Geometric theta-lifting for the dual pair SO_{2m}, Sp_{2n}

Let X be a smooth projective curve over an algebraically closed field of characteristic >2. Consider the dual pair H=SO_{2m}, G=Sp_{2n} over X with H split. Write Bun_G and Bun_H for the stacks of G-torsors and H-torsors on X. The theta-kernel on Bun_G\times Bun_H yields the theta-lifting functors between the derived categories of l-adic sheaves on Bun_G and Bun_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, for n=m the functor from the derived category on Bun_H to that on Bun_G commutes with Hecke functors with respect to the inclusion of the Langlands dual groups SO_{2n}\to SO_{2n+1}. For m=n+1 the functor from the derived category on Bun_G to that on Bun_H commutes with Hecke functors with respect to the inclusion of the Langlands dual groups SO_{2n+1}\to \SO_{2n+2}. In other cases the relation is more complicated and involves the SL_2 of Arthur. As a step of the proof, we establish the geometric theta-lifting for the dual pair GL_m, GL_n. Our global results are derived from the corresponding local ones, which provide a geometric analog of a theorem of Rallis.Comment: LaTeX2e, 76 pages, v7: this is a correction of the published version. In the published version the compatibility of the functoriality isomorphisms with the tensor structure on the category of representations of the dual group was not justified. We do this in the present version. We also pass from the formalism of derived categories to that of DG-categorie