Distinction of the Steinberg representation for inner forms of $GL(n)$

TL;DR

This paper characterizes when certain Steinberg representations of inner forms of GL(n) over non-archimedean fields are distinguished by a subgroup, establishing a criterion based on characters and confirming preservation under Jacquet-Langlands correspondence.

Contribution

It provides a precise criterion for the distinction of Steinberg representations of inner forms of GL(n) over non-archimedean fields, including multiplicity one results and preservation under Jacquet-Langlands.

Findings

Steinberg representation is distinguished iff the character restriction matches a specific quadratic character.

Multiplicity one for the space of invariant linear forms is established.

Jacquet-Langlands correspondence preserves distinction for Steinberg representations.

Abstract

Let $F$ be a non archimedean local field of characteristic not $2$. Let $D$ be a division algebra of dimension $d^2$ over its center $F$, and $E$ a quadratic extension of $F$. If $m$ is a positive integer, to a character $\chi$ of $E^*$, one can attach the Steinberg representation $St(\chi)$ of $G=GL(m,D\otimes_F E)$. Let $H$ be the group $GL(m,D)$, we prove that $St(\chi)$ is $H$-distinguished if and only if $\chi_{|F^*}$ is the quadratic character $\eta_{E/F}^{md-1}$, where $\eta_{E/F}$ is the character of $F^*$ with kernel the norms of $E^*$. We also get multiplicity one for the space of invariant linear forms. As a corollary, we see that the Jacquet-Langlands correspondence preserves distinction for Steinberg representations.