]]> We prove the equivalence of two conjectural constructions of unramified cuspidal automorphic functions on the adelic group associated to an irreducible --adic local system of rank on an algebraic curve over a finite field. The existence of such a function is predicted by the Langlands conjecture. The first construction, which was proposed by Shalika and Piatetski-Shapiro following Weil and Jacquet-Lang -lands (), is based on considering the Whittaker function. The second construction, which was proposed recently by Laumon following Drinfeld () and Deligne (), is geometric: the automorphic function is obtained via Grothendieck's faisceaux-fonctions'' correspondence from a complex of sheaves on an algebraic stack. Our proof of their equivalence is based on a local result about the spherical Hecke algebra, which we prove for an arbitrary reductive group. We also discuss a geometric interpretation of this result. BL1 1 A. Beauville, Y. Laszlo, Conformal blocks and generalized theta functions , Comm. Math. Phys. 164 (1994) 385-419. 95k:1401 BL 2 A. Beauville, Y. Laszlo, Un lemme de descente , C.R. Acad. Sci. Paris, Ser. I Math. 320 (1995) 335-340. 96a:14049 BZ 3 J. Bernstein, A. Zelevinsky, Representations of the group GL(n,F) where F is a local non-Archimedean field , Russ. Math. Surv. 31 (1976) 1-68. Br 4 R.K. Brylinski, Limits of weight spaces, Lusztig's q --analogs, and fiberings of adjoint orbits , Journal of AMS 2 (1989) 517-533. 90g:17011 CS 5 W. Casselman, J. Shalika, The unramified principal series of p --adic groups II. The Whittaker function , Comp. Math. 41 (1980) 207-231. 83i:22027 De 6 P. Deligne, Le conjecture de Weil II , Publ. IHES 52 (1980) 137-252 83c:14017 Dr 7 V.G. Drinfeld, Two-dimensional --adic representations of the fundamental group of a curve over a finite field and automorphic forms on GL(2) , Amer. J. Math. 105 (1983) 85-114. 84i:12011 DS 8 V.G. Drinfeld, C. Simpson, B -structures on G -bundles and local triviality , Math. Res. Lett. 2 (1995) 823--829. 96k:14013 GK 9 I.M. Gelfand, D. Kazhdan, Representations of the group GL(n,K) where K is a local field , in Lie Groups and Their Representations, ed. I.M. Gelfand, pp. 95-118, Adam Hilder Publ., 1975. 53:8334 Gi 10 V. Ginzburg, Perverse sheaves on a loop group and Langlands duality , Preprint alg-geom 9511007 (1995). 11 B.H. Gross, On the Satake isomorphism , Preprint (1996). GP 12 B.H. Gross, D. Prasad, On the decomposition of a representation of SO n when restricted to SO n-1 , Canadian J. Math. 44 (1992) 974-1002. 93j:22031 Gr 13 A. Grothendieck, Techniques de construction et theoremes d'existence en geometrie algebrique IV: les schemas de Hilbert , Seminaire Bourbaki 221 (1960 61), Benjamin (1966). He 14 W.H. Hesselink, Characters of the nullcone , Math. Ann. 252 (1980) 179-182. 82c:1700 JL 15 H. Jacquet, R.P. Langlands, Automorphic Forms on GL(2) , Lect. Notes in Math. 114 , Springer Verlag, 1970. 53:5481 JPSS 16 H. Jacquet, I.I. Piatetski-Shapiro, J. Shalika, Automorphic forms on GL(3) , I and II, Ann. Math. 109 (1979) 169-258. ; 80i:10034b 80i:10034a Ka 17 S. Kato, Spherical functions and a q --analogue of Kostant's weight multiplicity formula , Inv. Math. 66 (1982) 461-468. 84b:22030 Ko 18 B. Kostant, Lie group representations on polynomial rings , Amer. J. Math. 85 (1963) 327-404. 28:1252 La 19 R.P. Langlands, Problems in the theory of automorphic forms , in Lect. Notes in Math. 170 , pp. 18-61, Springer Verlag, 1970. 46:1758 LS 20 Y. Laszlo, Ch. Sorger, The line bundles on the moduli of parabolic G -bundles over curves and their sections , Ann. Sci. Ecole Norm. Sup. 30 (1997) 499--525. 97:14 La0 21 G. Laumon, Transformation de Fourier, constants d'equations fonctionnelles et conjecture de Weil , Publ. IHES 65 (1987) 131-210. 88g:14019 La1 22 G. Laumon, Correspondance de Langlands geometrique pour les corps de fonctions , Duke Math. J. 54 (1987) 309-359. 88g:11086 La:e 23 G. Laumon, Faisceaux automorphes lies aux series d'Eisenstein , in Automorphic Forms, Shimura varieties, and L --functions, vol. 1, pp. 227-281, Academic Press, 1990. 91k:11106 La:s 24 G. Laumon, L. Moret-Bailly, Champs algebriques , Preprint 92-42, Universite Paris Sud, 1992. La2 25 G. Laumon, Faisceaux automorphes pour GL n : la premiere construction de Drinfeld , Preprint alg-geom 9511004 (1995). Lu1 26 G. Lusztig, Green polynomials and singularities of unipotent classes , Adv. Math. 42 (1981) 169-178. 83c:20059 Lu2 27 G. Lusztig, Singularities, character formulas, and a q --analogue of weight multiplicities , Asterisque 101 (1983) 208-229. 85m:17005 Mac 28 I.G. Macdonald, Spherical Functions on a Group of p --adic Type, Ramanujan Inst. Publ., University of Madras, India, 1971. 55:8261 MV 29 I. Mirkovic, K. Vilonen Perverse sheaves on loop Grassmannians and Langlands duality , Preprint alg-geom 9703010, to appear in the proceedings of a conference on the geometric Langlands correspondence at Luminy, June 1995. PS 30 I.I. Piatetski-Shapiro, Euler subgroups , in Lie Groups and Their Representations, ed. I.M. Gelfand, pp. 597-620, Adam Hilder Publ., 1975. 53:10720 Sa 31 I. Satake, Theory of spherical functions on reductive algebraic groups over p --adic fields , Publ. IHES 18 (1963) 1-69. 33:4059 Sha 32 J.A. Shalika, The multiplicity one theorem for GL n , Ann. Math. 100 (1974) 171-193. 50:545 Shi 33 T. Shintani, On an explicit formula for class 1 Whittaker functions on GL n over P --adic fields , Proc. Japan Acad. 52 (1976) 180-182. 53:10991 34 A. Weil, Dirichlet Series and Automorphic Forms, Lect. Notes in Math. 189 , Springer Verlag, 1971. [1] A. Beauville, Y. Laszlo, Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994) 385-419. MR 95k:1401

[2] A. Beauville, Y. Laszlo, Un lemme de descente, C.R. Acad. Sci. Paris, Sér. I Math. 320 (1995) 335-340. MR 96a:14049

[3] J. Bernstein, A. Zelevinsky, Representations of the group where is a local non-Archimedean field, Russ. Math. Surv. 31 (1976) 1-68.

[4] R.K. Brylinski, Limits of weight spaces, Lusztig's -analogs, and fiberings of adjoint orbits, Journal of AMS 2 (1989) 517-533. MR 90g:17011

[5] W. Casselman, J. Shalika, The unramified principal series of -adic groups II. The Whittaker function, Comp. Math. 41 (1980) 207-231. MR 83i:22027

[6] P. Deligne, Le conjecture de Weil II, Publ. IHES 52 (1980) 137-252 MR 83c:14017

[7] V.G. Drinfeld, Two-dimensional -adic representations of the fundamental group of a curve over a finite field and automorphic forms on , Amer. J. Math. 105 (1983) 85-114. MR 84i:12011

[8] V.G. Drinfeld, C. Simpson, -structures on -bundles and local triviality, Math. Res. Lett. 2 (1995) 823-829. MR 96k:14013

[9] I.M. Gelfand, D. Kazhdan, Representations of the group where is a local field, in Lie Groups and Their Representations, ed. I.M. Gelfand, pp. 95-118, Adam Hilder Publ., 1975. MR 53:8334

[10] V. Ginzburg, Perverse sheaves on a loop group and Langlands duality, Preprint alg-geom/9511007 (1995).

[11] B.H. Gross, On the Satake isomorphism, Preprint (1996).

[12] B.H. Gross, D. Prasad, On the decomposition of a representation of when restricted to , Canadian J. Math. 44 (1992) 974-1002. MR 93j:22031

[13] A. Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique IV: les schémas de Hilbert, Séminaire Bourbaki 221 (1960/61), Benjamin (1966).

[14] W.H. Hesselink, Characters of the nullcone, Math. Ann. 252 (1980) 179-182. MR 82c:17004

[15] H. Jacquet, R.P. Langlands, Automorphic Forms on , Lect. Notes in Math. 114, Springer Verlag, 1970. MR 53:5481

[16] H. Jacquet, I.I. Piatetski-Shapiro, J. Shalika, Automorphic forms on , I and II, Ann. Math. 109 (1979) 169-258. MR 80i:10034a; MR 80i:10034b

[17] S. Kato, Spherical functions and a -analogue of Kostant's weight multiplicity formula, Inv. Math. 66 (1982) 461-468. MR 84b:22030

[18] B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963) 327-404. MR 28:1252

[19] R.P. Langlands, Problems in the theory of automorphic forms, in Lect. Notes in Math. 170, pp. 18-61, Springer Verlag, 1970. MR 46:1758

[20] Y. Laszlo, Ch. Sorger, The line bundles on the moduli of parabolic -bundles over curves and their sections, Ann. Sci. École Norm. Sup. 30 (1997) 499-525. CMP 97:14

[21] G. Laumon, Transformation de Fourier, constants d'équations fonctionnelles et conjecture de Weil, Publ. IHES 65 (1987) 131-210. MR 88g:14019

[22] G. Laumon, Correspondance de Langlands géométrique pour les corps de fonctions, Duke Math. J. 54 (1987) 309-359. MR 88g:11086

[23] G. Laumon, Faisceaux automorphes liés aux séries d'Eisenstein, in Automorphic Forms, Shimura varieties, and -functions, vol. 1, pp. 227-281, Academic Press, 1990. MR 91k:11106

[24] G. Laumon, L. Moret-Bailly, Champs algébriques, Preprint 92-42, Université Paris Sud, 1992.

[25] G. Laumon, Faisceaux automorphes pour : la première construction de Drinfeld, Preprint alg-geom/9511004 (1995).

