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Let X be a smooth projective curve over an algebraically closed field ofcharacteristic >2. Consider the dual pair H=GSO_{2m}, G=GSp_{2n} over X, whereH splits over an etale two-sheeted covering of X. Write Bun_G and Bun_H for thestacks 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 amorphism of the corresponding L-groups involving the SL_2 of Arthur. So, theyrealize the geometric Langlands functoriality for the corresponding morphismsof L-groups.  As an application, we prove a particular case of the geometric Langlandsconjectures for GSp_4. Namely, we construct the automorphic Hecke eigensheaveson Bun_{GSp_4} corresponding to the endoscopic local systems on X.

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