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On the Plancherel formula for almost algebraic real Lie groups. (English) Zbl 0546.22014
Lie group representations III, Proc. Spec. Year, College Park/Md. 1982-83, Lect. Notes Math. 1077, 101-165 (1984).
[For the entire collection see Zbl 0535.00011.]
Let G be a real separable Lie group, it is called almost algbraic if there exists a discrete central subgroup of G such that the quotient group is an open subgroup of a group of rational points of a linear algebraic group defined over \({\mathbb{R}}\). In an earlier paper [Construction de représentations unitaires d’un groupe de Lie, C.I.M.E. Cortona (1980)] the author constructed a set of irreducible unitary representations of a Lie group, which in the case of G is sufficiently large to decompose its regular representation. In this paper the author surveys some previous work on these representations with some improvements in the almost algebraic case. Finally he gives a rather explicit form of the Plancherel formula for G.
Reviewer: D.Miličić

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.