Distinguished Representations of ${\mathrm{GL}}_n({\mathbb{F}}_q)$

Abstract

Let $\mathrm{G}={\mathrm{Gl}}_n(K)$, and $\mathrm{H}={\mathrm{G}}^{\sigma }$ for $\sigma$ an involution of the form $g\to ag{a}^{-1}$, It is known that for $K={\mathbb{Q}}_q$ any irreducible representation of $\mathrm{G}$ with an $\mathrm{H}$ invariant functional, is self dual, we prove an analogous result for ${\mathbb{F}}_q$.