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 $E^*$, 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.