On the Geometric Langlands Conjecture

JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 15, Number 2, Pages 367–417 S 0894-0347(01)00388-5 Article electronically published on December 31, 2001

ON THE GEOMETRIC LANGLANDS CONJECTURE

E. FRENKEL, D. GAITSGORY, AND K. VILONEN

Introduction 0.1. Background. Let X be a smooth, complete, geometrically connected curve over the finite field Fq.DenotebyF the field of rational functions on X and by A the ring of adèles of F . The Langlands conjecture, recently proved by L. Lafforgue [Laf], establishes a correspondence between cuspidal automorphic forms on the group GLn(A) and irreducible, almost everywhere unramified, n–dimensional `– adic representations of the Galois group of F over F (more precisely, of the Weil group). An unramified automorphic form on the group GLn(A) can be viewed as a function on the set Bunn(Fq) of isomorphism classes of rank n bundles on the curve X. The set Bunn(Fq)isthesetofFq–points of Bunn, the algebraic stack of rank n bundles on X. According to Grothendieck’s “faisceaux–fonctions” correspondence, one can attach to an `–adic perverse sheaf on Bunn a function on Bunn(Fq)by taking the traces of the Frobenius on the stalks. V. Drinfeld’s geometric proof [Dr] of the Langlands conjecture for GL2 (and earlier geometric interpretation of the abelian class field theory by P. Deligne, see [Lau1]) opened the possibility that automorphic forms may be constructed as the functions associated to perverse sheaves on Bunn. Thus, one is led to a geometric version of the Langlands conjecture proposed by V. Drinfeld and G. Laumon: for each geometrically irreducible rank n local system E on X there exists a perverse sheaf AutE on Bunn (irreducible on each component), which is a Hecke eigensheaf with respect to E,inanappropriatesense (see [Lau1] or Sect. 1 below for the precise formulation). Moreover, the geometric Langlands conjecture can be made over an arbitrary field k. Building on the ideas of Drinfeld’s work [Dr], G. Laumon gave a conjectural construction of AutE in [Lau1, Lau2]. More precisely, he attached to each rank n 0 0 local system E on X a complex of perverse sheaves AutE on the moduli stack Bunn of pairs {M,s},whereM ∈ Bunn is a rank n bundle on X and s is a regular non- zero section of M. He conjectured that if E is geometrically irreducible, then this sheaf descends to a perverse sheaf AutE on Bunn (irreducible on each component), which is a Hecke eigensheaf with respect to E. In our previous work [FGKV], joint with D. Kazhdan, we have shown that the 0 F 0 function on Bunn( q) associated to AutE agrees with the function constructed previously by I.I. Piatetskii-Shapiro [PS1] and J.A. Shalika [Sha], as anticipated by Laumon [Lau2]. This provided a consistency check for Laumon’s construction.

Received by the editors February 14, 2001. 2000 Mathematics Subject Classiﬁcation. Primary 11R39, 11F70; Secondary 14H60, 22E55.

c 2001 by E. Frenkel, D. Gaitsgory, K. Vilonen 367

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In this paper we formulate a certain vanishing conjecture, and prove that Lau- mon’s construction indeed produces a perverse sheaf AutE on Bunn with desired properties, when the vanishing conjecture holds. In other words, the vanishing conjecture implies the geometric Langlands conjecture, over any field k. For the sake of definiteness, we work in this paper with a field k of characteristic p>0, but our results (with appropriate modifications, such as switching from perverse sheaves to D–modules) remain valid if char k =0. Moreover, in the case when k is a finite field, we derive the vanishing conjecture (and hence the geometric Langlands conjecture) from the results of L. Lafforgue [Laf]. To give the reader a feel for the vanishing conjecture, we give here one of its formulations (see Sect. 2 for more details). Let M and M0 be two vector bundles on X of rank k, such that deg(M0)−deg(M)=d>0. Consider the space Hom0(M, M0) M → M0 C d of injective sheaf homomorphisms , .Let oh0 be the algebraic stack which 0 M M0 → C d classifies torsion sheaves of length d,andletπ :Hom( , ) oh0 be the natural morphism sending M ,→ M0 to M0/M. Ld C d G. Laumon [Lau1] has defined a remarkable perverse sheaf E on oh0 for any local system E of rank n on X. The vanishing conjecture states that if E is irreducible and n>k,then

• 0 M M0 ∗ Ld ∀ − H (Hom ( , ),π ( E )) = 0, d>kn(2g 2).

0.2. Contents. The paper is organized as follows: In Sect. 1 we define Hecke functors and state the geometric Langlands conjecture. We want to draw the reader’s attention to the fact that our formulation is different from that given in [Lau1] in two respects. The Hecke property is defined here using only the first Hecke functor; according to Proposition 1.5, this implies the Hecke property with respect to the other Hecke functors. We also do not require the cuspidality property in the statement of the conjecture, because we show in Sect. 9 that the cuspidality of a Hecke eigensheaf follows from the vanishing conjecture. In Sect. 2 we recall the definition of Laumon’s sheaf and state our vanishing conjecture. In Sect. 3 we present two constructions of AutE following Laumon [Lau1, Lau2] (see also [FGKV]). A third construction, which uses the Whittaker sheaves is given in Sect. 4. This construction is the exact geometric analogue of the construction of Piatetskii-Shapiro [PS1] and Shalika [Sha] at the level of functions. The reader is referred to Sect. 3.10 for a summary of the relationship between the three constructions and the strategy of our proof. In Sects. 5–9 we derive the geometric Langlands conjecture assuming that the vanishing conjecture is true. Sect. 5 is devoted to the proof of the cleanness 0 property in Laumon’s construction. In Sect. 6 we prove that the sheaf AutE on 0 Bunn descends to a perverse sheaf AutE on Bunn. In Sects. 7 and 8 we give two alternative proofs of the Hecke property of AutE. We then show in Sect. 9 that the perverse sheaf AutE is cuspidal. In Sect. 10 we derive the vanishing conjecture from results of L. Lafforgue [Laf] when k is a finite field. The Appendix contains proofs of some results concerning the Whittaker sheaves, which are not necessary for our proof, but are conceptually important.

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0.3. Notation and conventions. Throughout this paper, k will be a ground field of characteristic p>0andX will be a smooth complete geometrically connected curve over k of genus g>1. Q This paper deals with `–adic perverse sheaves and complexes of perverse sheaves on various schemes over k,where` is a prime with (`, p) = 1. In particular, by a local system on X we will understand a smooth `–adic sheaf over X.Forbrevity, we will refer to a geometrically irreducible local system simply as an irreducible local system. When k = Fq we work with Weil sheaves (see [De]), instead of sheaves defined F Q over q. We choose a square root of q in `, which defines a half-integral Tate twist Q 1 `( 2 ). In addition to k–schemes, we will extensively use algebraic stacks in the smooth topology (over k); see [LMB]. If G is an algebraic group, we define BunG as a stack that classifies G–bundles on X. This means that Hom(S, BunG) is the groupoid whose objects are H–bundles on X × S and morphisms are isomorphisms of these bundles. The pull–back functor for a morphism S1 → S2 is defined in a natural way. When G = GLn,BunG coincides with Bunn, the moduli stack of rank n vector d bundles on X.WewriteBunn for the connected component of Bunn corresponding to rank n vector bundles of degree d. For an algebraic stack Y we will use the notation D(Y) for the derived category Q Y of `–adic perverse sheaves on . We refer the reader to Sect. 1.4 of [FGV] for our conventions regarding this category. When we discuss objects of the derived category, the cohomological grading should always be understood in the perverse ∗ ! sense. In addition, for a morphism f : Y1 → Y2, the functors f!, f∗, f and f should be understood “in the derived sense”. Y k F F F Y F If is a stack over = q and q1 is an extension of q,wedenoteby ( q1 ) F Y S the set of isomorphism classes of objects of the groupoid Hom(Spec q1 , ). If is a Y Y F perverse sheaf or a complex of perverse sheaves on ,then ( q1 ) is endowed with the function “alternating sum of traces of the Frobenius on stalks” (as in [De]). We S denote this function by fq1 ( ). For the general definitions related to the Langlands correspondence and the formulation of the Langlands conjecture we refer the reader to [Lau1], Sect. 1, and [FGKV], Sect. 2. In particular, the notions of cuspidal automorphic function or Hecke eigenfunction on GLn(A) may be found there.

1. Hecke eigensheaves In this section we introduce the Hecke functors and state the geometric Langlands conjecture.

1.1. Hecke functors. Consider the following correspondence:

← → ←−−h H1 −−−−−−→supp ×h × (1.1) Bunn n X Bunn, H1 M M0 M0 → M ∈ where the stack n classiﬁes quadruples (x, , ,β : , ), with x X, 0 0 M , M ∈ Bunn, such that M/M is the simple skyscraper sheaf supported at x, i.e., 0 ← → M/M is (non-canonically) isomorphic to OX (x)/OX . The morphisms h , h ,and supp are given by h←(x, M, M0)=M, h→(x, M, M0)=M0, and supp(x, M, M0)=x.

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1 → × The Hecke functor Hn :D(Bunn) D(X Bunn) is deﬁned by the formula n − 1 (1.2) H1 (K) = (supp ×h→) h←∗(K) ⊗ Q ( )[n − 1]. n ! ` 2 1 Consider the i-th iteration of Hn: 1 i → i × (Hn) :D(Bunn) D(X Bunn). Let ∆ denote the divisor in Xi consisting of the pairwise diagonals. Note that for 1 i K ∈ K | i any D(Bunn) the restriction (Hn) ( ) (X −∆)×Bunn is naturally equivariant i with respect to the action of the symmetric group Si on X − ∆. Consider a rank n local system E on X.WesaythatK ∈ D(Bunn)isaHecke eigensheaf,orthatithasaHecke property with respect to E,ifK =0andthere6 exists an isomorphism 1 K ' K (1.3) Hn( ) E , such that the resulting map 1 2 K | → K| (1.4) (Hn) ( ) (X×X−∆)×Bunn E E (X×X−∆)×Bunn

is S2–equivariant. This i mplies that 1 i i K | i → K| i (Hn) ( ) (X −∆)×Bunn E (X −∆)×Bunn

is Si–equivariant for any i. 1.2. Statement of the geometric Langlands conjecture. We are now ready to formulate the unramified geometric Langlands conjecture for GLn: 1.3. Conjecture. For each irreducible rank n local system E on X there exists d a perverse sheaf AutE on Bunn, irreducible on each connected component Bunn, which is a Hecke eigensheaf with respect to E. Conjecture 1.3 has been proved by Drinfeld [Dr] in the case when n =2(seealso [Ga]). In this paper we reduce Conjecture 1.3 to the Vanishing Conjecture 2.3. Then, in Sect. 10 we will show that when k is a finite field Fq, the Vanishing Conjecture follows from recent results of Lafforgue [Laf]. 1 1.4. Other Hecke functors. In addition to the functor Hn,wealsohaveHecke i → × functors Hn :D(Bunn) D(X Bunn)fori =2,...,n. To define them, consider Hi the stack n which classifies quadruples (x, M, M0,β : M0 ,→ M), 0 0 0 0 where x ∈ X, M , M ∈ Bunn such that M ⊂ M ⊂ M (x), and length(M/M )=i. H1 As in the case of n, we have a diagram ← → ←−−h Hi −−−−−−→supp ×h × Bunn n X Bunn, i and the functor Hn is defined by the formula (n − i)i Hi (K) = (supp ×h→) h←∗(K) ⊗ Q ( )[(n − i)i]. n ! ` 2 The following result is borrowed from [Ga]: 1.5. Proposition. Let K be a Hecke eigensheaf with respect to E. Then for i = i K ' i K 1,...,n we have isomorphisms Hn( ) Λ E .

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−i Proof. Consider the stack Modn of “lower modifications of length i”, which clas- 0 0 0 sifies the data of triples (M, M ,β : M ,→ M), where M, M ∈ Bunn and β is an embedding of coherent sheaves such that the quotient M/M0 is a torsion sheaf of length i. Let X(i) be the i-th symmetric power of X. We have a natural morphism supp : −i → (i) M0 M Modn X , which associates to ( , ,β) as above the divisor of zeros of the induced map det M0 → det M. Hi,+ −i ⊂ (i) Denote by n the preimage in Modn of the main diagonal X X .Note Hi Hi,+ that n is naturally a closed substack in n . ]−i M0 M ⊂ M ⊂ ⊂ Consider the stack Modn , which classifies the data ( = 0 1 ... Mi = M), where each Mj is a rank n vector bundle, and Mj/Mj−1 is a simple −i ] → −i skyscraper sheaf. There is a natural proper map p : Modn Modn ,which “forgets” the middle terms of the filtration. ] ]−i → i i → (i) There is also a natural map supp : Modn X such that if sym : X X −i ◦ ] ◦ ] → (i) denotes the symmetrization map, we have sym supp = supp p : Modn X . − ] 1 i − ]−i The open substack supp (X ∆) of Modn is isomorphic to the fiber product −i × i − Modn (X ∆). X(i) The map p is known to be small (see, e.g., [Lau1]). This implies that the complex i(n − 1) Spr := p (Q ( ))[i(n − 1)] ! ` 2 −i 2 · − on Modn is perverse (up to the cohomological shift by n (g 1) = dim(Bunn)) and is a Goresky-MacPherson extension of its restriction to supp−1(X(i) − ∆). In i(n−1) S S Si ' Q − particular, pr carries a canonical Si–action and ( pr) `( 2 )[i(n 1)]. ← → −i → Let h (resp., h ) denote the morphism Modn Bunn, which sends a triple 0 0 (M, M ,β)toM (resp., M ). By construction, for any K ∈ D(Bunn), × → ←∗ K ⊗ S ' × 1 i K (supp h )!(h ( ) pr) (sym id)!(Hn) ( ).

Thus, if K is a Hecke eigensheaf with respect to E,weobtainanSi–equivariant isomorphism × → ←∗ K ⊗ S ' i K (1.5) (supp h )!(h ( ) pr) sym!(E ) . To conclude the proof, we pass to the isotypic components of the sign representation of Si on both sides of formula (1.5) and restrict the resulting isotypic components to the main diagonal X ⊂ X(i). By this process the RHS of (1.5) tau- i K i K tologically yields Λ E . Thus, it remains to show that the LHS yields Hn( ). To see this, it suﬃces to note that Hom (sign, Spr)| i,+ is isomorphic to the constant Si Hn Hi Q (n−i)i − sheaf on n tensored by `( 2 )[(n i)i], by the Springer theory [BM, Sp]. The isomorphisms constructed in the above proposition have an additional property. To state it, let σ be the transposition acting on X ×X and let i, j ∈{1,...,n}. Clearly, the functors K 7→ i × ◦ j K | (Hn id) Hn( ) (X×X−∆)×Bunn and K 7→ ∗ ◦ j × ◦ i K | σ (Hn id) Hn( ) (X×X−∆)×Bunn

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from D(Bunn)toD((X × X − ∆) × Bunn) are naturally isomorphic. Hence, for a Hecke eigensheaf K, the following diagram is commutative: i j ∗ j i (H × id) ◦ H (K)| × − −−−−→ σ ◦ (H × id) ◦ H (K)| × − n n X X ∆ n  n X X ∆   (1.6) y y

i j ∗ j i Λ E Λ E K|X×X−∆ −−−−→ σ (Λ E Λ E) K|X×X−∆. n Finally, let us consider the Hecke functor Hn. By deﬁnition, this is the pull-back under the morphism mult : X × Bunn → Bunn given by (x, M) 7→ M(x). Hence if K is a Hecke eigensheaf with respect to E,then ∗ mult (K) ' ΛnE K.

2. The vanishing conjecture

Denote by Cohn the stack classifying coherent sheaves on X of generic rank n. More precisely, for each k–scheme S,Hom(S, Cohn) is the groupoid, whose objects are coherent sheaves MS on X × S, which are ﬂat over S, and such that over every ∈ M C d geometric point s S, s is generically of rank n.Wewrite ohn for the substack corresponding to coherent sheaves of generic rank n and degree d.

2.1. Laumon’s sheaf. In [Lau2] Laumon associated to an arbitrary local system E of rank n on X aperversesheafLE on Coh0. Let us recall his construction. Denote C rss C by oh0 the open substack of oh0 corresponding to regular semisimple torsion C C rss sheaves. Thus, a geometric point of oh0 belongs to oh0 if the corresponding coherent sheaf on X is a direct sum of skyscraper sheaves of length one supported C rss,d C rss ∩ C d at distinct points of X.Let oh0 = oh0 oh .Wehaveanaturalsmooth (d) − → C rss,d map (X ∆) oh0 . (d) (d) d Sd Let E be the d-th symmetric power of E, i.e., E =sym!(E ) ,where d → (d) (d) (d)| sym : X X . This is a perverse sheaf on X , and its restriction E X(d)−∆ ◦ Ld C rss,d is a local system, which descends to a local system E on oh0 .Theperverse ◦ Ld C d Ld sheaf E on oh0 is by definition the Goresky-MacPherson extension of E from C rss,d C d L C oh0 to oh0.Wedenoteby E the perverse sheaf on oh0, whose restriction C d Ld to oh0 equals E . Ld 2.2. The averaging functor. Using the perverse sheaf E we define the averaging d → functor Hk,E :D(Bunk) D(Bunk). We stress that the positive integer k is independent of n, the rank of the local system E. ≥ d M M0 For d 0, introduce the stack Modk, which classifies the data of triples ( , , 0 0 β : M ,→ M ), where M, M ∈ Bunk and β is an embedding of coherent sheaves such that the quotient M0/M is a torsion sheaf of length d, and the diagram

← → ←−h d −→h Bunk Modk Bunk, where h← (resp., h→) denotes the morphism sending a triple (M, M0,β)toM (resp., M0 d → C d ). In addition, we have a natural smooth morphism π :Modk oh0,which sends a triple (M, M0,β) to the torsion sheaf M0/M. d −d Note that Modn is isomorphic to the stack Modn which was used in the proof of Proposition 1.5. Under this isomorphism the maps h→ and h← are reversed.

