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Journal of the American Mathematical Society

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ISSN 1088-6834 (online) ISSN 0894-0347 (print)

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Geometric realization of Whittaker functions and the Langlands conjecture
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by E. Frenkel, D. Gaitsgory, D. Kazhdan and K. Vilonen PDF
J. Amer. Math. Soc. 11 (1998), 451-484 Request permission

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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Additional Information
  • E. Frenkel
  • Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
  • MR Author ID: 257624
  • ORCID: 0000-0001-6519-8132
  • D. Gaitsgory
  • Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
  • D. Kazhdan
  • Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
  • MR Author ID: 99580
  • K. Vilonen
  • Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02254
  • MR Author ID: 178620
  • Received by editor(s): March 31, 1997
  • Received by editor(s) in revised form: November 26, 1997
  • © Copyright 1998 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 11 (1998), 451-484
  • MSC (1991): Primary 11R39, 11F70; Secondary 14H60, 22E55
  • DOI: https://doi.org/10.1090/S0894-0347-98-00260-4
  • MathSciNet review: 1484882