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 GLn(AE ) 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 GLn,X′ is nonvanishing if and only if the corresponding local system E on X′ is conjugate selfdual 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.