On the classification of unitary representations of reductive Lie groups

Susana A. Salamanca-Riba and David A.jun. Vogan

Review from Zentralblatt MATH:

Let $G$ be a real reductive Lie group in Harish-Chandra's class. This paper presents a program for classifying unitary irreducible representations along the lines of the Vogan-Zuckerman classification of general irreducible representations. More precisely, the set $\Pi_u(G)$ of (equivalence classes of) irreducible unitary representations of $G$ is partitioned into disjoint subsets $\Pi_u^{\lambda_u}(G)$, where the parameter $\lambda_u$ runs over a very explicit set $\Lambda_u$. The representations in each subset $\Pi_u^{\lambda_u}$ are defined by a certain condition on their lowest $K$-types, but it is conjectured that they are all obtained by cohomological parabolic induction from unitary representations of a certain reductive subgroup $G(\lambda_u)$ of $G$. If the conjecture is true, then it yields an inductive framework for studying unitary representations of $G$ which is in particular powerful enough to classify those of sufficiently regular infinitesimal character. The authors reduce the proof of the conjecture to showing that certain Hermitian representations with large infinitesimal character cannot be unitary. They then present some supporting evidence for this assertion, using Parthasarathy's Dirac operator inequality, and show how a sharpening of that inequality would yield the desired conjecture. In the last section they show that their partition of the unitary dual behaves well with respect to the Fell topology and observe that, even if not all the representations in each piece are cohomologically induced from unitary representations (as required by the conjecture), then some closed subsets of them are so induced. The techniques are largely those of {\it D. A. Vogan}'s book [Representations of real reductive Lie groups, Prog. Math. 15 (Boston-Basel-Stuttgart 1981; Zbl 0863.22017)].

Reviewed by W.M.McGovern

Keywords: unitary representation; lowest $K$-type; cohomological parabolic induction; real reductive Lie group

Classification (MSC2000): 22E47 22E46

Full text of the article:

• PDF file (395 kilobytes)