Representations of complex semisimple Lie groups and Lie algebras

Abstract

1. Notation. The object of this note is to announce some results on representations of complex semisimple Lie groups and Lie algebras. © is a semisimple Lie algebra over C, the field of complex numbers. ®, considered over i?, the field of real numbers, is denoted by ® 0. ^ is a Cartan subalgebra of ®, W, the Weyl group of (®, I)). We use the standard terminology in the theory of semisimple Lie algebras (Jacobson [3] and Harish-Chandra [2(a)], [2(b)], [2(c)]). P0 is a positive system of roots, fixed once for all and £0= {#i, • • • , ou}, the associated fundamental system, n= ]C«ep 0 ®~a; tt, considered as a Lie algebra over JR, is denoted by tio. i)o = X)« R'Ha. Fix a square root ( —1)1/2 of — 1 in C. io is a compact form of ® containing ( —1)1/2 i)0. ®o = io+i)o+tto is an Iwasawa decomposition of ®o and G = K-A+-N the corresponding decomposition of G. c(X-*Xc) is the conjugation of ® corresponding to the compact form io. Let ® denote the Lie algebra ®X® over C, and let i:X-*(X°,X)