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

Sergey Lysenko

Arxiv ID: 0802.0457•Last updated: 12/6/2021

Let X be a smooth projective curve over an algebraically closed field of
characteristic >2. Consider the dual pair H=GSO_{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. So, they
realize the geometric Langlands functoriality for the corresponding morphisms
of L-groups.
As an application, we prove a particular case of the geometric Langlands
conjectures for GSp_4. Namely, we construct the automorphic Hecke eigensheaves
on Bun_{GSp_4} corresponding to the endoscopic local systems on X.

#### PaperStudio AI Chat

I'm your research assistant! Ask me anything about this paper.