Unipotent representations and reductive dual pairs over finite fields

Abstract

Consider the representation correspondence for a reductive dual pair ( G 1 , G 2 ) ({G_1},{G_2}) over a finite field. We consider the question of how the correspondence behaves for unipotent representations. In the special case of cuspidal unipotent representations, and a certain fundamental situation, that of "first occurrence", the representation correspondence takes a cuspidal unipotent representation of G 1 {G_1} to one of G 2 {G_2} . This should serve as a fundamental case in studying the correspondence in general over both finite and local fields.