[26] G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. Math. 42 (1981) 169-178. MR 83c:20059

[27] G. Lusztig, Singularities, character formulas, and a -analogue of weight multiplicities, Astérisque 101 (1983) 208-229. MR 85m:17005

[28] I.G. Macdonald, Spherical Functions on a Group of -adic Type, Ramanujan Inst. Publ., University of Madras, India, 1971. MR 55:8261

[29] I. Mirkovi\'{c}, K. Vilonen Perverse sheaves on loop Grassmannians and Langlands duality, Preprint alg-geom/9703010, to appear in the proceedings of a conference on the geometric Langlands correspondence at Luminy, June 1995.

[30] I.I. Piatetski-Shapiro, Euler subgroups, in Lie Groups and Their Representations, ed. I.M. Gelfand, pp. 597-620, Adam Hilder Publ., 1975. MR 53:10720

[31] I. Satake, Theory of spherical functions on reductive algebraic groups over -adic fields, Publ. IHES 18 (1963) 1-69. MR 33:4059

[32] J.A. Shalika, The multiplicity one theorem for , Ann. Math. 100 (1974) 171-193. MR 50:545

[33] T. Shintani, On an explicit formula for class 1 Whittaker functions on over -adic fields, Proc. Japan Acad. 52 (1976) 180-182. MR 53:10991

