The Unitary Dual of the Covering Groups of GL(n) over a Local Field

Abstract

A fundamental problem in harmonic analysis on a reductive group G over a local field F is to describe the equivalence classes of the irreducible representations of G, which is well known as the unitary dual of G. In the case F is an archimedean field, the big break through in last decade are David Vogan’s classification of the unitary dual of general line groups [V] and Dan Barbasch’s classification of the unitary dual for complex classical Lie groups [Ba]. In the case F is a non-archimedean local field, Marko Tadic classified the unitary dual of the general linear groups [T2]. Besides these groups the only groups whose unitary duals are known are those groups of small ranks. The unitary dual problem remains to be one of the most challenging questions in noncommutative harmonic analysis.