# Geometric theta-lifting for the dual pair GSp_{2n}, GO_{2m}

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## Abstract

Let X be a smooth projective curve over an algebraically closed field of characteristic >2. Consider the dual pair H=GO_{2m}, G=GSp_{2n} over X, where H splits over an etale two-sheeted covering of X. Write Bun_G and Bun_H for the stacks of G-torsors and H-torsors on X. We show that for m\le n (respectively, for m>n) the theta-lifting functor from D(Bun_H) to D(Bun_G) (respectively, from D(Bun_G) to D(Bun_H)) commutes with Hecke functors with respect to a morphism of the corresponding L-groups involving the SL_2 of Arthur. In two particular cases n=m and m=n+1 this becomes the geometric Langlands functoriality for the corresponding dual pair. As an application, we prove a particular case of the geometric Langlands conjectures. Namely, we construct the automorphic Hecke eigensheaves on Bun_{GSp_4} corresponding to the endoscopic local systems on X. See more