MATHEMATICS ANNALS OF. Localization of g-modules on the affine Grassmannian. November, anmaah. By Edward Frenkel and Dennis Gaitsgory

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1 ANNALS OF MATHEMATCS Localization of -modules on the affine Grassmannian By Edward Frenkel and Dennis Gaitsory SECOND SERES, VOL. 170, NO. 3 November, 2009 anmaah

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3 Annals of Mathematics, 170 (2009), Localization of -modules on the affine Grassmannian By EDWARD FRENKEL and DENNS GATSGORY Abstract We consider the cateory of modules over the affine Kac-Moody alebra b of ical level with reular central character. n our previous paper we conjectured that this cateory is equivalent to the cateory of Hecke eien-d-modules on the affine Grassmannian G..t//=GŒŒt. This conjecture was motivated by our proposal for a local eometric Lanlands correspondence. n this paper we prove this conjecture for the correspondin 0 equivariant cateories, where 0 is the radical of the wahori subroup of G..t//. Our result may be viewed as an affine analoue of the equivalence of cateories of -modules and D-modules on the fla variety G=B, due to Beilinson-Bernstein and Brylinski-Kashiwara. ntroduction 0.1. Let G be a simple complex alebraic roup and B its Borel subroup. Consider the cateory D.G=B/ -mod of left D-modules on the fla variety G=B. The Lie alebra of G, and hence its universal envelopin alebra U./, acts on the space.g=b; / of lobal sections of any D-module. The center Z./ of U./ acts on.g=b; / via the aumentation character 0 W Z./!. Let -mod 0 be the cateory of -modules on which Z./ acts via the character 0. Thus, we obtain a functor W D.G=B/ -mod! -mod 0 : n [BB81] A. Beilinson and J. Bernstein proved that this functor is an equivalence of cateories. Moreover, they eneralized this equivalence to the case of twisted D-modules, for twistins that correspond to dominant interal weihts 2 t. Let N be the unipotent radical of B. We can consider the N -equivariant subcateories on both sides of the above equivalence. On the D-module side this The research of E.F. was supported by the DARPA rant HR and by the NSF rant DMS

4 1340 EDWARD FRENKEL and DENNS GATSGORY is the cateory D.G=B/ -mod N of N -equivariant D-modules on G=B, and on the -module side this is the block of the cateory correspondin to the central character 0. The resultin equivalence of cateories, which follows from [BB81], and which was proved independently by J.-L. Brylinski and M. Kashiwara [BK81], is very important in applications to representation theory of. Now let y be the affine Kac-Moody alebra, the universal central extension of the formal loop aebra..t//. Representations of y have a parameter, an invariant bilinear form on, which is called the level. There is a unique inner product can which is normalized so that the square lenth of the maximal root of is equal to 2. Then any other inner product is equal to D k can, where k 2, and so a level corresponds to a complex number k. n particular, it makes sense to speak of interal levels. Representations, correspondin to the bilinear form which is equal to minus one half of the Killin form (for which k D h _, minus the dual Coxeter number of ) are called representations of ical level. This is really the middle point amonst all levels, and not the zero level, as one miht naively expect. There are several analoues of the fla variety in the affine case. n this paper (except in the Appendix) we will consider exclusively the affine Grassmannian Gr G D G..t//=GŒŒt : Another possibility is to consider the affine fla scheme Fl G D G..t//=, where is the wahori subroup of G..t//. Most of the results of this paper, that concern the ical level, have conjectural counterparts for the affine fla variety, but they are more difficult to formulate. n particular, one inevitably has to consider derived cateories, whereas for the affine Grassmannian abelian cateories suffice. We refer the reader to the ntroduction of our previous paper [FG06] for more details. There is a canonical line bundle can on Gr G such that the action of..t// on Gr G lifts to an action of y can on can. For each level we can consider the cateory D.Gr G / -mod of riht D-modules on Gr G twisted by k can, where D k can. (Recall that althouh the line bundle k can only makes sense when k is interal, the correspondin cateory of twisted D-modules is well-defined for an arbitrary k.) Since Gr G is an ind-scheme, the definition of these cateories requires some care (see [BD] and [FG04]). Let y -mod be the cateory of (discrete) representations of the affine Kac- Moody alebra of level. Usin the fact that the action of..t// on Gr G lifts to an action of y can on can, we obtain that for each level there is a naturally defined functor of lobal sections: (1) W D.Gr G / -mod! y -mod : The question that we address in this paper is if and when this functor is an equivalence of cateories, as in the finite-dimensional case.

5 LOCALZATON OF -MODULES ON THE AFFNE GRASSMANNAN The first results in this direction were obtained in [BD], [FG04]. Namely, in loc. cit. it was shown that if is such that D k can with k C h _ >0, then the functor of (1) is exact and faithful. (n contrast, it is known that this functor is not exact for k C h _ 2 >0.) The condition k C h _ >0 is analoous to the dominance condition of [BB81]. Let us call neative if k C h _ 0. n this case one can show that the functor of (1) is fully faithful. n fact, in this case it makes more sense to consider T -monodromic twisted D-modules on the enhanced affine fla scheme efl G D G..t//= 0 ; rather than simply twisted D-modules on Gr G, and the correspondin functor to y -mod. The above exactness and fully-faithfulness assertions are still valid in this context. However, the above functor is not an equivalence of cateories. Namely, the RHS of (1) has many more objects than the LHS. When is interal, A. Beilinson has proposed a conjectural intrinsic description of the imae of the cateory D.eFl G / -mod inside y -mod (see Remark (ii) in the ntroduction of [Bei06]). As far as we know, no such description was proposed when is not interal. t is possible, however, to establish a partial result in this direction. Namely, let 0 be the unipotent radical of the wahori subroup. We can consider the cateory D.eFl G / -mod 0 of 0 -equivariant twisted D-modules on efl G. The correspondin functor of lobal sections takes values in the affine version of cateory, i.e., in the subcateory y -mod 0 y -mod, whose objects are y - modules on which the action of the Lie alebra Lie. 0 / y interates to an alebraic action of the roup 0. One can show that the functor induces an equivalence between an appropriately defined subcateory of T -monodromic objects of D.eFl G / -mod 0 and a specific block of y -mod 0. This result, which is well-known to specialists, is unavailable in the published literature. For the sake of completeness, we sketch one of the possible proofs in the Appendix n this paper we shall concentrate on the case of the ical level, when k D h _. We will see that this case is dramatically different from the cases considered above. n [FG06] we made a precise conjecture describin the relationship between the correspondin cateories D.Gr G / -mod and y -mod. We shall now review the statement of this conjecture. First, let us note that the imae of the functor lies in a certain subcateory of y -mod, sinled out by the condition on the action of the center. Let Z denote the center of the cateory y -mod (which is the same as the center of the completed envelopin alebra of y ). The fact that this center is

