Classification of the irreducible representations of semisimple Lie groups.
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 mu in K, mu is a nonnegative real number. Then if pi in G, mu is called a lowest K-type of pi if mu is minimal among the K-types occurring in pi. We announce a parameterization of the set of representations containing mu 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 mu necessarily occurs with multiplicity one in such representations.
Vogan
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Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139.