Local Geometric Langlands Correspondence: The Spherical Case

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1 Local Geometric Lanlands Correspondence: The Spherical Case The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Frenkel, Edward, and Dennis Gaitsory Local eometric lanlands correspondence: The spherical case. In Alebraic analysis and around: In honor of Professor Masaki Kashiwara's 60th Birthday, ed. M. Kashiwara, T Miwa, et al, Tokyo : Mathematical Society of Japan. Published Version Citable link Terms of Use This article was downloaded from Harvard University s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at nrs.harvard.edu/urn-3:hul.instrepos:dash.current.terms-ofuse#laa

2 LOCAL GEOMETRIC LANGLANDS CORRESPONDENCE: THE SPHERICAL CASE arxiv: v1 [math.qa] 7 Nov 2007 EDWARD FRENKEL 1 AND DENNIS GAITSGORY 2 To Masaki Kashiwara on his 60th birthday Abstract. A module over an affine Kac Moody alebra b is called spherical if the action of the Lie subalebra [[t]] on it interates to an alebraic action of the correspondin roup G[[t]]. Consider the cateory of spherical b-modules of critical level. In this paper we prove that this cateory is equivalent to the cateory of quasi-coherent sheaves on the ind-scheme of opers on the punctured disc which are unramified as local systems. This result is a cateorical version of the well-known description of spherical vectors in representations of roups over local non-archimedian fields. It may be viewed as a special case of the local eometric Lanlands correspondence proposed in [FG2]. 1. Introduction A eneral framework for the local eometric Lanlands correspondence was proposed in our earlier work [FG2] (see also [FG3] [FG5] and [F2]). Accordin to our proposal, to each local Lanlands parameter σ, which is a local system on the punctured disc D = Spec C((t)) (or equivalently, a -bundle with a connection on D ), there should correspond a cateory C σ equipped with an action of the formal loop roup G((t)). Even more ambitiously, we expect that there exists a cateory C univ fibered over the stack Loc(D ) of -local systems, equipped with a fiberwise action of the ind-roup G((t)), whose fiber cateory at σ is C σ. Moreover, we expect C univ to be the universal cateory equipped with an action of G((t)). In other words, we expect that Loc(D ) is the universal parameter space for the cateorical representations of G((t)). The ultimate form of the local Lanlands correspondence for loop roups should be, rouhly, the followin statement: (1.1) cateories fiberin over Loc(D ) cateories equipped with action of G((t)) We should point out, however, that neither the notion of cateory fibered over a non-alebraic stack such as Loc(D ), nor the unversal property alluded to above are easy to formulate. So for now (1.1) should be understood heuristically, as a uidin principle. As we explained in [FG2], the local eometric Lanlands correspondence should be viewed as a cateorification of the local Lanlands correspondence for the roup G(F), where F is a local non-archimedian field. This means that the cateories C σ, equipped with an action of G((t)), that we wish to attach to the Lanlands parameters σ Loc(D ) should be viewed as Date: November Supported by DARPA and AFOSR throuh the rant FA Supported by NSF rant

3 2 EDWARD FRENKEL AND DENNIS GAITSGORY cateorifications of smooth representations of G(F) in the sense that we expect the Grothendieck roups of the cateories C σ to look like irreducible smooth representations of G(F) The spherical part. In the study of representations π of G(F), a standard tool is to consider the subspaces π K of vectors fixed by open compact subroups K of G(F). This procedure has a cateorical counterpart. Let K be a roup-scheme contained in G[[t]] and containin the Nth conruence subroup K N for some N (i.e., the subroup of G[[t]] consistin of elements conruent to 1 modulo t N C[[t]]). For example, K can be G[[t]] itself, or the Iwahori subroup I. Given a cateory C, acted on by G((t)), we can consider the correspondin K-equivariant cateory C K. Via (1.1), any such C K is also a cateory fibered over Loc(D ). This procedure applies in particular to C univ. Althouh at present, we do not know how to construct the entire cateory C univ, we do have a uess what C K univ for some choices of K. In this paper we specialize to the simplest case K = G[[t]]. (Another case, which can be explicitly analyzed is that of K = I, discussed in [FG2].) Based on the analoy with the classical local Lanlands correspondence for spherical representations, we propose: (1.2) C G[[t]] univ Rep(). Here Rep() is the cateory of (alebraic) representations of, which can be also thouht as the cateory of quasi-coherent sheaves on the stack pt/. The structure of cateory fibered over Loc(D ) comes from the maps of stacks (1.3) pt/ Locunr Loc(D ) correspondin to the inclusion of the stack Loc unr of unramified local systems (or, equivalently, local systems on the unpunctured disc D) into the stack Loc(D ) of all local systems Representations of critical level. In [FG2] we have considered a specific example of a cateory equipped with an action of G((t)); namely, the cateory ĝ crit mod of modules over the affine Kac Moody alebra ĝ of critical level. It carries a canonical action of the ind-roup G((t)) via its adjoint action on ĝ crit. What should be the relationship between ĝ crit mod and the conjectural universal cateory C univ? We note that the cateory ĝ crit mod naturally fibers over the ind-scheme Op(D ) of - opers on D introduced in [BD]. This is because, accordin to [FF, F1], the center Z of the cateory ĝ crit mod is isomorphic to the alebra of functions on Op(D ). The idea of [FG2] is that the latter fibration is a base chane of C univ, that is, there is a Cartesian diaram (1.4) ĝ crit mod Op(D ) C univ α Loc(D ) which commutes with the action of G((t)) alon the fibers of the two vertical maps. In other words, (1.5) ĝ crit mod C univ Op(D ). Loc(D )

