Distinguished Representations for $\rm{SL}_n(D)$ where $D$ is a quaternion division algebra over a $p$-adic field

TL;DR

This paper derives a multiplicity formula for distinguished representations of the special linear group over a quaternion division algebra, extending known results from split groups to non-split inner forms over p-adic fields.

Contribution

It provides the first multiplicity formula for $ m{SL}_n(D)$ distinguished by $ m{SL}_n^*(E)$, a non-split inner form analog of prior split group results.

Findings

Derived a multiplicity formula for distinguished representations.

Extended known results from split groups to quaternion division algebra cases.

Established a framework for analyzing distinguished representations in non-split inner forms.

Abstract

Let $D$ be a quaternion division algebra over a non-archimedean local field $F$ of characteristic zero. Let $E/F$ be a quadratic extension and $\rm{SL}_{n}^{*}(E) = {\rm{GL}}_{n}(E) \cap \rm{SL}_{n}(D)$. We study distinguished representations of $\rm{SL}_{n}(D)$ by the subgroup $\rm{SL}_{n}^{*}(E)$. Let $\pi$ be an irreducible admissible representation of $\rm{SL}_{n}(D)$ which is distinguished by $\rm{SL}_{n}^{*}(E)$. We give a multiplicity formula, i.e. a formula for the dimension of the $\mathbb{C}$-vector space ${\rm{Hom}}_{\rm{SL}_{n}^{*}(E)} (\pi, \mathbbm{1})$, where $\mathbbm{1}$ denotes the trivial representation of $\rm{SL}_{n}^{*}(E)$. This work is a non-split inner form analog of a work by Anandavardhanan-Prasad which gives a multiplicity formula for $\rm{SL}_{n}(F)$-distinguished irreducible admissible representation of $\rm{SL}_{n}(E)$.