Geometric realization of Whittaker functions and the Langlands conjecture

Frenkel, E.; Gaitsgory, D.; Kazhdan, D.; Vilonen, K.

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

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Geometric realization of Whittaker functions and the Langlands conjecture

Authors: E. Frenkel, D. Gaitsgory, D. Kazhdan and K. Vilonen
Journal: J. Amer. Math. Soc. 11 (1998), 451-484
MSC (1991): Primary 11R39, 11F70; Secondary 14H60, 22E55
MathSciNet review: 1484882
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Abstract: 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.

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