4 LOCAL GEOMETRIC LANGLANDS CORRESPONDENCE: THE SPHERICAL CASE 3 Given a ǧ-oper χ, let us consider it as a point of Spec(Z ), i.e., a character of Z. Let ĝ crit mod χ be the full subcateory ĝ crit mod χ of ĝ crit mod whose objects are ĝ crit -modules, on which the Z acts accordin to this character. This is the fiber cateory of the cateory ĝ crit mod over χ Op(D ). Let σ = α(χ) Loc(D ). By (1.5), we have: (1.6) C σ ĝ crit mod χ. As was mentioned above, at the moment we do not have an independent definition of C univ, and therefore we cannot make the equivalences (1.5) and (1.6) precise. But we use it as our uidin principle. This leads us to a number of interestin corollaries, some of which have been discussed in [FG2] [FG5]. For example, if χ, χ are two -opers, such that the correspondin local systems α(χ) and α(χ ) are isomorphic, for every choice of an isomorphism we are supposed to have an equivalence of cateories: (1.7) ĝ crit mod χ ĝ crit mod χ. This is a hihly non-trivial conjecture about representations of ĝ crit Harish-Chandra cateories. Let us return to the discussion of the cateory of K- equivariant objects in the context of C = ĝ crit mod. The correspondin cateory ĝ crit mod K identifies with the cateory of (ĝ crit, K) Harish-Chandra modules. When K is connected, this is a full abelian subcateory of ĝ crit mod, consistin of modules, on which the action of the Lie alebra Lie(K) ĝ crit is interable, i.e., comes from an alebraic action of K. Now specialize to the case K = G[[t]]. We call objects of the correspondin cateory ĝ crit mod G[[t]] of G[[t]]-equivariant ĝ crit -modules spherical. Combinin eqns. (1.2), (1.3) and (1.5), we arrive at the followin equivalence: (1.8) ĝ crit mod G[[t]] QCoh ( Loc unr Op(D ) Loc(D ) Here we should remark that althouh the stack Loc(D ) is a problematic object to work with, the fiber product Loc unr Loc(D ) Op(D ) appearin on the riht-hand side of (1.8) is a well-defined (non-reduced) ind-subscheme of Op(D ). This is the moduli ind-scheme of opers that are unramified as local systems. We denote this ind-scheme by Op unr. It is a disjoint union of formal schemes Opunr,λ, λ bein a dominant weiht, where the reduced scheme correspondin to each Op unr,λ is the scheme of λ-reular opers introduced in [FG2]. Op re,λ Thus, the heuristic uess iven by (1.8) leads to the followin precise statement, which is the main result of this paper: Main Theorem. The cateory ĝ crit mod G[[t]] of spherical ĝ crit -modules is equivalent to the cateory of quasi-coherent sheaves on the ind-scheme Op unr of -opers on D unramified as local systems. Moreover, we show that a functor from the former cateory to the latter one is an analoue of the Whittaker functor. ).