6 1342 EDWARD FRENKEL and DENNS GATSGORY nontrivial is what distinuishes the ical level from all other levels. Let Z re denote the quotient of Z, throuh which it acts on the vacuum module WD nd y ŒŒt. /: Let y -mod re be the full subcateory of y -mod, whose objects are y - modules on which the action of the center Z factors throuh Z re. t is known (see [FG04]) that for any 2 D.Gr G / -mod, the space of lobal sections.gr G ; / is an object of y -mod re. (Here and below we write M 2 if M is an object of a cateory.) Thus, y -mod re is the cateory that may be viewed as an analoue of the cateory -mod 0 appearin on the representation theory side of the Beilinson-Bernstein equivalence. However, the functor of lobal sections W D.Gr G / -mod! y -mod re is not full, and therefore cannot possibly be an equivalence. The oriin of the nonfullness of two-fold, with one inredient rather elementary, and another less so. First, the cateory y -mod re has a lare center, namely, the alebra Z re itself, while the center of the cateory D.Gr G / -mod is the roup alebra of the finite roup 1.G/ (i.e., it has a basis enumerated by the connected components of Gr G ). Second, the cateory D.Gr G / -mod carries an additional symmetry, namely, an action of the tensor cateory Rep. {G/ of the Lanlands dual roup {G, and this action trivializes under the functor. n more detail, let us recall that, accordin to [FF92], [Fre05], we have a canonical isomorphism between Spec.Z re / and the space Op L. / of L-opers on the formal disc (we refer the reader to 1 of [FG06] for the definition and a detailed review of opers). By construction, over the scheme Op L. / there exists a canonical principal {G-bundle, denoted G;Op {. Let {G;Z be the {G-bundle over Spec.Z re / correspondin to it under the above isomorphism. For an object V 2 Rep. {G/ let us denote by Z the associated vector bundle over Spec.Z re /, i.e., Z D G;Z { V. {G Consider now the cateory D.Gr G / -mod GŒŒt. By [MV07], this cateory has a canonical tensor structure, and as such it is equivalent to the cateory Rep. {G/ of alebraic representations of {G; we shall denote by V 7! V W Rep. {G/! D.Gr G / -mod GŒŒt the correspondin functor. Moreover, we have a canonical action of D.Gr G / - mod GŒŒt as a tensor cateory on D.Gr G / -mod by convolution functors, 7!? V :

7 LOCALZATON OF -MODULES ON THE AFFNE GRASSMANNAN 1343 A. Beilinson and V. Drinfeld [BD] have proved that there are functorial isomorphisms.gr G ;? V / '.Gr G ; / Z ; Z re V 2 Rep. {G/; compatible with the tensor structure. Thus, we see that there are nonisomorphic objects of D.Gr G / -mod that o under the functor to isomorphic objects of y -mod re n [FG06] we showed how to modify the cateory D.Gr G / -mod, by simultaneously addin to it Z re as a center, and dividin it by the above Rep. {G/-action, in order to obtain a cateory that can be equivalent to y -mod re. This procedure amounts to replacin D.Gr G / -mod by the appropriate cateory of Hecke eien-objects, denoted D.Gr G / Hecke Z -mod. By definition, an object of D.Gr G / Hecke Z -mod is an object 2 D.Gr G / -mod, equipped with an action of the alebra Z re by endomorphisms and a system of isomorphisms V W? V! Z ; V 2 Rep. {G/; compatible with the tensor structure. Z re We claim that the functor W D.Gr G / -mod! y -mod re naturally ives rise to a functor Hecke Z W D.Gr G / Hecke Z -mod! y -mod re. This is in fact a eneral property. Suppose for simplicity that we have an abelian cateory which is acted upon by the tensor cateory Rep.H /, where H is an alebraic roup; we denote this functor by 7!? V; V 2 Rep.H /. Let Hecke be the cateory whose objects are collections. ; f V V 2Rep.H / /, where 2 and f V is a compatible system of isomorphisms V W? V! V ; V 2 Rep.H /; where V is the vector space underlyin V. One may think of Hecke as the deequivariantized cateory with respect to the action of H. t carries a natural action of the roup H : for h 2 H, we have h. ; f V V 2Rep.H / / D. ; f.h id / ı V V 2Rep.H / /: The cateory may be reconstructed as the cateory of H -equivariant objects of Hecke with respect to this action, see [Gai]. Suppose that there is a riht-exact functor G W! 0, where 0 is another abelian cateory, such that we have functorial isomorphisms (2) G.? V / ' G. / V ; V 2 Rep.H /; compatible with the tensor structure. Then, accordin to [AG03], there exists a functor G Hecke W Hecke! 0 such that G ' G Hecke ı nd, where the functor

8 1344 EDWARD FRENKEL and DENNS GATSGORY nd W! Hecke sends to? H, where H is the reular representation of H. The functor G Hecke may be explicitly described as follows: the isomorphisms V and (2) ive rise to an action of the alebra H on G. /, and G Hecke. / is obtained by takin the fiber of G. / at 1 2 H. We take D D.Gr G / -mod, 0 D y -mod re, H D {G and G D. The only difference is that now we are workin over the base Z re, which we have taken into account The conjecture suested in [FG06] states that the resultin functor (3) Hecke Z W D.Gr G / Hecke Z -mod! y -mod re is an equivalence. n loc. cit. we have shown that the functor Hecke Z, when extended to the derived cateory, is fully faithful. This conjecture has a number of interestin corollaries pertainin to the structure of the cateory of representations at the ical level: Let us fix a point 2 Spec.Z re /, and let us choose a trivialization of the fiber G; { of {G;Z at. Let y -mod be the subcateory of y -mod, consistin of objects, on which the center acts accordin to the character correspondin to. Let D.Gr G / Hecke -mod be the cateory, obtained from D.Gr G / -mod, by the procedure 7! Hecke for H D {G, described above. Our conjecture implies that we have an equivalence (4) D.Gr G / Hecke -mod ' y -mod : n particular, we obtain that for every two points ; 0 2 Spec.Z re / and an isomorphism of {G-torsors G; { ' there exists a canonical equivalence y {G; 0 -mod ' y -mod 0. This may be viewed as an analoue of the translation principle that compares the subcateories -mod -mod for various central characters 2 Spec.Z.// in the finite-dimensional case. By takin D 0, we obtain that the roup {G, or, rather, its twist with respect to G; {, acts on y -mod. As explained in the ntroduction to [FG06], the conjectural equivalence of (4) fits into the eneral picture of local eometric Lanlands correspondence. Namely, for a point 2 Spec.Z re / ' Op L. / as above, both sides of the equivalence (4) are natural candidates for the conjectural Lanlands cateory associated to the trivial {G-local system on the disc. This cateory, equipped with an action of the loop roup G..t//, should be thouht of as a cateorification of an irreducible unramified representation of the roup G over a local non-archimedian field. Provin this conjecture would therefore be the first test of the local eometric Lanlands correspondence proposed in [FG06].

