Classification of the irreducible representations of semisimple Lie groups

摘要

We obtain a classification of the irreducible (nonunitary) representations of a connected semisimple Lie group G, in terms of their restriction to a maximal compact subgroup K of G. (A classification in terms of analytic properties of the representations has been given by R. P. Langlands [(1973), mimeographed notes, Institute for Advanced Study, Princeton, NJ] for linear groups.) We first define a norm on the representations of K: if μ ∈ $\widehat{K}$, ǁμǁ is a nonnegative real number. Then if π ∈ Ĝ, μ is called a lowest K-type of π if ǁμǁ is minimal among the K-types occurring in π. We announce a parameterization of the set of representations containing μ as a lowest K-type by the orbits of a finite group acting in a complex vector space (the dual of the vector part of a certain Cartan subgroup of G), and the result that μ necessarily occurs with multiplicity one in such representations.