5 4 EDWARD FRENKEL AND DENNIS GAITSGORY 1.4. Some corollaries. Let χ be a C-point of Op(D ), and let us consider the cateory ĝ crit mod G[[t]] χ. As an abelian cateory, this is a full subcateory of ĝ crit mod χ, consistin of G[[t]]-interable modules. One can show (see [FG3], Corollary 1.11) that this cateory is 0 unless χ Op unr. In the latter case, from the Main Theorem we obtain that the cateory ĝ crit mod G[[t]] χ is equivalent to the cateory of vector spaces. This result is the first test for our prediction that ĝ crit mod χ, as a cateory equipped with a G((t))-action, depends only on α(χ), as expected in (1.7). In addition, this equivalence is in areement with a classical fact that the space of spherical vectors in an irreducible representation of G(F) is either zero or one-dimensional. As another corollary of the Main Theorem, we obtain the followin description of the alebra of self-exts of the Weyl modules V λ in the derived cateory D(ĝ crit mod G[[t]] ) of (ĝ crit, G[[t]]) Harish-Chandra modules: Ext D(b crit mod G[[t]] ) (Vλ,crit, V λ,crit) Λ (N z re,λ re/ λ unr ), where Nre/unr λ is the bundle of Opre,λ in Op unr,λ. (In the above formula we identify the alebra of function on Op re,λ with the correspondin quotient of Z, denoted z re,λ.) For λ = 0 this isomorphism was previously established in [FT] by other methods Structure of the proof. The proof of the Main Theorem is quite simple. The main idea is that the cateory ĝ crit mod G[[t]] has a universal object, denoted D ch, which is the vacuum module of the chiral alebra of differential operators on G. The module D ch is in fact a ĝ crit -bimodule, and for any other object M ĝ crit mod G[[t]] we have M D ch 2 M ((t)) 2 (here stands for the semi-infinite Tor functor). ((t)) Therefore, in order to define functors and check isomorphisms on ĝ crit mod G[[t]], it is enouh to do so just for the module D ch. Thus, in Sect. 2 we prove a theorem that describes the structure of D ch as a bi-module over ĝ crit mod G[[t]], and in Sect. 3 we derive our Main Theorem from this structure theorem. 2. Chiral differential operators on G at the critical level In this section we describe the structure of the chiral alebra of differential operators (CADO) on a simple connected simply-connected alebraic roup G over C at the critical level, viewed as a bimodule over ĝ crit Notation. We will follow the notation of [FG2]. In particular, ĝ crit is the critical central extension of the formal loop alebra ((t)), ĝ crit mod is the cateory of discrete modules over ĝ crit, Z is the center of ĝ crit mod (or, equivalently, of the completion of the envelopin alebra of ĝ crit ). This is a topoloical commutative alebra. Accordin to a theorem of [FF, F1], the correspondin ind-scheme Spec(Z ) is canonically isomorphic to the moduli space Op(D ) of -opers on the formal punctured disc, where is the Lanlands dual roup to G (of adjoint type). For the definition of Op(D ), see [BD].

6 LOCAL GEOMETRIC LANGLANDS CORRESPONDENCE: THE SPHERICAL CASE 5 For λ Λ +, we let z re,λ denote the quotient of Z correspondin to the sub-scheme Op re,λ Op(D ) introduced in [FG2], Section 2.9. Let V λ,crit = Ind bcrit [[t]] C1 (V λ ) := U(ĝ crit ) V λ U([[t]] C1) be the Weyl module with dominant interal hihest weiht λ Λ +. Accordin to [FG6], Theorem 1, the action of Z on V λ,crit factors as follows: Furthermore, V λ,crit Z z re,λ End(V λ,crit ). is flat (and in fact, free) as a zre,λ -module. Let ĝ crit mod G[[t]] be the full abelian subcateory of ĝ crit mod. In this paper we will work with the naive derived cateory D(ĝ crit mod G[[t]] ). However, by eneralizin the arument of [FG2], Sect , one can identify D(ĝ crit mod G[[t]] ) with the G[[t]]-equivariant derived cateory correspondin to ĝ crit mod, as introduced in loc. cit., Sect In particular, for M ĝ crit mod G[[t]] we have: Ext i b crit mod G[[t]](Vλ,crit, M) Ext i G[[t]](V λ, M) Unramified opers. Let Op unr Op(D ) be the ind-subscheme of opers that are unramified as local systems. For any C-alebra A, the set of A-points of Op unr is by definition the set of opers on Spec A((t)), which are isomorphic, as local systems, to the trivial local system. We have: (2.1) Op unr, where Op unr,λ Op unr,λ λ Λ + are pairwise disjoint formal sub-schemes of Op(D ) with (Op unr,λ ) red Op re,λ. We will also use the notation Spec(Z unr ), Spec(Z unr,λ ) for these ind-schemes. Let ι λ re/unr denote the closed embeddin Spec(z re,λ ) Spec(Z unr,λ ); let I λ denote the (closed) ideal of Spec(z re,λ ) in Spec(Z unr,λ ); let Nre/ λ unr be the normal scheme to Spec(zre,λ ) in Spec(Z unr,λ ). It follows from [FG2], Section 4.6, that its sheaf of sections is a locally free z re,λ -module (in other words, Nre/unr λ is s vector bundle over Spec(zre,λ )). Moreover, Z carries a Poisson structure, which identifies Nre/ λ unr with Ω1 (z re,λ ). The followin fact was established in [FG3], Corollary 1.11 (note that Spec(Z unr ) was denoted by Spec(Z m.f. ) in [FG3]). Theorem 2.3. The support in Spec(Z ) of every M ĝ crit mod G[[t]] is contained in Spec(Z unr ). Thus, every G[[t]]-interable ĝ-module M splits as a direct sum λ M λ, where M λ is supported at Spec(Z unr,λ ) and has an increasin filtration whose sub-quotients are quotient modules of V λ,crit.