[34] A. Weil, Dirichlet Series and Automorphic Forms, Lect. Notes in Math. 189, Springer Verlag, 1971.
]]> 1998 American Mathematical Society Journal of the American Mathematical Society J. Amer. Math. Soc. jams 11 02 19980401 March 31, 1997 November 26, 1997 451 451-484 34 S0894-0347-98-00260-4 10.1090/S0894-0347-98-00260-4 0894-0347 1088-6834 English 11R39 11F70 14H60 22E55 1991 http://www.ams.org/jourcgi/jour-getitem?pii=S0894-0347-98-00260-4 Introduction Let X be a smooth, complete, geometrically connected curve over . Denote by F the field of rational functions on X , by A the ring of adeles of F , and by Gal ( F) the Galois group of F . The present paper may be considered as a step towards understanding the geometric Langlands correspondence between n --dimensional --adic representations of Gal ( F) and automorphic forms on the group GL n( A ) . We follow the approach initiated by V. Drinfeld Dr , who applied the theory of --adic sheaves to establish this correspondence in the case of GL 2 . The Langlands conjecture predicts that to any unramified irreducible n -dimen-sional representation of Gal ( F) , one can attach an unramified automorphic function f on GL n( A ) . The starting point of Drinfeld's approach is the observation that an unramified automorphic function on the group GL n( A ) can be viewed as a function on the set M n of isomorphism classes of rank n bundles on the curve X . The set M n is the set of --points of M n , the algebraic stack of rank n bundles on X . One may hope to construct the automorphic form associated to a Galois representation as a function corresponding to an --adic perverse sheaf on M n . This is essentially what Drinfeld did in Dr in the case of GL 2 . In abelian class field theory (the case of GL 1 ) this was done previously by P. Deligne (see La1 ). Let M' n denote the set of isomorphism classes of pairs L,s , where LM n is a rank n bundle on X and s is a regular non-zero section of L . Using a well-known construction due to Weil and Jacquet-Langlands JL for n 2 , and Shalika Sha and Piatetski-Shapiro PS for general n , one can associate a function f' on M' n to an unramified n --dimensional representation of Gal ( F) . The function f' is obtained from the Whittaker function W , a function canonically attached to . The Langlands conjecture prescribes that when is geometrically irreducible, the function f' should be constant along the fibers of the projection p: M' nM n . In other words, conjecturally, f' is the pull-back of a function f on M n ; the function f is then the automorphic function corresponding to . Thus, the key problem is to show that the function f' is constant along the fibers of p: M' nM n . Now let ' n be the moduli stack of pairs L,s , where L is a rank n bundle on X and s is a regular non-zero section of L . We have: M n n(),M' n ' n() . Each Galois representation gives rise to an --adic local system E on X of rank n . Drinfeld's idea, developed further by G. Laumon La1 , can be interpreted as follows. Suppose there exists an irreducible perverse sheaf S ' on ' n , with the property that the function S' associated to S ' on M' n ' n() equals f' . Then showing that the function f' is constant along the fibers of the projection p: M' nM n becomes a geometric problem of proving that S ' descends to a perverse sheaf S on n . In Dr , Drinfeld constructed such a sheaf S ' in the case of GL 2 . He started with a geometric realization of the Whittaker function as a perverse sheaf on the symmetric power of the curve X . Then he defined the sheaf S ' using a geometric version of the Weil-Jacquet-Langlands construction. Drinfeld showed that the sheaf S ' is locally constant along the fibers of p . Since the fibers of p are projective spaces, hence simply-connected, this implies that S ' is constant along the fibers of p . Drinfeld's construction raised hope that one can use a similar argument in the case of GL n . An analogue of Drinfeld's construction in the case of GL n has been proposed by Laumon in La1 , La2 . In order to construct S ' for GL n , he defined a sheaf L , which he considered as a geometric analogue of the Whittaker function W . However, the function L corresponding to L and the Whittaker function W are defined on different sets and their values are different; see La1 and Sect. 3.5 below. Using the sheaf L , Laumon La2 constructed a candidate for the sheaf S ' on ' n (this construction was independently found by D. Gaitsgory, unpublished). In order to justify this construction, one has to prove that the function S' on M' n , corresponding to the sheaf S ' , coincides with the function f' . This equality was conjectured by Laumon in La2 (Conjecture 3.2), and its proof is one of the main goals of this paper. To prove the equality S' f' , we reduce it to a local statement (see local ) which we make for an arbitrary reductive group. We prove local using the Casselman-Shalika formula for the Whittaker function CS (actually, local is equivalent to the Casselman-Shalika formula). local can be translated into a geometric statement about intersection cohomology sheaves on the affine Grassmannian (see second ). One essential difference between Laumon's approach and our approach is in interpretation of the sheaf L and the function L associated to this sheaf. Laumon interprets the local factors of L via the Springer sheaves and the Kostka-Foulkes polynomials (see 3). We interpret the local factors of L via the perverse sheaves on the affine Grassmannian and the Hecke algebra (see Sect. 4.2). Our interpretation, which was inspired by Lu1 , allows us to gain more insight into Laumon's construction. In particular, it helps to explain the apparent discrepancy between L and W : it turns out that L is related to W by a Fourier transform. Using this result, we demonstrate that the output of the two constructions -- S' and f' -- coincide. Let us now briefly describe the contents of the paper: In Sect. 2 we review some background material concerning the Langlands conjecture and the classical construction of the function f' together with its geometric interpretation. We follow closely Sect. 1 of La1 . In Sect. 3 we describe the construction of the sheaf S ' on ' n and the function S' on M' n . We state the main conjecture ( princ ) about the geometric Langlands correspondence for GL n and our main result ( prin ). In Sect. 4 we give an adelic interpretation of the construction of the function S' and reduce prin to a local statement, Fplus . In Sect. 5 we prove local for an arbitrary reductive group and derive from it Fplus . In Sect. 6 we interpret the results of Sect. 5 from the point of view of the spherical functions. In Sect. 7 we give a geometric interpretation of local and discuss a possible generalization of Laumon's construction to other groups. In this paper we work with algebraic stacks in the smooth topology in the sense of La:s . All stacks that we consider have locally the form Y G where Y is a scheme and G is an algebraic group acting on it. We will use the following notion of perverse sheaves on such algebraic stacks: for V Y G , a perverse sheaf on V is just a G --equivariant perverse sheaf on Y , appropriately shifted. Throughout this paper, for an --scheme (resp., for an --stack) V and for an algebra R over , V (R) will denote the set of R --points of V (resp., the set of isomorphism classes of R --points of V ). In most cases, schemes and stacks are denoted by script letters and their sets of F q --points are denoted by the corresponding italic letters (e.g., V and V ). We use the same notation for a morphism of stacks and for the corresponding map of sets. If S is a sheaf or a complex of sheaves on a stack V , then the corresponding set V is endowed with a function alternating sum of traces of the Frobenius'' (as in De ) which we denote by the corresponding italic letter S (we assume that a square root of q in is fixed throughout the paper). If V is a stack over F q , the set V is endowed with a canonical measure , which in the case when V is a scheme has the property (v) 1 , vV . For example, if G is a group, ((pt G )( F q)) G -1 . Background and the Shalika-Piatetski-Shapiro construction Langlands conjecture Let k F q be a finite field, and let X be a smooth complete geometrically connected curve over k . Denote by F the field of rational functions on X . For each closed point x of X , denote by x the completion of F at x , by x the ring of integers of x , and by x a generator of the maximal ideal of x . Let k x be the residue field x x x , and let q x q x be its cardinality. We denote by ' x X x the ring of adeles of F and by ' x X x its maximal compact subring. Consider the set G n of unramified and geometrically irreducible --adic representations of the Galois group Gal ( F) in GL n() , where is relatively prime to q , as in La1 , (1.1). Let A n be the set of cuspidal unramified automorphic functions on the group GL n( ) -- these are cuspidal functions on the set GL n(F)GL n( ) GL n() , which are eigenfunctions of the Hecke operators. Recall that for each x X and i 1,,n , one defines the Hecke operator T i x by the formula: ( T i x f )(g) M i n( x) f(gh) dh, where M i n( x) GL n( x) D x i GL n( x) GL n( x) GL n( ); here D x i is the diagonal matrix whose first i entries equal x , and the remaining n-i entries equal 1 , and dh stands for the Haar measure on GL n( x) normalized so that GL n( x) has measure 1 . Let B be the Borel subgroup of upper triangular matrices and T N its standard Levi decomposition. Cuspidality condition means that for each proper parabolic subgroup of GL n , whose unipotent radical V is contained in the upper unipotent subgroup N , V(F) V( ) f(vg) dv 0, g GL n( ). conj Lan For each G n , there exists a unique (up to a non-zero constant multiple) function f A n , such that for any x X T i x f q x -i(i-1) 2 Tr ( i ( Fr x)) f , i 1,,n, where T i x is the i th Hecke operator, and Fr x Gal ( F) is the geometric Frobenius element. conj Let P 1 GL n be the subgroup equation pp pmatrix g h 0 1 pmatrix g GL n-1 . equation Following Shalika and Piatetski-Shapiro, we construct a function f' on the double-quotient P 1(F)GL n( ) GL n(), which is cuspidal and which satisfies the Hecke eigenfunction property: equation hep T i x f' q x -i(i-1) 2 Tr ( i ( Fr x)) f' , i 1,,n, x X . equation The first step in the construction of f' is the construction of the Whittaker function. Whittaker functions Introduce the following notation: for a homomorphism Spec R X and an X --module M we denote by M R the R --module of sections of the pull-back of M to Spec R . Denote by the canonical bundle over X . Let GL n J(R) be the group of invertible n n matrices A (A ij ) 0i,j n-1 , where A ij R j-i . The group GL n J(R) is locally (for the Zariski topology) isomorphic to the corresponding non-twisted group GL n(R) . To establish such an isomorphism, one has to choose a non-vanishing regular section of , so the isomorphism is not canonical. It is easy to see that GL n J(R) is the group of R --points of a group scheme over X , but in this paper we will not use this fact. We denote by N J(R), T J(R), P 1 J(R) , etc., the corresponding subgroups of GL n J(R) . The twisted forms GL n J(R) have the following advantage. Let u i,i 1 be the i th component of the image of u N J(R) in N J(R) N J(R),N J(R) corresponding to the (i,i 1) entry of u . Then u i,i 1 R . Let us fix once and for all a non-trivial additive character : k . We define the character x of N J( x) by the formula x(u) i 1 n-1 ( Tr k x k ( Res x u i,i 1 )), and the character of N J( ) by ((u x)) x X x(u x). It follows that (u) 1 if u N J() or u N J(F) . For each x X consider the group GL n J( x) . Let be a semi-simple conjugacy class in GL n() . The following result is due to Shintani Shi , and Casselman and Shalika CS . thm wgamma (1) There exists a unique function W ,x on GL n J( x) that satisfies the following properties: itemize W ,x (gh) W ,x (g), h GL n J( x) , W ,x (1) 1 ; W ,x (ug) x(u) W ,x (g), u N J( x) ; W ,x is an eigenfunction with respect to the local Hecke-operators T i x , i 1,,n : T i x W ,x q x -i(i-1) 2 Tr ( i())W ,x . itemize (2) The function W ,x is given by the following formulas. For ( 1,, n) P n , the set of dominant weights of GL n (i.e., such that 1 2 n ), equation cassha W ,x ( diag ( x 1 ,, x n )) q x n() Tr (,V()), equation where V() is the irreducible representation of GL n() of highest weight , and n() i 1 n (i-1) i . For n - P n , W ,x ( diag ( x 1 ,, x n )) 0 . thm There is a bijection between the weight lattice of GL n and the double quotient N J( x)GL n J( x) GL n J( x) , which maps ( 1,, n) to the double coset of diag ( x 1 ,, x n ) . This explains the fact that W ,x is uniquely determined by its values at the points diag ( x 1 ,, x n ) . rem 1 The uniqueness of the Whittaker function is connected with the fact that an irreducible smooth representation of a reductive group G over a local non-archimedian field has at most one Whittaker model; see GK . There is an explicit formula for the Whittaker function associated to an arbitrary reductive group, due to Casselman and Shalika CS , which we will use in Sect. 5. rem Now we attach to A n the global Whittaker function W on GL n J( ) by the formula equation wsigma W ((g x)) x X W ( Fr x),x (g x). equation It satisfies: itemize W (gh) W (g), h GL n J() , W (1) 1 ; W (ug) (u) W (g), u N J( ) ; in particular, W is left invariant with respect to N J(F) ; For all i 1,,n and x X we have: T i x W q x -i(i-1) 2 Tr ( i (Fr x))W . itemize The construction of f' Let C (GL n J( )) N J( ) be the space of --valued smooth (see, e.g., BZ ) functions f on GL n J( ) , such that f(ug) (u) f(g), u N J( ) ; we call such functions (N J(A),) --equivariant. Let C (GL n J( )) P 1 J(F) cusp be the space of smooth functions f on GL n J( ) , which satisfy f(pg) f(g), p P 1 J(F) and are cuspidal, i.e., for each proper parabolic subgroup of GL n , whose unipotent radical V is contained in N , V J(F) V J( ) f(vg) dv 0, g GL n J( ). The following result is the main step in constructing automorphic functions for GL n . The existence of the subgroup P 1 plays a key role in this result, and this makes the case of GL n special. thm shaps There is a canonical isomorphism : C (GL n J( )) N J( ) C (GL n J( )) P 1 J(F) cusp given by the formula ((f))(g) y N n-1 J(F)GL n-1 J(F) f ( pmatrix y 0 0 1 pmatrix g ). This isomorphism commutes with the right action of GL n J( ) on both spaces. thm For the proof, see Sha and PS . Note that shaps is not stated exactly in this form in Sha but its proof can be extracted from the proof of Theorem 5.9 there. We remark that for each g GL n J( ) the sum above has finitely many non-zero terms. By construction, W C (GL n J( )) N J( ) . Let f' (W ) . The isomorphism of shaps clearly preserves the spaces of right GL n J() --invariant functions and commutes with the action of the Hecke operators on them. Therefore f' is right GL n J() --invariant and satisfies formula hep , i.e., it is an eigenfunction of the Hecke operators with the same eigenvalues as those prescribed by the Langlands conjecture. Furthermore, uniqueness of the Whittaker function W implies that the function f' is the unique function on GL n J( ) satisfying the above properties (up to a non-zero constant multiple). Thus, the Langlands Lan is equivalent to conj precise For each G n , the function f' is left GL n J(F) --invariant. conj rem There is an approach to proving this conjecture using analytic properties of the L --function of the Galois representation (see JL , JPSS , La0 ), which will not be discussed here. rem Interpretation in terms of vector bundles geomint We begin by fixing notation. Recall that B is the Borel subgroup in GL n (consisting of upper triangular matrices) and T B is the maximal torus (consisting of diagonal matrices). Let P be the maximal parabolic subgroup of GL n containing the subgroup P 1 of GL n defined by formula pp . Denote T J( x) T J( x) Diag ( x) , where Diag ( x) is the set of diagonal n n matrices with coefficients in x , B J( x) N J( x) T J( x) and P J( x) pmatrix a b 0 c pmatrix a GL n-1 J( x), c x , c 1 . Let B J( ) ' x X B J( x) and P J( ) ' x X P J( x) . Denote by Q the double quotient N J(F)B J( ) B J() . Note that since B J( ) B J() GL n J( ) GL n J(), Q is naturally a subset of N J(F)GL n J( ) GL n J() . Let M' n denote the double quotient P 1 J(F)P J( ) P J() . Note that since P J( ) P J() GL n J( ) GL n J(), M' n is naturally a subset of P 1 J(F)GL n J( ) GL n J() . Finally, set M n GL n J(F)GL n J( ) GL n J() . Let us denote by and p the obvious projections Q M' n and M' n M n , respectively. lem easy (1) There is a canonical bijection between the set Q and the set of isomorphism classes of the following data: L,(F i),(s i) , where L is a rank n vector bundle over X , 0 F n F n-1 F 1 F 0 L is a full flag of subbundles in L , and s i: i L i F i F i 1 is a non-zero X --module homomorphism. (2) There is a canonical bijection between the set M' n and the set of isomorphism classes of pairs L,s , where L is a rank n vector bundle over X , and s: n-1 L is a non-zero X --module homomorphism. The natural projection : QM' n sends L,(F i),(s i) Q to L,s n-1 M' n . (3) There is a canonical bijection between the set M n and the set of isomorphism classes of rank n vector bundles over X . The natural map p: M' n M n corresponds to forgetting the section s . lem proof Recall that for a morphism Spec R X and an X --module M we denote by M R the space of sections of the pull-back of M to Spec R . Denote by J 0 the vector bundle i 0 n-1 i . Let Bun be the set of data L, , x , where L is a rank n bundle on X , and : J 0 F L F and x: J 0 x L x , x X , are isomorphisms (generic and local trivializations'', respectively). We construct a map b: Bun GL n J( ) as follows. After the identifications of J 0 F F x J 0 x x x with J 0 x , and of L F F x L x x x with L x , x and give rise to homomorphisms J 0 x L x which we denote by the same characters. Let x ( x) -1 be the corresponding automorphism of J 0 x . To represent the element x by an n n matrix g x (g x,ij ) of the form given in Sect. 2.2, we set g x,ij to be equal to the element of j-i x corresponding to the map i x J 0 x x J 0 x j x . Thus, g x is the transpose Taking the transpose has some advantages, in particular, it agrees with the conventions adopted in La1 . of the matrix representing the action of x on J 0 x . The map b sends L, , x to (g x) x X GL n J( ) . It is easy to see that this map is a bijection. Now to prove part (1) of the lemma, let us observe that given a triple L,(F i),(s i) , we can choose and x 's in such a way that for each j 0,,n-1 , they map i j n-1 i R to F j,R L R and the associated maps j R F j,R F j 1,R coincide with restrictions of s j to Spec R (here R F or x ). With such a choice, g x B J( x) , x X , and the arbitrariness in the choice of (resp., x ) corresponds to left (resp., right) multiplication of (g x) x X by elements of N J(F) (resp., B J( x) ). This proves part (1) of the lemma. The proof of parts (2) and (3) is similar. proof Note that the function W (resp., f' ), which is defined on the set N J(F)GL n J( ) GL n J() (resp., P 1 J(F)GL n J( ) GL n J() ) is uniquely determined by its restriction to the subset Q (resp., M' n ), since it is an eigenfunction of the Hecke operators T n x . Now f' is a function on P 1 J(F)GL n J( ) GL n J() . Its restriction to M' n , which we also denote by f' , equals, by definition, (W ) , where denotes the operation of summing up a function along the fibers of the map (note that these fibers are finite). precise can now be stated in the following way. conj precise1 The function f' is constant along the fibers of the map p: M' n M n . conj In the next section we discuss a geometric version of this conjecture. Conjectural geometric construction of an automorphic sheaf Definitions of stacks Let M n be the moduli stack of rank n bundles on X . Recall that for an --scheme S , Hom (S, M n) is the groupoid whose objects are rank n bundles on X S and morphisms are isomorphisms of such bundles. Let M ' n be the moduli stack of pairs L,s , where L is a rank n bundle on X and s: n-1 M ' n is an embedding of X --modules. More precisely, Hom (S, M ' n) is the groupoid whose objects are pairs L S,s S , where L S Ob Hom (S, M n) and s S: X n-1 S L S is an embedding, such that the quotient L S Im s S is S --flat; morphisms are isomorphisms of such pairs which make the natural diagram commutative. The set M n (resp. M' n ) can be identified with the set of F q --points of n (resp. ' n ). As was explained in the introduction, we expect that f' is the function attached to a complex of --adic sheaves S ' on ' n . In this section we present the construction of a candidate for the complex S ' following Laumon La2 . At the level of --points, this construction is actually different from the construction of f' given in Sect. 2.3. The reason is the following. It is easy to define a naive'' stack Q classifying triples L,(F i),(s i) (as in easy ) with Q () Q and a morphism Q M ' n corresponding to the map of sets : Q M' n . But this Q is a disjoint union of connected components labeled by the n --tuples (d 0,,d n-1 ) , where d i is the degree of the divisor of zeros of the map s i: i F i F i 1 . On the other hand, the stack M ' n is a disjoint union of connected components corresponding to the degree of the divisor of zeros of s: n-1 L . Recall that under , s n-1 becomes s . This means that the fibers of are disconnected. Hence one cannot obtain an irreducible sheaf on M ' n as the direct image of a sheaf on Q . In this section we replace the naive'' stack Q by a stack Q , and the Whittaker function W by a perverse sheaf on Q . The pair ( Q , ) was first constructed by Laumon La2 . The stack Q The algebraic stack Q is defined as follows. For an --scheme S , Hom (S, Q ) is the groupoid whose objects are quintuples L S, S,J S,(J i,S ),(s i,S ) , where L S and J S are rank n bundles on X S , S: J S L S is an embedding of the corresponding X S --modules, such that the quotient is S --flat, (J i,S ) is a full flag of subbundles 0 J n,S J n-1,S J 1,S J 0,s J S, and s i,S is an isomorphism i X S J i,S J i 1,S , i 0,,n-1 . Morphisms are isomorphisms of the corresponding X S --modules making all natural diagrams commutative. We have a natural representable morphism of stacks : Q ' n , which for each --scheme S maps L S, S,J S,(J i,S ),(s i,S ) to the pair L S, S s n-1,S , where s n-1,S is viewed as an embedding of n X S into J S . Let Q ( F q) be the set of --points of Q (see Sect. 1.7). By definition, it consists of quintuples L,,J,(J i),(s i) , where L and J are rank n bundles on X , :J L is an embedding of the corresponding X --modules, (J i) is a full flag of subbundles 0 J n J n-1 J 1 J 0 J, and s i is an isomorphism s i: i J i J i 1 , i 0,,n-1 . There is a natural map of sets r:Q defined as follows. Given an object L,,J,(J i),(s i) , define F i to be the maximal locally free submodule of L of rank n-i , which contains the image of J i J under . Then (F i) is a full flag of subbundles of L . The composition of s i: i J i J i 1 with the natural map J i J i 1 F i F i 1 induced by is an --module homomorphism s' i: i F i F i 1 for each i 0,,n-1 . Then L,(F i),(s' i) is a point of Q . Thus we obtain a map r: Q . lem r1 The composition r:M' n coincides with the map . Moreover for every function f on we have (r (f)) (f) , the integrations being taken with respect to the canonical measures on each of the three sets. lem The sheaf L E In this section we recall Laumon's construction La1 of the sheaf L E . Let be the stack classifying torsion sheaves of finite length on X , i.e., for an --scheme S , Hom (S,) is the groupoid whose objects are coherent sheaves T S on X S , which are finite and flat over S (see La1 The notation used in La1 is 0 ; we suppress the upper index 0 to simplify notation. ). Let be the open substack of that classifies torsion sheaves that have at most n indecomposable summands supported at each point. The stack can be understood as follows. Let K be a field containing , and let T (K) . We have a (non-canonical) isomorphism equation torsion T O X K O X K (-D 1) O X K O X K (-D h), equation where X K Spec K Spec X , and D 1D 2 D h is a decreasing sequence (uniquely determined by T ) of effective divisors on X K . The torsion sheaf T belongs to (K) precisely when hn . Let S be an --scheme. Then a torsion sheaf T S(S) belongs to (S) if it does so at every closed point of S . The stack is a disjoint union of connected components m Z n,m , where the component n,m classifies torsion sheaves of degree m ; the degree of the torsion sheaf T in torsion is i deg (D i) . Each n,m has an open substack n,m rss classifying regular semi-simple torsion sheaves: an S --point T S of is said to be a point of n,m rss if for any --point of S the corresponding sheaf over X Spec Spec is isomorphic to X X(-x i) , where the points x i are distinct. Let X (m) denote the m th symmetric power of X and let X (m),rss be the complement to the divisor of diagonals in X (m) . We have a smooth map X (m),rss n,m rss . When we make base change from to , n,m rss can be identified with the quotient of X (m) by GL 1 m with respect to the trivial action. Let us consider the rank n local system E E on X corresponding to the Galois representation . Let us also write : X m X (m) for the natural projection. Define the m th symmetric power E (m) of E as the sheaf of invariants of E m under the natural