Irreducibility criterion for representations induced by essentially unitary ones (case of non-archimedean GL(n,A)

Abstract

Let A be a finite dimensional central division algebra over a local non-archimedean field F. Fix any parabolic subgroup P of GL(n,A) and a Levi factor M of P. Let \pi be an irreducible unitary representation of M and \phi (not necessarily unitary) character of M. We give an explicit necessary and sufficient condition for the parabolically induced representation Ind(\phi\pi) of GL(n,A) to be irreducible.