7 6 EDWARD FRENKEL AND DENNIS GAITSGORY 2.4. Chiral differential operators. Let X be a smooth alebraic curve. We will work with Lie-* alebras and chiral alebras on X and with modules over them supported at a fixed point x X (see [CHA] for the definitions). In this paper all chiral alebras will come from vertex alebras, and we will tacitly identify a chiral alebra with its vacuum module, i.e., its fiber at any point of a curve equipped with a coordinate. In fact, everythin may be rephrased in terms of the correspondin vertex (Lie) alebras and modules over them, but we will use the formalism of chiral alebras for the sake of consistency with [FG1] [FG3]. Recall from [AG, GMS] that for any level κ (i.e., an invariant bilinear form on ) we have the chiral alebra of differential operators (CADO), denoted by D ch G,κ. It comes equipped with two mutually commutin embeddins l (2.2) A,κ D ch r (G) κ A,κ, where κ = κ + 2κ crit. Recall that the fiber of A,κ at x is the vacuum Weyl module V,κ of level κ, and the fiber of D ch at x with Ind bκ [[t]] C1 (O G[[t]]) := U(ĝ κ ) O G[[t]]. U([[t]] C1) Here t is a formal coordinate at x and O G[[t]] is the alebra of functions on the roup G[[t]], on which [[t]] acts trivially and 1 acts as the identity. This is a module over ĝ κ ĝ κ. The action of ĝ κ on D ch (G) κ,x correspondin to the left arrow in (2.2) is the natural action on this induced module, and the action correspondin to the riht arrow in (2.2) was constructed in [AG, GMS]. We will refer to the two actions as the left and the riht actions, respectively CADO at the critical level. We now specialize to the critical level κ = κ crit. Then κ = κ crit, and so both left and riht actions of ĝ correspond to the critical level. We will describe the structure of D ch (G) crit,x as a ĝ crit -bimodule. From now on, when there is no confusion, we will skip the subscript x when describin the fiber of the chiral alebra at x. Let z denote the center of A,crit (note that z identifies with z re,0 ). The followin has been established in [FG1]: Lemma 2.6. The two embeddins l, r : z D ch differ by the automorphism of z, induced by Cartan involution τ of. As a bimodule over ĝ crit, D ch is G[[t]]-interable with respect to both actions. By The- ) (note that by Lemma 2.6, the differ by τ). Hence, we have a direct sum decomposition of Dch orem 2.3, its support over Spec(Z ) is contained in Spec(Z unr two actions of Z on D ch as a ĝ crit -bimodule: D ch λ Λ + D ch,λ where D ch,λ is the summand supported at Opunr,λ = Spec(Z unr,λ ) (see formula (2.1)). Recall that O G[[t]] denotes the alebra of functions on G[[t]]. It has a natural structure of commutative chiral alebra, and as such it is a chiral subalebra of D ch. The map O G[[t]] D ch respects the bimodule structure with respect to [[t]] ĝ crit.

8 LOCAL GEOMETRIC LANGLANDS CORRESPONDENCE: THE SPHERICAL CASE 7 For λ Λ + we have a natural map V λ V τ(λ) O G O G[[t]], compatible with the action of [[t]] [[t]]. Inducin, we obtain a map of bimodules over ĝ crit : From Lemma 2.6 we obtain: V λ,crit Vτ(λ),crit Dch,λ. Lemma 2.7. The above map factors throuh a map (2.3) V λ,crit V τ(λ),crit Dch,λ. z re,λ 2.8. A description of the CADO. Recall that I λ denotes the ideal of Spec(z re,λ ) in ). Consider the canonical increasin filtration on D ch,λ numbered by i = 0, 1,... Spec(Z unr,λ with F i (D ch,λ ) bein the ĝ crit sub-bimodule, annihilated by the i + 1-st power of the ideal I λ. By construction, the imae of the map (2.3) belons to F 0 (D ch,λ ). We are now ready to formulate the main result of this section: Theorem 2.9. (1) The map (2.3) defines an isomorphism (2) The canonical maps ive rise to isomorphisms V λ,crit V τ(λ),crit F 0 (D ch,λ ). z re,λ (I λ ) n /(I λ ) n+1 r n (D ch,λ ) r0 (D ch,λ ) z re,λ (2.4) r n (D ch,λ ) r0 (D ch,λ ) z re,λ of ĝ crit -bimodules, where N λ re/unr Sym n (N z re,λ re/unr λ ) is the normal bundle to Opre,λ The above theorem should be contrasted with the followin: in Op unr,λ. Lemma For a eneric κ (i.e., such that κ/κ c is not a rational number) we have an isomorphism (2.5) D ch G,κ of ĝ κ ĝ κ modules.,κ, λ Λ + V λ,κ V τ(λ) Proof. For any level κ we have a canonical non-zero homomorphism of ĝ κ ĝ κ modules (2.6) V λ,κ V τ(λ),κ Dch G,κ. If κ satisfies the conditions of the lemma, then both V λ,κ and Vτ(λ),κ are irreducible modules. Therefore the above maps are injective. The assertion of the lemma then follows from the obvious fact that the characters of the two sides of (2.5) are equal to each other.