action of the symmetric group S m , i.e., E (m) ( E m ) S m . The restriction E (m) X (m),rss clearly descends to a local system L E,m 0 on n,m rss , since it does over the algebraic closure of . We define the perverse sheaf L E as the sheaf on n , whose restriction to each n,m is the Goresky-MacPherson extension of L E,m 0 , i.e., L E n,m j L E,m 0 , where j: n,m rss n,m . We will need an explicit description of the function L E corresponding to L E on the set Coh n,m . Let x X be a closed point with residue field k x . Then we can regard x as a k x --rational point of X . Recall from Sect. 2.1 that we denote by q x the cardinality of k x , and that q x q x . We denote by n,m (x) the algebraic stack over k x that classifies torsion sheaves of degree m on X Spec Spec k x supported at x that have at most n indecomposable summands. Obviously, n,m (x) is a locally closed substack of n,m Spec Spec k x ; we denote by I m,x the corresponding embedding. Let L E,m,x be the pull-back to n,m (x) of the sheaf L E,m under the composition n,m (x) I m,x n,m Spec Spec k x n,m . We will denote the corresponding function by L E,m,x . In what follows, we fix, once and for all, a geometric point x over each closed point x X . We denote by E x the stalk of E at . Let P n,m be the set ( 1, 2,, n) i , 1 2 n 0, i i m . We can consider P n,m as a subset of the set P n of dominant weights of GL n . For P n,m , we write E x() for the representation of GL n()GL(E x) of highest weight . The stack n,m (x) has a stratification by locally closed substacks n,m (x) indexed by P n,m . The stratum corresponding to ( 1, 2,, n)P n,m parametrizes torsion sheaves of the form T X X(- 1 x) X X(- nx). Let B ,x denote the intersection cohomology sheaf associated to the constant sheaf on the stratum n,m (x) . Let T be an --valued point of n,m and let x X be a closed point with residue field k x . The pull-back of T to a sufficiently small Zariski neighborhood of x in X Spec Spec k x gives rise to an object T x of n,m x (x) , i.e., to a k x --rational point of n,m x (x) , for some m x . We have: x X m x(x) m . Moreover, there is a bijection: equation bijec () x X ' ; ; n(x)(k x). equation The explicit description of L E is given in the following proposition. Note that the shifts in degrees are due to the fact that the stack n,m (x) has dimension -m , whereas the dimension of n,m is zero. prop La1 , (3.3.8) descr (1) Let T be an --point of n,m . Then, using the notation above, we have: L E,m ( T ) x X L E,m x,x ( T x). (2) Furthermore, L E,m,x P n,m B ,x m ( -n() ) E x(), where n() i 1 n (i-1) i . prop rem 3 Let x be an -rational point of X . Consider now the variety N m gl m of nilpotent matrices. The stack m,m (x) is isomorphic to the stack N m GL m , where GL m acts on N m by conjugation. Let : N m N m denote the Springer resolution and let S p m R denote the Springer sheaf on N m . The sheaf S p m has a natural action of the symmetric group S m . It is shown in La1 that L E,m,x ( S p m (E x) m ) S m n,m (note that n,m is an open substack of m,m ). Hence the function L E associated to L E can be expressed via the Kostka-Foulkes polynomials; see La1 . However, it will be more convenient for us to use another interpretation of the sheaf L E,m,x , via the affine Grassmannian (see Sect. 4.2). This interpretation allows us to express L E in terms of the Hecke algebra H (GL n( ),GL n()) ; see Sect. 5.5. The fact that the two interpretations agree is due to Lusztig Lu1 . rem The sheaf Define a morphism of stacks : Q that sends a quintuple L S, S,J S,(J i,S ),(s i,S ) to the sheaf L S Im S . Now we define a morphism : Q , which at the level of --points sends L,,J,(J i),(s i) to the sum of n-1 classes in Ext 1( i, i-1 ) Ext 1(J i J i 1 ,J i-1 J i) that correspond to the successive extensions 0J i-1 J i J i-1 J i 1 J i J i 1 0. Given two coherent sheaves L and L' on X , consider the stack E xt 1(L',L) , such that the objects of the groupoid Hom (S, E xt 1(L',L)) are coherent sheaves L'' on X S together with a short exact sequence 0 L S L'' L' S 0, and morphisms are morphisms between such exact sequences inducing isomorphisms at the ends. There is a canonical morphism of stacks E xt 1(L',L) Ext 1(L',L) . We have for each i 1,,n-1 , a natural morphism i: Q E xt 1( i, i-1 ) , as above. Now is the composition Q i 1 n-1 E xt 1( i, i-1 ) i 1 n-1 Ext 1( i, i-1 ) n-1 . Let I be the Artin-Schreier sheaf on corresponding to the character . Recall that the Galois representation gives rise to a rank n local system E on X and to the sheaf L E on . Define the sheaf on Q as : ( L E) ( I ). Note that Q is an open substack in a vector bundle over the product of and a smooth stack that classifies extensions J as above. Hence the map is smooth, and is the Goresky-MacPherson extension from its restriction to the open substack -1 ( rss ) . Geometric Langlands conjecture for GL n Recall that we have a representable morphism of stacks : Q ' n that associates the pair L,s n-1 to an object L,,J,(J i),(s i) . We define the complex of --adic sheaves S ' on ' n to be the direct image S ': (). The following conjecture of Laumon is a geometric version of precise1 . conj La2 princ Let be in G n and E be the corresponding irreducible --adic local system on X . Then itemize The restriction of the complex S ' to each connected component of M ' n is an irreducible perverse sheaf up to a shift in degree. S 'p ( S ) , where p is the natural morphism ' n n , and S is a complex of sheaves on n , whose restriction to each connected component of M n is an irreducible perverse sheaf up to a shift. The sheaf S E is an eigensheaf of the Hecke correspondences in the sense of La1 , (2.1.1). itemize conj If this conjecture is true, then the function on M n associated to the sheaf S E is the automorphic function f corresponding to . The sheaf S E can therefore be called the automorphic sheaf corresponding to . The conjecture means that the sheaf S ' E is constant along the fibers of the morphism p . Thus, it is analogous to precise1 . The advantage of dealing with princ as compared to precise1 is that while the latter is a global statement, one could use local geometric information about the sheaf S ' E to tackle princ (as Drinfeld did in the case of GL 2 Dr ). rem The above conjecture is obviously false if one does not assume the irreducibility of (the complex S ' must be corrected by the corresponding constant terms'' in this case). However one can construct the automorphic sheaves S corresponding to 's, which are direct sums of one-dimensional representations, by means of the geometric Eisenstein series La:e . rem Main theorem Let S' denote the function on M' n associated to S ' . If princ is true, then this function has the same properties as the function f' defined in Sect. 2.3. Recall that these properties uniquely determine f' up to a non-zero factor. Therefore princ can be true only if the functions S' E and f' are proportional. This was conjectured by Laumon in La2 (Conjecture 3.2). One of our motivations was to prove this conjecture. More precisely, we prove the following: thm prin The functions S' and f' are equal. thm prin means that the function f' does come from a complex of --adic sheaves on M ' n . It also provides a consistency check for princ . Laumon has proved prin in La1 for GL 2 by a method different from the one we use below. We derive prin for GL n with arbitrary n from the following statement. prop Let F be the function on corresponding to . The function r (F) coincides with the restriction of the Whittaker function W to Q . prop prin immediately follows from because of r1 as shown in the diagram below. 1mm center picture (65,60)(-30,-55) (-4,0) (-1,0) 20 (4,0) (1,0) 20 (0,-3.5) (0,-1) 20 (0,-32) (0,-1) 20 (2.5,-2.5) (1,-1) 8.5 (11,-15.5) (-1,-1) 8.5 (-1.5,-1) (-30,-1) F q (-2,-29) M' n (-2,-56) M n (12,-14) Q (26,-1) Coh n (-15,2) (13,2) (-4,-14) (-4,-43) p (8,-6) r (8,-21) picture center The proof of will occupy Sects. 4 and 5 below. can be interpreted in the following way. Denote by Coh n (resp., Coh n,m (x) ) the set of --points (resp., k x --points) of n (resp., n,m (x) ). Set Coh n(x) m0 Coh n,m (x) . Then by formula bijec , Coh n ' x X Coh n(x) . We have: Coh n(x) P n m0 P n,m . Hence we can identify Coh n(x) with the set diag ( x 1 ,, x n ) 1 n 0 . The Whittaker function W can then be restricted to the Coh n , and it is uniquely determined by this restriction; see Sect. 2.2. Thus, both L E and W give rise to functions on Coh n . For each point t Coh n , the value of L E at t is given by taking the alternating sum of traces of the Frobenius on the stalk cohomologies of L E at t . On the other hand, the value of W is given by the trace of the Frobenius on the top stalk cohomology of L E at t (see La1 ). Therefore says that the contributions of all stalk cohomologies, other than the top one, are killed by the summation along the fibers of the projection r against the non-trivial character . rem As we mentioned in the introduction, Drinfeld has proved a version of princ for GL 2 . The case of GL 2 is special as explained below. Let M ' 2 be the open substack of M ' 2 , which parametrizes L,s , such that the image of s: L is a maximal invertible subsheaf of L . Due to the Hecke eigenfunction property of f' with respect to T 2 x , f' is uniquely determined by its restriction to ' 2 M ' 2() . But the map r is a bijection over -1 ( ' 2 ) , and hence -1 ( ' 2 ) can be considered as a subset of . Clearly, the map of Sect. 3.3 sends -1 ( ' 2 ) to Coh 1 Coh 2 . Furthermore, the restriction of L E to the stack 1 is simply a sheaf, i.e., it has stalk cohomology in only one degree. Therefore on Coh 1 the function L E equals the Whittaker function W . Hence, restricted to M ' 2 , the geometric construction evidently coincides with the construction described in Sect. 2.3. rem Reduction to a local statement Adelic interpretation of Recall that is the set of isomorphism classes of F q --points of Q , and in Sect. 3.1 we defined a map r:Q . Recall further that Coh n(x) P n m0 P n,m , and Coh n ' x X Coh n(x) . Denote GL n J( x) GL n J( x) Mat n J( x) . Let GL n J( ) be the restricted product ' x X GL n J( x) . We have bijections: equation iso1 GL n J( x) GL n J( x) GL n J( x) Coh n(x) equation and equation iso2 GL n J() GL n J( ) GL n J() Coh n. equation prop adelic (1) There is a bijection between the set and the set (N J(F)N J( )) N J() (GL n J( ) GL n J()). The group N J() acts on the product (N J(F)N J( ))(GL n J( ) GL n J()) according to the rule y(u,g) (uy -1 ,yg) . (2) The map r: Q identifies with the map (N J(F)N J( )) N J() (GL n J( ) GL n J()) Q N J(F)GL n J( ) GL n J() given by (u,g)ug . The map sends (u,g) to the image of g in GL n J() GL n J( ) GL n J() Coh n . The map : is the composition of the natural map N J(F) N J( ) N J() n-1 and the summation n-1 . prop proof We will use the notation introduced in the proof of easy . Let L,,J, (J i), (s i) be an element of . Then the triple J,(J i),(s i) is an element of Q . Hence we can associate to it homomorphisms J x: J 0 x J x , J : J 0 x J x and J x ( J x) -1 J : J 0 x J 0 x , as in the proof of easy . On the other hand, let x L be an isomorphism J 0 x L x . We extend it to a homomorphism J 0 x L x , which we denote by the same symbol. Denote by x the homomorphism J x L x induced by . Consider the automorphism x ( x L) -1 x x J of J 0 x . Now we assign to L,,J,(J i),(s i) the element ((u x),(g x)) of GL n J( ) GL n J( ) , where u x is the transpose of the matrix representing x , and g x is the transpose of the matrix representing x (see the proof of easy ). By construction, u x N J( x) and g x GL n J( x) . Furthermore, the arbitrariness in the choice of J corresponds to left multiplication of u x by elements of N J(F) , the arbitrariness in x L corresponds to right multiplication of g x by elements of GL n J( x) , and the arbitrariness in x J corresponds to the action of N J( x) on (u x,g x) according to the rule y (u x,g x) (u x y -1 , y g x) . This proves part (1) of the proposition. The proof of part (2) is now straightforward. proof Recall that L E is the function on Coh n associated to the sheaf L E . Denote by L E,x the function on Coh n(x) whose restriction to n,m (x)() is the function associated to L E,m,x . Part (1) of descr implies that L E(( x)) x X L E,x ( x), for all ( x) ' x X P n . Using the bijection iso1 (resp., iso2 ) we consider L E,x (resp., L E ) as a function on GL n J( x) (resp., GL n J( ) ). Let x: N J( x) be the character defined in Sect. 2.2. Note that the function x (resp., L E,x ) is right (resp., left) N J( x) --invariant. Recall further that F E is the function on associated to the sheaf F E . We conclude: lem ide Under the isomorphism of adelic , F E x X x L E,x . lem Let us