Unitary spectrum and the fundamental group for actions of semisimple Lie groups

Abstract

Suppose a locally compact group G acts on a separable metrizable space M, preserving a finite measure #. Then there is an associated unitary representation of G on LZ(M,/~)o, the square integrable functions orthogonal to the constants. This representation will in general be far from irreducible, and in fact, when G is not compact need not have any irreducible subrepresentations. Thus, in the decomposition of LZ(G)o into irreducible constituents, one not only requires a direct integral rather than a direct sum, but the associated spectral measure on the unitary dual G (say for type I groups) will, except in special circumstances, have no atoms. The point of this paper is to give natural and general conditions, mostly involving just rq(M), under which there will be an infinite atomic spectrum. (I.e., the spectral measure has infinitely many atoms.) The basic example of actions in which the spectrum is purely atomic, i.e., when the representation is a direct sum of irreducible representations, is the case of M = G/F, where F c G is a cocompact discrete subgroup. In this case, assuming 9 G to be simply connected, we have 7tl(M ) = F. The first results of this paper show that when G is a simple Lie group with R-rank(G)=> 2, we can deduce that any action of G on a compact manifold M with fundamental group isomorphic to such a F, and satisfying a natural hypothesis of either a geometric or ergodic theoretic nature, must also have an infinite atomic spectrum. In fact, under certain natural hypotheses the same is true as long as there is a surjection of hi(M) onto F. To formulate this more precisely, let F c Ad(G) be a cocompact lattice, and let Pr be the unitary representation of Ad(G) on L2(Ad(G)/F)o . We recall that an action of a group on a manifold (or more generally on a standard space in the sense of [4]) is called engaging if there is no loss of ergodicity in passing to finite covers. (See I'4] for discussion.)