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Geometric realization of Whittaker functions and the Langlands conjecture
Author(s):
E.
Frenkel;
D.
Gaitsgory;
D.
Kazhdan;
K.
Vilonen
Abstract | References | Similar articles | Additional information
Abstract:
We prove the equivalence of two conjectural constructions of unramified cuspidal automorphic functions on the adelic group
[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 [4] R.K. Brylinski, Limits of weight spaces, Lusztig's [5] W. Casselman, J. Shalika, The unramified principal series of [6] P. Deligne, Le conjecture de Weil II, Publ. IHES 52 (1980) 137-252 MR 83c:14017 [7] V.G. Drinfeld, Two-dimensional [8] V.G. Drinfeld, C. Simpson, [9] I.M. Gelfand, D. Kazhdan, Representations of the group [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 [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 [16] H. Jacquet, I.I. Piatetski-Shapiro, J. Shalika, Automorphic forms on [17] S. Kato, Spherical functions and a [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 [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 [24] G. Laumon, L. Moret-Bailly, Champs algébriques, Preprint 92-42, Université Paris Sud, 1992. [25] G. Laumon, Faisceaux automorphes pour [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 [28] I.G. Macdonald, Spherical Functions on a Group of [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 [32] J.A. Shalika, The multiplicity one theorem for [33] T. Shintani, On an explicit formula for class 1 Whittaker functions on [34] A. Weil, Dirichlet Series and Automorphic Forms, Lect. Notes in Math. 189, Springer Verlag, 1971. Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 11R39, 11F70, 14H60, 22E55 Retrieve articles in all Journals with MSC (1991): 11R39, 11F70, 14H60, 22E55
E.
Frenkel
D.
Gaitsgory
D.
Kazhdan
K.
Vilonen
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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-Langlands (
), 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. 
where 
is a local non-Archimedean field, Russ. Math. Surv. 31 (1976) 1-68. 
-analogs, and fiberings of adjoint orbits, Journal of AMS 2 (1989) 517-533. 
-adic groups II. The Whittaker function, Comp. Math. 41 (1980) 207-231. 
-adic representations of the fundamental group of a curve over a finite field and automorphic forms on 
, Amer. J. Math. 105 (1983) 85-114. 
-structures on 
-bundles and local triviality, Math. Res. Lett. 2 (1995) 823-829. 
where 
is a local field, in Lie Groups and Their Representations, ed. I.M. Gelfand, pp. 95-118, Adam Hilder Publ., 1975. 
when restricted to 
, Canadian J. Math. 44 (1992) 974-1002. 
, Lect. Notes in Math. 114, Springer Verlag, 1970. 
, I and II, Ann. Math. 109 (1979) 169-258. 
-functions, vol. 1, pp. 227-281, Academic Press, 1990. 
: la première construction de Drinfeld, Preprint alg-geom/9511004 (1995). 
, Ann. Math. 100 (1974) 171-193. 
over 
-adic fields, Proc. Japan Acad. 52 (1976) 180-182.