1997/03/18 by Edward Frenkel, Dennis Gaitsgory, Frenkel, E. +5
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.alg-geom/9703022
openalex publication_date 1997/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove the equivalence of two conjectural constructions of unramified cuspidal automorphic functions on the adelic group GLn(A) associated to an irreducible l-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.