Distinction of the Steinberg representation for inner forms of \(\mathrm{GL}(n)\) (Q1686783)

Let \(F\) be a nonarchimedean local field of characteristic \(\neq 2\). Let \(D\) be a division algebra of dimension \(d^2\) over its center \(F\) and let \(E/F\) be a quadratic extension. Consider the groups \(G=\mathrm{GL}_m(D\otimes_FE)\) and its subgroup \(H=\mathrm{GL}_m(D)\). The main result of the present paper asserts that for a character \(\chi\) of \(H\) and the Steinberg representation \(St(1)\) of \(G\) the space \(\mathrm{Hom}_H(St(1),\chi)\) is zero, unless \(\chi=\eta_{E/F}^{md-1}\circ N_H\), where \(\eta_{E/F}\) is the non-trivial character of \(F^\times\) which is trivial on the norms of \(E^\times\) and \(N_H\) is the reduced norm of the matrix algebra \(\mathrm{Mat}_m(D)\) which contains \(H\). This is a particular case of some conjectures of \textit{D. Prasad} [``A `relative' Langlands correspondence'', Preprint, \url{arXiv:1512.04347}].