9 LOCALZATON OF -MODULES ON THE AFFNE GRASSMANNAN Unfortunately, at the moment we are unable to prove the equivalence (3) in eneral. n this paper we will treat the followin particular case: Recall that 0 denotes the unipotent radical of the wahori subroup, and let us consider the correspondin 0 -equivariant subcateories on both sides of (3). On the D-module side, we obtain the cateory D.Gr G / Hecke Z -mod 0, defined in the same way as D.Gr G / Hecke Z -mod, but with the requirement that the underlyin D-module be stronly 0 -equivariant. On the representation side, we obtain the cateory y -mod 0 re, correspondin to the condition that the action of Lie. 0 / y interates to an alebraic action of 0. We shall prove that the functor Hecke Z defines an equivalence (5) D.Gr G / Hecke Z -mod 0! y -mod 0 re : This equivalence implies an equivalence (6) D.Gr G / Hecke -mod 0 ' y -mod 0 for any fixed character 2 Spec.Z re / and a trivialization of G; { as above. n particular, we obtain the corollaries concernin the translation principle and the action of {G on y -mod 0. We remark that from the point of view of the local eometric Lanlands correspondence the cateories appearin in the equivalence (6) should be viewed as cateorifications of the space of -fixed vectors in an irreducible unramified representation of the roup G over a local non-archimedian field (which is a principal series representation of the correspondin affine Hecke alebra). Let us briefly describe the stratey of the proof. Due to the fact [FG06] that the functor in one direction in (5) is fully-faithful at the level of the derived cateories, the statement of the theorem is essentially equivalent to the fact that for every object 2 y -mod 0 re there exists an object 2 D.Gr G/ Hecke Z -mod 0 and a nonzero map Hecke Z.Gr G ; /!, to be explained in detail in Section 3. We exhibit a collection of objects w;re, numbered by elements w 2 W, where W is the Weyl roup, which are quotients of Verma modules, such that for every 2 y -mod 0 re, for at least one w, we have Hom. w;re; / 0. We then show (see Theorem 3.2) that each such w;re is isomorphic to Hecke Z.Gr G ; Z w/ for some explicit object Z w 2 D.Gr G / Hecke Z -mod 0, thereby provin the equivalence (5) t is instructive to put our results in the context of other closely related equivalences of cateories. Usin the (tautoloical) equivalence: D.Gr G / -mod 0 ' D.eFl G / -mod GŒŒt

10 1346 EDWARD FRENKEL and DENNS GATSGORY (here and below we omit the subscript when D 0) and the equivalence of Theorem 5.5, we obtain that for every neative interal level D k can there exists an equivalence between D.Gr G / -mod 0 and the reular block of the cateory y -mod GŒŒt, studied in [KL93], [KL94]. The latter cateory is equivalent, accordin to loc. cit., to the cateory of modules over the quantum roup Uq res./, where q D exp i=.k C h _ /. Usin these equivalences, it has been shown in [AG03] that the cateory D.Gr G / Hecke -mod 0, defined as above, is equivalent to the reular block u q./-mod 0 of the cateory of modules over the small quantum roup u q./. The tensor product by the line bundle can h_ defines an equivalence D.Gr G / Hecke -mod 0! D.Gr G / Hecke -mod 0 (but this equivalence does not, of course, respect the functor of lobal sections). Combinin this with the equivalence of (6), we obtain the followin diaram of equivalent cateories: (7) y -mod 0 D.GrG / Hecke -mod 0! u q./ -mod : Recall that in [ABB C 05] it was shown that the cateory D.Gr G / Hecke -mod 0 is equivalent to an appropriately defined cateory D. l 1 2 / 0 of 0 -equivariant D-modules on the semi-infinite fla variety (it is defined in terms of the Drinfeld compactification Bun N ). Hence, we obtain another diaram of equivalent cateories: (8) y -mod 0 n particular, we obtain a functor D.GrG / Hecke -mod 0! D. l 1 2 / 0 : D. l 1 2 / 0! y -mod 0 ; which is, moreover, an equivalence. ts existence had been predicted by B. Feiin and the first named author. n fact, one would like to be able to define the cateory D. l 1 2 / without imposin the 0 -equivariance condition, and extend the equivalence of [ABB C 05] to this more eneral context. Toether with the equivalence of (3), this would imply the existence of the diaram y -mod D.GrG / Hecke -mod! D. l 1 2 /; but we are far from bein able to achieve this oal at present. Finally, let us mention one more closely related cateory, namely, the derived cateory D QCoh.. {G= {B/ DG -mod/ of complexes of quasi-coherent sheaves over the DG-scheme. {G= {B/ DG WD Spec Sym { G= {B.1. {G= {B/Œ1 / :

11 LOCALZATON OF -MODULES ON THE AFFNE GRASSMANNAN 1347 This DG-scheme can be realized as the derived Cartesian product e L pt; L where pt! L corresponds to the point 0 2 L, and e L D f.x; Lb/jx 2 Lb L is Grothendieck s alteration. From the results of [ABG04] one can obtain an equivalence of the derived cateories D b QCoh.. {G= {B/ DG -mod/ ' D b D.Gr G / Hecke -mod 0 : Hence we obtain an equivalence: (9) D b QCoh.. {G= {B/ DG -mod/ ' D b y -mod 0 : The existence of such an equivalence follows from the Main Conjecture 6.11 of [FG06]. Note that, unlike the other equivalences mentioned above, it does not preserve the t-structures, and so is inherently an equivalence of derived cateories Contents. Let us briefly describe how this paper is oranized: n Section 1, after recallin some previous results, we state the main result of this paper, Theorem 1.7. n Section 2 we review representation-theoretic corollaries of Theorem 1.7. n Section 3 we show how to derive Theorem 1.7 from Theorem 3.2, and in Section 4 we prove Theorem 3.2. Finally, in the Appendix, we prove a partial localization result at the neative level referred to in Section 0.2. The notation in this paper follows that of [FG06]. 1. The Hecke cateory n this section we recall the main definitions and state our main result. We will rely on the concepts introduced in our previous paper [FG06] Recollections. Let be a simple finite-dimensional Lie alebra, and G the connected alebraic roup of adjoint type with Lie alebra. We shall fix a Borel subroup B G. Let {G denote the Lanlands dual roup of G, and by L its Lie alebra. Let Gr G D G..t//=GŒŒt be the affine Grassmannian associated to G. We denote by D.Gr G / -mod the cateory of ically twisted riht D-modules on the affine Grassmannian and by D.Gr G / -mod GŒŒt

