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Geometric Waldspurger periods

Published online by Cambridge University Press:  01 March 2008

Sergey Lysenko*
Affiliation:
Université Paris 6, Institut de Mathématiques, Analyse Algébrique, 175 rue du Chevaleret, F-75013 Paris, France (email: lysenko@math.jussieu.fr)
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Abstract

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Let X be a smooth projective curve. We consider the dual reductive pair , over X, where H splits on an étale two-sheeted covering . Let BunG (respectively, BunH) be the stack of G-torsors (respectively, H-torsors) on X. We study the functors FG and FH between the derived categories D(BunG) and D(BunH), which are analogs of the classical theta-lifting operators in the framework of the geometric Langlands program. Assume n=m=1 and H nonsplit, that is, with connected. We establish the geometric Langlands functoriality for this pair. Namely, we show that FG :D(BunH)→D(BunG) commutes with Hecke operators with respect to the corresponding map of Langlands L-groups LHLG. As an application, we calculate Waldspurger periods of cuspidal automorphic sheaves on BunGL2 and Bessel periods of theta-lifts from to . Based on these calculations, we give three conjectural constructions of certain automorphic sheaves on (one of them makes sense for -modules only).

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2008