Distinguished representations for $\mathrm{SL}(\mathrm{n},\mathrm{F})$

Abstract

Let $F$ be a finite field, and let $\mathbb{E}$ be either a quadratic field extension $E/F$ or the split algebra $F\oplus F$. We study distinguished representations of ${\mathrm{SL}}_{2n}(\mathrm{F})$ by the subgroup $H_{♭}:={\mathrm{SL}}_{2n}(\mathrm{F})\cap {\mathrm{GL}}_{\mathrm{n}}(\mathbb{E})$, which is a variation of the work of Anandavardhanan and Prasad on distinguished representations of ${\mathrm{SL}}_{\mathrm{n}}(\mathbb{E})$ by the subgroup ${\mathrm{SL}}_{\mathrm{n}}(\mathrm{F})$. This is in a similar framework of our earlier work of a $p$-adic non-split variation of Anandavardhanan-Prasad over finite fields. We give a formula for the dimension of the complex vector space ${\mathrm{Hom}}_{{\mathrm{H}}_{♭}}(\ddot{\mathrm{I}}{€}_{♭},1)$ in terms of certain characters of $F^{\times }$, where $\ddot{I}{€}_{♭}$ is an irreducible representation which is also distinguished by $H_{♭}$.

9 pages