Test vectors for local periods

Abstract

Let E/F be a quadratic extension of non-archimedean local fields of characteristic zero. An irreducible admissible representation π of GL(n,E) is said to be distinguished with respect to GL(n,F) if it admits a non-trivial linear form that is invariant under the action of GL(n,F). It is known that there is exactly one such invariant linear form up to multiplication by scalars, and an explicit linear form is given by integrating Whittaker functions over the F-points of the mirabolic subgroup when π is unitary and generic. In this paper, we prove that the essential vector of [JPSS81] is a test vector for this standard distinguishing linear form and that the value of this form at the essential vector is a local L-value. As an application we determine the value of a certain proportionality constant between two explicit distinguishing linear forms. We then extend all our results to the non-unitary generic case.