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

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 ${\mathrm{SL}}_{\mathrm{n}}^{\ast }(\mathrm{E})={\mathrm{GL}}_{\mathrm{n}}(\mathrm{E})\cap {\mathrm{SL}}_{\mathrm{n}}(\mathrm{D})$. We study distinguished representations of ${\mathrm{SL}}_{\mathrm{n}}(\mathrm{D})$ by the subgroup ${\mathrm{SL}}_{\mathrm{n}}^{\ast }(\mathrm{E})$. Let $\ddot{I}€$ be an irreducible admissible representation of ${\mathrm{SL}}_{\mathrm{n}}(\mathrm{D})$ which is distinguished by ${\mathrm{SL}}_{\mathrm{n}}^{\ast }(\mathrm{E})$. We give a multiplicity formula, i.e. a formula for the dimension of the $\mathbb{C}$-vector space ${\rm{Hom}}_{\rm{SL}_{n}^{*}(E)} (π, \mathbbm{1})$, where $\mathbbm{1}$ denotes the trivial representation of ${\mathrm{SL}}_{\mathrm{n}}^{\ast }(\mathrm{E})$. This work is a non-split inner form analog of a work by Anandavardhanan-Prasad which gives a multiplicity formula for ${\mathrm{SL}}_{\mathrm{n}}(\mathrm{F})$-distinguished irreducible admissible representation of ${\mathrm{SL}}_{\mathrm{n}}(\mathrm{E})$.