The action of a mirabolic subgroup on a symmetric variety

Abstract

Let F be a local field of characteristic zero. Let E be a quadratic field extension of F. We show that any PE-invariant linear functional on a GLn(E)-distinguished irreducible smooth admissible representation π of GL2n(F) is also GLn(E)-invariant, where PE is the standard mirabolic subgroup of GLn(E). Then we give a simple proof to show that the exterior square L-function has a pole at s=0 is a necessary condition such that a unitary and unramified principal series representation π is distinguished by GLn(E) when E/F is unramified.