9 8 EDWARD FRENKEL AND DENNIS GAITSGORY For special values of κ, when κ/κ c Q, the modules V λ,κ and V τ(λ),κ may become reducible, and so the structure of D ch G,κ may become more complicated. Theorem 2.9 describes what happens at the critical level κ = κ crit. In this case the imae of the homomorphism (2.6) is equal to V λ,crit V τ(λ),crit. In other words, we observe the collapse of the derees of freedom z λ correspondin to z λ. But these derees of freedom are restored by the second factor in (2.4) Proof of part (2). We shall first prove part (2) of Theorem 2.9. Recall the chiral alebroid A ren,τ,crit of [FG1], Section 4, whose chiral envelopin alebra is Dch,0, by Lemma 9.7 of loc. cit. (note that the assertion of Theorem 2.9 for λ = 0 follows in fact from this isomorphism and Lemma 7.4 of [FG1]). A version of Kashiwara s theorem proved in Section 7 of loc. cit. implies the followin: Proposition Let M be a chiral A ren,τ,crit -module, whose support over Spec(Z ) is contained in Spec(Z unr,λ ). Let F i (M), i = 1, 2,... be the canonical increasin filtration on M by the powers of I λ. Then (a) R i (ι λ )! (M) = 0 for i > 0 and R 0 (ι λ )! (M) F 0 (M). (b) The canonical maps (I λ ) n /(I λ ) n+1 r n (M) r 0 (M) z re,λ ive rise to isomorphisms r n (M) r 0 (M) z re,λ Sym n (N z re,λ re/ λ unr )., which is a chiral module over Dch,0 We apply this proposition to D ch,λ, and the assertion of point (2) of Theorem 2.9 follows. A ren,τ,crit, and hence over Proof of part (1). To prove part (1) of Theorem 2.9, let us first show that the map (2.3) is injective. Indeed, let K denote its kernel; this is a bimodule over ĝ crit, supported at Spec(Z unr,λ ) and G[[t]]-interable with respect to both actions. Hence, if K 0, there exists a non-zero map of V λ,crit K of ĝ crit-modules, with respect to the left action. However, we claim that the map (2.7) Hom bcrit (V λ,crit, V λ,crit V τ(λ),crit ) Hom b crit (V λ,crit, D ch,λ ) z re,λ is injective, and in fact an isomorphism. This would lead to a contradiction, implyin that K = 0. To show that (2.7) is an isomorphism, consider the composition (2.8) V τ(λ),crit Hom b crit (V λ,crit, Vλ,crit V τ(λ),crit ) Hom b crit (V λ,crit, Dch,λ ). z re,λ We claim that the first arrow in (2.8) is an isomorphism. Indeed, since V τ(λ),crit is flat over z re,λ, we have Hom bcrit (V λ,crit, V λ,crit V τ(λ) z re,λ and by the main result of [FG6], the natural map z re,λ,crit ) End b crit (V λ,crit) End bcrit (V λ,crit) z re,λ V τ(λ),crit,

10 LOCAL GEOMETRIC LANGLANDS CORRESPONDENCE: THE SPHERICAL CASE 9 is an isomorphism. Now we claim that the composition in (2.8) is an isomorphism. The latter is equivalent to point (a) of the followin assertion, established in [AG]: Lemma (a) (b) For i > 0, Hom bcrit (V λ,crit, D ch ) Hom G[[t]] (V λ, D ch ) V τ(λ),crit. Ext i G[[t]](V λ, D ch ) = Computation of characters. We are now ready to finish the proof of Theorem 2.9. Usin the coordinate on the formal disc, we will view D ch as acted on by G G G m, where the latter acts by loop rotations. It is easy to see that the isotypic components for the above action are finite-dimensional. Usin point (2) of Theorem 2.9 and the above injectivity result, we obtain that the theorem would follow once we show that for each µ 1, µ 2, d, dim ( Hom G G Gm (V µ1 V µ2 C d, D ch )) = ( dim (Hom G G Gm V µ1 V µ2 C d, (V λ,crit V τ(λ) z re,λ λ,crit ) z re,λ )) Sym z re,λ(n λ re/ unr ). Since Nre/ λ unr Ω1 (z re,λ ), and since each z re,λ is isomorphic to a polynomial alebra, the multiplicities of the irreducibles in the G G G m -modules are the same. (V λ,crit V τ(λ) z re,λ,crit ) z re,λ Hence, it suffices to show that for each µ 1, µ 2, d, Sym z re,λ(n λ re/unr ) and Vλ,crit V τ(λ),crit dim ( Hom G G Gm (V µ1 V µ2 C d, D ch ) ) = ( )) dim (Hom G G Gm V µ1 V µ2 C d, V λ,crit Vτ(λ),crit. λ However, the one-parameter families of G G G m -modules iven by V λ,κ ħ +κ crit V τ(λ), κ ħ +κ crit and D ch G,ħ, where κ ħ = ħκ 0 for some non-zero invariant inner product κ 0, are ħ-flat. Hence, it is sufficient to check the equality dim ( Hom G G Gm (V µ1 V µ2 C d, D ch G,ħ) ) = = ( )) dim (Hom G G Gm V µ1 V µ2 C d, V λ,κ ħ +κ crit V τ(λ), κ ħ +κ crit λ for a eneric ħ. The latter equality indeed holds, since for ħ irrational we have an isomorphism of ĝ crit -bimodules: by Lemma 2.5. D ch G,ħ λ V λ,κ ħ +κ crit V τ(λ), κ ħ +κ crit

