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

TL;DR

This paper investigates distinguished representations of the special linear group over finite fields, providing a formula for their multiplicity in terms of characters, extending previous work to a new subgroup setting.

Contribution

It introduces a new framework for understanding distinguished representations of $ m{SL}(2n,F)$ with respect to a subgroup involving a quadratic extension or split algebra, and derives a dimension formula.

Findings

Derived a formula for the dimension of Hom spaces for distinguished representations.

Extended the theory of distinguished representations to a new subgroup setting.

Connected finite field results with previous p-adic and complex field work.

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 $\rm{SL}_{2n}(F)$ by the subgroup $H_{\flat} := \rm{SL}_{2n}(F) \cap \rm{GL}_{n}(\mathbb{E})$, which is a variation of the work of Anandavardhanan and Prasad on distinguished representations of $\rm{SL}_{n}(\mathbb{E})$ by the subgroup $\rm{SL}_n(F)$. 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 $\rm{Hom}_{H_{\flat}}(\pi_{\flat}, 1)$ in terms of certain characters of $F^{\times}$, where $\pi_{\flat}$ is an irreducible representation which is also distinguished by $H_{\flat}$.