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d → The averaging functor Hk,E :D(Bunk) D(Bunk) is defined by the formula d · k K 7→ h→(h←∗(K) ⊗ π∗(Ld )) ⊗ Q ( )[d · k]. ! E ` 2 2.3. Vanishing Conjecture. Assume that E is an irreducible local system of rank n. Then for all k =1,... ,n− 1 and all d satisfying d>kn(2g − 2),thefunctor d Hk,E is identically equal to 0. The statement of the Vanishing Conjecture is known to be true for k =1(see below). The goal of this paper is to show that if Conjecture 2.3 holds for any given irreducible rank n local system E, then the geometric Langlands Conjecture 1.3 holds for E. In addition, in Sect. 10 we will prove Conjecture 2.3 in the case when k is a finite field Fq if the following statements are true (see [BBD] for the definition of a pure local system): (a) E is pure (up to a twist by a one-dimensional representation of the Weil group of Fq), and either (b) there exists a cuspidal Hecke eigenfunction associated to the pull-back of E × F F F to X q1 for any finite extension q1 of q;or Fq 0 (b ) the space of unramified cuspidal automorphic functions on the group GLk over the adèles is spanned by the Hecke eigenfunctions associated to rank k × F local systems on X q1 , for all kcohomology • 0 M0 M ∗ Ld H (Hom ( , ),π ( E )). 2.6. ProofofthestatementofConjecture2.3inthecasek =1. Recall the Deligne vanishing theorem (see the Appendix of [Dr]): Let AJ : X(d) → Picd(X) be the Abel-Jacobi map, and let E be an irreducible local system of rank n>1. Then for d>n(2g − 2) (d) (AJ)!(E )=0.

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This theorem implies the case k = 1 of Conjecture 2.3. Indeed, consider the mor- (d) → C d O O phism X oh0 that associates to a divisor D the torsion sheaf X (D)/ X . C r,d C d This morphism is smooth and its image is the open substack oh0 of oh0 corresponding to those torsion sheaves T on X for which dimk(End(T)) = length(T) C rss,d ⊂ C r,d (such torsion sheaves are called regular). Clearly, oh0 oh0 .SincetheLau- Ld C d (d) mon sheaf E is an irreducible perverse sheaf on oh0,andE is an irreducible (d) Ld perverse sheaf on X , we obtain that the pull-back of E under the morphism (d) → C d (d) X oh0 is isomorphic to E . Observe now that the diagram of stacks ← d → Bun1 Mod1 Bun1 may be identiﬁed with Pic(X) ← Pic(X) × X(d) → Pic(X), where the left arrow is the projection on the ﬁrst factor and the right arrow is the composition × Pic(X) × X(d) −−−−→id AJ Pic(X) × Picd(X) −mult−−→ Pic(X). Therefore, for K ∈ D(Pic(X)) we have: → ←∗ K ⊗ ∗ Ld ' K (d) h! (h ( ) π ( E )) mult!( (AJ)!(E )) = 0, by Deligne’s theorem.

3. The construction of AutE C 0 M M ∈ C Let ohn denote the stack classifying pairs ( ,s), where ohn and s is an injective map Ωn−1 → M. Here Ω stands for the canonical bundle of X and we k ⊗k 0 C 0 write Ω for Ω .WedenotebyBunn the preimage in ohn of the open substack ⊂ C C 0 → C Bunn ohn.Let%n : ohn ohn be the forgetful map; we use the same 0 → notation for the forgetful map Bunn Bunn. In this section, starting with a local system E on X of an arbitrary rank, we will S0 C 0 construct a complex E on ohn. Later we will show that if E is an irreducible S0 local system of rank n which satisﬁes Conjecture 2.3, then E descends to a perverse sheaf SE on Cohn. The restriction of SE to Bunn will then be the Hecke eigensheaf S0 AutE. We present below three constructions of E (two of them in this section, and one more in the next section).

3.1. The first construction. The following is a version of the construction presented in [Lau2, FGKV]. Define an algebraic stack Qe as follows. For a k–scheme S,Hom(S, Qe)isthe M M0 e M groupoid whose objects are quadruples ( S,βS, ( i,S ), (si,S )), where S is a × M0 × coherent sheaf on X S of generic rank n, S is a rank n bundle on X S, M0 → M O βS : S S is an embedding of the corresponding X×S–modules, such that M0 the quotient is S–flat, ( i,S ) is a full flag of subbundles M0 ⊂ M0 ⊂ ⊂ M0 ⊂ M0 M0 (3.1) 0= 0,S 1,S ... n−1,S n,s = S, e n−i O ' M0 M0 and si,S is an isomorphism Ω S i,S / i−1,S,i=1,... ,n. The morphisms are the isomorphisms of the corresponding OX×S–modules making all diagrams commutative (we remark that in [FGKV] we used the notation J instead of M0).

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e Qe → C 0 k There is a representable morphism of stacks ν : ohn,whichforeach – M M0 e M ◦ e scheme S maps ( S,βS, ( i,S ), (si,S )) to the pair ( S,βS s1,S), where s1,S is n−1 O M0 viewed as an embedding of Ω S into S. Qe → C M M0 e We also define the morphism α : oh0 sending ( S,βS, ( i,S ), (si,S )) to e the sheaf MS/ Im βS, and the morphism ev : Q → Ga defined as follows. Given two coherent sheaves L and L0 on X, consider the stack Ext1(L0, L). The objects of the groupoid Hom(S, Ext1(L0, L)) are coherent sheaves L00 on X × S together with a short exact sequence 00 0 0 → L OS → L → L OS → 0, and morphisms are maps between such exact sequences inducing the identity isomorphisms at the ends. There is a canonical morphism from the stack Ext1(L0, L) to the scheme Ext1(L0, L). We have for each i =1,... ,n− 1 a natural morphism Qe → E 1 i i−1 M M0 e evi : xt (Ω , Ω ), which sends the data of ( S,βS, ( i,S ), (si,S )) to → M0 M0 → M0 M0 → M0 M0 → 0 i,S / i−1,S i+1,S/ i−1,S i+1,S/ i,S 0. Now ev is the composition nY−1 nY−1 Qe → E 1 i i−1 → 1 i i−1 → Gn−1 sum→ G (3.2) ev : xt (Ω , Ω ) Ext (Ω , Ω ) a a. i=1 i=1 F → Q We fix a non-trivial character ψ : q `, which gives rise to the Artin-Shreier e e sheaf Iψ on the additive group Ga. Define the complex WE on Q by the formula dim We := α∗(L ) ⊗ ev∗(I ) ⊗ Q ( )[dim], E E ψ ` 2 where dim is the dimension of the corresponding connected component of Qe. e Since the morphism α is smooth, WE is a perverse sheaf. Finally, we define the S0 C 0 complex E on ohn by S0 e We E := ν!( E ). 3.2. The second construction, via Fourier transforms. This construction is due to Laumon [Lau2]. It amounts to expressing the first construction as a series of Fourier transforms. Thus, we obtain an alternative construction of the restriction S0 C 0 C C of E to the preimage in ohn of a certain open substack n of ohn. For technical reasons, which will become clear in the course of the proof, we choose a slightly smaller open subset of Bunn than in [Lau2]. We will need the following result: We call a vector bundle M very unstable if M can be decomposed into a direct 1 sum M ' M1 ⊕ M2, such that Mi =0andExt6 (M1, M2) = 0. It is clear that very vuns unstable vector bundles form a constructible subset Bunn of Bunn. Let Lest be any fixed line bundle.

3.3. Lemma. There exists an integer cg,n with the following property: if d ≥ cg,n M ∈ d k M Lest 6 M and Bunn( ) is such that Hom( , ) =0,then is very unstable. Proof. We will prove a slightly stronger statement. Namely, for each n we will M ∈ d ≥ M Lest 6 ﬁnd an integer cg,n such that for Bunn, d cg,n,withHom( , ) =0 1 there exists a decomposition M ' M1 ⊕ M2 with Mi =0,Ext6 (M1, M2) = 0, and M M deg( 2) ≥ deg( ) . rank(M2) n

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est Set cg,1 =deg(L ). By induction, we can assume that cg,i, i ≤ n − 1, satisfying the above properties have been found. Let us show that any integer cg,n such that i (3.3) (c − dest − (n − 1)(2g − 2)) · >c , ∀i =1,...,n− 1, g,n n − 1 g,i will do. M ∈ d ≥ Indeed, let cg,n be such an integer. Suppose that for some Bunn, d cg,n, we have Hom(M, Lest) =6 0. Then there exists a short exact sequence 0 → M0 → M → L0 → 0,

0 L0 L0 Lest 6 deg(M ) ≥ where is a line bundle such that Hom( , ) =0.By(3.3)wehave: n−1 deg(M) 1 L0 M0 M ' M0 ⊕ L0 n . Hence, if Ext ( , ) vanishes, the decomposition satisfies our requirements and we are done. Thus, it remains to consider the case Ext1(L0, M0) =6 0. Then, by Serre duality, we obtain that Hom(M0 ⊗ Ω−1, L0) =06 and hence Hom(M0 ⊗ Ω−1, Lest) =6 0. From the definition of M0 and (3.3) we con- 0 −1 clude that deg(M ⊗ Ω ) >cg,n−1. We first observe that this forces n>2. For n =2wegetdeg(M0 ⊗ Ω−1) > deg(Lest), forcing Hom(M0 ⊗ Ω−1, Lest)tovanish, a contradiction. Using our induction hypothesis, we can find a direct sum decomposition M0 ⊗ − 0 0 0 0 deg(M0 ) M0⊗ −1 1 ' M ⊕ M 1 M M 2 ≥ deg( Ω ) Ω 1 2 with Ext ( 1, 2)=0and i n−1 ,wherei = M0 M0 rank( 2). Moreover, without loss of generality we can assume that 2 admits no further decomposition satisfying the above condition (indeed, if it does, we simply M0 M0 ≥ M0 ⊗ −1 · i ≥ M0 Lest split 2 further). Since deg( 2) deg( Ω ) n−1 cg,i,Hom( 2, ) must vanish by the induction hypothesis. 1 Lest M0 ⊗ M M0 ⊗ By Serre duality, it follows that Ext ( , 2 Ω) = 0, and hence 2 := 2 Ω is a direct summand in M. More precisely, M ' M2 ⊕ M1,whereM1 fits into a short exact sequence → M0 ⊗ → M → L0 → 0 1 Ω 1 0. 1 Therefore, Ext (M1, M2)=0and deg(M ) deg(M0) deg(M) 2 ≥ ≥ . rank(M2) n − 1 n This completes the proof. Notational convention. For notational convenience, by degree of a coherent sheaf of generic rank k we will understand its usual degree −k(k − 1)(g − 1), so that the bundle O ⊕ Ω ⊕ ...⊕ Ωk−1 is of degree zero.

est To deﬁne Cn, we choose the line bundle L of a suﬃciently large degree such that for any bundle M on X of rank k ≤ n,Hom(M, Lest) = 0 implies that (a) deg(M) >nk(2g − 2), (b) Ext1(Ωk−1, M)=0. For example, any line bundle Lest of degree > (2n +2)(g − 1) will do. Thus, let cg,n be an integer satisfying the requirements of Lemma 3.3. For ≥ Cd C d M ∈ C d d cg,n,let k be the open substack ofS ohk consisting of ohk such that Hom(M, Lest) = 0. Finally, we set C = Cd. k d≥cg,n k M ∈ d −Cd ∩ d k ≥ Note that by construction any (Bunn n Bunn)( ), for d cg,n,isvery unstable. This property of Cn will be crucial in Sect. 6.

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3.4. The fundamental diagram. Let k−1 Ek = the stack classifying pairs (Mk,sk), Mk ∈ Ck,sk ∈ Hom(Ω , Mk), E∨ → k → M → M → M ∈ C k = the stack classifying extensions 0 Ω k+1 k 0, with k k . E → C ∨ E∨ → C We have natural projections ρk : k k and ρk : k k, which form dual C Lest | vector bundles over k, due to the above conditions on .Wehave:ρk = %k Ek . Next, we set: E0 { M ∈ E | }⊂E (3.4) k = ( k,sk) k sk is injective k, E∨0 { → k → M → M → ∈ E∨ | M ∈ C }⊂E∨ (3.5) k = (0 Ω k+1 k 0) k k+1 k+1 k . E0 ' E∨0 E0 → E E0 Clearly, k k−1.Denotebyjk the embedding k , k.Notealsothat k is C 0 an open substack in ohn. Consider the following diagram:

j E ←n E0 ' E∨0 → E∨ E n - n n−1 , n−1 n−1 ∨ ρn ρn−1 ρn−1 . & . ... Cn Cn−1

j E∨ E ←1 E0 ' E∨0 → E∨ 1 1 - 1 0 , 0 ∨ ∨ ρ1 ρ1 ρ0 ... & . & C1 C0 F E0 L We set E,1 to be a complex on 1 equal to the pull-back of Laumon’s sheaf E under ρ∨ E0 ' E∨0 → E∨ −→0 C ' C 1 0 , 0 0 oh0. ∨ E0 → E∨ Since ρ0 is a smooth morphism and 1 0 is an open embedding, the restric- F E0 tion of E,1[d] to the connected component of 1 corresponding to coherent sheaves of degree d is a perverse sheaf. F E0 Next, we deﬁne the complexes E,k on k by the formula:

F =Four(j (F )) |E0 , E,k+1 k! E,k k+1 where Four is the Fourier transform functor. Unraveling the second construction we obtain (see [Lau2]):

3.5. Lemma. The complex FE,n coincides (up to a cohomological shift and Tate’s S0 E0 ⊂ C 0 twist) with the restriction of E to n ohn.

3.6. The cleanness property of FE,k. Let us now assume that E is an irreducible rank n local system and that Conjecture 2.3 holds for E. In Sect. 5 we prove the following theorem, which was conjectured by Laumon in [Lau2], Expos´e I, Conjecture 3.2.

3.7. Theorem. For k =1,... ,n− 1, the canonical maps jk!(FE,k) → jk∗(FE,k) are isomorphisms.

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Recall that a complex K on Y is called clean with respect to an embedding →j K → K K | K Y , Y if j!( ) j∗( ) is an isomorphism, i.e., j∗( ) Y −Y =0.When is a perverse sheaf, cleanness implies that j!(K) ' j!∗(K) ' j∗(K). In this language F E0 E0 → E Theorem 3.7 states that the sheaf E,k on k is clean with respect to jk : k , k. Ld By construction, Laumon’s sheaf E is perverse and irreducible. As was men- F E0 tioned earlier, the restriction of E,1 to each connected component of 1 is, therefore, also an irreducible perverse sheaf, up to a cohomological shift. Since the Fourier transform functor preserves perversity and irreducibility, we obtain by induction: F E0 3.8. Corollary. The restriction of E,n to each connected component of n is an irreducible perverse sheaf, up to a cohomological shift. In Sect. 6 we will derive from Corollary 3.8 the following theorem, which was 0 conjectured by Laumon in [Lau2], Expos´e I, Conjecture 3.1. Denote by ρn the E0 → C morphism n n obtained by restriction from ρn.

3.9. Theorem. The complex FE,n descends to Cn, i.e., there exists a perverse S0 C sheaf E on n, such that n2 · (g − 1) F ⊗ Q ( )[n2 · (g − 1)] ' ρ0∗(S0 ). E,n ` 2 n E S0 C Moreover, the restriction of E to each connected component of n is a non-zero irreducible perverse sheaf. 3.10. A summary. Above we have described two constructions of a Hecke eigensheaf associated to a local system E. In the next section we will describe the third construction. Before doing that, we wish to summarize the relations between the three constructions and to indicate the strategy of the proof of the main result of this paper that the Vanishing Conjecture 2.3 for E implies the geometric Langlands Conjecture 1.3. S0 In the ﬁrst construction given in Sect. 3.1 we constructed a sheaf E on the C 0 M n−1 → M M moduli stack ohn of pairs ( , Ω , ), where is a coherent sheaf on X of generic rank n. In the second construction given in Sect. 3.2 we deﬁned a sheaf FE,n on an open E0 C 0 C ⊂ C substack n of ohn (recall that this open substack is the preimage of n ohn C 0 → C under %n : ohn ohn). Finally, in the third construction given in the next section we produce a sheaf 0 0 M n−1 → M M AutE on the stack Bunn of pairs ( , Ω , ), where is a rank n vector 0 −1 ⊂ C 0 bundle on X (obviously, Bunn is the preimage %n (Bunn) ohn). The relations between these sheaves are as follows: 0 S |E0 ' F E n E,n and 0 0 S | 0 ' Aut , E Bunn E up to cohomological shifts and Tate’s twists. These isomorphisms are established in Lemma 3.5 and Lemma 4.4, respectively. In particular, 0 F | −1 ' | −1 E,n % (Bunn ∩Cn) AutE % (Bunn ∩Cn).

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Thus, the primary role of the ﬁrst construction is to establish a link between the second and the third ones. Our ﬁrst goal is to prove Corollary 3.8 that if E is irreducible, then FE,n (obtained as the result of the second construction) is irreducible and perverse when restricted to each connected component (up to a cohomological shift and a twist). This complex is obtained by iterating the operations of Fourier transform, which are known to preserve irreducibility and perversity, and of !–extensions with respect to the open embeddings jk. Hence we need to show that the !–extensions of the intermediate sheaves FE,k coincide with their Goresky-MacPherson extensions, i.e., that FE,k are clean with respect to jk. This is the content of Theorem 3.7, which is derived in Sect. 5 from the Vanishing Conjecture 2.3. Next, we will prove that FE,n descends to a perverse sheaf on the open subset C ∩ ⊂ C 0 E0 → C n Bunn n under a natural smooth morphism ρn : n n (see Theorem 3.9). This will be done using the perversity and irreducibility of FE,n and some infor- 0 mation about the Euler characteristics of the stalks of the sheaf AutE.Inorderto compute these Euler characteristics, we use the third construction (and its relation to the second construction) in an essential way. S0 C ∩ Having obtained the perverse sheaf E on n Bunn, we take its Goresky- MacPherson extension to the union of those connectedS components of Bunn which have a non-empty intersection with C , i.e., to Bund . This gives us a n d≥cg,n n perverse sheaf AutE on this stack, irreducible on each connected component. Then we prove that AutE may be extended to the entire stack Bunn in such a way that it is a Hecke eigensheaf. We give two independent proofs of this statement, in Sect. 7 and Sect. 8, using the third and the second constructions, respectively. Finally, in Sect. 9 we prove that the Hecke property of AutE and Conjecture 2.3 imply that AutE is automatically cuspidal. This summarizes the main steps in our proof of Conjecture 1.3. In addition, we will prove the following result. Let us denote by SE the Goresky-MacPherson extension of AutE from Bunn to Cohn. In Sect. 8 we will show that the sheaves ∗ S S0 C 0 %n( E)and E on ohn are isomorphic, up to a cohomological shift and Tate’s ∗ 0 0 twist.Thesameistrueforthesheaves%n(AutE)andAutE on Bunn. 0 0 0 Note, however, that neither S nor Aut ' S | 0 is perverse on the entire E E E Bunn C 0 0 stack ohn and BunE, respectively. This does not contradict the above assertions: although the morphism %n is smooth over Cn, it is not smooth over the entire Cohn.

