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

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.