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 functio..