4. The construction via the Whittaker sheaf S0 In this section we will present another construction of the sheaf E (more pre- 0 1 cisely, of its restriction to Bunn). Conceptually, this construction should be viewed as a geometric counterpart of the construction of automorphic functions for GLn from the Whittaker functions due to Piatetskii-Shapiro [PS1] and Shalika [Sha] (see [FGKV] and Sect. 4.13 below for more details).

4.1. Drinfeld’s compactiﬁcation. We introduce the stack Q, which classiﬁes the data (M, (si)), where M is a rank n bundle and si, i =1,... ,n, are injective

1This construction was independently found by I. Mirkovi´c.

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homomorphisms of coherent sheaves Ωn−1 −−→s1 M, Ω(n−1)+(n−2) −−→s2 Λ2M, (4.1) ... − n(n 1) sn−1 n−1 Ω 2 −−−−→ Λ M,

n(n−1) sn n Ω 2 −−→ Λ M

satisfying the requirement that at the generic point of X the collection (si) defines a complete flag of subbundles in M (see [FGV]). In concrete terms, this requirement may be phrased as follows: the transposed ∗ iM∗ → −(n−1)−...−(n−i) maps si :Λ Ω should satisfy the Plücker relations ∗ ⊗ ∗ k ≤ ≤ ≤ ≤ − (4.2) (sp sq)(Φp,q)=0, 1 k p q n 1. k pM ⊗ qM Here Φp,q is the subsheaf of Λ Λ spanned by elements of the form

(4.3) (v ∧ ...∧ v ) ⊗ (w ∧ ...∧ w ) 1 q X1 q − ⊗ ∧ (v1 ...w1 ...wk ...vp) (vi1 ...vik wk+1 ...wq)

i1<...

(i.e., we exchange k–tuples of elements of the set {vi} with the first k elements of the set {wj} preserving the order). To motivate this definition, denote by Fl(V ) the variety of full flags in an n– dimensional vector space V over k. We have a natural embedding nY−1 i (s1,... ,sn−1):Fl(V ) ,→ PΛ V. i=1 According to the results of [T] and [Fu], Sect. 9.1, the ideal of the image of Fl(V ) under this map is generated by the elements of the form (4.3).2 Thus, if all of the above homomorphisms

(n−1)+...+(n−i) i si :Ω → Λ M,i=1,... ,n− 1,

are maximal embeddings (i.e., bundle maps), then the data of (M, (si)) determine a full ﬂag of subbundles of M. We denote the open substack of Q classifying the data (M, (si)) satisfying this condition by Q and the open embedding Q ,→ Q by j. d We will denote by Q (resp., Qd, Qed) the connected component of Q (resp., Q, Qe) corresponding to vector bundles M of degree d (recall our notational convention d 0 from Sect. 3.2). Note that dim(Q )=dn +dim(Q ). There is a morphism d (4.4) τ : Q −→ X(d)

− n(n 1) n sending (M, (si)) to D, the divisor of zeros of the last map sn :Ω 2 −→ Λ M D d in (4.1). Denote by Q (resp., QD) the preimage of D under τ in Q (resp., QD).

2If char k = 0, then the generators with k = 1 suﬃce, but not always so if char k > 0; see [T].

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F QD T 4.2. Remark. The stack is the stack BunN defined in [FGV], Sect. 2.2.2, where FT is the T –bundle on X, which corresponds to the n–tuple of line bundles F n−1 O QD T (Ω ,...,Ω, (D)). The stack is the stack BunN from [FGV], Sect. 2.2.1. FT We recall from [FGV, BG] that the Drinfeld compactification BunN classifies the λˇ data (M, (κ )), where λˇ runs over the set of dominant weights of GLn.Further, κλˇ is a homomorphism of coherent sheaves Lλˇ → Vλˇ ,whereLλˇ is the line FT FT FT λˇ bundle FT × λˇ,andVF is the vector bundle corresponding to M and the Weyl T T λˇ representation of GLn of highest weight λ. In addition, the homomorphisms κ have to satisfy a set of Plücker type relations described in Sect. 2.2.2 of [FGV]. λˇ ωˇi These relations determine all κ ’s from si := κ ,i =1,... ,n− 1. Equivalently, these relations may be described in the form (4.2).

d 4.3. The Whittaker sheaf. Observe that the substack Qd of Q embeds as an Qed M M0 open substack into , which classiﬁes those quadruples ( ,β,( i ), (si)), for which M M0 → M is torsion-free and i , are maximal embeddings (i.e., bundle maps) for e i =1,... ,n− 1. Recall the morphism ev : Q → Ga. Denote its restriction to 0 Qd ⊂ Qed also by ev. First, we deﬁne the complex Ψ0 on Q as dim(Q0) Ψ0 = j ev∗(I ) ⊗ Q ( )[dim(Q0)]. ! ψ ` 2

We have a natural morphism q : Q → Bunn taking (M, (si)) to M. Recall the d stack Modn from Sect. 2.2 and consider the Cartesian product d Q0 × d Z := Modn . Bunn

0 → d d 0 0 Let h : Z → Q be the morphism that sends (M, (si), M ,β : M → M )to M0 0 0 ( , (si)), where si is the composition

β Ω(n−1)+···+(n−i) −→si ΛiM −→ ΛiM0.

It is clear that 0h→ is a proper morphism of stacks, which makes the following diagram commutative:

0 0 ← 0 → d Q ←−−−h − Zd −−−−h → Q       (4.5) qy 0qy qy

← → ←−−−h − d −−−−h → Bunn Modn Bunn Wd Qd 3 The Whittaker sheaf E on is deﬁned by the formula d · n (4.6) Wd := 0h→(0h←∗(Ψ0) ⊗ (π ◦ 0q)∗(Ld )) ⊗ Q ( )[d · n]. E ! E ` 2 W0 ' 0 Wd W0 Q In other words, E Ψ , and in general E is obtained from E via a –version d 0 d Q0 → Qd of the averaging functor Hn,E. Namely, deﬁne the functor Hn,E :D( ) D( )

3 In the case of GL2 this sheaf was studied in [Ly1].

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by the formula d · n (4.7) K 7→ 0h→(0h←∗(K) ⊗ (π ◦ 0q)∗(Ld )) ⊗ Q ( )[d · n]. ! E ` 2 Wd 0 d 0 Then E = Hn,E(Ψ ). W Q Qd Wd Let E be the complex on , whose restriction to equals E.Denoteby Q → 0 M M n−1 → M ν : Bunn the morphism which sends ( , (si)) to ( ,s1 :Ω ). Set 0 W AutE := ν!( E ). 0 4.4. Lemma. The complex AutE is canonically isomorphic to the restriction of S0 C 0 0 E from ohn to Bunn. Qed Qed M M0 e Proof. Let tf be the locus in corresponding to those data ( ,β,( i ), (si)) ◦ for which M is torsion-free. Observe that Zd := (0h←)−1(Q0) ⊂ Zd is canonically Qed 0 Q0 Q0 identified with tf .SinceΨ is extended by zero from to , the proposition follows from the commutativity of the diagram ◦ ∼ Zd −−−−→ Qed  tf   0h→y νey Qd −−−−ν → 0 Bunn which is verified directly from the definitions. Wd 4.5. The structure of E. In the rest of this section we describe the structure of the Whittaker sheaf using the results of our previous work [FGV]. Strictly speaking, these results are not necessary in our proof of the geometric Langlands Conjecture 1.3. They are used only in the proof of the Hecke eigensheaf property presented in Sects. 7.2–7.6, for which we give an alternative proof in Sect. 8, which does not use the Whittaker sheaf. d To state our results, we recall first that the substack Qd of Q embeds as an open Qed Qd → Qd →α C d C r,d substack into and the composition oh0 takes values in oh (see Sect. 2.6 for this notation). Hence, Qd d e d ∗ (d) ∗ dim( ) d (4.8) W |Qd ' W |Qd ' τ (E )|Qd ⊗ ev (I ) ⊗ Q ( )[dim(Q )], E E ψ ` 2 where τ is the morphism in (4.4). We will prove the following statement: Wd Qd 4.6. Proposition. The complex E on is an irreducible perverse sheaf which is the Goresky-MacPherson extension of its restriction to Qd. The first step in the proof of Proposition 4.6 is provided by the following result of [FGV]: 0 ∗ 0 4.7. Lemma. The canonical map Ψ → j!∗j (Ψ ) is an isomorphism. 0 In other words, the perverse sheaf Ψ0 on Q0 is clean with respect to j : Q0 ,→ Q . Wd ' 0 d ∗ 0 Therefore, since E Hn,E(j!∗j (Ψ )) (cf. formula (4.7)), we obtain that D Wd ' Wd ( E ) E∗ , where D is the Verdier duality functor, and E∗ is the dual local system to E.

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Wd ∗ Thus, E is Verdier self-dual up to replacing E by E and to prove Propo- d sition 4.6 it suffices to introduce a stratification of Q and to show that the ∗– Wd Qd − Qd restriction of E to each stratum in appears in strictly negative cohomological degrees (with respect to the perverse t–structure). This stratification {Qµ} Wd is introduced in Sect. 4.9. The description of the restriction of E to each stratum given in Proposition 4.12 will imply that it appears in strictly negative cohomological degrees (except for the open stratum).

4.8. Substacks of Q defined by orders of zeros of sections. The Langlands Q dual group to GLn is GLn( `). In what follows we represent each weight of Q GLn( `) as a string of integers (d1,... ,dn), so that dominant weights satisfy ≥ ≥ ≥ + d1 d2 ... dn. We denote the set of dominant weights by Pn . The irre- Q ∈ + ducible finite-dimensional representation of GLn( `) with highest weight µ Pn µ will be denoted by V . Wedenotebyw0 the longest element of the Weyl group of GLn, which acts on the weights by the formula w0 · (d1,... ,dn)=(dn,... ,d1). For an anti-dominant weight µ,wedenotebyVµ the irreducible finite-dimensional −w (µ) representation of GLn with lowest weight µ, i.e., Vµ ' V 0 . { 1 m} Q Let µ = µ ,... ,µ be a collection of weights of GLn( `),wheresomeofthe µj’s may coincide. We will denote by Xµ the corresponding partially symmetrized power of X with all the diagonals removed. In other words, if m = m1 + ...+ ms r is such that a given weight µ appears in the collection exactly mr times, then Xµ = X(m1) × ...× X(ms) − ∆. We will think of a point x of Xµ as a collection of pairwise distinct points xj, j j j j =1,... ,m,toeachofwhichthereisanassignedweightµ =(d1,... ,dn). µ We associate to µ astackQ , which classifies the data (M, (si), x), where M is a vector bundle of rank n, x is a point of Xµ represented by a collection of distinct points xj ∈ X,and (n−1)+...+(n−i) j j · j → iM si :Ω Σ (d1 + ...+ d ) x , Λ j i are injective homomorphisms of coherent sheaves satisfying the Plücker relations from [FGV, BG]. The locus where all the maps si are maximal embeddings (i.e., are bundle µ maps) is an open substack Qµ of Q .Inotherwords,Qµ classifies the data (M, (Mi), (sei), x), where M is a rank n bundle, 0 = M0 ⊂ M1 ⊂ ... ⊂ Mn = M is a full flag of subbundles of M,andse ,i=1,... ,n, is an isomorphism i iM ' (n−1)+...+(n−i) j j · j Λ i Ω Σ (d1 + ...+ d ) x . j i

µ,x For a ﬁxed point x ∈ Xµ, we will denote by Q and Qµ,x the corresponding µ closed substacks of Q and Qµ, respectively. For the reader’s convenience, we identify the above stacks with those studied in F µ,x F Qµ,x ' T Q ' T F [FGV]: BunN and BunN ,where T is the T –bundle on X,which corresponds to the n–tuple of line bundles n−1 j · j j · j O j · j Ω Σ d1 x ,...,Ω Σ dn−1 x , Σ dn x . j j j

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d 4.9. Stratiﬁcation of Q . If the collection µ satisﬁes the conditions X j j ≥ ∀ j (4.9) d1 + ...+ di 0, i, j, di = d, i,j

Qµ → Qd Qµ then we have a natural closed embedding , ,andsoP is a locally closed Qd m j · j ∈ (d) substack of .Inparticular,foradivisorD = j=1 d x X we have D µ,x isomorphisms Q ' Q and QD ' Qµ,x,whereµj =(0,... ,dj). The following statement follows from [FGV], Corollary 2.2.9. 4.10. Lemma. The locally closed substacks Qµ with µ satisfying the condition d (4.9) form a stratiﬁcation of Q .

4.11. Restrictions of WE to the strata. The collection µ is called anti-dominant if all the weights µj are anti-dominant. For an anti-dominant µ,wehaveamap µ µ ev : Q → Ga deﬁned as in [FGV]. Namely,

nY−1 µ 1 ev : Q → Ext (Mi/Mi+1, Mi−1/Mi)) i=1 nY−1 → H1 X, Ω Σ (dj − dj ) · xj j i i+1 i=1 → 1 ⊕n−1 ' Gn−1 sum→ G H (X, Ω) a a (compare with formula (3.2)). We then set ◦ dim(Qµ) (4.10) Ψµ := evµ ∗(I ) ⊗ Q ( )[dim(Qµ)]. ψ ` 2 µ Denote by jµ the embedding Qµ → Q . AccordingtoTheorem2of[FGV],the ◦ ◦ µ µ µ µ sheaf Ψ is clean, i.e., Ψ := j! (Ψ ) is an irreducible perverse sheaf isomorphic to ◦ ◦ µ µ µ,x µ,x Qµ,x j!∗(Ψ ). In a similar way we define the perverse sheaves Ψ and Ψ on and µ,x Q , respectively. Next,toalocalsystemE on X and an anti-dominant collection µ we associate µ a local system Eµ on Q as follows. Recall that we denote by Vµ the irreducible ∈− + representation of GLn with lowest weight µ Pn .LetEµ be the local system on X associated to E and Vµ. For µ corresponding to the partition m = m1 + ... + ms consider the sheaf (m1) (ms) (m1) × × (ms) (Eµ1 ) ... (Eµs ) on X ... X .DenotebyEµ its restriction µ to the compliment of all diagonals, i.e., to X . Let us denote by τµ the natural Qµ µ 0 ∗ morphism from to X and set Eµ := τµ (Eµ). 0 1 m Thus, Eµ depends only onN the positions of the points x ,... ,x ,anditsfiber M e 1 m m at ( , (si), (x ,... ,x )) is j=1 Eµj ,xj ,whereEµj ,xj denotes the fiber of Eµj at xj.

∗ Wd Qµ ⊂ Qd 4.12. Proposition. The –restriction of E to is zero unless all weights µj are anti-dominant. When they are, this restriction is canonically identiﬁed with ◦ µ ⊗ 0 ⊗ Q d−m − Ψ Eµ `( 2 )[d m].

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As was explained earlier, this proposition implies Proposition 4.6. Indeed, we ∗ Wd Qµ ⊂ Qd − Qd had to show that –restriction of E to each stratum in appears in negative cohomological degrees. According to Proposition 4.12, this restriction − − j j j lives in the cohomological degree (d m). However, since each µ =(d1,... ,dn) j ≤ ≤ j j j ≥ − ≥ satisfies d1 ... dn and d1 + ...+ dn 0, we obtain that d m 0andthe equality takes place only when every µj is of the form (0,... ,0, 1). However, the stratum corresponding to this µ is contained in Q. The proof of Proposition 4.12 is given in the Appendix. A similar result is proved in [Ly2]. 4.13. The Whittaker function. In this subsection we will assume that k is the finite field Fq. We will show that the function associated with the Whittaker sheaf may be identified with the restriction of the Whittaker function. First we briefly recall the definition of the Whittaker function (see [FGKV], Sect. 2, for more details). Consider the group GLn(A) over the ring of adèles of F = Fq(X), and let N(A) be its upper unipotent subgroup. Denote by ui,i+1 the i-th component of the image of u ∈ N(A)inN(A)/[N(A),N(A)] corresponding to the (i, i +1)entryofu. Recall that we have fixed a non-trivial additive character × F → Q A 4 ψ : q ` . We define the character Ψ of N( ) by the formula nY−1 Y

Ψ (ux)x∈|X| = ψ(Trkx/Fq (Resx ux,i,i+1)). i=1 x∈|X| It follows from the residue theorem that Ψ(u)=1ifu ∈ N(F ). Now let E be a rank n local system on X. Then there exists a unique (up to a non-zero scalar multiple) function WE on GLn(A), which is right GLn(O)– invariant, left (N(A), Ψ)–equivariant, and is a Hecke eigenfunction associated to E. This function is called the Whittaker function corresponding to E. Casselman– Shalika [CS] and Shintani [Shi] have given an explicit formula for WE (see, e.g., Theorem 2.1 of [FGKV]). The left (N(A), Ψ)–equivariance of WE implies that it is left N(F )–invariant, where F = Fq(X). Therefore we obtain a function on the double quotient Q = N(F )\GLn(A)/GLn(O). We denote this function also by WE. In the same way as in the the proof of Lemma 2.1 from [FGKV], we identify the F Qµ,x j j j set of q–points of ,whereµ =(d1,... ,dn),j =1,... ,m, with the projection onto Q of the subset A · dn(x) d1(x) · O ⊂ A N( ) diag(πx ,... ,πx ) GLn( ) GLn( ), x∈|X| j where di(x)=di ,ifx = xj ,anddi(x) = 0, otherwise. Thus, we may embed the set d d Q (Fq) of isomorphism classes of Fq–points of Q into Q for all d ≥ 0. Comparing Proposition 4.12 with the Casselman-Shalika-Shintani formula, we obtain:

Wd Qd F 4.14. Proposition. The function fq( E ) on ( q) corresponding to the sheaf Wd Qd F ⊂ E equals the restriction of the Whittaker function WE to ( q) Q. 0 Furthermore, the geometric construction of the sheaf AutE describedinthissec- tion translates at the level of functions into the construction due to Shalika [Sha]

4 J A A For this formula to be well-deﬁned, we should consider a twisted version GLn( )ofGLn( ) introduced in [FGKV], Sect. 2; then ui,i+1 is naturally a diﬀerential.

