Representations of Semisimple Lie

The theory of minimal types for representations of complex semisimple Lie groups [K. R. Parthasarathy, R. Ranga Rao and V. S. Varadarajan, Ann. of Math. (2) 85 (1967), 383-429, Chapters 1, 2 and 3] is reformulated so that it can be generalized, at least partially, to real semisimple Lie groups. A rather complete extension of the complex theory is obtained for the semisimple Lie groups of real rank 1. More specifically, let G-NAK be an Iwasawa decomposition of a connected real semisimple Lie group with finite center, and let M be the centralizer of A in K. Suppose that G has real rank 1. Let a E (^ denotes the set of equivalence classes of continuous finite dimensional complex irreducible representations), and let YE be the class under which the highest restricted weight space of any member of a transforms. It is proved by means of an unpublished general formula of B. Kostant that there exists $ K such that m(a,8) = m(8,y) = 1 (m denotes multiplicity). Moreover, a can be chosen so that it depends only on y, and not on a. The corresponding complex-valued homomorphism on the centralizer of K in the complex enveloping algebra of the Lie algebra of G is computed. A similar approach is used to study a certain series of infinite dimensional irreducible representations of G related to a series of representations studied by Harish-Chandra. The computation of the above-mentioned homomorphism is embedded in a general theory (for all real groups G) based on a certain enveloping algebra decomposition which generalizes a decomposition used to study the classical class 1 infinitesimal spherical functions. The general theory deals with arbitrary elements of A and R in the same sense that the class 1 theory deals with the trivial elements of M and K. Furthermore, the general theory handles arbitrary -2IIICSL~1; '-~-rc