11 10 EDWARD FRENKEL AND DENNIS GAITSGORY 3. The cateory of spherical modules In this section we use the results on the CADO obtained in the previous section to prove the Main Theorem stated in the Introduction Semi-infinite cohomoloy functor. Define the character (3.1) χ 0 : n + ((t)) C by the formula χ 0 (e α,n ) = { 1, if α = α ı, n = 1, 0, otherwise. We have the functors of semi-infinite cohomoloy (the + quantum Drinfeld Sokolov reduction) from the cateory of ĝ crit -modules to the cateory of raded vector spaces, (3.2) M H 2 +i (n + ((t)), n + [[t]], M χ 0 ), introduced in [FF, FKW] (see also [FB], Ch. 15, and [FG2], Sect. 18; we follow the notation of the latter). More enerally, for a complex M of ĝ crit -modules (or of n((t))-modules), the correspondin semi-infinite Chevalley complex C 2 (n((t)), M χ 0 ) ives rise to a well-defined trianulated functor D + (ĝ crit mod) D(Vect). This is an analoue of the Whittaker functor in representation theory of reductive roups over local fields. Since Z maps to the center of the cateory ĝ crit mod, the above functor naturally lifts to a functor D + (ĝ crit mod) D(Z mod). By Theorem 2.3, the composed functor factors throuh a functor D + (ĝ crit mod G[[t]] ) D + (ĝ crit mod) D(Z mod), Ψ : D + (ĝ crit mod G[[t]] ) D ( QCoh(Spec(Z unr )) ). The main result of this paper is the followin: Theorem 3.2. The functor Ψ is exact (with respect to the natural t-structures) and defines an equivalence of abelian cateories (3.3) ĝ crit mod G[[t]] QCoh(Spec(Z unr )) Stratey of the proof. We will derive Theorem 3.2 from the followin two statements. Proposition 3.4. There exists an isomorphism of alebras (3.4) R Hom bcrit mod G[[t]](Vλ,crit, V λ,crit) R Hom QCoh(Spec(Z unr )) (z re,λ, z re,λ ). Let ĝ crit mod re,λ be the full subcateory of ĝ crit mod, consistin of modules, whose support over Z is contained in Spec(z re,λ ). Let ĝ crit mod G[[t]] re,λ denote the intersection ĝ crit mod re,λ ĝ crit mod G[[t]].

