Analysis on Homogeneous Spaces and Representations of Lie Groupspp

The Langlands classiication theorem describes all admissible representations of a reductive group G in terms of the tempered representations of Levi subgroups of G. I will describe work with Susana Salamanca-Riba that provides (conjecturally) a similar description of the unitary representations of G in terms of certain very special unitary representations of Levi subgroups. x1. Introduction Suppose G is a real reductive Lie group in Harish-Chandra's class (see HC], section 3). There are two powerful general techniques for constructing irreducible unitary representations of G. Parabolic induction is based on real analysis and geometry on certain compact homogeneous spaces G=P. Cohomological induction is based on complex analysis on certain indeenite KK ahler homogeneous spaces G=L. When G is SL(2; R), parabolic induction gives rise to Bargmann's unitary principal series representations , and cohomological induction to Bargmann's discrete series. We are concerned here not with the details of these constructions, but rather with the question of classiication: which unitary representations can be found (and which cannot be found) by these methods. In the case of SL(2; R), what is missing are the complementary series representations , the two \limits of discrete series representations," and the trivial representation. In general a precise answer is diicult to obtain, and from some perspectives even undesirable. By deformation arguments, one can push either construction to yield larger sets of unitary representations. Thus for example the complementary series representations of SL(2; R) may be regarded as parabolically induced, and the limits of discrete series as cohomologically induced. The trivial representation emerges from either construction as a kind of very singular limiting case.