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and Piatetskii-Shapiro [PS1] (see Sect. 2 of [FGKV] for a review of this construction).

5. Cleanness in Laumon’s construction Let E be an irreducible rank n local system for which Conjecture 2.3 holds. In this section we derive Theorem 3.7 for E, i.e., we prove that the complex FE,k on E0 E0 → E k is clean with respect to jk : k , k. We will begin by stating a well-known lemma, which will serve as one of the key ingredients in the proof. Consider the following situation: let E be a vector bundle over a scheme (or stack) Y . Let us denote by ρ : E → Y the projection, by j i : Y → E the zero section, and by E0 ,→ E the complement of the zero section. 0 Assume that K is a complex on E , equivariant with respect to the Gm–action.

5.1. Lemma. The complex K is clean with respect to j if and only if (ρ◦j)!(K)=0. To prove the lemma, it suﬃces to note that cleanness of K is equivalent to the ! 0 statement that i j!(K) = 0. But for any Gm–equivariant complex K on E,wehave: ! 0 0 i (K ) ' ρ!(K ). Our proof of Theorem 3.7 will proceed by induction on the length of torsion in Ck. Let us ﬁrst consider the case where there is no torsion at all, i.e., we will show F → F −1 C ∩ ⊂ C that jk!( E,k) jk∗( E,k) is an isomorphism on the open set ρk ( k Bunk) k. 0 E → C E0 → C Recall that ρk (resp., ρk) denotes the projection k k (resp., k k). −1 C ∩ E0 E F On ρk ( k Bunk), k is the complement of the zero section in k and E,k is Gm–equivariant. Thus, by Lemma 5.1, we are reduced to showing that 0 F | ρk!( E,k) Ck ∩Bunk =0. By Lemma 4.4, with n replaced by k, we obtain that up to a cohomological shift and Tate’s twist 0 d 0 ρ !(FE,k)|C ∩ d ' H (q!(Ψ ))|C ∩ d . k k Bunk k,E k Bunk

The definition of Ck in Sect. 3.4 implies that if Ck ∩ Bunk =6 ∅,thend>nk(2g − 2). 0 F | Thus, the Vanishing Conjecture 2.3 implies ρk!( E,k) Ck ∩Bunk =0. 5.2. Induction on the length of torsion. To set up the induction, we fix some notation. For an integer `,letuswriteCk,≤` for the open substack of Ck consisting of coherent sheaves whose torsion is of length ≤ `.SetCk,<` = Ck,≤`−1 to be the open substack in Ck,≤` that corresponds to the locus where the torsion is of length <`. Finally, let Ck,` be the closed substack of Ck,≤` corresponding to coherent sheaves whose torsion is precisely of length `. E −1 C E −1 C E −1 C Set k,≤` = ρk ( k,≤`), k,<` = ρk ( k,<`), and k,` = ρk ( k,`). Furthermore, let us write E0 E0 ∩ E E0 E0 ∩ E k,≤` = k k,≤`, k,<` = k k,<`, Et E − E0 Et Et ∩ E k = k k, k,` = k k,`, etc. F E0 → We assume, by induction, that E,k is clean with respect to the inclusion k,<` , E F E0 → E k,<`. To show cleanness of E,k with respect to the inclusion k,≤` , k,≤`,it F E − Et → E suffices to prove cleanness of E,k with respect to ( k,≤` k,`) , k,≤`.We would like to argue in the same manner as we did above in the case when ` was zero and using Lemma 5.1. Unfortunately, we cannot apply this lemma directly,

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E Et because k,≤` is not a vector bundle over k,`. However, it will become a vector bundle after a smooth base change. Consider the stack Cohk,≤` which classiﬁes coherent sheaves of generic rank k g with torsion of length ≤ `, and the stack Cohk,≤`, which classiﬁes the following M ∈ T ∈ C ` data: 0 Bunk, oh0, and a short exact sequence

0 → M0 → M → T → 0. g There is a canonical morphism r : Cohk,≤` → Cohk,≤` which associates to a triple as above the coherent sheaf M ∈ Cohk. 5.3. Lemma. The morphism r is smooth.

Proof. First, the stack Cohk is known to be smooth. One proves this simultaneously with the fact that Cohk is indeed an algebraic stack in the smooth topology by covering it by a Hilbert scheme. g Therefore both stacks Cohk,≤` and Cohk,≤` are smooth, the former being an open substack in Cohk, and the latter being a vector bundle over Bunk. Hence in order to show that r is smooth, it suffices to show that the fiber of r is smooth over any field-valued point M ∈ Cohk,≤`. By definition, the tangent space to the fiber of r at the point 0 → M0 → M → T → 0isHom(M0, T). The dimension of Hom(M0, T)isk · ` because M0 is torsion–free. As r is separable and the dimensions of the tangent spaces to the fibers are constant, we conclude that r is smooth. Since r is smooth by Lemma 5.3, it is sufficient to prove cleanness after this base g g change r : Cohk,≤` → Cohk,≤`. Consider the stack Cohk,≤` × Ek,≤`. It classifies Cohk,≤` the following data: M ∈ C T ∈ C ` M ∈ k, oh0, tf Bunk, (5.1) k−1 s 0 → Mtf → M → T → 0, Ω → M. Its substack E˜t Cg × Et k,` = ohk,≤` k,` Cohk,≤` consists of the data (5.1) such that the extension is split and the image of s belongs M E˜t to the torsion part of .Inotherwords, k,` classifies the data M ∈ T ∈ C ` M ⊕ T ∈ C k−1 −→κ T (5.2) tf Bunk, oh0 (such that tf k), Ω . ˜ g Denote by Ek,≤` the open substack of Cohk,≤` × Ek,≤` defined by the con- Cohk,≤` dition est Hom(Mtf , L )=0; here Lest is the line bundle of Sect. 3.2. The above condition guarantees that Mtf ⊕ T ∈ Ck.Set E˜0 E˜ ∩ Cg × E0 k,≤` = k,≤` ohk,≤` k,≤` . Cohk,≤` E˜t E˜ Obviously, k,` is contained in k,≤`.

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E˜ → E˜t There is a natural morphism ρk,` : k,≤` k,` which maps the quadruple in the ˜ k−1 s definition of Ek,≤` to the data (Mtf , T,κ), where κ is the composition Ω → M → T E˜ E˜t M T .Inthisway k,≤` becomes a vector bundle over k,`; the fiber over ( tf , ,κ) can be canonically identified with the vector space 1 k−1 κ Ext (Cone(Ω −→ T), Mtf ). 0 E˜0 → E˜ −→ρk,` E˜t Let ρk,` denote the composition k,≤` , k,≤` k,`. Fe F E˜0 Fe Let E,k denote the pull-back of E,k to k,≤`. One readily verifies that E,k is G E˜ → E˜t equivariant with respect to the m–action along the fibers of ρk,` : k,≤` k,`. e The assertion of the theorem reduces to the fact that FE,k is clean with respect E˜0 → E˜ to the open embedding k,≤` , k,≤` and we already know this assertion on the E˜ − E˜t open substack k,≤` k,`. Fe E˜0 E˜ − E˜t By applying Lemma 5.1 to E,k extended by 0 from k,≤` to k,≤` k,`,we reduce our assertion to showing that 0 Fe (5.3) ρk,`!( E,k)=0.

5.4. Proofofformula(5.3). Recall the stack Qe of Sect. 3.1 and let us denote by Qe Q Q0 Q Q0 k (resp., k, k, k) its version with n replaced by k.Inparticular, k classifies M0 M0 e M0 → M0 M0 points of the form ( ,β,( i ), (si)), where the map β : = n is the 0 e identity. We will denote such a point simply by (M , (sei)). Denote by WE,k the e perverse sheaf on Qk defined as in Sect. 3.1. Qe × E˜0 Consider the Cartesian product k k,≤`. This is the stack that classifies C 0 ohk the data of M ∈ M Lest T ∈ C ` tf Bunk with Hom( tf , )=0, oh0, 0 (5.4) 0 → Mtf → M → T → 0, M ,→ M, M0 e e ∈ Q0 ( , s1,... ,sk) k. e By Lemma 3.5, FE,k is the direct image (with compact supports) under the map Qe × E˜0 → E˜0 We Qe Qe × E˜0 k k,≤` k,≤` of the pull-back of E,k from k to k k,≤`. Hence, in C 0 C 0 ohk ohk order to prove (5.3), it suffices to show that the compactly supported cohomology of Qe × E˜0 → E˜0 → E˜t We the fiber of k k,≤` k,≤` k,` with coefficients in the pull-back of E,k C 0 ohk M ∈ T ∈ C ` k−1 → T vanishes. To this end, let us fix a point ( tf Bunk, oh0,κ :Ω )in E˜t k,` and analyze the fiber over this point. Let us write Y for the closed substack M0 e e ∈ Q0 of the the fiber which lies over a fixed point ( , s1,... ,sk) k and where the composition φ : M0 → M → T is also fixed. From the discussion above we conclude that (5.3) follows if we show that for all M ∈ T ∈ C ` M0 e e ∈ Q0 M0 → T ( tf Bunk, oh0, ( , s1,... ,sk) k,φ: ) as above, we have • We | (5.5) Hc (Y, E,k Y )=0. To prove this, we will first reduce to the case when φ : M0 → T is surjective. Let us denote by T0 the image of φ and by T00 the cokernel of κ;write`0 (resp., `00)

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for the length of T0 (resp., T00). Let Y 0 be the scheme deﬁned in the same way as Y , but for M T0 M0 e e ∈ Q0 0 M0 → T0 ( tf , , ( , s1,... ,sk) k,φ: ). We have a natural map v : Y → Y 0, which associates to a point 0 (5.6) (0 → Mtf → M → T → 0, M ,→ M) the point 0 0 0 0 0 (5.7) (0 → Mtf → M → T → 0,φ: M ,→ M ), where M0 is the preimage of T0 under M → T. e e 00 W | W | 0 ⊗ L` 00 5.5. Lemma. The complexes v!( E,k Y ) and E,k Y ( E )T are isomorphic `00 up to a cohomological shift and Tate’s twist; here (L )T00 is the stalk of Laumon’s 00 E T00 ∈ C ` sheaf at oh0 .

Proof. Let us recall the following basic property of Laumon’s sheaf LE (cf. [Lau2]): d0,d00 Consider the stack Fl0 that classiﬁes short exact sequences → Te0 → Te → Te00 → Te0 ∈ C d0 Te00 ∈ C d00 0 0, oh0 , oh0 . 0 00 ` ,` → C d 0 00 Let p denote the natural projection Fl0 oh0 (here d = d +d )thatassociates 0 00 0 d ,d → C d × to a short exact sequence as above its middle term, and let q :Fl oh0 00 0 C d oh0 denote the other natural projection. In [Lau2] Laumon proved that ◦ ∗ Ld ' Ld0 Ld00 (5.8) q! p ( E ) E E . 0 → C d0 × C d00 We have a natural map Y oh0 oh0 that sends the data of 0 0 0 0 (0 → Mtf → M → T → 0, M ,→ M )

0 00 M0 M0 T00 0 M0 − M0 00 00 → d ,d to ( / , )withd =deg( ) deg( ),d = ` and a map Y Fl0 that sends 0 (0 → Mtf → M → T → 0, M ,→ M) to 0 → M0/M0 → M/M0 → T00 → 0. M0 We | ∗ Ld Note that since is ﬁxed, E,k Y is isomorphic to the pull-back of p ( E ) under e `00 d0 d00 this map. Similarly, WE,k|Y 0 ⊗ (L )T00 is isomorphic to the pull-back of L L 0 00 E E E 0 → C d × C d under Y oh0 oh0 . We have the following diagram: 0 00 Y 0 ←−−−− Y 0 × Fld ,d ←−−−− Y 0 00 0 Cohd ×Cohd  0  0   y y

0 00 C d0 × C d00 ←−−−q − d ,d oh0 oh0 Fl0 , in which the composed upper horizontal map is v. Moreover, it is easy to see that 0 00 0 × d ,d ← the map Y Fl0 Y is a fibration with fibers being affine spaces C d0 ×C d00 oh0 oh0

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of the same dimension. Therefore, up to Tate’s twist and a cohomological shift, 0 00 We | 0 → C d ×C d ◦ ∗ Ld v!( E,k Y ) is isomorphic to the pull-back under Y oh0 oh0 of q! p ( E ). Hence, the assertion of the lemma follows from (5.8).

5.6. Endoftheproofofformula(5.3). The above considerations show that it suﬃces to treat the case when φ : M0 → T is surjective. Let M1 denote the kernel of φ. Let us observe that the scheme Y can be identiﬁed with the scheme 0 1 1 Hom (M , Mtf ) of injective maps M → Mtf . Indeed, to

0 0 → Mtf → M → T → 0, M ,→ M

1 0 we associate M ,→ M ,→ M, which maps into Mtf by assumption. And vice 1 0 versa: to an embedding M → Mtf we associate M = Mtf ⊕ M . M1 We | ∗ Ld M Moreover, the sheaf E,k Y becomes isomorphic to π ( E ), where d =deg( tf ) 1 e − deg(M ). Therefore, the cohomology Hc(Y,WE,k|Y ) equals the cohomology appearing in Conjecture 2.5. est By assumption, the vector bundle Mtf satisﬁes: Hom(Mtf , L ) = 0. Hence, by est the condition on L (cf. Sect. 3.2), deg(Mtf ) >nk(2g −2), and so d =deg(Mtf )+ `>nk(2g − 2). The required vanishing statement now follows from Conjecture 2.5 applied to E. This completes the proof of formula (5.3) and Theorem 3.7.

6. Descent of the sheaf FE,n As in the previous section, we keep the assumption that the local system E is irreducible and that Conjecture 2.3 holds for E. Our goal here is to prove Theorem 3.9. Having established Theorem 3.7 for all k =1,... ,n− 1 we know, according to Cd F ⊗ Q d Corollary 3.8, that over n the complex E,n `( 2 )[d] is an irreducible perverse sheaf.

0 E0 → C 6.1. Euler characteristics. The morphism ρn : n n is smooth of relative 2 d dimension d − n (g − 1), and the sheaf F ⊗ Q ( )[d]|E0 ∩ 0 −1 Cd is perverse E,n ` 2 n (ρn) ( n) and irreducible. Hence in order to prove Theorem 3.9 it suﬃces to show that when

d ≥ c , the restriction F | 0 −1 C ∩ d is non-zero and that it descends to a g,n E,n (ρn) ( n Bunn) C ∩ d perverse sheaf on n Bunn. Recall from Lemma 4.4 that

0 F | 0 −1 C ∩ d ' Aut | 0 −1 C ∩ d , E,n (ρn) ( n Bunn) E (ρn) ( n Bunn) up to a cohomological shift. The proof will be based on the following proposition:

0 6.2. Proposition. The Euler characteristics of the stalks of Aut | 0 are con- E Bunn 0 → stant along the ﬁbers of the projection %n :Bunn Bunn. Moreover, they are not −1 C ∩ d ≥ identically equal to zero over (%n) ( n Bunn)(for d cg,n).

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6.3. Derivation of descent from Proposition 6.2. Suppose d ≥ cg,n. Then the d perverse sheaf F |E0 ∩ 0 −1 Cd ⊗ Q ( )[d] is the Goresky-MacPherson extension E,n n (ρn) ( n) ` 2 0 E0 of a local system on a locally closed substack U of n, contained inside the open 0 −1 C ∩ d ⊂ E0 substack (ρn) ( n Bunn) n. ⊂ C ∩ d There exists a smooth locally closed substack U1 n Bunn, such that if 0 0 −1 0 ∩ 0 0 we set U1 =(ρn) (U1), the intersection U1 U is open and dense in U and 0 0 ∩ 0 → 0 ρn : U1 U U1 is surjective. Since ρn is smooth, in order to prove Theorem 3.9 d it suﬃces to show that F ⊗ Q ( )[d]| 0 is a pull-back of a local system on U . E,n ` 2 U1 1 We have the following general result: 6.4. Lemma. Let Y beasmoothscheme(or stack) and let K be an irreducible perverse sheaf on Y . If the Euler characteristics of the stalks of K are the same at all k–points of Y ,thenK is a local system. If these Euler characteristics are not identically equal to 0,thenK =06 .

Proof. Let Y0 ⊂ Y be the maximal open subset over which K is a local system. By the irreducibility assumption, K is the Goresky-MacPherson extension of a local system on Y0.SinceY is smooth, it is enough to show that Y −Y0 is of codimension ≥ 2. Suppose this is not so. Then Y − Y0 contains a divisor. Let A denote the strict Henselization of the local ring at the generic point of this divisor and let η (resp., s) be the generic (resp., closed) point of Spec(A). By our assumptions, K|Spec(A) is the Goresky-MacPherson extension of a local system on η.ThestalkKη is a representation of the Galois group Γ of the field of Γ fractions of A, and we have Ks ' (Kη) . By the assumption on the Euler characteristics, dim(Ks)=dim(Kη), i.e., the representation of Γ on Kη is trivial. But this means that K extends as a local system to the entire Spec(A), which is a contradiction. 0 d Let us show that this lemma is applicable for Y = U and K = F ⊗Q ( )[d]| 0 . 1 E,n ` 2 U1 Indeed, the Euler characteristics of stalks of FE,n|U 0 are constant along the fibers 0 → 1 0 ∩ 0 F of U1 U1, by Proposition 6.2. Moreover, they are constant on U1 U ,since E,n 0 is a local system there. Hence, the Euler characteristics are constant on all of U1, 0 ∩ 0 → since U1 U U1 is surjective. d Thus, we obtain that F ⊗ Q ( )[d]| 0 is a non-zero local system. E,n ` 2 U1 0 E | By definition, U1 is a complement to the zero section in the vector bundle n U1 . By construction, FE,n is equivariant with respect to the natural Gm–action along 0 0 → the fibers of the projection ρn : U1 U1. Therefore, the assertion of Theorem 3.9 d follows from the next lemma applied to E := E | , K := F ⊗ Q ( )[d]| 0 . n U1 E,n ` 2 U1 6.5. Lemma. Let E → Y be a vector bundle and let us denote by E0 the complement to the zero section. Let K be a local system on E0, equivariant with respect to the Gm–action along the fibers. Then K descends to a local system on Y . Proof. This follows from the fact that any local system on a projective space is isomorphic to the trivial local system. Now we prove Proposition 6.2. The first step is the following statement. 6.6. Lemma. Let E0 be another rank n local system, not necessarily irreducible. 0 0 k Then the Euler characteristics of AutE and AutE0 are equal at any given –point 0 of Bunn.

