On the plancherel formula for almost algebraic real lie groups

Abstract

Introduction. In this paper I consider some problems in harmonic analysis for a class of real Lie groups which I call algebraic. More precisely, an almost algebraic group is a triple where G is a real separable Lie group, r a discrete subgroup of the center of G, a linear algebraic Lie group defined over JR, such that G/r is open (for the Hausdorff topology) in the group g(JR) of real points of (1) There are two main reasons to consider this class of groups. First, let H be a connected, simply connected Lie group. Then its derived group G can be given a structure of an almost algebraic group and Pukanszky has shown in [Pu] that several questions in harmonic analysis can be reduced (however in a nontrivial way) to questions about G. Second, this class is very convenient: it is stable under all the constructions which occur in the Mackey inductive process, and the groups G are type I (by Harish Chandra and Dixmier [Di 1]). In my paper [Du3] I constructed a set of unitary irreducible representations of a Lie group G which, when G has an almost algebraic structure, is sufficiently large to decompose the regular representation. Here, I present a survey of some work done in [Du 3,5], {Li 1,2,3], [BOU], [Kha 1,2] 0n tnese representations, including some improvements in the almost algebraic case. I then write a rather explicit form of the Plancherel formula for G.