12 1348 EDWARD FRENKEL and DENNS GATSGORY the correspondin GŒŒt -equivariant cateory. Recall that via the eometric Satake equivalence (see [MV07]) the cateory D.Gr G / -mod GŒŒt has a natural structure of tensor cateory under convolution, and as such it is equivalent to Rep. {G/. We shall denote by V 7! V the correspondin tensor functor Rep. {G/! D.Gr G / -mod GŒŒt. We have the convolution product functors 2 D.Gr G / -mod; V 2 D.Gr G / -mod GŒŒt 7!? V 2 D.Gr G / -mod : These functors define an action of Rep. {G/, on the cateory D.Gr G / -mod. Thus, in the terminoloy of [Gai], D.Gr G / -mod GŒŒt has a structure of the cateory over the stack pt = {G. Now let y -mod denote the cateory of (discrete) representations of the affine Kac-Moody alebra at the ical level (see [FG06]). Let 2 y -mod be the vacuum module nd y ŒŒt. /. Denote by Z the topoloical commutative alebra that is the center of y -mod. Let Z re denote its reular quotient, i.e., the quotient modulo the annihilator of. We denote by y -mod re the full subcateory of y -mod, consistin of objects, on which the action of the center Z factors throuh Z re. Recall that via the Feiin-Frenkel isomorphism [FF92], [Fre05], the alebra Z re identifies with the alebra of reular functions on the scheme Op L. / of L-opers on the formal disc. n particular, Spec.Z re / carries a canonical {G-torsor, denoted G;Z {, whose fiber at 2 Spec.Zre {G; / ' Op L. / is the fiber of the {G-torsor underlyin the oper at the oriin of the disc. The {G-torsor G;Z { ives rise to a morphism Spec.Z re /! pt = {G. We shall denote by V 7! Z the resultin tensor functor from Rep. {G/ to the cateory of locally free Z re - modules. We define D.Gr G / Hecke Z -mod as the fiber product cateory D.Gr G / -mod Spec.Z re /; pt = {G in the terminoloy of [Gai]. Explicitly, D.Gr G / Hecke Z-mod has as objects the data of. ; V; 8 V2 Rep. {G//, where is an object of D.Gr G / -mod, endowed with an action of the alebra by endomorphisms, and V are isomorphisms of D-modules Z re compatible with the action of Z re conditions are satisfied:? V ' Z ; Z re on both sides, and such that the followin two

13 LOCALZATON OF -MODULES ON THE AFFNE GRASSMANNAN 1349 For V bein the trivial representations, the morphism V is the identity map. For V; W 2 Rep. {G/ and U WD V W, the diaram is commutative..? V /? W!? U? V?id?y? W U y. Z /? W Z re? y Z Z re? y Z.? W / id Z W! Z Z Z re Z re Z re Morphisms in this cateory between. ; V / and. 0 ; 0V / are maps of D-modules W! 0 that are compatible with the actions of Z re on both sides, and such that.id Z / ı V D 0V ı.? id V /: 1.2. Definition of the functor. Recall that accordin to [FG04], the functor of lobal sections 7!.Gr G ; / defines an exact and faithful functor D.Gr G / -mod! y -mod re. Let us recall, followin [FG06], the construction of the functor Hecke Z W D.Gr G / Hecke Z! y -mod re : First, let us recall the followin result of [BD] (combined with an observation of [FG06, Lemma 8.4.3]): THEOREM 1.3. functorial isomorphism (1) For 2 D.Gr G / -mod and V 2 Rep. {G/ there is a ˇV W.Gr G ;? V / '.Gr G ; / Z : Z re

14 1350 EDWARD FRENKEL and DENNS GATSGORY (2) For ; V as above and W 2 Rep. {G/, U WD V W the diaram is commutative. Gr G ;.? V /? W? ˇW y! GrG ;?. V? W / Gr G ;.? V / Z.Gr G ;? U / ˇV? y Z re.gr G ; / Z Z!.GrG ; / Z re Z re Consider the scheme som Z W Spec.Z re pt = {G ˇU? y? y Z re Z Z re /. Let 1 somz denote the unit section Spec.Z re /! som Z. We denote by R Z the direct imae of the structure sheaf under Spec.Z re pt = {G, viewed as an object of Rep. {G/. t carries an action of Z re /! by endomorphisms. /; by Let Z be the associated (infinite-dimensional) vector bundle over Spec.Z re definition, we have a canonical isomorphism Z ' Fun.som Z /: We will think of the projection p r W som Z! Spec.Z re / as correspondin to the oriinal Z re -action on R Z, and hence on Z, by the transport of structure. We will think of the other projection p l W som Z! Spec.Z re /, as correspondin to the Z re -module structure on Z comin from the fact that this is a vector bundle associated to a {G-representation. We claim that for every object 2 D.Gr G / Hecke Z -mod, the y -module.gr G ; / carries a natural action of the alebra Fun.som Z / by endomorphisms. First, note that.gr G ; / is a Z re -bimodule: we shall refer to the Z re -action, comin from its action on any object of y -mod re, as riht, and to the one, comin from the Z re -action on, as left. On the one hand, we have: and on the other hand,.gr G ;? RZ / ˇR Z '.Gr G ; /.Gr G ;? RZ / R Z ' Fun.som Z / r;z re ;l l;z re ;l Fun.som Z /;.Gr G ; /:

15 LOCALZATON OF -MODULES ON THE AFFNE GRASSMANNAN 1351 By composin we obtain the desired action map RZ ıˇr 1 Z.Gr G ; / Fun.som Z / r;z re ;l! Fun.som Z / l;z re ;l 1.Gr G ; / som Z!.Gr G ; /: The fact that it is associative follows from the second condition on V and Theorem 1.3(2). We define the functor Hecke Z by 7!.Gr G ; / Fun.som Z /;1 som Z Z re : Since the functor is exact, the functor Hecke Z is evidently riht-exact, and we will denote by L Hecke Z its left derived functor D.D.Gr G / Hecke Z -mod/! D.y -mod re / The followin was established in [FG06, Th ]: THEOREM 1.4. The functor L Hecke Z, restricted to is fully faithful. D b.d.gr G / Hecke Z n [FG06] we formulated the followin: -mod/; CONJECTURE 1.5. The functor Hecke Z is exact and defines an equivalence of cateories D.Gr G / Hecke Z -mod and y -mod re The statement of the main result. Recall that both cateories y -mod re and D.Gr G / Hecke Z -mod carry a natural action of the roup G..t// (see [FG06, 22], where this is discussed in detail). Let GŒŒt be the wahori subroup, the preimae of the Borel subroup B G in GŒŒt under the evaluation map GŒŒt!G. Let 0 be the unipotent radical of. Let us denote by D.Gr G / Hecke Z -mod 0 and y -mod 0 re the correspondin cateories of 0 -equivariant objects. Since 0 is connected, these are full subcateories in D.Gr G / Hecke Z -mod and y -mod re, respectively. The functor Hecke Z induces a functor D.Gr G / Hecke Z -mod 0! y -mod 0 re. The oal of this paper is to prove the followin special case of Conjecture 1.5: THEOREM 1.7. (1) For any 2 D.Gr G / Hecke Z (2) The functor is an equivalence of cateories. L i Hecke Z.Gr G ; / D 0 for all i > 0: -mod 0 we have Hecke Z W D.Gr G / Hecke Z -mod 0! y -mod 0 re