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In order to prove the lemma, we will use the following corollary of a theorem of Deligne from [Il], Corollary 2.10:

6.7. Theorem. Let f : Y1 → Y2 be a proper morphism of schemes (or a proper 0 representable map of stacks).LetK and K be two complexes on Y1, which are locally isomorphic, by which we mean that they can be represented as inverse limits of ´etale-locally isomorphic complexes with torsion coeﬃcients. Then f!(K) and 0 f!(K ) have equal Euler characteristics at every k–point of Y2.

n Recall the stack Q and note that the group (Gm) acts on it by the rule

(c1,... ,cn) · (M,s1,... ,sn)=(M,c1 · s1,... ,cn · sn).

n−1 n−1 n Consider the quotient Qr := Q/(Gm) ,where(Gm) ⊂ (Gm) corresponds to G Q → 0 the omission of the ﬁrst copy of m. Then the morphism ν : Bunn factors as Q → Q →νr 0 r Bunn . The following is proved in [BG], Proposition 1.2.2:

n 6.8. Lemma. The morphism qr : Q/(Gm) → Bunn is representable and proper. Q → 0 We obtain from this lemma that νr : r Bunn is also proper. n−1 Now let us take the quotient of the diagram (4.5) by (Gm) :

0 ← 0 → 0 h h d Q ←−−−r − Zd −−−−r → Q r r r  0   qry qr y qr y

← → 0 ←−−−h − d −−−−h → d Bunn Modn Bunn 0 0 Q0 → Q0 0 Denote by Ψr the !–direct image of Ψ under r. It is clear that AutE can be written as d · n (6.1) Aut0 = ν 0h→ (0h←∗(Ψ0) ⊗ 0q∗π∗(Ld )) ⊗ Q ( )[d · n], E r! r ! r r r E ` 2 and similarly for E0. This formula and Theorem 6.7 readily imply the equality of the Euler character- 0 0 ◦ 0 → d → 0 istics of AutE and AutE0 . Indeed, the morphism νr hr : Zr Bunn is proper and the complexes 0 ←∗ 0 ⊗ 0 ∗ ∗ Ld 0 ←∗ 0 ⊗ 0 ∗ ∗ Ld hr (Ψr) qr π ( E ), hr (Ψr) qr π ( E0 ) Ld are locally isomorphic, since so are the corresponding Laumon’s sheaves E and Ld E0 . This completes the proof of Lemma 6.6. We remark that formula (6.1) is the generalization of the Radon transform construction (as opposed to the Fourier transform construction from Sect. 3.2) in Drin- feld’s original proof [Dr] of the Langlands conjecture in the case of GL2.

6.9. Conclusion of the proof of Proposition 6.2. According to Lemma 6.6, in order to prove Proposition 6.2 it suﬃces to show that there exists at least one local system E, for which the statement of Proposition 6.2 is true. Hence it suﬃces to prove it for the trivial local system. Using the reduction technique of [BBD], Sect. 6.1.7, we obtain:

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6.10. Lemma. Suppose that Proposition 6.2 holds when E is the trivial local system in the case when the ground field k is a finite field of characteristic p.Then Proposition 6.2 holds when E is the trivial local system in the case of an arbitrary field k of the same characteristic. k 0 Proof. Let s1 and s2 be two –points of Bunn, which project to the same point of Bunn. First, we can assume that all our data are defined over an algebra A finitely generated over a finite field; A ⊂ k. In other words, we have the stacks (Bunn)A 0 → 0 and (Bunn)A and sections si :SpecA (Bunn)A. ∗ 0 Consider si (AutE0 ),i =1, 2, where E0 is the trivial local system (so that it is defined over A), as `–adic complexes on Spec A. By localizing A we may assume that it is smooth over a finite field, and that the above complexes are locally constant. Let η :Speck → Spec A be the canonical generic geometric point of Spec A,and let a :SpecFq → Spec A be some closed geometric point of Spec A. We need to ∗ 0 compare the Euler characteristics of the stacks (si (AutE0 ))η for i =1, 2. Since our ∗ 0 complexes are locally constant, we may instead compare the stalks (si (AutE))a.In other words, we can make the comparison over the finite field, as required.5

Thus, it suffices to prove Proposition 6.2 for the trivial local system in the case when k is a finite field. Let us apply Lemma 6.6 again and obtain that it suffices to find just one local system Es in the case when k is a finite field, for which Proposition 6.2 is true. We will take as Es any irreducible local system, which satisfies the following conditions: (a) Es is pure, and (b) there exists a cuspidal Hecke eigenfunction associated to the pull-back of Es × F F F to X q1 for any finite extension q1 of q. Fq For example, such a local system can be constructed as follows: pick a cyclic n–sheeted étale cover Xe → X,andletEs be the direct image of a generic rank one local system on Xe of finite order. Then Es is pure, so condition (a) is satisfied. Moreover, according to Theorem 6.2 of [AC] (see also [K]), this local system also satisfies condition (b).6 Thus, we have at our disposal at least one irreducible rank n local system Es, for which the above conditions (a), (b), as well as Conjecture 2.3, are true. We now prove that then Proposition 6.2 also holds for this Es. To prove the first assertion 0 of Proposition 6.2, it suffices to show that the function fq1 (AutEs )(obtainedby 0 0 F taking the traces of Frobenius on the stalks of AutEs ; see Sect. 0.3) on Bunn( q1 ) is constant along the fibers of the projection 0 F → F Bunn( q1 ) Bunn( q1 ) F F for all finite extensions q1 of q. But Theorem 3.1 of [FGKV] states that if a cuspidal Hecke eigenfunction asso- e F ciated to any given rank n local system E exists on Bunn( q1 ), then its pull-back 0 F 0 to Bunn( q1 )equalsfq1 (AutEe ) up to a non-zero scalar.

5Note that if the ground field k is of characteristic 0, we can use a similar argument by choosing A to be a finitely generated algebra over Z. 6Actually, Lafforgue’s results [Laf] imply that any irreducible local system E satisfies conditions (a) and (b), up to a twist with a rank one local system.

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Applying these results to our local system Es, we obtain that the function 0 0 F → F fq1 (AutEs ) is constant along the fibers of %n :Bunn( q1 ) Bunn( q1 ). This proves the first assertion of Proposition 6.2 for Es. It remains to prove the non-vanishing assertion of Proposition 6.2 for Es.Ac- cording to Proposition 10.1, if E is an irreducible local system on a curve X over a finite field, which satisfies the above conditions (a) and (b), then Conjecture 2.3 holds for E. Hence by our assumptions on Es, Conjecture 2.3 holds for Es.There- 0 C ∩ d 0 fore by Theorem 3.7 the restriction of AutEs to the preimage of n Bunn in Bunn is a perverse sheaf, up to a cohomological shift. Hence it suffices to show that this restriction is non-zero (for if a perverse sheaf has zero Euler characteristics everywhere, then this sheaf is zero). For that, it is enough to show that the corresponding function does not vanish C ∩ d F ≥ C ∩ d 6 ∅ identically on ( n Bunn)( q), if d cg,n (note that in this case n Bunn = ). However, by assumption, d − C ∩ d ⊂ vuns ≥ Bunn ( n Bunn) Bunn ,dcg,n vuns (see Sect. 3.2 for the definition of Bunn ). The definition of cuspidal function implies the following

6.11. Lemma. Let f be a cuspidal function on Bunn(Fq).Thenitsrestrictionto vuns F Bunn ( q) is identically zero.

Proof. Let M ∈ Bunn(Fq) be a very unstable bundle, and let M ' M1 ⊕ M2 be the corresponding decomposition, with rk(Mi)=ni. n F → F × F Let rn2,n1 : Funct(Bunn( q)) Funct(Bunn2 ( q) Bunn1 ( q)) be the corre- n sponding constant term operator. Since f is cuspidal, we have rn2,n1 (f)=0. n n However, by applying the deﬁnition of rn2,n1 and evaluating rn2,n1 (f)atthepoint M2 × M1 ∈ Bunn (Fq) × Bunn (Fq), we obtain that it is equal to the integral 2 1 Z f(M 0),

0 0→M2→M →M1→0 1 over the ﬁnite set Ext (M1, M2)(Fq) (the measure on this set is a non-zero multiple of the tautological measure). 1 M M n M M However, by our assumption, Ext ( 1, 2) = 0, therefore rn2,n1 (f)( 2, 1)= f(M), up to a non-zero constant.

Thus, we obtain the second assertion of Proposition 6.2 for our local system Es. This completes the proof of Theorem 3.9.

7. The Hecke property of AutE S0 C In the previous section we constructed a perverse sheaf E on n, whose pull-back 2· − to E0 is F ⊗ Q ( n (g 1) )[n2 · (g − 1)]. n E,n ` 2 S Let S be the Goresky-MacPherson extension of S0 to Cohd . Finally, E E d≥cg,n n S |S set AutE := E Bund . d≥cg,n n Our goal is to prove the following

7.1. Theorem. The perverse sheaf AutE can be uniquely extended to the entire stack Bunn, so that it becomes a Hecke eigensheaf with respect to E.

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Theorem 7.1 will follow from Proposition 7.7, as will be explained in Sect. 7.8. We will give two independent proofs of Proposition 7.7. The ﬁrst one, presented in Sects. 7.2–7.6, uses the Whittaker sheaf WE . The second proof, given in Sect. 8, uses the Hecke-Laumon property of the Laumon sheaf LE. 0 7.2. The Hecke property on the stack Bunn. Consider the Cartesian product 0 × H1 H1 → → Bunn n,wherethemap n Bunn is h . We have a commutative diagram, Bunn in which the right square is Cartesian: 00 ← ×00 → 0 ←−−−h − 0 × H1 −supp−−−−−−h→ × 0 Bunn Bunn n X Bunn  Bun n     (7.1) y y y

← × → ←−−−h − H1 −−−−−−→supp h × Bunn n X Bunn, where the morphisms 00h← and 00h→ are given by 00h← :(x, M, M0,β : M0 ,→ M,s0 :Ωn−1 → M0) 7→ (M,s= β ◦ s0 :Ωn−1 → M), 00h→ :(x, M, M0,β : M0 ,→ M,s0 :Ωn−1 → M0) 7→ (M0,s0). 7.3. Proposition. For any local system E of rank n, n − 2 (7.2) (supp ×00h→) 00h←∗(Aut0 ) ⊗ Q ( )[n − 2] ' E Aut0 . ! E ` 2 E d First, we will reformulate this proposition in terms of the stack Q , introduced in Sect. 4.1. Qd Qd 7.4. A reformulation. We need to introduce two more stacks + and ++ closely Qd Qd M related to .Thestack ++ classifies the data of (x, , (si)) as in the definition d of Q , but with (n−1)+...+(n−i) i si :Ω → (Λ M)(x),i=1,... ,n. Qd The stack + classifies the same data with the additional condition that the image of s1 is contained in M (and not just M(x)). We have tautological closed embeddings × Qd → Qd → Qd X , + , ++. Qd+1 → 0 Recall that we have a forgetful morphism Bunn, and the morphism 00 ← 0 × H1 → 0 Hd+1 h :Bunn n Bunn.DenotebyQ the corresponding fiber product. Bunn Consider the following commutative diagram: e← e → Qd+1 ←−−−h − Qd+1 × H1 −−−−h → Qd n ++ x Bunx n x      

d+1 d (7.3) Q ←−−−− QHd+1 −−−−→ Q   +    νy y id ×νy

00 ← ×00 → 0 ←−−−h − 0 × H1 −supp−−−−−−h→ × 0 Bunn Bunn n X Bunn Bunn The bottom left and the top right squares in this diagram are Cartesian.

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0 W Q → 0 By deﬁnition, AutE = ν!( E), where ν : Bunn is the forgetful morphism Qd → 0 Qd → deﬁned in Sect. 4.3. Note that ν : Bunn extends to a morphism + 0 Bunn, which we also denote by ν. Using the diagram (7.3), we obtain that the 0 LHS of formula (7.2), restricted to the degree d connected component of Bunn,is × Wd isomorphic to the complex (id ν)!( E,+), where − Wd e→ e←∗ Wd+1 | ⊗ Q n 2 − (7.4) E,+ := h! h ( E ) Qd `( )[n 2]. + 2 Therefore Proposition 7.3 follows from Proposition 7.5, which is proved in the Appendix.

Wd × Qd ⊂ Qd 7.5. Proposition. The complex E,+ is supported on X +,andits × Qd Wd restriction to X is isomorphic to E E.

7.6. The Hecke property on Bunn. Observe that in the diagram (1.1) deﬁning 1 the Hecke functor Hn we have: → −1 ← −1 (supp ×h ) (X × (Cn ∩ Bunn)) ⊂ (h ) (Cn ∩ Bunn).

Therefore we can define a functor D(Cn ∩Bunn) → D(X × (Cn ∩Bunn)) by formula 1 1 i (1.2). We denote this functor also by Hn and consider its iterations (Hn) and i the corresponding functors Hn. The notion of Hecke eigensheaf also makes sense in this context. We now derive from Proposition 7.3 the following | 7.7. Proposition. The perverse sheaf AutE Cn∩Bunn is a Hecke eigensheaf with respect to E. E0 ∩ −1 d Proof. Recall from Sect. 3.2 and Lemma 4.4 that over n (%n) (Bunn)wehave an isomorphism −d + c F ' Aut0 ⊗Q ( )[−d + c], E,n E ` 2 where c is a constant depending only on g and n. 1 C ∩ The isomorphism of Hn(AutE)andE AutE over n Bunn now follows 0 from Proposition 7.3 via diagram (7.1), using the fact that the morphism ρn : 0 ∩E0 → ∩C (Bunn n) (Bunn n) is smooth, representable and has connected fibers. Moreover, it follows from the construction of the isomorphism of Proposition 7.5 that this isomorphism satisfies condition (1.4).

Now we derive Theorem 7.1 from the above proposition.

7.8. ProofofTheorem7.1.Recall the morphism mult : X × Bunn → Bunn given by (x, M) 7→ M(x). In the same way as in the proof of Proposition 1.5, we obtain from Proposition 7.7 that there is an isomorphism ∗ | ' n | (7.5) mult (AutE) X×(Cn∩Bunn) Λ E AutE X×(Cn∩Bunn).

Since the morphism mult : X × Bunn → Bunn is smooth, the isomorphism of d ≥ formula (7.5) holds over the entire component Bunn for d cg,n (and not only over C ∩ d n Bunn). Now we extend AutE to all other connected components of Bunn as follows: for ⊂ d0 00 every open substack U Bunn of ﬁnite type, there exists an integer d such that

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00 for any x ∈ X, the morphism multd00·x :Bunn → Bunn sending M to M(d · x) maps U into Cn ∩ Bunn.WesetAutE |U to be ∗ ⊗ n ⊗−d00 multd00·x(AutE) (Λ Ex) .

According to formula (7.5), this gives a well-deﬁned sheaf AutE ontheentireBunn, ∗ n together with an isomorphism mult (AutE) ' Λ E AutE. Proposition 7.7 then implies that AutE is a Hecke eigensheaf. Indeed, the exis- tence and uniqueness of the isomorphism (1.3) satisfying (1.4) over the entire Bunn follow from the construction, using the fact that formula (1.6) holds over Cn ∩Bunn. This completes the proof of Theorem 7.1.

0 0 7.9. Lifting of AutE to Bunn. We have the sheaves AutE on Bunn and AutE on 0 Bunn. Consider the commutative diagram: Bun0 ∩E0 −−−−→ Bun0 n n  n   0 % ρny ny

Bunn ∩Cn −−−−→ Bunn ≥ ∗ 0 ⊗ −d+c − By construction, for d cn,g,thesheaves%n (AutE)andAutE ( 2 )[ d + c] E0 ∩ −1 d are isomorphic over n (%n) (Bunn), where c is a constant independent of d.In this subsection we will address the following question, posed by V. Drinfeld: ∗ 0 ⊗ −d+c − Are the sheaves %n (AutE) and AutE ( )[ d + c] isomorphic on the entire 0 2 Bunn? The answer is aﬃrmative. Indeed, consider the diagram

00 ← ×00 → 0 ←−−−h − 0 × Hi −supp−−−−−−h→ × 0 Bunn Bunn n X Bunn  Bun n     y y y

← × → ←−−−h − Hi −−−−−−→supp h Bunn n Bunn, deﬁned in the same way as diagram (7.1). From Proposition 7.3 we derive, in the same way as in Proposition 1.5, that i(n − i − 1) (supp ×00h→) 00h←∗(Aut0 ) ⊗ Q ( )[i(n − i − 1)] ' ΛiE Aut0 . ! E ` 2 E

In addition, from the Hecke property of AutE it follows that i(n − i) (supp ×00h→) 00h←∗(% ∗(Aut )) ⊗ Q ( )[i(n − i)] ' ΛiE ρ0 ∗(Aut ). ! n E ` 2 n E K 7→ ×00 → 00 ←∗ 0 ∗ K As before, for i = n, the functor (supp h )! h (ρn ( )) amounts to the pull-back under the map 0 × 0 → 0 mult : X Bunn Bunn given by (x, M,s:Ωn−1 → M) 7→ (M(x),s0 :Ωn−1 → M → M(x)).

