On the Geometric Langlands Functoriality for the Dual Pair Sp_{2n}, SO_{2m}

MEMBERS SEMINAR
Topic:	On the Geometric Langlands Functoriality for the Dual Pair Sp_{2n}, SO_{2m}
Speaker:	Sergey Lysenko
Affiliation:	Université Paris 6, France and Member, School of Mathematics
Date:	Monday, December 11
Time/Room:	4:00pm - 5:00pm/S-101

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I will report on a the following work in progress. Let X be a smooth connected curve over an algebraically closed field. Consider the dual pair H=SO_{2m}, G=Sp_{2n} over X with H split. Let Bun_G and Bun_H be the stacks of G-torsors and H-torsors on X. The theta-sheaf on Bun_G\times Bun_H yields the theta-lifting functors between the derived categories D(Bun_H) and D(Bun_H).

Assuming the purity of the above theta-sheaf, we prove that these functors realize the geometric Langlands functoriality for this dual pair (in the everywhere nonramified case). Its correct formulation involves the SL_2 of Arthur (or rather its maximal torus). The local part of the prove is unconditional and provides a geometric analog of a theorem of Rallis.