Representations of Lie algebras by normal operators

Abstract

Recently I. E. Segal [2] has proposed a study of the unitary representations of a complex semisimple Lie group G by studying holomorphic representations of G by normal operators. To this end he proved that every unitary representation U of G may be written U(g) = R(g)R(g-) * (g E G) where R is an analytic holomorphic representation of G by normal operators such that if R(g1) and R(g2) are defined then R(gi) commutes with R(g2)*. Now a representation of a group by normal operators is a peculiar phenomenon because the product of two normal operators is usually not normal. In this paper we study this peculiarity in Lie algebraic terms. We achieve a decomposition of a representation of a semisimple Lie algebra by normal operators into the sum of two representations which commute with each other. One of these is by skewadjoint operators, and the other is a representation (necessarily by normal opeiators) which commutes with its contragredient. We begin by establishing our results in the following generalised setting: Let ut be a Lie algebra over some fixed field of characteristic other than 2. On ut we assume the existence of a linear mapping sending an element x of ut into x* such that (x*)*=x and [x*, y*] = [x, y]*. In other words our mapping is an anti-automorphism of order 2. If a and b are subsets of ut let [a, b ] =the linear span of I [x, y] I xCa, yEb} and a* = {x*I x(E}. We will say that an element x of u is nrml if [x, x*] =0 and that x is skew if x*= -x.