Spectra in representations of semisimple Lie group

Abstract

The spectrum of the infinitesimal generator of a one-parameter group of unitary operators arising from a representation of a semisimple Lie group is determined. The support of the spectral measure depends only on whether the group is a group of automorphisms of a bounded symmetric domain. 1. Let [I be a (strongly) continuous unitary representation of a semisimple Lie group G on a separable Hilbert space H. Let C' (ll) denote the set of all vectors v EC H such that HI()v is a C' function on G. Then C'(T) is a dense linear subspace of H, and contains the dense set of analytic vectors for the representation [6]. We denote the Lie algebra of G by g0. If x E g0, (II(exp(tx))) is a oneparameter group of unitary operators and has a skew-adjoint infinitesimal generator dll(x). Thus IH(exp(tx))=exp(t dJl(x)), t E R. Also C'(1I) is contained in the domain of dHl(x), and is invariant under dJl(x) for all x e g0. Moreover, x-dr1(x)IC'([J) defines a representation of g0 by essentially skew-adjoint operators. We shall examine the spectrum of the selfadjoint operator -idf(x) whenever exp(tx) satisfies a certain noncompactness condition explained below. The information obtained will depend on G but not on HI or x. More precisely, if f A dE, is the spectral resolution of i dJJ(x), C. C. Moore [5] has shown that the projection valued measure E is absolutely continuous with respect to Lebesgue measure, and its support is either (-oo, oo), (0, xo) or (o, 0). In the first instance, we shall say -i dll(x) has two-sided spectrum, while in the second and third instances we shall say idfl (x) has one-sided spectrum. One's feeling, and this is borne out by our investigations, is that most of the time we can expect a two-sided spectrum, and that a one-sided Received by the editors February 6, 1973. AMS (MOS) subject classifications (1970). Primary 22D10, 22E45, 47B1 5; Secondary 32M15, 81A78.