Test vectors for local periods

TL;DR

This paper proves that the essential vector serves as a test vector for the distinguished linear form in local periods, linking its value to a local L-value and extending results to non-unitary cases.

Contribution

It establishes the essential vector as a test vector for local periods and relates its value to local L-values, extending previous results to non-unitary generic representations.

Findings

Essential vector is a test vector for the distinguished linear form.

Value of the linear form at the essential vector equals a local L-value.

Results extended to non-unitary generic representations.

Abstract

Let $E/F$ be a quadratic extension of non-archimedean local fields of characteristic zero. An irreducible admissible representation $\pi$ 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 $\pi$ 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.