Geometric realization of Whittaker functions and the Langlands conjecture

Abstract

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