Distinguished Representations and Poles of Twisted Tensor L-Functions

Abstract

Let E/F be a quadratic extension of p-adic fields. If π is an admissible representation of GL_n(E) that is parabolically induced from discrete series representations, then we prove that the space of P_n(F)-invariant linear functionals on π has dimension one, where P_n(F) is the mirabolic subgroup. As a corollary, it is deduced that if π is distinguished by GL_n(F), then the twisted tensor L-function associated to π has a pole at s = 0. It follows that if π is a discrete series representation, then at most one of the representations π and π otimes χ is distinguished, where χ is an extension of the local class field theory character associated to E/F. This is in agreement with a conjecture of Flicker and Rallis that relates the set of distinguished representations with the image of base change from a suitable unitary group.