0 −1 d E0 ∩ Any open substack U of ﬁnite type in (%n) (Bunn )canbemappedinto n −1 d ≥ 0 0 → 0 %n (Bunn)withd cg,n by means of multd00·x :Bunn Bunn. Hence, over

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−1 (%n) (U)wehave: 0 0 −d + c Aut0 ⊗(ΛnE )⊗d ⊗ Q ( )[−d0 + c] E x ` 2 0 00 0 ∗ 0 −d − n · d + c 0 00 ' mult 00· (Aut ) ⊗ Q ( )[−d − n · d + c] d x E ` 2 ' 0 ∗ 0 ∗ ' ∗ ⊗ n ⊗d00 multd00·x (ρn (AutE)) %n (AutE) (Λ Ex) . The fact that the constructed isomorphism does not depend on the choice of x 0 and d follows in the same way as the corresponding assertion for AutE in the proof of Theorem 7.1.

8. The Hecke-Laumon property of SE In this section we give an alternative proof of Proposition 7.7, and hence of Theorem 7.1. Consider the diagram

← → h h C ←−−l HLd −−l→ C d × C (8.1) ohn n oh0 ohn, HLd → M0 → M → T → where the stack n classiﬁes short exact sequences 0 0with M0 ∈ C T ∈ C d ← → M ohn, oh0. The projections hl and hl send such data to and (M0, T), respectively. (Recall that in Sect. 5.4 we encountered this stack for n =0 d0,d00 andcalleditFld .) d C → C d × C The Hecke-Laumon functor HLn :D( ohn) D( oh0 ohn) is deﬁned by the formula ( · HLd (K)=h→ h←∗(K) ⊗ Q ( d (n+1) )[d · (n +1)],n≥ 1, (8.2) n l ! l ` 2 d K → ←∗ K HL0( )=hl !hl ( ) (see [Lau2]). Note that for d = d1 + d2 there is a natural isomorphism of functors

× d2 ◦ d1 ' d1 × ◦ d (id HLn ) HLn (HL0 id) HLn . Finally, let us note that (5.8) stated in Sect. 5.4 reads as

d1 Ld ' Ld1 Ld2 HL0 ( E ) E E .

8.1. Deﬁnition. We say that a complex K ∈ D(Cohn)hasaHecke-Laumon property (or is a Hecke-Laumon eigensheaf) with respect to E if for each d we are given an isomorphism d K ' Ld K (8.3) HLn( ) E

such that for d = d1 + d2 the diagram ∼ (id × HLd2 ) ◦ HLd1 (K) −−−−→ Ld1 Ld2 K n  n E E   (8.4) ∼y ∼y ∼ d1 × ◦ d K −−−−→ d1 Ld K (HL0 id) HLn( ) HL0 ( E) is commutative.

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8.2. Restriction to Cn. Note that the deﬁnition of the Hecke-Laumon property C C HLd 0 makes sense not only on ohn, but also on n. Consider the stack ( n) := HLd × C d × E0 n oh0 n and note that it ﬁts into the diagram: C d×C oh0 ohn

0h← 0h→ E0 ←−−−l − (HLd )0 −−−−l → Cohd × E0 n n 0 n    %n y y id ×%ny

h← h→ C ←−−−l − HLd −−−−l → C d × C ohn n oh0 ohn It is shown in [Lau2] (by induction on k)that d · (n +1) 0h→ 0h←(F ) ⊗ Q ( )[d · (n +1)]' Ld F . l ! l E,n ` 2 E E,n 0 E0 → C Therefore, since ρn : n n is smooth, representable and with connected ﬁbers, we obtain: S0 C 8.3. Corollary. The perverse sheaf E on n is a Hecke-Laumon eigensheaf with respect to E. We will now prove the following result:

8.4. Proposition. Let S be a perverse sheaf on Cohn,andletD(S) be its Verdier dual sheaf. Suppose that S and D(S) satisfy the Hecke-Laumon property with respect ∗ K S| to local systems E and E , respectively. Then := Bunn is a Hecke eigensheaf with respect to E. Proof. We start with the following general observation. Let ρ : E → Y be a vector bundle, let i : Y → E be the zero section, and let j : E0 → E be its complement. Let us denote by ρe : PE → Y the corresponding projectivized bundle. 0 Suppose that F is a Gm–equivariant perverse sheaf on E,andsetF := F|E0 .We will denote by Fe the perverse sheaf on PE corresponding to F0, i.e., the pull-back Fe E0 F0 ⊗ Q −1 − of to is `( 2 )[ 1]. We have the following assertion (see [Ga]).

8.5. Lemma. Assume that ρ∗(F)[−1] and ρ!(F)[1] are perverse sheaves. Then e Fe F ⊗Q 1 ' e Fe ' F ⊗Q −1 − ρ!( ) is a perverse sheaf as well, and ρ!( ) `( 2 )[1] ρ!( ) ρ∗( ) `( 2 )[ 1]. ! ∗ ! Proof. Since F is Gm–equivariant, ρ!(F) ' i (F)andρ∗(F) ' i (F). By applying i to the triangle 0 ∗ j!F → F → i∗i (F), 0 ! 0 we obtain that ρ!j!(F ) ' i j!(F ) has perverse cohomology only in cohomological degrees 0 and 1. Using the Leray spectral sequence of the composition E0 → PE → Y ,weobtain e e 0 0 that ρe!(F) must be perverse. In addition, we obtain that ρe!(F) ' h (ρ!j!(F )) ⊗ Q −1 e Fe ∗ F ⊗ Q −1 − `( 2 ), which identiﬁes ρ!( )withi ( ) `( 2 )[ 1]. Similarly, we obtain: e Fe ' ! F ⊗ Q 1 ρ!( ) i ( ) `( 2 )[1]. We will reduce the assertion of Proposition 8.4 to the above lemma. Set Y = C 1 × ⊂ C 1 × C E oh0 Bunn oh0 ohn and take to be → −1 C 1 × ⊂ HL1 (hl ) ( oh0 Bunn) n.

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→ −1 C d × ⊂ HLd (Note that in general the preimage (hl ) ( oh0 Bunn) n is the same as h← Cg Cg → HLd →l C the stack ohn,≤d introduced in Sect. 5.2 and the map ohn,≤d , n ohn g becomes the map r : Cohn,≤d → Cohn,≤d ⊂ Cohn.) F ←∗ S ⊗ Q n F G Set = hl ( ) `( 2 )[n]. Then is m–equivariant and perverse, according E0 ← ⊂ C to Lemma 5.3. In addition, the image of under hl lies in Bunn ohn. By the assumption of Proposition 8.4, S is a Hecke-Laumon eigensheaf. This F ⊗Q 1 ' L1 K F implies that ρ!( ) `( 2 )[1] E ,andsoρ!( )[1] is a perverse sheaf. Applying Verdier duality and using the assumptions of Proposition 8.4 regarding D(S), we obtain that ρ∗(F)[−1] is a perverse sheaf too. Hence, we can apply Lemma 8.5. → C 1 × PE Let us perform a base change with respect to X oh0.ThenX C 1 oh0 H1 identiﬁes naturally with the Hecke correspondence n in such a way that

ρe : X × PE → X × Bunn C 1 oh0 → 1 K ' K becomes the projection h . Therefore, Lemma 8.5 implies that Hn( ) E . The fact that this isomorphism indeed satisfies condition (1.4) follows from property (8.4) in the case d = 2. This completes the proof of Proposition 8.4. 8.6. Remark. V. Drinfeld has asked the following question about the possibility of proving a theorem converse to Proposition 8.4: Let K be a perverse sheaf on Bunn, which is a Hecke eigensheaf with respect to E. Is it true that the Goresky-MacPherson extension of K to Cohn has the Hecke-Laumon property with respect to E? We conjecture that the answer to this question is affirmative. 8.7. Second proof of Proposition 7.7. It is clear from the above proof that Proposition 8.4 is still valid if we replace the stacks Cohn and Bunn by their substacks Cn and Cn ∩ Bunn, respectively. Now we apply this modification of Propo- S S0 K | sition 8.4 in the situation when = E ,and =AutE Cn∩Bunn . It follows from the definitions that all conditions of Proposition 8.4 are satisfied (in particular, we D S0 ' S0 have: ( E) E∗ ). The statement of Proposition 7.7 (and hence Theorem 7.1) now follows directly from Proposition 8.4. S C 0 8.8. Lifting of E to ohn. We have the diagram: 0 E0 −−−−→ Coh n  n   0 % ρn y ny

Cn −−−−→ Cohn S d Recall the deﬁnition of the perverse sheaf SE on ≥ Coh given in the begin- d cn,g n S ning of Sect. 7. By Theorem 3.9 and Lemma 3.5, over (% )−1(C ∩( Cohd )) n n d≥cn,g n ∗ S S0 ⊗ Q −d+c − the complexes %n( E )and E `( 2 )[ d + c] are isomorphic, where c is a constant independent of d. Consider the Goresky-MacPherson extension of AutE (which by now is deﬁned on the whole of Bunn)toCohn. By abuse of notation we still denote this extension by SE. Now we prove the following assertion: ∗ The sheaf SE has the Hecke-Laumon property with respect to E,and%n (SE) ' S0 ⊗ Q −d+c − E `( 2 )[ d + c].

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0 Let us denote by (X × Cohn) the open substack of X × Cohn corresponding to those pairs (x, M)forwhichM has no torsion supported at x. In a similar way we i 0 i deﬁne the substack (X × Cohn) of X × Cohn. We deﬁne the functors i C → × C 0 (8.5) Hn :D( ohn) D((X ohn) )

in the same way as before. Since we already know AutE is a Hecke eigensheaf, we obtain that n n S ' S | i 0 (8.6) Hn( E) Λ (E) E (X ×Cohn) .

By arguing as in Sect. 7.8, we deduce the Hecke-Laumon property of SE on the C S | entire ohn from (8.6) and the fact that E Cn is a Hecke-Laumon sheaf. ∗ S ' S0 ⊗Q −d+c − Similarly, the isomorphism %n ( E ) E `( 2 )[ d+c] follows in the same way as in Sect. 7.9.

9. Cuspidality

9.1. Constant term functors. Let P ⊂ GLn be the standard (upper) parabolic subgroup corresponding to a partition (n1,... ,nk)ofn, with the Levi quotient ' × × → M GLn1 ... GLnk . The embedding of P in GLn and the projection P M induce morphisms p and q in the diagram p q (9.1) Bunn ←−−− BunP −−−→ BunM . G → The constant term functor RM :D(Bunn) D(BunM ) is deﬁned by the formula G K ∗ K K ∈ G F RM ( )=q!p ( ). We say that D(Bunn)iscuspidal if RM ( ) = 0 for all proper parabolic subgroups P of G. For a partition n = n1+n2 let P (n1,n2) be the corresponding parabolic subgroup × in GLn with the Levi factor GLn1 GLn2 . In this case diagram (9.1) is ←p −→q × Bunn BunP (n1,n2) Bunn1 Bunn2 . → × We denote the corresponding constant term functor D(Bunn) D(Bunn1 Bunn2 ) n K by Rn1,n2 ( ). K n K It is easy to see that a complex is cuspidal if and only if Rn1,n2 ( ) = 0 for all partitions n = n1 + n2,withn1,n2 > 0. In this section we prove the following

9.2. Theorem. Let AutE be a Hecke eigensheaf on Bunn with respect to an irreducible rank n local system E, which satisﬁes Conjecture 2.3. Then AutE is cuspidal. As a corollary we obtain the following statement:

9.3. Corollary. The perverse sheaf AutE is the extension by zero from an open substack of finite type on every connected component of Bunn. Proof. The proof of the corollary will rely on the following well-known assertion: 9.4. Lemma. For a fixed line bundle L and an integer d, consider the open sub- d M M L stack U of Bunn which classifies vector bundles with Hom( , )=0.ThenU is of finite type. d To prove the corollary, let us consider a connected component Bunn. Without est loss of generality we may assume that d ≥ cg,n.TakeL = L and let U be as in d − vuns the lemma. By definition, Bunn U is contained in Bunn .

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However, by arguing as in Lemma 6.11 we obtain that if a complex K is cuspidal, then it has zero stalks at all very unstable bundles. Therefore, K| d is extended by zero from U. This completes the proof of Bunn Corollary 9.3.

The proof of Theorem 9.2 as well as other results of this section relies on the following computation: 9.5. Proposition. Let K be a Hecke eigensheaf with respect to some rank n local system E0,andletE be another local system, of an arbitrary rank. Then d Hd (K) ' K ⊗ H•(X(d), (E ⊗ E0∗)(d)) ⊗ Q ( )[d]. n,E ` 2 Proof. Consider the diagram

← → ←−−h d −−−−−−→supp ×h (d) × Bunn Modn X Bunn . We need to prove that 1 (9.2) (supp ×h→) (h←∗(K) ⊗ π∗(Ld )) ⊗ (Q ( )[1])⊗d·(n−1) ' (E ⊗ E0∗)(d) K. ! E ` 2 ]d M M ⊂ M ⊂ ⊂ Consider the stack Modn, which classiﬁes the data ( = 0 1 ... 0 Mi = M ), where each Mj is a rank n vector bundle, and Mj/Mj−1 is a simple ]d skyscraper sheaf. (Note that there is a canonical isomorphism between Modn and ]−d the stack Modn , introduced in the proof of Theorem 1.5, under which the roles of the projections h← and h→ get reversed.) ]d → d Let p : Modn Modn be the forgetful map. We also have a natural morphism ] ]d → d supp : Modn X . Consider the corresponding diagram:

g← supp]×hg→ Bun ←−−−h − Mod]d −−−−−−→ Xd × Bun  n  n  n    idy py sym × idy

← × → ←−−−h − d −−−−−−→supp h (d) × Bunn Modn X Bunn ∗ d Since p is small, the complex p! supp] (E ) is a perverse sheaf (up to a cohomological shift), which is the Goresky-MacPherson extension of its own restriction to supp−1(X(d) − ∆). In particular, it carries a canonical action of the symmetric group Sd and ∗ ∗ ] d Sd ' Ld (p! supp (E )) π ( E ). 1 H1 As was noted before, the stack Modn is isomorphic to the stack n, but under this isomorphism the maps h← and h→ become interchanged. By iterating the deﬁnition of the Hecke property, we obtain that the fact that K is a Hecke eigensheaf with respect to E0 then implies that n − 1 (supp] × hf→) hf←∗(K) ⊗ Q ( )[n − 1] ' (E0∗)d K. ! ` 2 Hence we obtain: ∗ 1 (supp] × hf→) (hf←∗(K) ⊗ supp] (Ed)) ⊗ (Q ( )[1])⊗d·(n−1) ' (E ⊗ E0∗)d K. ! ` 2

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By taking the direct image of the last isomorphism under sym : Xd → X(d) we obtain ∗ 1 (supp ×h→) (h←∗(K) ⊗ p supp] (Ed)) ⊗ (Q ( )[1])⊗d·(n−1) ! ! ` 2 ' ⊗ 0∗ d K sym!((E E ) ) .

Moreover, this isomorphism is compatible with the Sd–action on both sides. By passing to the Sd–invariants we obtain formula (9.2) and hence the statement of the proposition. 9.6. Remark. Let us see what the isomorphism of formula (9.2) looks like at the level of fibers over a given point D ∈ X(d), in terms of the general Hecke functors λ · ∈ Hn introduced in Sect. A.1. To simplify notation we take D = d x,forsomex X. By formula (A.4) below, the stalk of the LHS of formula (9.2) at D can be identified with M − w0(λ) K ⊗ λ x Hn ( ) Ex . ∈ + λ Pn,d Since K is a Hecke eigensheaf with respect to E0, this is isomorphic to M K ⊗ 0∗ λ ⊗ λ ' K ⊗ d ⊗ 0∗ (E )x Ex Sym (Ex E x), ∈ + λ Pn,d which is the stalk of the RHS of formula (9.2) at d × x. −d 9.7. Remark. Recall the stack Modk introduced in the proof of Theorem 1.5. −d Denote by Hk,E the functor d · n K 7→ h→(h←∗(K) ⊗ π∗(Ld )) ⊗ Q ( )[d · n] ! E ` 2 ← → −d (where h and h are taken according to the definition of Modk ). It follows from −d d the definition that the functor Hk,E is both left and right adjoint to Hk,E∗ . In the same way as in the proof of Proposition 9.5 we obtain: d (9.3) H−d (K) ' K ⊗ H•(X(d), (E ⊗ E0)(d)) ⊗ Q ( )[d]. n,E ` 2 Now we are ready to prove Theorem 9.2.

9.8. Lemma. For each d, n = n1 + n2,alocalsystemE and K ∈ D(Bunn), n ◦ d K ∈ × the object Rn1,n2 Hn,E( ) D(Bunn1 Bunn2 ) has a canonical ﬁltration by the − · objects (Hd1 × Hd2 )◦ Rn (K)⊗ Q ( n1 d2 )[−n · d ] for all possible partitions n1,E n2,E n1,n2 ` 2 1 2 d = d1 + d2 with d1,d2 ≥ 0. 9.9. ProofofTheorem9.2. Theorem 9.2 follows from Lemma 9.8. Indeed, take d>2n2(2g − 2). On the one hand, according to Proposition 9.5, d ' ⊗ • (d) ⊗ ∗ (d) Hn,E(AutE) AutE H (X , (E E ) ), hence

n d n • (d) ∗ (d) d R ◦ H (AutE) ' R (AutE) ⊗ H (X , (E ⊗ E ) ) ⊗ Q ( )[d]. n1,n2 n,E n1,n2 ` 2 On the other hand, Conjecture 2.3 implies that all (Hd1 × Hd2 )◦Rn (Aut ) n1,E n2,E n1,n2 E 2 must vanish, because either d1 or d2 must be greater than n (2g − 2).

