Distinguished Representations with respect to Symmetric Subgroups of $GL_{n}(\mathbb{F}_{q})$

TL;DR

This paper investigates certain distinguished representations of the general linear group over finite fields, proving they satisfy a specific symmetry property, thereby confirming a conjecture related to these representations.

Contribution

It establishes that representations distinguished by symmetric subgroups of $GL_{n}(F_q)$ satisfy a particular involution symmetry, confirming a version of the Prasad-Lapid conjecture.

Findings

Representations are isomorphic to their involution duals.

Confirmed a version of the Prasad-Lapid conjecture.

Provides structural insights into distinguished representations.

Abstract

We study representations of $GL_{n}(\mathbb{F}_{q})$ that are distinguished with respect to a symmetric subgroup $H=GL_{n}(\mathbb{F}_{q})^{\sigma}$, where $\sigma$ is an involution. We prove that those representations satisfy $\pi \cong \pi ^{*,\sigma}$, thus positively answering a version of the Prasad-Lapid conjecture.