extend the function L E,x by zero from GL n J( x) to GL n J( x) . Then adelic and ide imply: (r F E)((g x)) x X ; ; u x N J( x) N J( x) L E,x (u x -1 g x) x(u x), for all g x GL n J( x) (each sum is actually finite). Let du x be the Haar measure on N J( x) normalized so that N J( x) du x 1 . Using left GL n J( x) --invariance of L E,x , we can rewrite the last formula as equation globalint (r F E)((g x)) x X N J( x) L E,x (u x -1 g x) x(u x) ; du x. equation states that (r F E)((g x)) W ((g x)) Q . According to formulas wsigma and globalint , this is equivalent to the formula equation show N J( x) L E,x (u x -1 g x) x(u x) ; du x W ( Fr x) (g x), equation for all g x GL n J( x) . Since both the left and the right hand sides of show are left (N J( x), x) --equivariant and right GL n J() --invariant, it suffices to check formula show when g x diag ( 1 ,, n ) , where ( 1,, n) P n . Using the explicit formula cassha , we reduce to the following local statement. prop Fplus equation oh N J( x) L E,x (u x diag ( x 1 ,, x n )) -1 (u x) ; du x q x n() Tr (( Fr x),E x()), equation where P n . prop Positive part of the affine Grassmannian Recall that J 0 i 0 n-1 i . From now on we work in the local setting. Hence we choose once and for all a trivialization of J 0 on the formal neighborhood of x X and identify GL n J( x) with GL n( x) . For this reason we suppress the index J in what follows. According to part (2) of descr , equation le L E,x P n Tr (( Fr x),E x()) B , equation where B is the function on Coh n(x) associated to the sheaf B ,x m (-n()) (see Sect. 3.2). We view B as a GL n( x) --invariant function on GL n( x) GL n( x) . Let x be a closed point of X . To simplify notation, from now on we will assume that x is an --rational point of X (otherwise, we simply make a base change from to the residue field k x of x ). Define the functor which sends an --scheme S to the set of isomorphism classes of pairs L,t , where L is a rank n bundle on X S and t is its trivialization on (X S) - ( x S) . This functor is representable by an ind--scheme x (see BL1 ), which we call the affine Grassmannian x (for the group GL n ). The ind--scheme x splits into a disjoint union of connected components: x m Z x m indexed by the degree of L for L,t x . There is a bijection between the set Gr x of F q --points of x and the quotient GL n( x) GL n( x) ; see BL . The analogous quotient over the field of complex numbers is known as the affine, or periodic, Grassmannian. This explains the name that we use. There exists a proalgebraic group ( x) whose set of --points is GL n( x) . The group ( x) acts on x , and its orbits stratify x by locally closed finite-dimensional subschemes x indexed by the set P n of dominant weights of GL n . The stratum x is the ( x) --orbit of the coset diag ( x 1 ,, x n ) GL n( x) Gr x . Recall that there is an inner product on the set of GL n weights defined by the formula (,) i 1 n i i , and that x 2(,) , where is the half sum of the positive roots of GL n , ((n-1) 2,(n-3) 2,,-(n-1) 2) . Let Q , be the constant sheaf supported on the stratum x . Denote by A ,x the intersection cohomology sheaf on the closure of x , which is the Goresky-MacPherson extension of the sheaf Q , 2(,) ((,)) . Let A be the GL n( x) --invariant function on GL n( x) GL n( x) , which is the extension by zero of the function associated to A ,x . Define now the closed subscheme x of x which at the level of points corresponds to pairs L,t , for which t extends to an embedding of X --modules X n L . The scheme x also splits into a disjoint union of connected components x m x m, , where x m, x x m . It is clear that x m, is a union of the strata x , for P n,m (see Sect. 3.2). The set x () identifies with the quotient GL n( x) GL n( x) . Consider the morphism q m,x : x m, m,n (x) , which sends a pair L,t to the quotient X n Im t , where t : L X n . This morphism can be described as follows. There is a natural vector bundle m,n over the stack m,n , whose fiber at T is Hom ( X n , T ) ; x m, is an open substack of the total space of the bundle n , which corresponds to epimorphic elements of Hom ( X n , T ) . The map q m,x is simply the projection from this substack to the base. Hence q m,x is a smooth morphism of algebraic stacks. It is clear that it preserves the stratification (compare with Lu1 ). Note that x m, m(n-1) and n,m (x) -m . Therefore lem equation smooth q m,x B ,x m (-n()) A ,x -m(n-1) (-(,)-n()). equation lem Recall that m i 1 n i . The lemma implies that as functions on GL n( x) GL n( x) , equation equ B (-1) (n-1) q x (,) n() A . equation Hence, according to le , equation le1 L E,x P n (-1) (n-1) q x (,) n() Tr (( Fr x),E x()) A . equation This formula will be used in the next section in the proof of Fplus . Proof of the local statement In this section we will state and prove a general result concerning reductive groups over the local non-archimedian field of positive characteristic. In the case of GL n this result implies Fplus . General set-up Let G G( K ) be a connected, reductive, split algebraic group over the field K (()) , T -- its split maximal torus contained in a Borel subgroup B , and N -- the unipotent radical of B . We again denote by O the ring of integers of K , by a generator of its maximal ideal, and by q the cardinality of the residue field k F q . Let K be the compact subgroup G( O ) of G . We fix a Haar measure of G , such that K has measure 1 . Let H (G,K) denote the Hecke algebra of G with respect to K , i.e., H (G,K) is the algebra of --valued compactly supported K --bi-invariant functions on G with the convolution product: equation conv (f 1 f 2)(g) G f 1(x) f 2(gx -1 ) ; dx. equation Let be the Langlands dual group of G (without the Weil group of ), and (resp., ) be the set of weights (resp., dominant weights) of . Each can be viewed as a one-parameter subgroup of G( ) , and hence () is a well-defined element of G( x) . We denote by c the characteristic function of the double coset K () K G . The functions c form a basis of H (G,K) . Let Rep () denote the Grothendieck ring of the category of finite-dimen-sional representations of () . We consider it as a --algebra. If V is a finite-dimensional representation of , denote by V the corresponding element of Rep () . In particular, for each , let V() be the finite-dimensional representation of with highest weight . The following statement, often referred to as the Satake isomorphism, is well-known; see Sa , La , Mac , . thm Satake There is a unique isomorphism S: Rep () H (G,K) , which maps V() to H q -(,) ( c ; a c ), a . thm rem character Each semi-simple conjugacy class of the group () defines a homomorphism : Rep () , which maps V to Tr (,V) . We denote the corresponding homomorphism H by the same symbol . This allows us to identify the spectrum of the commutative algebra H with the set of semi-simple conjugacy classes of () . In particular, we have: (H ) Tr (,V()) . rem Hecke algebra and the affine Grassmannian he Let again X be as in Sect. 1.1, and x be its --point. We define, in the same way as in Sect. 4.2 for G GL n , the ind--scheme G r (G) (G) x that classifies pairs P ,t , where P is a principal G --bundle on X and t is its trivialization over X-x . The ind--scheme structure on (G) is described, e.g., in LS . Note that due to the results of BL , DS , the global curve X is inessential in the above definition. We could simply take X Spec . In particular, there is a bijection between the set of F q --points of (G) and the set G K . There is a proalgebraic group () , whose set of --points is G() . This group acts on (G) , and its orbits stratify (G) by locally closed finite-dimensional subschemes (G) indexed by the set of dominant weights of . The stratum (G) is the G() --orbit of the coset () G() , where () T( ) G( ) is defined above. Denote by (,) the pairing between and the sum of the fundamental coweights of . Let Q , be the constant sheaf supported on the stratum (G) . Denote by A A ,x the intersection cohomology sheaf on the closure of (G) , which is the Goresky-MacPherson extension of the sheaf Q , 2(,) ((,)) (note that (G) 2(,) ). Let A be the function associated to A . We use the same notation for its extension by zero to the whole (G) . Clearly, the functions A , , form a basis in the --vector space (G K) K of K --invariant functions on G K with compact support. We have an isomorphism of vector spaces: H (G K) K , which commutes with the action of H . Therefore H H , , can also be considered as elements of (G K) K . prop hla H (-1) 2(,) A . prop This result is due to Lusztig Lu1 , Lu2 and Kato Ka (see, e.g., Theorem 1.8, Lemma 2.7, formula (3.5) of Ka ). It implies that equation stalk H (y) (-1) 2(,) i i( A ) y ; q i 2 , equation where j( A ) y is the j th stalk cohomology of A at y Gr(G) . Fourier transform Let us denote by Res : the map defined by the formula Res ( n f i i ) f -1 . We define a character of N in the following way: (u) i 1 l ( Res (u i)), where u i, i 1,,l N N,N , are natural coordinates on N N,N corresponding to the simple roots and : is a fixed non-trivial character. Consider the space (G K) N of left (N,) --equivariant and right K --invariant functions on G that have a compact support modulo N . This space is a module over H (G,K) , with the action defined by formula conv . For , let be the function from (G K) N , which vanishes outside the N --orbit of () and equals q -(,) at () . The elements , , provide a --basis for (G K) N . Define the linear map : (G K) K (G K) N by the formula equation check1 ((f))(g) N f(ug) -1 (u) ; du, equation where du stands for the Haar measure on N normalized so that N( O ) ; du 1 . lem The map defines an isomorphism of H --modules (G K) K (G K) N . lem proof Let c (G K) K be the characteristic function of the K --orbit of the coset () K G K . It follows from the definition that (c ) q (,) ; b , . This implies the lemma. proof It is natural to call the Fourier transform. We are now ready to state our main local theorem. thm local The map sends H to . thm This theorem is equivalent to the formula equation formula N H (u()) -1 (u) ; du q -(,) , . equation Proof of local proof Our proof relies on the result of Casselman-Shalika (and Shintani for G GL n ), which describes the values of the Whittaker function at the points () (cf. Theorem 2.1 and 1). Let be a semi-simple conjugacy class in and let W be a -valued function on G with the following three properties: itemize W (gh) W (g), h K , W (1) 1 ; W (ug) -1 (u) W (g), u N ; equation hecke G f(x) W (gx) dx (f) W (g), g G, f H equation (see character for the definition of ). itemize thm CS , Shi prime The function W satisfying these properties exists, and it is unique. For , the value of this function at () is equation css W (()) q -(,) Tr (,V()) equation if is a dominant weight, and 0 otherwise. thm The function W is called the Whittaker function corresponding to . Now we can prove local . Let be as above and let s be a linear functional (G K) N given by the formula equation sgamma s () NG W (g) (g) ; dg, equation where dg is the measure on N G induced by the Haar measure on G from Sect. 5.1 and the Haar measure on N from Sect. 5.3. By construction, the function W (u) (u) is left N --invariant. The integral sgamma converges, because, by definition, has compact support modulo N . lem mapp The map s is a homomorphism of H (G,K) --modules (G K) N Q , , where Q , is the one-dimensional representation of H (G,K) corresponding to its character . lem proof Each f H acts on (G K) N by mapping (G K) N to f . By definition of the convolution product (see formula conv ), we have: (f )(y) G f(x) (yx -1 ) ; dx. Hence s (f ) NG W (g) ( G f(x) (g x -1 ) ; dx ) ; dg. Changing the order of integration and using the invariance of the Haar measure, we obtain s (f ) NG ( G W (gx) f(x) ; dx ) (g) ; dg. By hecke , s (f ) (f) s (). proof By formula css and the definition of the function , the function W equals q -2(,) Tr (, V()) times the characteristic function of the double coset N () K . Hence s ( ) N G W (g) (g) dg equals q -(,) Tr (, V()) times the measure of the right K --orbit () K in N G . This measure equals (K Ad () (N())) (N()) (Ad () (N())) due to our normalization. The latter equals q 2(,) . Therefore s ( ) Tr (, V()) for each . Any (G K) N can be written as a finite sum a , where a . We can identify the vector space (G K) N with Rep () , by mapping to V() . Let be the image of in Rep () under this identification. Then s () a Tr (,V()) is simply the value of at Spec Rep () . Since the algebra Rep () has no nilpotents, ' , if and only if s () s (') for all semi-simple conjugacy classes in () , Therefore to prove local , it is sufficient to check that for each semi-simple conjugacy class in () and , s (H ) Tr (,V()). But the