On the classification of unitary representations of reductive Lie groups

Abstract

Suppose G is a real reductive Lie group in Harish-Chandra's class. We propose here a structure for the set \Pi_u(G) of equivalence classes of irreducible unitary representations of G. (The subscript u will be used throughout to indicate structures related to unitary representations.) We decompose \Pi_{u}(G) into disjoint subsets with a (very explicit) discrete parameter set \Lambda_u: \Pi_u(G) = \bigcup_{\lambda_u \in \Lambda_u} \Pi_u^{\lambda_u}(G). Each subset is identified conjecturally with a collection of unitary representations of a certain subgroup G(\lambda_u) of G. (We will give strong evidence and partial results for this conjecture.) In this way the problem of classifying \Pi_u(G) would be reduced (by induction on the dimension of G) to the case G(\lambda_u) = G.