Members’ Seminar

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.