16 1352 EDWARD FRENKEL and DENNS GATSGORY 2. Corollaries of the main theorem We shall now discuss some applications of Theorem 1.7. Note that both sides of the equivalence stated in Theorem 1.7 are cateories over the alebra Z re Specialization to a fixed central character. We fix a point 2 Spec.Z re /, i.e., a character of Z re, and consider the subcateories on both sides of the equivalence of Theorem 1.7(2), correspondin to objects on which the center acts accordin to this character. Let us denote the resultin subcateory of y -mod 0 re by y -mod 0. The resultin subcateory of D.Gr G/ Hecke Z -mod 0 can be described as follows. Let us denote by D.Gr G / Hecke -mod the cateory, whose objects are the data of. ; V /, where 2 D.Gr G / -mod and V are isomorphisms of D-modules defined for every V 2 Rep. {G/,? V ' V ; where V denotes the vector space underlyin the representation V. These isomorphisms must be compatible with tensor products of objects of Rep. {G/ in the same sense as in the definition of D.Gr G / Hecke Z -mod. Note that D.Gr G / Hecke -mod carries a natural weak action of the alebraic roup {G: 1 Given an S-point of {G and an S-family of objects. ; V / of D.Gr G / Hecke -mod we obtain a new S-family by keepin the same, but replacin V by V, where acts naturally on V S. n addition, D.Gr G / Hecke -mod carries a commutin Harish-Chandra action of the roup G..t//; in particular, the subcateory D.Gr G / Hecke -mod 0 makes sense. Let G; { be the fiber of the {G-torsor G;Z { at. Tautoloically we have: LEMMA 2.2. (1) For every trivialization W G; { ' 0 there exists a canonical {G equivalence respectin the action of G..t//; D.Gr G / Hecke Z -mod ' D.Gr G/ Hecke -mod; where the LHS denotes the fiber of D.Gr G / Hecke Z -mod at. (2) f 0 D for 2 {G, the above equivalence is modified by the self-functor of D.Gr G / Hecke -mod, iven by the action of. Hence, from Theorem 1.7 we obtain: COROLLARY 2.3. For every trivialization W G; { ' 0 there exists a {G canonical equivalence y -mod 0 ' D.Gr G/ Hecke -mod 0 : 1 We refer the reader to [FG06, 20.1], where this notion is introduced.

17 LOCALZATON OF -MODULES ON THE AFFNE GRASSMANNAN 1353 From Corollary 2.3 we obtain: COROLLARY 2.4. (1) For any two points 1 ; 2 2 Spec.Z re / and an isomorphism of {G-torsors G; { 1 ' G; { 2 there exists a canonical equivalence y -mod 0 1 ' y -mod 0 2 : (2) For every 2 Spec.Z re /, the roup of automorphisms of the {G-torsor G; { acts on the cateory y -mod 0. More enerally, let S be an affine scheme, and let 1;S and 2;S be two S-points of Spec.Z re /. Let y -mod 0 S;1 and y -mod 0 S;2 be the correspondin base-chaned cateories. By definition, the objects of y -mod i;s are the objects of y -mod re, endowed with an action of S compatible with the initial action of Z re on via the homomorphism Z re! S, correspondin to i;s. Morphisms in this cateory are y -morphisms compatible with the action of S. COROLLARY 2.5. For every lift of the map. 1;S 2;S / W S! Spec.Z re / Spec.Z re / to a map S! som Z, there exists a canonical equivalence y -mod 0 S;1 ' y -mod 0 S;2 : 2.6. Description of irreducibles. Corollary 2.3 allows us to describe explicitly the set of irreducible objects in y -mod 0 re. For that we will need to recall some more notation related to the cateories D.Gr G / Hecke -mod and D.Gr G / Hecke Z -mod. Consider the foretful functor D.Gr G / Hecke -mod!d.gr G / -mod. t admits a left adjoint, denoted nd Hecke, which can be described as follows. Let R be the object of Rep. {G/ equal to G { under the left reular action; let R denote the correspondin object of D.Gr G / -mod GŒŒt. Then for 2 D.Gr G / -mod, the convolution? R is naturally an object of D.Gr G / Hecke -mod, and it is easy to see that nd Hecke. / WD? R is the desired left adjoint. Similarly, the foretful functor D.Gr G / Hecke Z -mod! D.Gr G / -mod admits a left adjoint functor nd Hecke Z iven by 7!? RZ. The next assertion follows from the definitions: LEMMA 2.7. (1) For 2 D.Gr G / -mod there exist canonical isomorphisms:.gr G ; nd Hecke Z. // '.Gr; / Fun.som Z /; where Fun.som Z / is a module over Z re via either of the projections som Z! Spec.Z re /. Z re

18 1354 EDWARD FRENKEL and DENNS GATSGORY (2) For as above, Hecke Z Gr G ; nd Hecke Z. / '.Gr; /: Let us now recall the description of irreducible objects of D.Gr G / Hecke -mod 0, established in [ABB C 05, Cor ]. Recall that -orbits on Gr G are parametrized by the set W aff =W, where W aff denotes the extended affine Weyl roup. For an element zw 2 W aff let us denote by C zw;grg the correspondin irreducible object of D.Gr G / -mod. For an element w 2 W, let L w 2 W aff denote the unique dominant coweiht satisfyin: h {; i L 0 D if w. {/ is positive, and 1 if w. {/ is neative, for { runnin over the set of vertices of the Dynkin diaram. t was shown in loc. cit. that the objects nd Hecke.C ww / for w 2 W are the irreducibles of D.Gr G / Hecke -mod 0. Combinin this with Lemma 2.7 and Corollary 2.3, we obtain: THEOREM 2.8. somorphism classes of irreducible objects of y -mod 0 re are parametrized by pairs. 2 Spec.Z re /; w 2 W /. For each such pair the correspondin irreducible object is iven by.gr G ; C ww / : 2.9. The alebroid action. Let isom Z be the Lie alebroid of the roupoid som Z. Accordin to [BD] (see also [FG06, 7.4] for a review), we have a canonical action of isom Z on zu re.y/ by outer derivations, where zu re.y/ is the topoloical associative alebra correspondin to the cateory y -mod re and its tautoloical foretful functor to vector spaces. n more detail, there exists a topoloical associative alebra, denoted by U ren;re.y / and called the renormalized universal envelopin alebra at the ical level. t is endowed with a natural filtration, with the 0-th term U ren;re.y / 0 bein U re.y /, and Z re U ren;re.y / 1 =U ren;re.y / 0 ' U re.y / b isom Z : The action of isom Z on zu re.y/ is iven by the adjoint action of isom Z, rearded as a subset of U ren;re.y / 1 =U ren;re.y / 0. Let S be an affine scheme, and let S be an S-point of Spec.Z re /. Let S be a section of isom Z j S. Set S 0 WD S Spec. Œ" =" 2 /; then the imae of S in T.Spec.Z re //j S ives rise to an S 0 -point, denoted, 0 S, of Spec.Zre /. Z re