composition s : (G K) K Q , is a homomorphism of H -modules, by and mapp . It is easy to check directly that the value of this homomorphism on the element H 0 ch K (G K) K equals 1 . Therefore s (H ) s (H H 0) (H ) s (H 0) (H ) Tr (,V()) (see character ), and local follows. rem equi Our proof shows that local is equivalent to prime . rem Proof of Fplus Note that (-1) 2(,) (-1) (n-1) . Hence we obtain from hla and formula le1 : equation lenew L E,x P n q x (,) n() Tr (( Fr x),E x()) H . equation Therefore we find: align N J( x) L E,x (u x diag ( x 1 ,, x n )) -1 (u x) ; du x P n q x (,) n() Tr (( Fr x),E x()) N J( x) H (u x diag ( x 1 ,, x n )) -1 (u x) ; dx. align According to formula , the latter sum equals q x n() Tr (( Fr x),E x()) , which is the right hand side of formula oh . Now Fplus is proved, and this finishes the proof of and prin . Whittaker functions and spherical functions In this section we give an interpretation of local from the point of view of the theory of spherical functions. Throughout this section we will work over the field of complex numbers instead of . In particular, all functions will be --valued, and H will be a --algebra. The map Denote by C (G K) K (resp., C (G K) N ) the space of smooth left K --invariant (resp., (N,) --equivariant) and right K --invariant functions on G . We also denote by (G K) K (resp., (G K) N ) the subspace of compactly supported (resp., compactly supported modulo N ) functions. Each element of C (G K) K can be written as an infinite sum a c , where c is the characteristic function of the G() --orbit Gr(G) . lem fini For each g G , ((c ))(g) 0 for all but finitely many . lem proof It suffices to prove the statement for g () . In this case, it is easy to see that for all but finitely many , there exists an element v N (depending on ) with (v) 1 , such that u N , u () Gr(G) if and only if (vu) () Gr(G) . But then ((c ))(()) (v) ((c ))(()) , and hence ((c ))(g) 0 . proof Therefore defines a map C (G K) K C (G K) N , f (f) , which is equivariant with respect to the action of Hecke operators. Now we define a map : C (G K) N C (G K) K by the formula equation inverse ((f))(g) K f(kg) dk, equation where dk stands for the Haar measure on K of volume 1 . This map is also equivariant with respect to the action of Hecke operators. We define a as the element of (G K) K equal to ()( ch K) ( 0) . The same argument as in the proof of fini shows that the map sends functions from (G K) N to (G K) K . Hence a (G K) K H . Introduce the notation (a f)(g) G a(x) f(gx) ; dx. Then we obtain: equation star ()(f) a f, f C (G K) K. equation In the next section we will use the element a to clarify the connection between Whittaker functions and spherical functions. Connection between a and the Plancherel measure Let be a semi-simple conjugacy class in the group () . Recall Sa , Mac that the spherical function S is the unique K --bi-invariant function on G , such that itemize f S (f) S , f H , where : H is the character corresponding to defined in character ; S (1) 1 . itemize These properties imply that equation vazh G f(x) S (x) ; dx (f). equation Now let W be the Whittaker function on G as defined in Sect. 5.4 but with the character -1 of N replaced with the character . It is straightforward to check that the function (S ) satisfies all the properties of the function W from proof , except for the normalization condition W (1) 1 . By prime , (S ) is proportional to W . lem whereitgoes (S ) (a) W . lem proof Introduce a() by the formula (S ) a() W . Since (W ) S by definition, we obtain, using formula star and the properties of S : a() S ()(S ) a S (a) S . proof According to Mac , (1.5.1), there exists a unimodular measure d() (Plancherel measure) on the maximal compact subtorus u of , which satisfies equation plancherel G f 1(g) f 2(g) dg u (f 1) (f 2) ; d(), equation for all f 1, f 2 H . Setting f 2 ch K , we obtain: equation raz f(1) u (f) ; d(), f H . equation By local , (H ) . But it is clear that (( ))(1) ,0 . Therefore, using star , we see that (a H )(1) ,0 . Substituting this into formula raz and using the formula (H ) Tr (,V()) , we obtain: equation del u Tr (,V()) a() ; d() ,0 . equation There exists a unique measure d () on u (induced by the Haar measure on u ), such that equation orth u Tr (,V()) Tr (,V()) ; d () , . equation Formula del then implies prop aga a() d () d() . prop Now whereitgoes gives us: equation proportion (S ) d () d() W . equation Another proof of local In this subsection, which is independent from the previous one, we use spherical functions to give another proof of local . Substituting f 1 ch KgK into formula plancherel and using formula vazh , we obtain that for any f H , equation fourier f(g) u S (g) (f) ; d(). equation Since (H ) Tr (,V()) , we have: equation alambda H (g) u S (g) Tr (,V()) ; d(). equation According to formula alambda , equation a0 H 0(g) u S (g) ; d(). equation Hence (H 0)(g) u (S )(g) ; d() u W (g) a() ; d(). On the other hand, it is clear from definition that (H 0) 0 . Therefore, substituting g () and using formula css , we obtain formula del . Repeating the argument with the Haar measure given above, we obtain proportion . Now formulas alambda , proportion and css give: ((H ))(()) u W (()) Tr (,V()) ; d () q -(,) u Tr (,V()) Tr (,V()) ; d () q -(,) , . This proves formula formula and local over the field of complex numbers. Since H takes values in rational numbers and takes values in the roots of unity, the validity of formula over implies its validity over . The function L The Whittaker function can be written as a series W Tr (,V()) . This series obviously makes sense, since the supports of the functions do not intersect. In view of local , it is natural to consider the series equation Fgamma L Tr (,V()) H . equation However, the convergence of this series is not at all automatic, because the supports of functions H do intersect; for instance, each H has a non-zero value at 1 . In this section we study the question of convergence of L . Let us write: H q -(,) P (q -1 ) c , where q -(,) P is a polynomial in q -1 (recall that c is the characteristic function of the G() --orbit (G) ). Formula alambda can be rewritten as follows: equation al1 H (g) u S (g) a() -1 Tr (,V()) ; d (). equation Using the defining properties of the spherical function S , we can write it as a series equation sga S s (q -1 ) c , equation where s (q -1 ) is a rational function in q -1 of the form Q(q -1 ) (q -1 ) . Here Q(q -1 ) i 1 l 1-q -m i-1 1-q -1 ( l is the rank of G , the m i 's are the exponents of G ; note that Q(q) G B() ), and is a polynomial in q 1 . Its coefficients are finite integral linear combinations of characters of irreducible representations of (for an explicit formula, see Mac ). It follows from formula whereitgoes that a() has the same structure as a function of q -1 . Hence both s (q -1 ) and a() -1 can be viewed as formal Laurent power series in q -1 and formula al1 can be viewed as an identity on such power series. We have: s m -M Tr (,R m) q -m and a() -1 m -M' Tr (,U m) q -m , where R m and U m are finite linear combinations of irreducible representations of (the summation is actually only over m ). Then formula al1 can be written as follows: equation f1 q -(,) P (q -1 ) N q -N u Tr (, m R m U N-m ) Tr (,V()) ; d (). equation By formula orth , the q -N coefficient of P (q -1 ) equals the multiplicity of V() in m R m U N-m . But by construction the latter is a finite linear combination of irreducible representations of . Hence we obtain the following lem For each and N there are only finitely many , such that q -(,) P has a non-zero coefficient in front of q -N . lem rem This can also be seen from the explicit formula for P obtained in Lu2 , Ka . rem Therefore for each g G , L (g) given by formula Fgamma makes sense as a formal power series in q -1 , each coefficient being a finite linear combination of characters. Furthermore, we see, by reversing the argument above that as formal power series in q -1 , equation propor L (g) a() -1 S (g), g G. equation In order to estimate the convergence of this series, we have to compute a() explicitly. According to Mac , (5.1.2), d() Q(q -1 ) W (1-()) (1-q -1 ()) d, where W is the number of elements in the Weyl group, and is the set of roots of G . The notation d means the Haar measure on u , which gives it volume 1 . On the other hand, d () 1 W (1-()) d. Now aga gives: equation propsg a() (1-q -1 ()) Q(q -1 ) . equation Let L g be the adjoint representation of , and i( L g ) be its i th exterior power. Formula propsg means that a H is the image under the Satake isomorphism S of the following element in Rep : i 1 l (1-q -m i-1 ) -1 i 0 L g i( L g ) (-1) i q -i . Formula propor implies the following result. prop converge If the conjugacy class satisfies: q -1 () q, , then L (g) converges absolutely to Q(q -1 ) (1-q -1 ()) S (g) for all g G . prop Note that when G GL n , formula Fgamma looks similar to formula lenew for the function L E,x . Besides powers of q x , the difference is that in lenew the summation is restricted to the subset P n of the set P n of all dominant weights of GL n . However, L E,x is not the restriction of Fgamma to the union of strata Gr(GL n) with P n , because the functions H with P n - P n do not vanish on those strata. While L given by Fgamma is manifestly an eigenfunction of the Hecke operators, L E,x is not. It is actually an eigenfunction of some other operators, similar to the Hecke operators, which were defined by Laumon La1 . For general G there is no analogue of the subset , and so the function L seems to be the closest analogue of L E,x in the general setting. According to local and formula css , (L ) equals the Whittaker function W . rem lfunct Let be a semi-simple conjugacy class of () and r: () Aut V a finite-dimensional representation of () . Recall that the local L --function associated to the pair (,V) is defined by the formula equation lfun L(,V;s) det ( 1 - r() q -s ) -1 . equation In particular, if V , L g is the adjoint representation, then L(, L g ;s) (1-q -s ) -l (1 - () q -s ) -1 . Hence a() can be written as a() L(, L g ;1) i 1 l (1-q -m i-1 ). Thus, we obtain: equation vani (S ) L(, L g ;1) -1 i 1 l (1-q -m i-1 ) -1 W . equation Using arguments similar to those of Casselman and Shalika CS , one can show that the irreducible unramified representation corresponding to has a Whittaker model if and only if (S ) 0 . Formula vani means that (S ) 0 if and only if L(, L g ;s) is regular at s 1 . We conclude that the irreducible unramified representation of G with the Langlands parameter has a Whittaker model if and only if L(, L g ;s) is regular at s 1 (in that case the Whittaker model is actually unique). This agrees with a special case of a conjecture of Gross and Prasad GP (Conjecture 2.6). rem Identities on P According to formula propor , for each we have the following equality of power series in q -1 : equation id1 : Tr (,V()) q -(,) P (q -1 ) i 1 l 1-q -m i-1 1-q -1 (1-q -1 ()) -1 S (()). equation Recall that the coefficients of the polynomial P (which can be interpreted as a Kazhdan-Lusztig polynomial for the affine Weyl group Lu2 , Ka ) are given by dimensions of stalk cohomologies of the perverse sheaf A : equation P q (,) i i( A ) () ; q i 2 . equation Thus, formula id1 is an identity which connects these dimensions with the values of the spherical functions. The latter are known explicitly; they can be expressed via the Hall-Littlewood polynomials Mac . For example, let us apply formula id1 when G SL 2 and 0 . In this case, , and the set of dominant weights of the dual group PGL 2 can be identified with the set of non-negative even integers. To weight 2m corresponds the 2m 1 --dimensional representation V(2m) of PGL 2 , and Tr (,V(2m)) i -m m i . Formula id1 then gives: m ( j -m m j ) i i( A 2m ) 1 ; q i 2 1 q -1 (1-q -1 )(1-q -1 -1 ) . This is easy to see directly, because A 2m is known to be the constant sheaf on the closure of the stratum (SL 2) 2m placed in degree -2m . For general G , formula id1 with 0 can be interpreted as follows. Let R() be the graded ring of polynomials on , J() be its subring of --invariants, and H() be the subspace of --harmonic polynomials on . For a graded space V , we denote by V j its j th homogeneous component. If each V j is a representation of , we denote by Ch (,V) the graded character of V : Ch (,V) j 0 Tr (,V j) q -j . Note that the graded character of R() equals equation tozh1 Ch (,R()) i 1 l (1-q -1 ) -l (1-q -1 ()) -1 equation (compare with lfunct ), while Ch (,J()) i 1 l (1-q -m i-1 ) -1 . Now let H() be the (graded) space of --harmonic polynomials on . According to Theorem 0.2 of B. Kostant Ko , R() J() H() . Hence Ch (,H()) i 1 l 1-q -m i-1 1-q -1 (1-q -1 ()) -1 