Geometric theta-lifting for the dual pair SO2m, Sp2n

Let X be a smooth projective curve over an algebraically closed field of characteristic > 2. Consider the dual pair H = SO2m, G = Sp2n over X with H split. Write BunG and BunH for the stacks of G-torsors and H-torsors on X . The theta-kernel AutG,H on BunG ×BunH yields the theta-lifting functors FG : D(BunH) → D(BunG) and FH : D(BunG) → D(BunH) between the corresponding derived categories. Assuming the purity of AutG,H , we describe the relation of these functors with Hecke operators. In two particular cases it becomes the geometric Langlands functoriality for this pair (in the nonramified case). Namely, we show that for n = m the functor FG : D(BunH) → D(BunG) commutes with Hecke operators with respect to the inclusion of the Langlands dual groups Ȟ →̃ SO2n →֒ SO2n+1 →̃ Ǧ. For m = n + 1 we show that the functor FH : D(BunG) → D(BunH) commutes with Hecke operators with respect to the inclusion of the Langlands dual groups Ǧ →̃ SO2n+1 →֒ SO2n+2 →̃ Ȟ . In other cases the relation is more complicated and involves the SL2 of Arthur. As a step of the proof, we establish the geometric theta-lifting for the dual pair GLm,GLn. The latter result as well as our local results (providing a geometric analog of a theorem of Rallis) are unconditional.