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

Nadir Matringe

Published 2016-02-16Version 1

Let $F$ be a non archimedean local field of characteristic not $2$. Let $D$ be a finite dimensional division algebra with center $F$, $E$ a quadratic extension of $F$, and $m$ 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 if $d$ is even, then $St(\chi)$ is $H$-distinguished if and only if $\chi_{|F^*}$ is the quadratic character $\eta_{E/F}$ with kernel the norms of $E^*$, whereas if $d$ is odd, then $St(\chi)$ is $H$-distinguished if and only if $\chi_{|F^*}=\eta_{E/F}^{m-1}$. 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.