a() -1 . Thus, we obtain another interpretation of a() -1 : it is equal to the graded character of the space of --harmonic polynomials. Note that it also equals the graded character of the ring of regular functions on the nilpotent cone N in . Formula id1 together with this interpretation give us the following result. prop equation new P 0, (q -1 ) q (,) j 0 q -j mult (V(),H() j), equation where mult (V(),H() j) is the multiplicity of V() in H() j . prop A complete description of these multiplicities has been given by Kostant in Ko . In fact, applying Theorem 0.11 of Ko to formula new , we obtain: equation tozh2 P 0, q (,) i 1 l q -m i() , equation where m i() are the generalized exponents associated to the representation V() , defined in Ko . In the special case when V() is the adjoint representation L g , these are just the exponents of , and formula tozh2 specializes to Lusztig's formula (see Lu2 , p. 226) P 0, adj i 1 l q m i-1 . Formula tozh2 is not new: R. Brylinski Br observed that it immediately follows if one compares the Lusztig-Kato formula Lu2 , Ka for P 0, and the Hesselink-Peterson formula He for the right hand side of tozh2 . Note that in contrast to her argument, our proof of formula tozh2 does not use the Lusztig-Kato formula. Using f1 it is easy to derive a formula analogous to new for P in terms of Hall-Littlewood polynomials. Geometric analogue of local and some open problems In this subsection we will formulate a geometric analogue of local . Recall the Grassmannian (G) of section 5.2. This is a strict ind--scheme over , i.e., an inductive system of --schemes (G) k, k0 , where all maps i k,m : (G) k (G) m, k m , are closed embeddings. For more details, see, e.g., LS , MV . By a --sheaf on (G) we will understand a system of --sheaves k on each (G) k and a compatible system of isomorphisms k i k,m m for all k m . There exists an ind--group scheme N ( ) , whose set of --points is N( ) . This ind-group scheme acts on the Grassmannian (G) , and its orbits stratify (G) by ind-schemes S , . The stratum S is the --orbit of () Gr(G) . Denote by j the embedding S (G) . We choose a generic additive character : G a,K and define a homomorphism : by the formula Res ,: ,, where Res is the geometric analogue of the residue map of Sect. 5.3. As before, let : denote a non-trivial character and let I denote the corresponding Artin-Schreier sheaf on . Consider the category () ( (G)) of () --equivariant perverse sheaves on (G) with finite-dimensional support. This category is a geometric analogue of the Hecke algebra H (G K) K (see cate below). Now we define an abelian category Sh ( (G)) (a geometric analogue of (G K) N ) and a collection of cohomology functors i: () ( (G)) Sh ( (G)) , which are a geometric analogue of the map of Sect. 5.3. The objects of the category Sh ( (G)) are --sheaves E on (G) which satisfy the following conditions: (1) j E 0 except for finitely many ; (2) t j E I -1 are trivial local systems of finite rank for all , where t : S is the map u u (t) . Morphisms in this category are defined in an obvious way. lem first If is not dominant, then for every E Ob ( Sh (G)) , j ( E ) 0 . lem The proof is analogous to the proof of the corresponding statement for functions. Thus, E not only satisfies property (1) above, but also satisfies the stronger property (1 ' ) j E 0 except for finitely many . Now we construct the functors i: () ( (G)) Sh ( (G)) . Consider the sequence of maps: CD a q p ,, CD where a is given by acting with on and p,q are projections. For () ( (G)) we then set: i() R i p ( I ), with q a ,. To formulate the geometric analogue of local , recall that for LP we denote by A the Goresky-MacPherson extension of the sheaf Q , 2(,) ((,)) associated to the () --orbit (G) . For each , denote by the isotropy group of () . Since is dominant, the restriction of to equals 0 . Therefore the map restricts to a map on S , which we continue to denote by the same letter : S . With this notation, the sheaf I is a sheaf on S . Recall that j denotes the embedding S (G) . Now we are ready to state the geometric analogue of local . conj general2 i( A ) cases j I ((,)) if i 2(,), 0 if i2(,) ,. cases conj We will now formulate second describing the stalk cohomologies of the sheaves i( A ) . The statement of second does not explicitly involve the category Sh ( (G)) and the functors i . However, general2 can be derived from second . Note that the support of the restriction of A to S , i.e., (G) S , is finite-dimensional. To work in a geometric setting, let us extend the base field from to F q and use Weil sheaves. Denote by , the restriction of to (G) S . conj second For dominant c k( (G) S , A , I ) cases (-(,)) if and k 2(,), 0 otherwise ,. cases conj Proving second would yield an alternative proof of local , and hence of the Casselman-Shalika formula css (see equi ). One sees readily that second holds when . Let us, then, assume that is dominant and . The projection formula implies that H c (S , A I ) H c (, R A I ). Therefore second follows from the following conj third For , dominant and the sheaves R k A are constant on . conj By Theorem 4.3a of MV we see that if third holds, then R k A 0 unless k 2((,)-1) . Here we are using the fact that the results of MV , stated there over C , extend to our current context. In this subsection we speculate about what could be the analogue of Laumon's sheaf L E in the case of an arbitrary reductive group. Recall from Sect. 3.2 that L E is a sheaf on the stack n canonically attached to a rank n local system E on X . This sheaf is used as the starting point of the conjectural construction of the automorphic sheaf on M n associated to E ; see La2 and Sect. 3 above. First we revisit the case of GL n and introduce a scheme X () with a smooth morphism q: X () n , and take the pull-back of L E to X () . The scheme X () classifies pairs L,t , where L is a rank n bundle on X and t: X n L is an embedding of X --modules. The scheme X () is a disjoint union of the smooth schemes ,m X () , m0 , corresponding to bundles of degree m . The scheme ,m X () is isomorphic to the Grothendieck Quot--scheme Quot m n X X . Recall Gr that Quot m n X X k classifies the quotients of n X that are torsion sheaves of length m ; at the level of points, L,t corresponds to the quotient of n X by the image of L under the transpose homomorphism t : L n X . The morphism q: X () n sends L,t to n X Im t . In the same way as in Sect. 4.2, one can show that q is smooth. We denote by the same character L E the pull-back of L E by q . It is the pair ( X () , L E) that we would like to generalize to other groups. Let us describe the basic structure of X () . For each k1 , we introduce the scheme X (k) over X (k),rss (see Sect. 3.2), which parametrizes the objects (x 1,,x k), L,t , where (x 1,,x k) is a set of k non-ordered distinct points of X , L is a rank n bundle over X , and t is its trivialization over X - x 1,,x k , which extends to an embedding of X --modules X n L . The fiber of this scheme over (x 1,,x k) X (k),rss is the product of the x i . It is easy to describe the pull-back L X (k) E, of L E under the natural morphism k: X (k) X () . In particular, the restriction of L X (k) E, to the fiber of X (k) over (x 1,,x k) is i 1 k L x i E, , where equation box1 L x E, P n A ,x (n-1) ( (n-1) 2) E x(). equation The set of --points of X () is isomorphic to the quotient GL n( ) GL n() . For groups other than GL n we do not have analogues of the subset GL n( ) GL n( ) , the subset P n , and the subscheme of the affine Grassmannian. For this reason, we cannot avoid considering a substantially larger object in place of X () . Thus, for a reductive group G , we consider the set G( ) G() . This set, which we denote by Gr(G) X () , is isomorphic to the set of isomorphism classes of pairs P ,t , where P is a principal G --bundle over X and t is an isomorphism between P and the trivial bundle on a Zariski open subset of X . It is not difficult to define a functor (G) X () from the category of --schemes to the category of sets, whose set of --points is Gr(G) X () . It would be desirable to have a notion of perverse sheaf on (G) X () . A --local system E on X should give rise to a perverse sheaf L E on (G) X () , analogous to the sheaf L E in the case of GL n ; this sheaf should be irreducible if E is irreducible. Although we do not know how to define such an object, we describe below what its pull-backs should be under certain natural morphisms. For each k1 , following Beilinson and Drinfeld, we introduce the ind--scheme (G) X (k) over X (k),rss , which parametrizes the objects (x 1,,x k), P ,t , where (x 1,,x k) is a set of non-ordered distinct points of X , P is a principal G --bundle over X , and t is its isomorphism with the trivial bundle over X - x 1,,x k . The fiber of this scheme over (x 1,,x k) X (k),rss is the product of the x i (see MV , Sect. 3). We have an obvious set-theoretic map k: Gr(G) X (k) G(G) X () , which can also be defined on the level of functors: schemes sets. The pull-back of L E to (G) X (k) should be the sheaf L X (k) E on (G) X (k) (inductive limit of perverse sheaves), such that its restriction to the fiber i 1 k (G) x i over (x 1,,x k) is i 1 k L x i E , where equation rx L x E A ,x E x() equation (up to shifts in degree). Here E x() has the same meaning as in the case of GL n . The sheaves L X (k) E have been previously considered by Beilinson and Drinfeld in the context of the geometric Langlands correspondence. Formula rx is analogous to formula box1 . The main difference is that in box1 the summation is restricted to the subset P n of the set P n of all dominant weights of GL n , which does not have an analogue for general G (compare with Sect. 6.4). rem cate Let ( x) ( (G) x) be the category of ( x) --equivariant perverse sheaves on (G) x (we consider objects of ( x) ( (G) x) as pure of weight 0 ). This category is a tensor category, and as such, it is equivalent to the tensor category R ep of finite-dimensional representations of . To be precise, this result has been proved in Gi , MV over the ground field (in this setting, this isomorphism was conjectured by V. Drinfeld; see also Lu2 ). But the proof outlined in MV can be generalized to the --case, so here we assume the result to be true over as well. Note that there is a small error in MV . The tensor structure (or, more precisely, the commutativity constraint), which is given by the convolution product, should be altered slightly. This alteration does not affect the structure of ( x) ( (G) x) as a monoidal category. We simply replace the perverse sheaf A ,x with (-1) 2(,) A ,x , where the sign (-1) 2(,) is to be viewed as a formal symbol. The symbol (-1) 2(,) has the effect of making the cohomology of (-1) 2(,) A ,x lie in even degrees only. Then the equivalence of categories R ep ( x) ( (G) x) takes the irreducible representation V() to the perverse sheaf (-1) 2(,) A ,x . With this adjustment the sign in hla disappears and the equivalence above can be viewed as a categorical version of the Satake isomorphism Rep H (see Satake ). Indeed, an equivalence of two categories induces an isomorphism of their Grothendieck rings. But the Grothendieck ring of ( x) ( (G) x) is canonically isomorphic to the Hecke algebra H via the faisceaux--fonctions'' correspondence. Now consider the left regular representation of , V() V() , as an ind--object of the category R ep . The corresponding ind--object of the category ( x) ( (G) x) is L E x given by formula rx , adjusted for the formal signs, i.e., equation rxx L x E (-1) 2(,) A ,x E x() . equation rem rem Let L X E be the function associated to the sheaf L X E , and let L x E be the restriction of L X E to Gr(G) x Gr(G) X . Using hla we obtain: equation series L x E Tr ( x,V() ) H ,x , equation where x ( Fr x) . Hence the function L x E coincides with the function L x -1 given by formula Fgamma . According to converge , the series series converges absolutely if q x -1 ( x) q x, , and is proportional to the spherical function S x -1 . rem Acknowledgments We express our gratitude to J. Bernstein and I. Mirkovic for valuable discussions and to B. Gross for useful remarks concerning the Whittaker models. We thank A. Beilinson and V. Drinfeld for sharing with us their ideas and unpublished results about the affine Grassmannian, which we used in Sect. 7.2. We are indebted to the referee for valuable comments and suggestions. The research of E. Frenkel was supported by grants from the Packard and Sloan Foundations, and by NSF grants DMS 9501414 and DMS 9304580. The research of D. Gaitsgory was supported by NSF grant DMS 9304580. D. Kazhdan was supported by NSF grant DMS 9622742. K. Vilonen was supported by NSF grant DMS 9504299 and by NSA grant MDA 90495H103.