Geometric realization of Whittaker functions and the Langlands conjecture

Frenkel, E.¹ ; Gaitsgory, D.² ; Kazhdan, D.¹ ; Vilonen, K.³

¹ Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
² School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
³ Department of Mathematics, Brandeis University, Waltham, Massachusetts 02254

Journal of the American Mathematical Society, Tome 11 (1998) no. 2, pp. 451-484

Résumé

We prove the equivalence of two conjectural constructions of unramified cuspidal automorphic functions on the adelic group $GL_n(\mathbb A)$ associated to an irreducible $\ell$–adic local system of rank $n$ on an algebraic curve $X$ 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 ($n=2$), is based on considering the Whittaker function. The second construction, which was proposed recently by Laumon following Drinfeld ($n=2$) and Deligne ($n=1$), 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.

Détail

DOI : 10.1090/S0894-0347-98-00260-4

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