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Geometric Weil representation: local field case

Part of: Lie groups Algebraic number theory: global fields

Published online by Cambridge University Press:  01 January 2009

Vincent Lafforgue  and
Sergey Lysenko
Vincent Lafforgue
Affiliation:
Institut de Mathématiques, Université Paris 6, 175, rue du Chevaleret, 75013 Paris, France (email: vlafforg@math.jussieu.fr)
Sergey Lysenko
Affiliation:
Institut de Mathématiques, Université Paris 6, 175, rue du Chevaleret, 75013 Paris, France (email: vlafforg@math.jussieu.fr)
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Abstract

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Let k be an algebraically closed field of characteristic greater than 2, and let F=k((t)) and G=𝕊p2d. In this paper we propose a geometric analog of the Weil representation of the metaplectic group . This is a category of certain perverse sheaves on some stack, on which acts by functors. This construction will be used by Lysenko (in [Geometric theta-lifting for the dual pair S𝕆2m, 𝕊p2n, math.RT/0701170] and subsequent publications) for the proof of the geometric Langlands functoriality for some dual reductive pairs.

Keywords

Weil representationmetaplectic groupgeometric Langlands program

MSC classification

Secondary: 11R39: Langlands-Weil conjectures, nonabelian class field theory 22E50: Representations of Lie and linear algebraic groups over local fields

Information

Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 1 , January 2009 , pp. 56 - 88
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Drinfeld, B., Quantization of the Hitchin integrable system and Hecke eigensheaves, Preprint, downloadable from http://www.math.utexas.edu/∼benzvi/Langlands.html.Google Scholar
[2]Delign, P., Métaplectique, A letter to Kazhdan (1982).Google Scholar
[3]Gerardin, P., Weil representations associated to finite fields, J. Algebra 46 (1977), 54–102.Google Scholar
[4]Gurevich, S. and Hadani, R., Proof of the Kurlberg–Rudnick rate conjecture, C. R. Math. Acad. Sci. Paris 342 (2006), 69–72, math-ph/0404074.Google Scholar
[5]Gurevich, S. and Hadani, R., Heisenberg realizations, eigenfunctions and proof of the Kurlberg–Rudnik supremum conjecture, math-ph/0511036..Google Scholar
[6]Gurevich, S. and Hadani, R., Quantization of symplectic vector spaces over finite fields, arXiv: 0705.4556.Google Scholar
[7]Howe, R., θ-series and invariant theory, Proc. Symp. Pure Math. 33 (1979), 275–285, part 1.Google Scholar
[8]Lion, G. and Vergne, M., The Weil representation, in Maslov index and theta series, Progress in Mathematics, vol. 6 (Birkhäuser, Boston, MA, 1980).Google Scholar
[9]Lysenko, S., Moduli of metaplectic bundles on curves and theta-sheaves, Ann. Sci. École Norm. Sup. (4) 39 (2006), 415–466.CrossRefGoogle Scholar
[10]Lysenko, S., Geometric theta-lifting for the dual pair S𝕆2m,𝕊p2n, math.RT/0701170.Google Scholar
[11]Moeglin, C., Vignéras, M.-F. and Waldspurger, J.L., Correspondances de Howe sur un corps p-adique, Lecture Notes in Mathematics, vol. 1291 (Springer, Berlin, 1987).Google Scholar
[12]Prasad, D., A brief survey on the theta correspondence, in Number theory (Tiruchirapalli, 1996), Contemporary Mathematics, vol. 210 (American Mathematical Society, Providence, RI, 1998), 171–193.CrossRefGoogle Scholar
[13]Weil, A., Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143–211.CrossRefGoogle Scholar