19 LOCALZATON OF -MODULES ON THE AFFNE GRASSMANNAN 1355 Let y -mod S (resp., y -mod S 0) be the correspondin base-chaned cateory, where the latter identifies with the cateory of discrete modules over zu re.y/ S (resp., zu re.y/ re S 0). Then the above action of isom Z on y -mod re ives Z Z re rise to the followin construction: To every 2 y -mod S we can functorially attach an extension (10) 0!! 0!! 0; 0 2 y -mod S 0 : The module 0 is defined as follows. The above action of isom Z by outer derivations of zu re.y/ allows us to lift S to an isomorphism A. S / W zu re We set 0 to be the zu re.y/ Z re ; S 0.y/ Z re ; S 0 S 0! zu re.y/ Z re S Œ" =" 2 : ; S S 0-module, correspondin via A. S / to Œ" =" 2. The isomorphism A. S / is defined up to conjuation by an element of the form 1 C " u, u 2 zu re.y/ S. Since this automorphism can be canonically Z re ; S lifted onto Œ" =" 2, we obtain that 0 is well-defined. By construction, the functor 7! 0 respects the Harish-Chandra G..t//- actions on the cateories y -mod S and y -mod S 0, respectively. Let us note now that data. S W S! Spec.Z re be rearded as a map S 0! som Z, where first and second projections are equal, respectively, to S 0! som Z Spec.Z re / S 0! S S! Spec.Z re / and S 0 0 S! Spec.Z re /: Hence, Corollary 2.5 ives rise to an equivalence and, in particular, to a functor y -mod 0 S Œ" ="2 ' y -mod 0 S 0 ; (11) y -mod 0 S! y -mod 0 S 0 : /; S 2 isom Z j S / as above can PROPOSTON The functor 7! 0 W y -mod S! y -mod S 0 of (10), restricted to y -mod 0 S, is canonically isomorphic to the functor (11). Proof. The assertion follows from the fact that for 2 D.Gr G / -mod, the y -action on.gr G ; / lifts canonically to an action of U ren;re.y / (see [FG06, 7.4]), so that for.s; S ; S / as above we have a canonical trivialization W.Gr G ; / 0 '.Gr G ; /Œ" =" 2 ;

20 1356 EDWARD FRENKEL and DENNS GATSGORY in the notation of (10). Moreover, this functorial isomorphism is compatible with that of Theorem 1.3 in the sense that for every V 2 Rep. {G/, the diaram.gr G ;? V / 0? V!.Gr G ; V /Œ" =" 2?? ˇV y ˇV idy 0 S.Gr G ; /!.Gr G ; / Œ" =" 2 ; Z re commutes, where the bottom arrow comprises the isomorphism and the canonical action of S on Z. The latter compatibility follows assertion (b) in Theorem of [FG06] Relation to semi-infinite cohomoloy. Let us consider the functor of semi-infinite cohomoloy on the cateory y -mod 0 re : Z re 7! H 1 2 C.n..t//; nœœt ; 0 / (see [FG06, 18] for details concernin this functor). For an S-point S of Spec.Z re / and 2 y -mod S, each H 1 2 Ci.n..t//, nœœt, 0 / is naturally an S -module. Let now. 1;S ; 2;S / be a pair of S-points of Spec.Z re /, equipped with a lift S! som Z, and let 1 2 y -mod 0 S;1 and 2 2 y -mod 0 S;2 be two objects correspondin to each other under the equivalence of Corollary 2.5. PROPOSTON Under the above circumstances the S -modules H 1 2 Ci.n..t//; nœœt ; 1 0 / and H 1 2 Ci.n..t//; nœœt ; 2 0 / are canonically isomorphic. Proof. The assertion of the proposition can be tautoloically translated as follows: The functor D.Gr G / -mod! H 1 2 Ci.n..t//;nŒŒt ; 0 / y -mod re! Z re -mod factors throuh a functor H 1 2 Ci {G W D.Gr G / -mod! Rep. {G/; followed by the pull-back functor, correspondin to the morphism Spec.Z re /! pt = {G. Moreover, for V 2 Rep. {G/ we have a functorial isomorphism (12) H 1 2 Ci.? V / ' H 1 2 Ci. / V; {G {G compatible with the isomorphism of Theorem 1.3(1).

21 LOCALZATON OF -MODULES ON THE AFFNE GRASSMANNAN 1357 The souht-after functor H 1 2 Ci {G 18.3]. Namely, has been essentially constructed in [FG06, Hom G { V L ; H 1 2 Ci. / WD H i.n..t//; j {G N..t//t L 0 /; in the notation of loc. cit. The isomorphisms (12) follow from the definitions. Finally, we would like to compare the isomorphisms of Proposition 2.12 and Proposition Let be an object of y -mod 0 re ; let S be an S-point of Spec.Z re / and S a section of isom Z j S. On the one hand, in Proposition of [FG06] we have shown that there exists a canonical isomorphism: a W H 1 2 Ci.n..t//; nœœt ; 0 0 / ' H 1 2 Ci.n..t//; nœœt ; 0 /Œ" =" 2 ; valid for any 2 y -mod re. On the other hand, combinin Proposition 2.10 and Proposition 2.12 we obtain another isomorphism b W H 1 2 Ci.n..t//; nœœt ; 0 0 / ' H 1 2 Ci.n..t//; nœœt ; 0 /Œ" =" 2 : Unravelin the two constructions, we obtain the followin: LEMMA The isomorphisms a and b coincide. 3. Proof of the main theorem n Section 1.6 we constructed a functor Hecke Z W D.Gr G / Hecke Z -mod 0! y -mod 0 re : Now we wish to show that this functor is an equivalence of cateories. This will prove Theorem 1.7. We start by considerin in Section 3.1 certain objects Z w; w 2 W, of the cateory D.Gr G / Hecke Z -mod 0 such that Hecke Z. Z w/ ' w;re, the latter bein the standard modules of the cateory y -mod 0 re. The main result of Section 3.1, Theorem 3.2, which states the existence of Z w, will be proved in Section 4. Next, in Section 3.4 we prove part (1) of Theorem 1.7 that the functor Hecke Z is exact. We then outline in Section 3.9 a eneral framework for provin that it is an equivalence. Usin this framework, we prove Theorem 1.7 modulo Theorem 3.2. n Section 3.14 we explain what needs to be done in order to prove our stroner Conjecture 1.5. Finally, in Sections we ive an alternative proof of part (1) of Theorem 1.7.