12 LOCAL GEOMETRIC LANGLANDS CORRESPONDENCE: THE SPHERICAL CASE 11 Proposition 3.5. The functors Ψ and L V λ,crit L define mutually quasi-inverse equivalences z re,λ ĝ crit mod G[[t]] re,λ zre,λ mod. For λ = 0 this was proved in [FG1] (see also Conjecture of [F2]). We remark that, conversely, both of these propositions follow from Theorem 3.2 and the isomorphism (3.5) Ψ(V λ,crit ) zre,λ established in [FG6]. Note that Spec(z re,λ ) Spec(Z unr ) is a reular embeddin. Therefore we have R Hom QCoh(Spec(Z unr )) (z re,λ Combinin this with Proposition 3.4, we obtain: Corollary 3.6., z re,λ ) Λ (N λ z re,λ re/unr ). (3.6) R Hom bcrit mod G[[t]](Vλ,crit, Vλ,crit ) Λ (N z re,λ re/ λ unr ). For λ = 0 the isomorphism (3.6) was established in [FT] by other methods. The above proof is independent of [FT] and therefore provides an alternative arument. Let χ be a C-point of Op unr, that is, a λ-reular oper in Opre,λ for some λ Λ +. We denote by ĝ crit mod χ the cateory of ĝ crit -modules on which the center Z acts accordin to the character associated to χ. Let ĝ crit mod G[[t]] χ be the correspondin G[[t]]-equivariant cateory. This cateory contains the quotient V λ,crit (χ) of the Weyl module Vλ,crit by the central character χ. Theorem 3.2 then has the followin corollary (see Conjecture of [F2]): Corollary 3.7. For any χ Op re,λ, λ Λ+, the cateory ĝ crit mod G[[t]] χ is equivalent to the cateory of vector spaces: its unique, up to isomorphism, irreducible object is V λ,crit (χ) and any other object is isomorphic to a direct sum of copies of V λ,crit (χ). This equivalence is iven by the functor Ψ. This provides a non-trivial test of our conjecture, described in the Introduction (see formula (1.7)), that the cateories ĝ crit mod K χ and ĝ crit mod K χ are equivalent whenever the local systems underlyin χ and χ are isomorphic to each other Computation of Ψ. The first step is to compute the functor Ψ on the objects D ch,λ. Since the functor Ψ commutes with direct limits, from Theorem 2.9 and (3.5) we obtain that B λ G := Ψ(Dch,λ ) is acyclic off cohomoloical deree 0 (here we view Dch,λ ĝ crit mod G[[t]] via the left action l). Proposition 3.9. The functor Ψ defines an isomorphism Hom bcrit mod(v λ,crit, Dch,λ ) Hom Z mod(z re,λ, B λ G ) as an object of (here we consider the left action l of ĝ crit on D ch,λ ). Furthermore, the hiher Ri Hom s, vanish. R i Hom bcrit mod G[[t]](Vλ,crit, Dch,λ ) and Ri Hom D(QCoh(Spec(Z unr ))) (z re,λ, B λ G ),

13 12 EDWARD FRENKEL AND DENNIS GAITSGORY Proof. From Lemma 2.14, we know that for i > 0 and R i Hom bcrit mod G[[t]](Vλ,crit, D ch,λ ) = 0 Hom bcrit mod(v λ,crit, D ch,λ ) Vτ(λ),crit. By Theorem 2.9(2) and formula (3.5), B λ G has a filtration with the associated raded quotients iven by r n (B λ G) V τ(λ),crit Sym n (N z re,λ re/unr λ ). z re,λ Moreover, it follows from the definition of the filtration on D ch that this filtration is the canonical one, iven by the powers of annihilation by I λ. This implies that R i (ι λ )! (Bˇλ G ) = 0 for i > 0 and that the natural map V τ(λ),crit R0 (ι λ )! (Bˇλ G ) is an isomorphism Proof of Proposition 3.4. Consider the relative Chevalley complex C ([[t]];, D ch,λ V λ ) taken with respect to the riht action of ĝ crit on D ch,λ, as a complex of objects of ĝ crit mod G[[t]]. By Lemma 2.14, it is quasi-isomorphic to V λ,crit itself. We need to show that the functor Ψ induces isomorphisms ( ) R i Hom bcrit mod V λ,crit,c ([[t]];, D ch,λ G[[t]] V λ ) ( R i Hom D(QCoh(Spec(Z unr ))) z re,λ, Ψ ( )) C ([[t]];, D ch,λ V λ ). Takin into account Proposition 3.9, it remains to show that the natural map ( ) (3.7) Ψ C ([[t]];, D ch,λ V λ ) C ([[t]];, B λ G V λ ) is an isomorphism, i.e., that the correspondin spectral sequences converes. The latter is established as follows: we endow the bi-complex in the LHS of (3.7) with an additional Z-radin, as in [FG6], Section 4 (see also [FG2], Section 18.11). We obtain that in each raded deree, the correspondin bi-complex is concentrated in a shift of a positive quadrant, hence the converence Proof of Proposition 3.5. We are now ready to derive Proposition 3.5. The fact that functor z re,λ mod ĝ crit mod G[[t]] re,λ, iven by L V λ,crit L, z re,λ is an equivalence, follows from Proposition 3.4 by repeatin verbatim the arument in [FG1], Section 8. It remains to show that Ψ(V λ,crit z re,λ L) is acyclic away from the cohomoloical deree 0, and that the 0-th cohomoloy is isomorphic to L.

