On quadratic distinction of automorphic sheaves

We prove a geometric version of a classical result on the characterization of an irreducible cuspidal automorphic representation of being the base change of a stable cuspidal representation of the quasi-split unitary group associated to the quadratic extension E/F, via the nonvanishing of certain period integrals, called being distinguished. We show that certain cohomology of an automorphic sheaf of GL _n,X ^′ is nonvanishing if and only if the corresponding local system E on X ^′ is conjugate self-dual with respect to an étale double cover X′/X of curves, which directly relates to the base change from the associated unitary group. In particular, the geometric setting makes sense over any base field.