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

Geometric realization of Whittaker functions and the Langlands conjecture
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by E. Frenkel, D. Gaitsgory, D. Kazhdan and K. Vilonen

J. Amer. Math. Soc. 11 (1998), 451-484

DOI: https://doi.org/10.1090/S0894-0347-98-00260-4

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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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