14 LOCAL GEOMETRIC LANGLANDS CORRESPONDENCE: THE SPHERICAL CASE 13 Since the functors appearin above commute with direct limits, we can assume that L is finitely presented. Since z re,λ is isomorphic to a polynomial alebra, we can further assume that L admits a finite resolution by free z re,λ -modules. This reduces the assertion to the formula (3.5) Exactness. We are now ready to show that the functor Ψ is exact, i.e., that for M ĝ crit mod G[[t]], the object Ψ(M) is acyclic away from the cohomoloical deree 0. Indeed, since Ψ commutes with direct limits, we can assume that M is supported at the k-th infinitesimal neihborhood of Spec(z re,λ ) inside Spec(Z unr,λ ). By (finite) devissae, that is, by representin M as a k-iterated successive extension of modules supported at Spec(z re,λ ), we may further assume that M belons to ĝ crit mod G[[t]] re,λ. In the latter case, the assertion follows from Proposition Completion of the proof of Theorem 3.2. Let us now show that the functor Ψ induces isomorphisms (3.8) R i ( Hom bcrit mod V λ G[[t]],crit, M ) R i ( Hom D(QCoh(Spec(Z unr ))) z re,λ, Ψ(M) ) for any i and M ĝ crit mod G[[t]]. Both sides commute with direct limits in M, so we can aain assume that M is supported at the k-th infinitesimal neihborhood of Spec(z re,λ ) inside Spec(Z unr,λ ), and further that it is on object of ĝ crit mod G[[t]] re,λ, i.e., M V λ,crit z re,λ L -module L. Usin commutation with direct limits aain, we can assume that L is finitely presented, and hence admits a finite resolution by free z re,λ -modules. In the latter for some z re,λ case, the isomorphism of (3.8) follows from Proposition 3.4. By the same devissae procedure we conclude that the functor Ψ induces isomorphisms R i Hom bcrit mod G[[t]](M 1, M 2 ) R i Hom D(QCoh(Spec(Z unr ))) (Ψ(M 1 ), Ψ(M 2 )) for any i and M 1, M 2 ĝ crit mod G[[t]]. Finally, it remains to see that Ψ is essentially surjective. Aain, by commutation with direct limits, it is sufficient to see that any L QCoh(Spec(Z unr )) supported at the k-th infinitesimal neihborhood of Spec(z re,λ ) lies in the imae of Ψ. Since Ψ induces an isomorphism on the level of Ext 1, by induction, we can assume that k = 0, i.e., L z re,λ mod. In the latter case, the assertion follows from Proposition 3.5. References [AG] S. Arkhipov and D. Gaitsory, Differential operators on the loop roup via chiral alebras, IMRN 2002, no. 4, [BD] A. Beilinson and V. Drinfeld, Quantization of Hitchin s interable system and Hecke eiensheaves, available at arinkin/lanlands/ [CHA] A. Beilinson and V. Drinfeld, Chiral alebras, American Mathematical Society Colloquium Publications 51, AMS, [FF] B. Feiin and E. Frenkel, Affine Kac Moody alebras at the critical level and Gelfand Dikii alebras, In: Infinite Analysis, eds. A. Tsuchiya, T. Euchi, M. Jimbo, Adv. Ser. in Math. Phys. 16, , Sinapore: World Scientific, [F1] E. Frenkel, Wakimoto modules, opers and the center at the critical level, Adv. Math. 195 (2005)

15 14 EDWARD FRENKEL AND DENNIS GAITSGORY [F2] E. Frenkel, Lanlands Correspondence for Loop Groups, Cambride Studies in Advanced Mathematics 103, Cambride University Press, [FB] E. Frenkel and D. Ben-Zvi, Vertex Alebras and Alebraic Curves, Mathematical Surveys and Monoraphs 88, Second Edition, AMS, [FG1] E. Frenkel and D. Gaitsory, D-modules on the affine Grassmannian and representations of affine Kac Moody alebras, Duke Math. J. 125 (2004) [FG2] E. Frenkel and D. Gaitsory, Local eometric Lanlands correspondence and affine Kac Moody alebras, In: Alebraic Geometry and Number Theory, Proress in Math. 253, pp , Birkhäuser Boston, [FG3] E. Frenkel and D. Gaitsory, Fusion and convolution: applications to affine Kac Moody alebras at the critical level, Pure and Applied Math Quart. 2, (2006) [FG4] E. Frenkel and D. Gaitsory, Localization of b-modules on the affine Grassmannian, Preprint math.rt/ [FG5] E. Frenkel and D. Gaitsory, Geometric realizations of Wakimoto modules at the critical level, Preprint math.rt/ , to appear in Duke Math. Journal. [FG6] E. Frenkel and D. Gaitsory, Weyl modules and opers without monodromy, Preprint arxiv: [FKW] E. Frenkel, V. Kac, and M. Wakimoto, Characters and fusion rules for W alebras via quantized Drinfeld-Sokolov reduction, Comm. Math. Phys. 147 (1992) [FT] E. Frenkel and C. Teleman, Self-extensions of Verma modules and differential forms on opers, Compos. Math. 142 (2006) [Ga] D. Gaitsory, The notion of cateory over an alebraic stack, Preprint math.ag/ [GMS] V. Gorbounov, F. Malikov and V. Schechtman, On chiral differential operators over homoeneous spaces, Int. J. Math. Math. Sci. 26 (2001) Department of Mathematics, University of California, Berkeley, CA 94720, USA address: frenkel@math.berkeley.edu Department of Mathematics, Harvard University, Cambride, MA 02138, USA address: aitsde@math.harvard.edu