Linear periods for unitary representations

TL;DR

This paper classifies irreducible unitary representations of GL(2n) over non-Archimedean fields with nonzero linear periods, using Tadic's classification and Speh representations, providing criteria for their existence.

Contribution

It provides a complete classification of unitary representations with nonzero linear periods and establishes necessary and sufficient conditions for Speh representations.

Findings

Classification of unitary representations with nonzero linear periods

Criteria for existence of nonzero linear periods in Speh representations

Utilization of Tadic's classification for the analysis

Abstract

Let $F$ be a local non-Archimedean field of characteristic zero with a finite residue field. Based on Tadi\'{c}'s classification of the unitary dual of $\mathrm{GL}_{2n}(F)$, we classify irreducible unitary representations of $\mathrm{GL}_{2n}(F)$ that have nonzero linear periods, in terms of Speh representations that have nonzero periods. We also give a necessary and sufficient condition for the existence of a nonzero linear period for a Speh representation.