Geometric theta-lifting for the dual pair GSp2n, GO2m

Let X be a smooth projective curve over an algebraically closed field of characteristic > 2. Consider the dual pair H = GO2m, G = GSp2n over X , where H splits over an étale two-sheeted covering π : X̃ → X . Write BunG and BunH for the stacks of G-torsors and H-torsors on X . We show that for m ≤ n (respectively, for m > n) the theta-lifting functor FG : D(BunH) → D(BunG) (respectively, FH : D(BunG) → D(BunH)) commutes with Hecke functors with respect to a morphism of the corresponding L-groups involving the SL2 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 BunGSp 4 corresponding to the endoscopic local systems on X .