Distinguished Representations with respect to Symmetric Subgroups of $G{L}_n({\mathbb{F}}_q)$

Abstract

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