# 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

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.