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

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 $E^{\ast }$, 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 ${\chi }_{|F^{\ast }}$ is the quadratic character ${\eta }_{E/F}^{md-1}$, where ${\eta }_{E/F}$ is the character of $F^{\ast }$ with kernel the norms of $E^{\ast }$. 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.

arXiv commentWe corrected a mistake in Section 3.1