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However, since E ⊗ E∗ contains the trivial rank one local system, H•(X(d), (E ⊗ E∗)(d)) =06 foranyd. n Hence, Rn1,n2 (AutE) = 0. This completes the proof of Theorem 9.2. d × 9.10. Proof of Lemma 9.8. Let Modn BunP (n1,n2) be the Cartesian product Bunn → d → defined using the projection h :Modn Bunn. Our task is to calculate the direct image under → d × h−→×id →q × Modn BunP (n1,n2) BunP (n1,n2) Bunn1 Bunn2 Bunn d × id−→×p ←∗ K ⊗ of the pull-back under Modn BunP (n1,n2) Modn of the complex h ( ) Bunn ∗ Ld π ( E ). By definition, the above Cartesian product classifies the data of 0 0 M ∈ Bunn, M ∈ Bunn,β: M ,→ M, (9.4) M ∈ M ∈ → M → M → M → 1 Bunn1 , 2 Bunn2 , 0 1 2 0. d × First, we decompose Modn BunP (n1,n2) into locally closed substacks, which Bunn we will denote by

d × d1,d2 (Modn BunP (n1,n2)) . Bunn d × By definition, a point of Modn BunP (n1,n2) as in formula (9.4) belongs to Bunn 0 d × d1,d2 M ∩ M M − (Modn BunP (n1,n2)) if we have deg( 1)=deg( 1) d1.Fromeach Bunn d × d1,d2 d1 × d2 (Modn BunP (n1,n2)) there is a natural map to Modn1 Modn2 ,which Bunn M0 M0 ∩ M → M M0 M0 M0 → M sends a point as above to ( 1 := 1 , 1, 2 := / 1 , 2). To prove the proposition it suffices to show that the direct image under this map d × d1,d2 of the complex that we obtain on (Modn BunP (n1,n2)) by restriction from Bunn d × ← × ← ∗ n K ⊗ Modn BunP (n1,n2) can be canonically identified with (h h ) (Rn1,n2 ( )) Bunn ∗ − · × Ld1 Ld2 ⊗ Q n1 d2 − · (π π) ( E E ) `( 2 )[ n1 d2] in the diagram h←×h← π×π × ←− d1 × d2 −→ C d1 × C d2 Bunn1 Bunn2 Modn1 Modn2 oh0 oh0 . For that purpose, we decompose the map

d × d1,d2 → d1 × d2 (Modn BunP (n1,n2)) Modn1 Modn2 Bunn

as a composition of several ones. First, we introduce the stack Y1, which classiﬁes the data of 0 → M → M → T → M ∈ T ∈ C d1 0 1 1 1 0, 1 Bunn1 , 1 oh0 , 0 → M → M → T → M ∈ T ∈ C d2 0 2 2 2 0, 2 Bunn2 , 1 oh0 , → T → T → T → → M0 → M0 → M0 → 0 1 2 0, 0 1 2 0.

d × d1,d2 → Y It is easy to see that the natural map (Modn BunP (n1,n2)) 1 is a Bunn fibration into affine spaces, with each fiber being a principal homogeneous space 1 T2 M0 for Ext ( , 1). Therefore, by the projection formula, the direct image of our

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Y Y → ×C d complex to 1 is the pull-back under the map 1 Bunn oh0 (which sends a 0 − · M T K Ld ⊗ Q n1 d2 − · point as above to ( , )) of E `( 2 )[ n1 d2]. Now, let Y2 be the stack classifying the data of 0 → M → M → T → M ∈ T ∈ C d1 0 1 1 1 0, 1 Bunn1 , 1 oh0 , 0 → M → M → T → M ∈ T ∈ C d2 0 2 2 2 0, 2 Bunn2 , 2 oh0 , → M0 → M0 → M0 → 0 1 2 0.

The projection Y1 → Y2 corresponds to forgetting the class of the extension 0 → T1 → T → T2 → 0. Moreover, we have a Cartesian square: Y −−−−→ HLd1 1 0   y y

Y −−−−→ C d1 × C d2 2 oh0 oh0

d1 Ld ' Ld1 Ld2 Using the projection formula and the fact that HL0 ( E ) E E ,weobtain Y → Y K Ld that direct image under 1 2 of the pull-back of E is the tensor product of the pull-back of K under the map Y2 → Bunn, which sends a point as above to M0 Ld1 Ld2 Y C d1 ×C d2 and the pull-back of E E under the natural map from 2 to oh0 oh0 . Finally,notethatwehaveaCartesiansquare: Bun ←−−−− Y P(n1,n2) 2   qy y

h←×h← × ←−−−−− d1 × d2 Bunn1 Bunn2 Modn1 Modn2 Y → M0 → M0 → where the upper horizontal arrow sends a point of 2 as above to 0 1 M0 → 2 0. The assertion follows now by the projection formula.

10. Proof of the Vanishing Conjecture over Fq In this section we prove Conjecture 2.3 in the case when the ground field k is F d → a finite field q, i.e., that the functor Hk,E :D(Bunk) D(Bunk) introduced in Sect. 2.2 is identically zero if E is an irreducible local system of rank n,andk and d satisfy the inequalities kkn(2g − 2). Namely, we will prove the following proposition:

10.1. Proposition. Let E be a rank n local system on X over the finite field Fq, which is (a) pure up to a twist by a one-dimensional representation of the Weil group of Fq, and satisfies one of the following conditions: (b) there exists a cuspidal Hecke eigenfunction associated to the pull-back of E to × F F F X q1 for any finite extension q1 of q;or Fq 0 (b ) the space of unramified cuspidal automorphic functions on the group GLk over the adèles is spanned by the Hecke eigenfunctions corresponding to rank × F k local systems on X q1 , for all k

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According to [Laf], Theorem VII.6, any irreducible local system E, such that det E is of finite order, is pure. Therefore condition (a) of Proposition 10.1 is satisfied for any irreducible rank n local system E. Moreover, both statements (b) and (b0) hold for such E, according to the main theorem of Lafforgue’s work. Hence we obtain that the Vanishing Conjecture 2.3 is true for all irreducible local systems if the ground field k is finite. Our proof of Proposition 10.1 proceeds as follows. We first show vanishing of d Hk,E at the level of functions. Using the purity property conjectured by Deligne d K and proved by Lafforgue [Laf], we will then deduce that Hk,E( ) = 0 for any K ∈ D(Bunk),k =1,... ,n− 1,d>kn(2g − 2).

(1) (1) (2) (2) 10.2. L–functions. Let Γ =(γx )x∈|X| and Γ =(γx )x∈|X| be two collec- Q Q tions of semi-simple conjugacy classes in GLk( `)andGLn( `), respectively. We attach to it the L–function Y (1) (2) − (1) ⊗ (2) deg x −1 L(Γ , Γ ,t)= det(Idkn (γx γx ) t ) , x∈|X| viewed as a formal power series in t. N 0 A A To an unramified irreducible representation π = x∈X πx of GLk( ), where is the ring of adèles of F = Fq(X), we attach the collection Γπ =(sx)x∈|X|,where sx is the Satake parameter of πx. 0 If π and π are unramified irreducible representations π of GLk(A)andGLn(A), respectively, we write 0 L(π × π ,t):=L(Γπ, Γπ0 ,t). If π and π0 are in addition cuspidal automorphic representations, then L(π × π0,t) is the Rankin–Selberg L–function of the pair π, π0. The following statement follows from results of [PS2, CPS] (see [Laf], Appendice B, for a review).7 10.3. Theorem. If π, π0 are cuspidal automorphic representations and kdim H1(X(d),E0 ⊗ E)=kn(2g − 2).

7We are grateful to V. Drinfeld for pointing this out to us.

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If E is a rank n local system on X,andπ is an irreducible unramiﬁed representation of GLk(A), we will write

L(π × E,t):=L(Γπ, ΓE,t).

d d 10.5. Computation of Hk,E at the level of functions. The functor Hk,E gives rise to a linear map on the space of functions on the set Bunk(Fq)ofFq–points d of Bunk.WedenotethisoperatorbyHk,E . In this subsection we will prove that d ≡ − − Hk,E 0 for all k =1,... ,n 1andd>kn(2g 2). Recall that Bunk(Fq) is naturally identiﬁed with the double quotient

GLk(F )\GLn(A)/GLn(O) (see, e.g., [FGKV], Sect. 2). Let π be a cuspidal unramified automorphic representation of GLk(A). Attached to it is a cuspidal automorphic function on Bunk(Fq), unique up to a non-zero scalar multiple. We normalize it in some way and denote the result by fπ. −d In Remark 9.7 we defined the functor Hk,E, which is left and right adjoint to d −d Hk,E∗ .DenotebyHk,E the corresponding linear map on the space of functions on Bunk(Fq). We have the following analogue of formula (9.3), which is proved using a calculation similar to the one presented in the proof of Proposition 9.5: X −d · d × · (10.1) Hk,E(fπ) t = L(π E,t) fπ. d≥0 It is clear from the definition that h d 0i h −d 0 i (10.2) Hk,E(f),f = f,Hk,E (f ) ,

where the inner product of two automorphic functions f1,f2 on GLk(A) is deﬁned by the formula Z

hf1,f2i = f1(g)f2(g)dg. GLk(F )\GLk(A) Formula (10.1) then implies: h d i × h i (10.3) Hk,E(f),fπ = L(π E,t) f,fπ . 10.6. Lemma. Let E be a rank n local system such that L(π×E,t) is a polynomial of degree kn(2g − 2) for all irreducible cuspidal automorphic representations π of A − d F GLk( ),k =1,... ,n 1.ThenHk,E(f)=0for any function f on Bunk( q). Proof. By induction, we may assume that the assertion is known for k0 kn(2g 2). Therefore Hk,E(f)= 0. d − − Thus, in order to prove vanishing of Hk,E for k =1,... ,n 1andd>kn(2g 2), we need to show that L(π × E,t) is a polynomial of degree kn(2g − 2) for all irreducible cuspidal automorphic representations π of GLk(A)fork =1,... ,n− 1. Thiscanbedoneintwoways:byidentifyingL(π × E,t)withL(π × π0,t)orwith 0 × d ≡ − L(E E,t). As the result, we obtain that Hk,E 0,k =1,... ,n 1, if either of the statements (b) or (b0) listed in Proposition 10.1 is true.

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Indeed, if the statement (b) is true, then there exists an unramiﬁed cuspidal 0 0 automorphic representation π of GLn(A), such that L(π × E,t)=L(π × π ,t). d − Vanishing of Hk,E for kkn(2g 2) then follows from Theorem 10.3. If the statement (b0) is true, then vanishing follows from Lemma 10.4.

10.7. Conclusion of the proof of Proposition 10.1. To complete the proof d of Proposition 10.1 we need to show that vanishing of the operator Hk,E at the d level of functions implies the vanishing of the operator Hk,E at the level of sheaves, provided that E is pure. In order to do that, we proceed as follows: for each x :SpecFq → Bunk,denotebyδx the direct image with compact support of the F d constant sheaf on Spec q. Proving that the functor Hk,E vanishes is equivalent to d showing that Hk,E (δx)=0,forallx. Since Bunk is a stack (and not a scheme), δx is not necessarily an irreducible perverse sheaf, but it is a mixed complex. Therefore, it suﬃces to show that d K K d K Hk,E( ) = 0, for any mixed complex . Decomposing Hk,E ( ) in the derived d K K category, we obtain that it is enough to show that Hk,E( )=0,when is a pure perverse sheaf. Now let E be a pure irreducible rank n local system on X. Then Laumon’s sheaf LE is also pure. The pull-back with respect to a smooth morphism preserves purity, and so does the push-forward with respect to a proper representable morphism (see ← × d → ×C d [BBD]). But the morphism (h π):Modn Bunn oh0 is smooth, and the → d → d K morphism h :Modn Bunn is proper and representable. Hence Hk,E ( ) is pure, if K is pure. d K d K d K The function fq1 (Hk,E ( )) associated to the sheaf Hk,E ( )equalsHk,E(fq1 ( )) r for any for q1 = q ,r ∈ Z>0 (here we use the notation introduced in Sect. 0.3). But d K − according to the computation of Sect. 10.5, Hk,E(fq1 ( )) = 0 for all k =1,... ,n 1 and d>kn(2g − 2) if either of the conditions (b) or (b0) of Proposition 10.1 holds for E. In addition, we have: 10.8. Lemma. A pure complex F vanishes if and only if the corresponding func- F r ∈ Z tion fq1 ( ) is zero, for all q1 = q ,r >0.

Proof. Since F is non-zero, there exists a locally closed subset U such that F|U is locally constant and non-zero. Since F is pure, F|U is pointwise pure. But for a F| pointwise pure non-zero locally constant complex, all functions fq1 ( U ) cannot be r identically equal to zero for all q1 = q ,r ∈ Z>0, by the condition on the absolute values of the Frobenius eigenvalues. Therefore the statement of Conjecture 2.3 holds for E. This completes the proof of Proposition 10.1. 10.9. Remark. Formula (10.2) has a geometric counterpart: h d K K0i'hK −d K0 i K K0 ∈ Hk,E ( ), , Hk,E∗ ( ) , , D(Bunk), where hK, K0i := RHom(K, K0). Note that a priori RHom(K, K0)makessenseifK 0 is the !–extension from a substack of Bunk of ﬁnite type, or K is the ∗–extension from a substack of Bunk, whose intersection with every connected component is of ﬁnite type. Let E and E0 be two irreducible local systems on X,ofranksn and k, respectively, where k

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In particular, AutE0 exists and is cuspidal, according to Theorem 9.2. Therefore, for every d,AutE0 | d is extended by zero from an open substack of ﬁnite type of Bunk d Bunn. From formula (9.3) we obtain the following analogue of formula (10.3):

d d (d) ∗ 0 (d) d (10.4) hH (K), Aut 0 i'H (X , (E ⊗ E ) ) ⊗hK, Aut 0 i⊗Q ( )[d]. k,E E E ` 2 Since Hd(X(d), (E∗ ⊗ E0)(d))=0ford>kn(2g − 2) (see the proof of Lemma 10.4), we ﬁnd that d h K 0 i Hk,E( ), AutE =0,

for all K ∈ D(Bunk), if d>kn(2g − 2). Thus we obtain a geometric analogue of Proposition 10.1: 10.10. Proposition. Suppose that for k =1,... ,n − 1 the Vanishing Conjec- ture 2.3 is true for rank k local systems on X and in addition the following statement holds: 00 (b ) If F ∈ D(Bunk) is cuspidal and satisﬁes hF, AutE0 i =0for all irreducible rank k local system E0 on X,thenF =0. Then the Vanishing Conjecture 2.3 is true for any irreducible local system on X of rank n. The above statement (b00) is known to be true for k =1inthecasewhen char k = 0, by the Fourier-Mukai transform [Lau3, R].

Appendix A. Hecke functors and Whittaker sheaves A.1. General Hecke functors. We recall some results from Sect. 5 of [FGV]. Let xHn be the full Hecke correspondence stack at x ∈|X|.Inotherwords, 0 0 xHn classifies triples (M, M ,β), where M and M are rank n bundles on X and M| →∼ M0| Q β is an isomorphism X−x X−x. To a dominant weight λ of GLn( `) Hλ H we associate a closed finite-dimensional substack x n of x n, which classifies the 0 triples (M, M ,β), such that for an algebraic representation V of GLn(k), whose weights are ≤ νˇ, we have the following embeddings induced by β on the entire X:

VM0 (hw0(λ), νˇi·x) ⊂ VM ⊂ VM0 (hλ, νˇi·x),

where VM is the vector bundle on X associated with V and the principal GLn– bundle on X corresponding to M (recall that w0 stands for the permutation (d1,d2,... ,dn) 7→ (dn,... ,d2,d1)). λ → Using this stack, we deﬁne the Hecke functor x Hn :D(Bunn) D(Bunn)by the formula dim(Bun ) K 7→ h→(h←∗(K) ⊗ IC ) ⊗ Q ( n )[dim(Bun )], ! λ ` 2 n ← → 0 0 where h (resp., h ) sends (M, M ,β)toM (resp., M ), and ICλ is the intersection Hλ cohomology sheaf on x n. Hωi In particular, if λ is the i-th fundamental weight ωi, then the stack x n is ∈ Hi Hi → ωi nothing but the preimage of x X in n under supp : n X. Hence x Hn is i × ' ⊂ × the composition of Hn followed by the restriction to x Bunn Bunn X Bunn.

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The results of [Lu, Gi, MV] imply the following formula: M λ ◦ µ ' ν ⊗ ν λ ⊗ µ (A.1) x Hn x Hn x Hn HomGLn (V ,V V ), + ν∈Pn where the notation V λ is as in Sect. 4.8. λ,x Q0 × Hλ Consider the ﬁber product Z := x n. It was proved in [FGV], 0 Bunn Prop. 5.3.4, that there exists a commutative diagram 0 0 ← 0 → −λ,x Q ←−−−h − Zλ,x −−−−h → Q       qy 0qy qy

← → ←−−−h − Hλ −−−−h → Bunn x n Bunn −λ,x −λ,x where we write Q for Q when the collection λ (resp., x) consists of just one element λ (resp., x), in the notation of Sect. 4.8. According to Theorems 3 and 4 of [FGV], adapted to our present notation, we have: dim(Bun ) (A.2) 0h→(0h←∗(Ψ0) ⊗ 0q∗(IC )) ⊗ Q ( n )[dim(Bun )] ' Ψ−λ,x. ! λ ` 2 n More generally, let x1,... ,xm be a set of distinct points, diﬀerent from x,and let µ =(µ0,µ1,... ,µm) be a collection of dominant weights. Set − λ,µ,x Q µ,x × Hλ Z := x n, Bunn where x =(x, x1,... ,xm). Denote λ =(λ, 0,... ,0). We have a commutative diagram: −µ,x 0 ← 0 → −µ−λ,x Q ←−−−h − Zλ,µ,x −−−−h → Q       qy 0qy qy

← → ←−−−h − Hλ −−−−h → Bunn x n Bunn − − − 0 λ Q µ,x → Q µ λ,x Denote by xHn the functor D( ) D( ), dim(Bun ) K 7→ 0h→(0h←∗(K) ⊗ 0q∗(IC )) ⊗ Q ( n )[dim(Bun )]. ! λ ` 2 n Then Corollary 5.4.3 of [FGV] gives: M 0 λ −µ,x ' −(ν,µ1,... ,µm),x ⊗ ν λ ⊗ µ0 (A.3) xHn(Ψ ) Ψ HomGLn (V ,V V ). + ν∈Pn A.2. Proof of Proposition 4.12. To simplify the notation, we consider the case when m = 1, i.e., µ = µ and x = x.Form>1 the proof is essentially the same. µ,x d·x d By construction, Qµ,x and Q are substacks of Q ⊂ Q . · ∈ (d) d → (d) Observe that the preimage of d x X under supp : Modn X can be 0 identiﬁed with the closed substack of xHn, which classiﬁes those triples (M, M ,β), ∼ 0 0 for which β : M|X−x → M |X−x extends to an embedding M ,→ M over the entire X, and deg(M0) − deg(M)=d. d → C d ∗ Ld ⊗ Recall the morphism π :Modn oh0. Let us denote the pull-back π ( E ) Q d·n · Pd `( 2 )[d n]by E.

