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

TL;DR

This paper establishes a precise criterion for the irreducibility of certain induced representations of non-archimedean GL(n,A), extending understanding of representation theory in this setting.

Contribution

It provides an explicit necessary and sufficient condition for irreducibility of induced representations from essentially unitary representations over non-archimedean division algebras.

Findings

Derived an irreducibility criterion for induced representations

Extended irreducibility conditions to non-unitary characters

Applied results to representations of GL(n,A) over division algebras

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.