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

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 $χ$ of $E^*$, one can attach the Steinberg representation $St(χ)$ of $G=GL(m,D\otimes_F E)$. Let $H$ be the group $GL(m,D)$, we prove that $St(χ)$ is $H$-distinguished if and only if $χ_{|F^*}$ is the quadratic character $η_{E/F}^{md-1}$, where $η_{E/F}$ 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.

No reviews yet.

No comments yet.