22 1358 EDWARD FRENKEL and DENNS GATSGORY 3.1. Standard modules. For an element w 2 W, let w D nd y ŒŒt.M w./ / be the Verma module over y, where for a weiht we denote by M the Verma module over with hihest weiht. Let w;re D w Z re Z be the maximal quotient module that belons to y -mod re. n fact, it was shown in [FG06, Cor ], that as modules over Z, all w are supported over a quotient alebra Z nilp, and are flat as Z nilp -modules. The subscheme Spec.Z re / Spec.Z / is contained in Spec.Z nilp /, so that the definition of w;re does not nelect any lower cohomoloy. The main inredient in the remainin steps of our proof of Theorem 1.7 is the followin: THEOREM 3.2. For each w 2 W there exists an object Z w 2 D.Gr G / Hecke Z - mod 0, such that Hecke Z.Gr G ; w / is isomorphic to w;re. The proof of this theorem will consist of an explicit construction of the objects Z w, which will be carried out in Section 4. The proof of Theorem 1.7 will only use a part of the assertion of Theorem 3.2: namely, that there exist objects Z w 2 D.Gr G / Hecke Z -mod 0, endowed with a surjection (13) Hecke Z.Gr G ; Z w / w;re: What we will actually use is the followin corollary of this statement: COROLLARY 3.3. For every 2 y -mod 0 re there exists an object 2 D.Gr G / Hecke Z -mod 0 and a nonzero map Hecke Z.Gr G ; /!. Proof. By [FG06, Lemma 7.8.1], for every object 2 y -mod 0 re there exist w 2 W and a nonzero map w;re! Exactness. Let us recall from Section 2.6 the left adjoint functor nd Hecke Z to the obvious foretful functor D.Gr G / Hecke Z -mod! D.Gr G / -mod. t is clear that every object of D.Gr G / Hecke Z -mod can be covered by one of the form nd Hecke Z. /. From Lemma 2.7(1) we obtain that we can use boundedfrom-above complexes, whose terms consist of objects of the form nd Hecke Z. /, in order to compute L Hecke Z. Thus, we obtain: LEMMA 3.5. For 2 D.Gr G / Hecke Z -mod, L i Hecke Z.Gr G ; / ' Tor Fun.som Z/ i.gr G ; /; Z re :

23 LOCALZATON OF -MODULES ON THE AFFNE GRASSMANNAN 1359 We shall call an object of D.Gr G / Hecke Z -mod finitely enerated if it can be obtained as a quotient of an object of the form nd Hecke Z. /, where is a finitely enerated object of D.Gr G / -mod. t is easy to see that an object 2 D.Gr G / Hecke Z -mod is finitely enerated if and only if the functor Hom D.GrG / Hecke Z -mod. ; / commutes with direct sums. We shall call an object of D.Gr G / Hecke Z -mod finitely presented, if it is isomorphic to coker nd Hecke Z. 1 /! nd Hecke Z. 2 /, where 1 ; 2 are both finitely enerated objects of D.Gr G / -mod. The followin lemma is straihtforward. LEMMA 3.6. (1) An object 2 D.Gr G / Hecke Z -mod is finitely presented if and only if the functor Hom D.GrG / Hecke Z -mod. ; / commutes with filterin direct limits. (2) Every object of D.Gr G / Hecke Z -mod is isomorphic to a filterin direct limit of finitely presented ones. The proof of the followin proposition will be iven in Section PROPOSTON 3.7. For every finitely presented object of D.Gr G / Hecke Z -mod, the correspondin object L Hecke Z.Gr G ; / 2 D.y -mod re / belons to D b.y -mod re /. The crucial step in the proof of part (1) of Theorem 1.7 is the followin: PROPOSTON 3.8. f 2D.Gr G / Hecke Z -mod 0 is such that L Hecke Z.Gr G ; / belons to D b.y -mod re / 0, then L i Hecke Z.Gr G ; / D 0; i > 0: Proof. Let be the lowest cohomoloy of L Hecke Z.Gr G ; /, which lives, for example, in deree k. By Corollary 3.3 there exist another object 1 2 D.Gr G / Hecke Z -mod 0 and a nonzero map Hecke Z.Gr G ; 1 /!. Hence, we obtain a nonzero map in D.y -mod re / L Hecke Z.Gr G ; 1 /Œk! L Hecke Z.Gr G ; /: But by Theorem 1.4, such a map comes from a map 1 Œk!, which is impossible if k > 0. Proof of part (1) of Theorem 1.7. Combinin Proposition 3.7 and Proposition 3.8, we obtain that L i Hecke Z.Gr G ; / D 0 for any i > 0 and any 2 D.Gr G / Hecke Z -mod 0, which is finitely presented. However, by Lemma 3.5, the functors 7! L i Hecke Z.Gr G ; / commute with direct limits, and our assertion follows from Lemma 3.6(2).

24 1360 EDWARD FRENKEL and DENNS GATSGORY 3.9. Proof of the equivalence. Consider the followin eneral cateorical framework. Let G W 1! 2 be an exact functor between abelian cateories. Assume that for X; Y 2 1 the maps and are isomorphisms. Hom 1.X; Y /! Hom 2.G.X/; G.Y // Ext 1 1.X; Y /! Ext 1 2.G.X/; G.Y // LEMMA f G admits a riht adjoint functor F which is conservative, then G is an equivalence. 2 Proof. The fully faithfulness assumption on G implies that the adjunction map induces an isomorphism between the composition F ı G and the identity functor on 1. We have to show that the second adjunction map is also an isomorphism. For X let Y 0 and Z 0 be the kernel and cokernel, respectively, of the adjunction map G ı F.X 0 /! X 0 : Bein a riht adjoint functor, F is left-exact, hence we obtain an exact sequence 0! F.Y 0 /! F ı G ı F.X 0 /! F.X 0 /: But since F.X 0 /! F ı G.F.X 0 // is an isomorphism, we obtain that F.Y 0 / D 0. Since F is conservative, this implies that Y 0 D 0. Suppose that Z 0 0. Since F.Z 0 / 0, there exists an object Z 2 1 with a nonzero map G.Z/! Z 0. Consider the induced extension 0! G ı F.X 0 /! W 0! G.Z/! 0: Since G induces a bijection on Ext 1, this extension can be obtained from an extension 0! F.X 0 /! W! Z! 0 in 1. n other words, we obtain a map G.W /! X 0, which does not factor throuh G ı F.X 0 / X 0, which contradicts the.g; F/ adjunction. Thus, in order to prove part (2) of Theorem 1.7 it remains to show that the functor Hecke Z W D.Gr G / Hecke Z -mod 0! y -mod 0 re admits a riht adjoint. (The fact that it is conservative will then follow immediately from Corollary 3.3.) Recall from [FG06, 20.7], that the tautoloical functor D.Gr G / Hecke Z -mod 0,! D.Gr G / Hecke Z -mod admits a riht adjoint, iven by Av 0. Hence, it suffices to prove the followin: 2 Recall that a functor F is called conservative if for any X 0 we have F.X/ 0.