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It follows from the results of [Lau1], Sect. 3, and [FGKV], Sect. 4.2, that the ∗ Pd −1 · ⊂ d –restriction of E to supp (d x) Modn can be canonically identified with M λ d d (A.4) IC− ⊗ Hom (V , Sym (V ⊗ E )) ⊗ Q ( )[d]. w0(λ) GLn x ` 2 ∈ + λ Pn,d Here Xn + { | ≥ ≥ ≥ ≥ } (A.5) Pn,d = (d1,... ,dn) d1 d2 ... dn 0, di = d , i=1 (1,0,... ,0) and V = V stands for the defining representation of GLn.Further,we have: M d ⊗ ' λ ⊗ λ (A.6) Sym (V Ex) V Ex . ∈ + λ Pn,d Wd By comparing the definition of E with formulas (A.2), (A.4) and (A.6), we · Wd Qd x obtain that the restriction of E to can be identified with M d (A.7) Ψw0(λ),x ⊗ E ⊗ Q ( )[d], w0(λ),x ` 2 ∈ + λ Pn,d which is what we had to prove. A.3. Proof of Proposition 7.5: Local computation. Consider the morphism Qd → × (d+1) M τ : ++ X X , sending (x, , (si)) to (x, D), where D is the divisor of n(n−1)/2 zeroes of the map sn :Ω → (det M)(x). We start by describing explicitly Wd Qd ∩ −1 × the restriction of the complex E,+ to + τ (x D). Let us write D = d0 · x + d1 · x1 + ...+ dm · xm,wherexi are pairwise distinct and different from x,andd0 + d1 + ...+ dm = d +1. Qd ∩ −1 × It follows from the definitions given in Sect. 4.8 that the stack ++ τ (x D) 00 ν ,x is identified with Q ,wherex =(x, x1,... ,xm)andν00 =(ν000,ν1,... ,νm), with 000 − 0 j j Qd ∩ ν =( 1, 0,... ,0,d )andν =(0,... ,0,d ),j =1,... ,m.Furthermore, + 0 ν ,x ν,x τ −1(x×D) is identified with the substack Q of Q ,whereν0 =(ν00,ν1,... ,νm), with ν00 =(0, −1, 0,... ,0,d0)andνj,j =1,... ,m,asabove. d+1 ν,x We also have a morphism τ : Q → X(d+1),andweidentifyτ −1(D)withQ , where ν =(ν0,ν1,... ,νm), with ν0 =(0,... ,0,d0)andνj,j =1,... ,m,asabove. We have a commutative diagram: 00 ν,x 0h← 0h→ ν ,x Q ←−−−− Zω1,µ,x −−−−→ Q       y y y

e ← e → Qd+1 ←−−−h − Qd+1 × H1 −−−−h → Qd n ++ Bunn Moreover, it follows from the deﬁnitions that both squares of this diagram are Cartesian. Therefore we obtain the following formula for the restriction of the 0 Wd e→ e←∗ Wd+1 | ⊗ Q n−2 − Qd ∩ −1 × Qν ,x sheaf E,+ = h! h ( E ) Qd `( 2 )[n 2] to + τ (x D)= : + 0 ω d+1 Wd | 0 ' 1 W | | 0 (A.8) E,+ Qν ,x xHn E Qν,x Qν ,x .

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By Proposition 4.12, M Wd+1| ' µ,x ⊗ ⊗ ⊗ ⊗ (A.9) E Qν,x Ψ Eµ0,x Eµ1,x1 ... Eµm,xm , µ 0 m j ∈ + where the summation is over all µ =(µ ,... ,µ )withµ w0(Pn,dj ),j = 0,... ,m. Applying formula (A.3), we obtain M 0 0 0 ω1 d+1 µ ,x d ∗ W | ν,x ' ⊗ 00 ⊗ ⊗ (A.10) xHn E Q Ψ HomGLn (Vµ , Sym (V Ex) V ) µ0 d ⊗ E 1 1 ⊗ ...⊗ E m m ⊗ Q ( )[d], µ ,x µ ,x ` 2 0 00 1 m 00 + j where µ =(µ ,µ ,... ,µ )withµ running over the set w0(Pn ), and µ running + ∗ Q over the set w0(Pn,dj )forj =1,... ,m.HereV is the representation of GLn( `) dual to V . ν00,x We have a stratification of the stack Q analogous to the stratification described in Lemma 4.10. We list only the strata that can possibly support a sheaf 0 0 of the form Ψµ ,x.ThoseareQµ ,x,whereµ0 =(µ00,... ,µm)aresuchthat µ00 ≥ ν000,µj ≥ νj,j =1,... ,m. (We recall that the inequality λ ≥ λ0 means 0 that λ belongs to the set λ + R+,whereR+ is the set of all linear combinations of simple roots αi,i=1,... ,n− 1, of GLn with non-negative integer coefficients.) 0 0 ν ,x The stratum Qµ ,x belongs to the substack Q if and only if in addition µ00 ≥ 00 000 ν = ν + α1. 0 Recall that each sheaf Ψµ ,x is the extension by zero of its restriction to the 0 stratum Qµ ,x.Thestratumwithµ0 appearing in the summation of the RHS of 0 Qν ,x 0 00 ∈ + formula (A.10) belongs to if and only if d > 1, w0(µ ) Pn,d0−1,and j ∈ + × Qd ⊂ Qd w0(µ ) Pn,dj for all j =1,... ,m. All of these strata belong to X +. Wd × Qd ⊂ Qd Therefore E,+ is supported on X +. This proves the first assertion of Proposition 7.5. 0 0 ∈ + Furthermore, we have for d > 1andanyµ w0(Pn,d0−1): d0 ∗ 0 0 ⊗ ⊗ ' ⊗ 00 HomGLn (Vµ , Sym (V Ex) V ) Ex Eµ ,x. Combining this with formulas (A.8), (A.10) and (A.9), we obtain the desired iso- Wd | ' Wd × ⊂ × (d+1) morphism E,+ X×Qd E E over the preimage of each x D X X Qd in +. A.4. Proof of Proposition 7.5: Global computation. To complete the proof Wd | ' Wd of Proposition 7.5, we need to show that the isomorphism E,+ X×Qd E E holds globally, and not only on each fiber τ −1(x × D). Qd In order to show that, we introduce one more stack, +−. This is a closed Qd × (d) ⊂ × substack of ++ which is the preimage of the incidence divisor X X X (d+1) Qd → × (d+1) Qd X under the morphism τ : ++ X X . Equivalently, the stack +− may be defined by the condition that the image of sn is contained in det M ⊂ (det M)(x). d M M M → Let us consider the stack Modn,++ which classifies the data (x, 0, , 0 , M(x)), where x ∈ X,deg(M) − deg(M0)=d,andthemapM0 → M(x)issuch

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that the image of det M0 is contained in (det M)(x) (and not just in (det M)(n·x)). d d M Let Modn,+− be the closed substack of Modn,++, where the image of det 0 is containedindetM. d+1 × H1 Consider the Cartesian product Modn n, which classiﬁes the data Bunn 0 0 0 (A.11) (x, M0, M, M , M0 ,→ M , M ,→ M ), 0 0 where deg(M ) − deg(M0)=d +1andM /M is the simple skyscraper sheaf supported at x. We have a natural proper morphism d+1 × H1 → d c :Modn n Modn,++, Bunn M0 d+1 × H1 which corresponds to “forgetting” , and a natural projection b :Modn n Bunn → d+1 M Pd+1 Modn , which corresponds to “forgetting” . We deﬁne the complex E,++ d on Modn,++ as n Pd+1 := c (b∗(Pd+1)) ⊗ Q ( )[n] E,++ ! E ` 2 Pd ∗ Ld ⊗ Q d·n · Pd+1 Pd+1 (recall that E := π ( E ) `( 2 )[d n]). Let E,+− be the restriction of E,++ d Q −1 − to Modn,+− tensored with `( 2 )[ 1]. Now form a commutative diagram, in which the left square is Cartesian: 0 ← 0 → 0 h − h − d Q ←−−−+ − Zd −−−−+ → Q  +− +−    qy 0qy qy

← → ←−−−h − d −−−−h → × Bunn Modn,+− X Bunn Consider the complex − d 0 → 0 ← ∗ 0 0 ∗ d+1 1 W − := h − ( h − (Ψ ) ⊗ q (P )) ⊗ Q ( )[−1]. E,+ + ! + E,+− ` 2 Wd Qd − Qd ∩ Qd Since we already know that E,+ vanishes on + ( + +−), it suffices to show Wd Qd ∩ Qd × Qd that the restriction of E,+− to + +− is supported on X , where it is Wd isomorphic to E E. × d d Observe that X Modn is naturally a closed substack in Modn,+−. The following result completes the proof of Proposition 7.5: Pd+1 A.5. Lemma. (1) The complex E,+− is a perverse sheaf, and there is a natural surjection 1 (A.12) Pd+1 (E ⊗ Q ( )[1]) Pd . E,+− ` 2 E (2) Let KE be the kernel of the map (A.12).The*–restriction of the complex 0 → 0 ← ∗ 0 ⊗ 0 ∗ K Qd Qd ∩ Qd h+−!( h+− (Ψ ) q ( E)) from +− to + +− vanishes identically. A.6. An informal explanation. Before giving a formal proof of Lemma A.5, we explain the main idea behind it. In this discussion we will assume that our ground field k is algebraically closed. Recall the full Hecke correspondence stack xHn. Observe that the fiber → −1 0 0 → (h ) (M )ofxHn over M ∈ Bunn under the map h is isomorphic to the affine Grassmannian Grx (see, e.g., [FGKV, FGV]). Let Ox be the complete local ring at

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x. The group GLn(Ox) acts naturally on Grx and we say that a perverse sheaf on Grx is spherical if it is GLn(Ox)–equivariant. (In particular, every spherical per- λ ∩ Hλ ∈ + verse sheaf is smooth along the stratification of Grx by Grx := Grx x n, λ Pn .) The category of spherical perverse sheaves is known to be semi-simple and equiva- Q lent as a tensor category to the category of representations of GLn( `), with the fiber functor being the functor of (total) global cohomology (see [Lu, Gi, MV, BD]). It is possible to generalize this equivalence to the case when x and M0 are allowed to “move” along X × Bunn.Namely,letHn be the stack classifying quadruples (M, M0,β,x), where M and M0 are rank n bundles on X,andβ is an isomorphism 0 M|X−x ' M |X−x. Then there is an equivalence between the category of perverse Q sheaves on X equipped with GLn( `)–action and a certain subcategory of the category of perverse sheaves on Hn.Foreachx ∈ X this equivalence “restricts” to the equivalence of the previous paragraph. Moreover, one can generalize this construction to the case of several points. For ◦ 1 any partition d =(d1,... ,dk)ofd, consider the open subset Xd of X(d ) × ...× (dk) X consisting of k–tuples of divisors (D1,... ,Dk), such that supp Di∩supp Dj = ◦ d (d) ∅,ifi =6 j.DenotethemapX → X by pd. We introduce an abelian category Ad Ad F (d) n as follows. The objects of n are perverse sheaves on X equipped with a Q GLn( `)–action, together with the following extra structure: for each partition d, ∗ F Q the sheaf pd( ) should carry an action of k copies of GLn( `), compatible with the Q F Q → original GLn( `)–action on with respect to the diagonal embedding GLn( `) Q ×k (GLn( `)) . For different partitions, these actions should be compatible in the obvious sense. In addition, it is required that whenever di = dj,i =6 j, the action Q ∗ F of the i-th and j-th copies of GLn( `)onpd( ) commutes with the corresponding ◦ Z d Ad natural 2–action on X . The definition of morphisms in n is clear. HBD,d Let now n be the symmetrized version of the Beilinson-Drinfeld affine Grass- HBD,d mannian (see [BD]). By definition, n is the ind-stack classifying quadruples (M, M0,D,β), where M, M are as above, D ∈ X(d) and β is an isomorphism be- M M0 d tween and away from the support of the divisor D.Inparticular,Modn HBD,d is naturally a closed substack of n , corresponding to the condition that the meromorphic map M0 → M defined by β is regular. HBD,d One can introduce the notion of a spherical perverse sheaf on n and con- Ad struct an equivalence between the above category n and the category of spherical HBD,d d ⊗ ⊗Q d perverse sheaves on n . For example, the perverse sheaf Sym (V E) `( 2 )[d], Ad Pd which is naturally on object of n,goestothesheaf E (considered as a sheaf on HBD,d d n supportedonModn). Ad One can also define categories analogous to n over partially symmetrized powers d of X. From this point of view, the sheaves on Modn,+− that we are interested in × (d) Q correspond to perverse sheaves on X X equipped with GLn( `)–action and an additional structure as above. Pd+1 × (d) ⊂ In particular, the sheaf E,+− corresponds to the restriction to X X X × X(d+1) of the sheaf

d +1 (V ∗ ⊗ Q ) Symd+1(V ⊗ E) ⊗ Q ( )[d +1] ` ` 2

on X × X(d+1).

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The ﬁrst assertion of Lemma A.5 then translates into the statement that this restriction is perverse, and that there is a map ∗ ⊗ Q d+1 ⊗ | → Q ⊗ d ⊗ (A.13) (V `) Sym (V E) X×X(d) ( ` E) Sym (V E), which becomes after a cohomological shift by d + 1 a surjection of perverse sheaves. The required map is induced by the obvious map Q d+1 ⊗ | → ⊗ d ⊗ ` Sym (V E) X×X(d) (V E) Sym (V E). e d+1 e | → e d e (In fact, for any local system E on X,themapSym (E) X×X(d) E Sym (E), whichisaninjectionofsheaves, becomes after a cohomological shift a surjection of e d e d+1 e | perverse sheaves such that E Sym (E)[d + 1] is the cosocle of Sym (E) X×X(d) [d + 1]; see [Dr]). Moreover, the kernel of the map (A.13) is supported on the incidence divisor X × X(d) ⊂ X × X(d+1), and there it satisﬁes the following property. Its stalk at a (d) point (x, D1,... ,Dk)ofX ×X (assuming that supp Di ∩supp Dj = ∅,deg(Di)= i Q ×k d )isaGLn( `) –module which decomposes into irreducible components of the λ1 ⊗ ⊗ λk i + form V ... V , where at least one λ does not belong to Pn,di .Thisproves the second assertion of Lemma A.5. In the proof of Lemma A.5 given below we simply perform the same manipula- d tions as above directly in the category of sheaves on Modn,+−.

Pd+1 A.7. ProofofLemmaA.5.First observe that E,++ is a perverse sheaf on d Modn,++, by a standard smallness result in the theory of the affine Grassmannian. Pd+1 Pd+1 The assertion about E,+− follows because E,++ has no subquotients supported over the incidence divisor X × X(d) → X × X(d+1). Pd+1 ⊗ Q −1 − Pd To construct the surjection E,+− (E `( 2 )[ 1]) E we introduce the 0 d+1 d+1 × × (d) stack Modn =Modn (X X ). X(d+1) We consider two perverse sheaves on it. The first one, denoted by F1, is the pull- Pd+1 0 d+1 → d+1 back of E under Modn Modn . To construct the other sheaf, consider d × H1 → 0 d+1 M M M0 M ⊂ the morphism a :Modn n Modn defined by sending ( 0, , , 0 Bunn 0 0 M, M ⊂ M )to(M0 ⊂ M ) × (x, D), where D = τ(M0 ⊂ M). The second perverse sheaf F2 is by definition n a (id × supp)∗(Pd E) ⊗ Q ( )[n], ! E ` 2 × d × H1 → d × F where (id supp) : Modn n Modn X.(Thus, 1 corresponds to the Bunn Bunn Q d+1 ⊗ | ⊗Q d+1 F sheaf ` Sym (V E) X×X(d) `( 2 )[d+1]and 2 corresponds to the sheaf ⊗ d ⊗ ⊗ Q d+1 × (d) (V E) Sym (V E) `( 2 )[d +1]onX X .) F F Pd+1 → There is a natural surjective map 1 2. Now the desired map E,+− ⊗ Q 1 Pd F → F (E `( 2 )[1]) E is obtained from the map 1 2 by adjunction. It is ⊗ Q 1 Pd surjective, because (E `( 2 )[1]) E has no subquotients supported on proper closed substacks. This completes the proof of part (1) of the lemma. To prove part (2), we choose x × D ∈ X × X(d) and calculate the restriction K d of E to its preimage in Modn,+−. To simplify notation, assume that D is of the form d · x.

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× · ∈ × (d) d supp−→ × (d) The preimage of x d x X X under Modn,+− X X is naturally a closed substack of xHn. Using formula (A.4), we obtain that the restriction to −1 × · Pd+1 ⊗ Q 1 Pd τ (x d x) of the surjection E,+− (E `( 2 )[1]) E can be identiﬁed with the map M λ d+1 ∗ d +1 IC− ⊗ Hom (V , Sym (V ⊗ E ) ⊗ V ) ⊗ Q ( )[d +1] w0(λ) GLn x ` 2 + λ∈Pn M λ d d +1 IC− ⊗ Hom (V ,E ⊗ Sym (V ⊗ E )) ⊗ Q ( )[d +1]. w0(λ) GLn x x ` 2 + λ∈Pn

The kernel of this map is nothing but the restriction of KE to the preimage of × · ∈ × (d) d x d x X X in Modn,+−. It is clear that if the summand corresponding to K 6∈ + the sheaf IC−w0(λ) appears in E ,thenλ Pn,d. Therefore, the required vanishing follows from formula (A.2) and Lemma 4.10.

Acknowledgments We express our gratitude to D. Kazhdan for his collaboration in [FGKV], which has inﬂuenced this work. We also thank V. Drinfeld, D. Kazhdan, and I. Mirkovi´c for valuable discussions.

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Department of Mathematics, University of California, Berkeley, California 94720

Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

Department of Mathematics, Northwestern University, Evanston, Illinois 60208

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