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.