25 LOCALZATON OF -MODULES ON THE AFFNE GRASSMANNAN 1361 PROPOSTON The functor admits a riht adjoint. Hecke Z W D.Gr G / Hecke Z Proof. First, we will show the followin: -mod! y -mod re LEMMA The functor W D.Gr G / -mod! y -mod re admits a riht adjoint. Proof. We will prove that for any level k the functor W D.Gr G / k -mod! y -mod k admits a riht adjoint (see the ntroduction for the definition of these cateories). That is, we have to prove the representability of the functor (14) 7! Hom yk -mod.gr G ; /; for every iven 2 y k -mod. Consider the followin eneral set-up. Let be an abelian cateory, and let 0 be a full (but not necessarily abelian) subcateory, such that the followin holds: 0 is equivalent to a small cateory. The cokernel of any surjection X 00 X 0 with X 0 ; X , also belons to 0. is closed under filterin direct limits. For X 2 0, the functor Hom.X; / commutes with filterin direct limits. Every object of is isomorphic to a filterin direct limit of objects of 0. Then we claim that any contravariant left-exact functor F! Vect, which maps direct sums to direct products (and, hence, direct limits to inverse limits, by the previous assumption), is representable. ndeed, iven such F, consider the cateory of pairs.x; f /, where X 2 0 and f 2 F.X/. Morphisms between.x; f / and.x 0 ; f 0 / are maps W X! X 0, such that.f 0 / D f. By the first assumption on 0, this cateory is small. By the second assumption on 0 and the left-exactness of F, this cateory is filterin. t is easy to see that the object lim X!.X;f / represents the functor F. We apply this lemma to D D.Gr G / k -mod with 0 bein the subcateory of finitely-enerated D-modules. We set F to be the functor (14), and the representability assertion follows. Note that we could have applied the above eneral principle to D D.Gr G / Hecke Z -mod, where 0 is the subcateory of finitely presented objects, and obtain the assertion of Proposition 3.11 riht away.

26 1362 EDWARD FRENKEL and DENNS GATSGORY Thus, for, let be the object of D.Gr G / -mod that represents the functor 1 7! Hom y -mod re.gr G ; 1 /; for a iven 2 y -mod re. We claim that is naturally an object of D.Gr G / Hecke Z -mod and that it represents the functor (15) 1 7! Hom y -mod re Hecke Z.Gr G ; 1 /; : First, since the alebra Z re acts on by endomorphisms, the object carries an action of Z re by functoriality. Let us now construct the morphisms V. Evidently, it is sufficient to do so for V finite-dimensional. Let V denote its dual. For a test object 1 2 D.Gr G / -mod we have: Hom D.GrG / -mod. 1 ;? V / ' Hom D.GrG / -mod. 1? V ; / ' Hom y -mod re.gr G ; 1? V /; ' Hom y -mod re ' Hom y -mod re.gr G ; 1 / Z re Z re;.gr G ; 1 /; Z ; where the last isomorphism takes place since Z is locally free. For the same reason, Hom D.GrG / -mod. 1 ; Z / ' Hom y -mod re.gr G ; 1 /; Z ; which implies that there exists a canonical isomorphism V Z re? V ' Z ; as required. That these isomorphisms are compatible with tensor products of objects of Rep. {G/ follows from Theorem 1.3(2). Finally, the fact that. ; V /, thus defined, represents the functor (15), follows from the construction. This completes the proof of Proposition Thus, we obtain that the functor Hecke Z admits a riht adjoint functor. Moreover, this riht adjoint functor is conservative by Corollary 3.3. Therefore part (2) of Theorem 1.7 now follows from part (1), proved in Section 3.4, and Lemma 3.10, modulo Proposition 3.7 and Theorem 3.2. t remains to prove those two statements. Proposition 3.7 will be proved in the next subsection and Theorem 3.2 will be proved in Section Proof of Proposition 3.7. Recall the cateory D.Gr G / Hecke -mod, introduced in Section 2.6. Recall also that the {G-torsor G;Z { on Spec.Zre / is noncanonically trivial, and let us fix such a trivialization. This choice identifies the Z re Z re Z re

27 LOCALZATON OF -MODULES ON THE AFFNE GRASSMANNAN 1363 cateory D.Gr G / Hecke Z -mod with D.Gr G / Hecke -mod Z re, i.e., with the cateory of objects of D.Gr G / Hecke -mod endowed with an action of Z re by endomorphisms. Under this equivalence, the functor 7! nd Hecke Z. / oes over to 7! nd Hecke. / Z re : Note also that the trivialization of G;Z { identifies som Z with Spec.Z re / {G Spec.Z re /, so that the map 1 somz corresponds to re Spec.Z / 1. For as above, {G we have an identification Gr G ; nd Hecke Z. / '.Gr G ; / { G Zre : Let be a finitely presented object of D.Gr G / Hecke Z -mod equal to the cokernel of a map W nd Hecke. 1 / Z re! nd Hecke. 2 / Z re : Recall that Z re is isomorphic to a polynomial alebra Œx 1 ; : : : ; x n ; : : :. Since 1 was assumed finitely enerated, a map as above has the form m id ŒxmC1 ;x mc2 ;:::, where m is a map nd Hecke. 1 / Œx 1 ; : : : ; x m! nd Hecke. 2 / Œx 1 ; : : : ; x m defined for some m. Hence, as a module over Fun.som Z / ' Z re { G Zre, (16).Gr G ; / ' Œx mc1 ; x mc2 ; : : : ; where is some module over Z re G { Œx 1; : : : ; x m. We can compute.gr G ; / L Z re Fun.som Z / in two steps, by first restrictin to the preimae of the diaonal under Spec.Z re / {G Spec.Z re / and then by further restriction to sittin inside Spec. Œx mc1 ; x mc2 ; : : : / Spec. Œx mc1 ; x mc2 ; : : : /; Spec. Œx 1 ; : : : ; x m / Spec. Œx mc1 ; x mc2 ; : : : / Spec. Œx 1 ; : : : ; x m / {G Spec. Œx 1 ; : : : ; x m / Spec. Œx mc1 ; x mc2 ; : : : /: When we apply the first step to the module appearin in (16), it is acyclic of cohomoloical deree 0. The second step has a cohomoloical amplitude bounded