom by “killing” (= taking co-invariantsB×B→B×B of) the monodromy· along the first Mw factore (very much the same way as thee Vermae module Mµ is obtained from Mµ). Similarly, the projective w has a flag formed by the various y’s, hence has free P ↔ M pro-unipotent monodromy; again, ∆(Pw 0 ) is obtained from w by “killing” that monodromy. In particular, one has an equality· of multiplicities:P

↔ ↔ (6.9) [ w : y]=[∆(Pw 0 ):∆(M y 0 )] . P M · · We will also need Harish-Chandra modules ∨ which are in a sense dual ana- Mw logues of w. We first define the corresponding Ug-modules as follows. For w W , M n ∈ and each n =1,2,... , we have a well defined object Mµ/ µ Mµ 0,where µ n J ∈O J ⊂ Shµ is the maximal ideal. Let D(Mµ/ µ Mµ) be its dual in 0. There are natural J n+1 On projections (not imbeddings!): D(Mµ/ µ Mµ) D(Mµ/ µ Mµ) ,n=1,2,... , J J n ↔ and we set Mµ∨ := lim D ( M µ / µ M µ ). Let w∨ := ∆(Mw∨ 0 ) be the corresponding J M · ←−n Harish-Chandra modules. We note that for w = w we have = is the 0 w∨0 2ρ object introduced in (5.9.2); see also (5.23). M R− It was shown in [BeGe] that, for each w W , there is a unique indecomposable ∈ projective functor Φw (see [BeGe, Thm. 3.3]) such that Φw(M0)=Pw0. Hence, for each w W , there is, by Propositions 6.1 and 6.2, a unique projective· , ∈ Pw ∈HC0 WALL-CROSSING FUNCTORS AND -MODULES 29 D such that the functor Φ corresponds on -modules to the convolution functor w D w ,where := 2ρ is as above. It is not difficult to see that w M7→P ∗R∗M R R− P is the indecomposable projective cover of e (considered 2 paragraphs above), Mw equivalently, the projective cover of the simple -module supported on the G- diagonal orbit in , corresponding to the elementD w. B×B Recall that an object of the category is callede a tilting module, if it is self- dual and has a Verma-flag. Observe thatO projective functors take self-dual objects into self-dual ones, and modules with Verma-flag into modules with Verma-flag; cf. [BeGe]. Since M 2ρ is a simple, hence, self-dual Verma module, it follows that, for − each w W , the module Qw 0 := Φw(M 2ρ) is tilting. In fact, the Qw 0 ,w W , are exactly∈ all the indecomposable· tilting− modules in the category{ .· ∈ } O0 Definition. A finite direct sum of objects of D 0 of the form w := w ,w W (cf. (5.10)) will be called a tilting Harish-ChandraHC module. Q P ∗R ∈ Note that this definition gives a natural mixed structure on ; cf. [BGS, 4]. Qw § Theorem 6.10. (i) For any w W , the complex := is actually a ∈ Qw Pw ∗R -module (not just an object of the derived category), w 0. D (ii) For any y,w W we have is againQ ∈HC a tilting module. ∈ Qy ∗Qw ∈HC0 e (iii) For any w, y W the following multiplicity formula holds: ∈

[Qw : y∨]=[Pw0 :Myw0 0]. M · · Moreover,

Hom ( w, y) Hom ( w, y), HC0 Q Q ' HC0 P P and

Hom (Qw 0,Qy 0) Hom (Pw 0,Py 0). O · · ' O · · ↔ Before going into the proof we note that Proposition 6.2 yields w ∆(M 2ρ )= − ↔ ↔ ↔ ↔ Q ∗ w ∆(M 2ρ )= w?∆(M 2ρ )=∆Φw(M 2ρ) =∆(Q w 0 ). Furthermore, P ∗R∗ − P − − · ↔ one shows as above that the Harish-Chandra module ∆(Q w 0 ) is obtained from the tilting Harish-Chandra module by taking coinvariants of· the monodromy along Qw the ﬁber of the projection (the ﬁrst factor of the map ). B→B B×B→B×B Sketch of the proof of Theorem 6.10. To prove (i) we exploit an interpretation of e e e the functor , as of a projective functor. Such functors take - M7→Py∗R∗M D modules into -modules, because of the global interpretation via bi-modules over D the extended enveloping algebra U. Hence, using the notation of (5.19) we gete = e= = ? , and part (i) is proved. Further, for Qy Qy ∗L∆ Pw ∗R∗L∆ Py L∆ ∈HC0 any M Mod 0 (Ug), one has ( y e w) M = y w M =( y? w)?M . Part (ii)∈ now follows| | from PropositionQ ∗Q 6.1(iii)∗ andP ∗R∗P Remark∗R∗ 5.7. P P 1 Observe next that, for any y W ,wehave`(y− )+`(yw )=`(w ). It follows ∈ 0 0 that ∨ 1 yw∨ = w∨ = . Also, one knows that y ∨ 1 = ∆;cf. My− ∗M 0 M 0 R M ∗My− L (5.23). Therefore, we deduce:

y = y ( ∨ 1 ∨ )=( y ∨ 1) ∨ = ∆ ∨ = ∨ . M ∗R M ∗ My− ∗Myw0 M ∗My− ∗Myw0 L ∗Myw0 Myw0 It follows that the functor of -convolution with takes modules with y-flag to modules (not just complexes∗ in the derived category)R with -flagM and, more- yw∨ 0 over, the multiplicities in the two flags correspond. Since M has a -flag, we Pw My 30 ALEXANDER BEILINSON AND VICTOR GINZBURG deduce that = is an actual module. Furthermore, has a -flag, w w w yw∨ 0 and we haveQ [ : P ∗R]=[ : ].This, together with (6.9)Q and theM remark w yw∨ 0 w y preceding theQ proofM of the theorem,P M implies the multiplicity formula of part (iii). Finally, we observe that the functor of -convolution with is an equivalence of derived categories, for it has -convolution∗ with as its inverse.R Therefore, we ∗ R! have Hom 0 ( w, y) Hom 0 ( w, y). The last equation of part (iii) is proved similarly (cf.HC discussionQ Q ' precedingHC theP proofP of the theorem).

Remarks. (i) The multiplicity formula of Theorem 6.10(iii) may be viewed as a character formular for tilting modules. An analogous formula for tilting modules over an affine Lie algebra is the main result of [S2]. We observe further that, in the affine setup there are two affine flag manifolds, + and , with strata of finite co-dimension and strata of finite dimension, respectively.B CategoryB− at a positive O level has enough projectives and corresponds -modules on , while category D B+ at a negative level corresponds -modules on . The latter category has no Oprojectives, but has well defined tiltingD modulese instead.B− In the affine setup, the “kernel” lives naturally on + e . Our formula w := w says that the R B ×B− Q P ∗R tiltings on are obtained from the projectives on + by convolving the latter B− B with , the most natural way to “transport” projectives from + to . (ii)R It would be very interesting to prove an affine analogueB of CorollaryB− 6.4 saying that Γ( + , )isthesemi-infinitely induced module that played a crucial role in [S2].B ×B− R (iii) The same argument as the one used in the proof of the Hom-equality of Theorem 6.10(iii) also yields i i Ext ( y, w)=Ext ( y, w)=0, i>0. HC0 Q Q HC0 P P ∀ References

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Department of Mathematics, University of Chicago, Chicago, Illinois 60637 E-mail address: [email protected]

E-mail address: [email protected]