A unified approach to concrete Plancherel theory of homogeneous spaces
type: Item
instance of: scholarly article
title: A unified approach to concrete Plancherel theory of homogeneous spaces
zbMATH Open document ID: 0940.43008
DOI: 10.1007/BF02677843
author: Ronald L. Lipsman
published in: Manuscripta Mathematica
publication date: 1998-06-16T00:00:00Z
full work available at URL: https://eudml.org/doc/156328
review text: Let \(G\) be a Lie group, \(H\subset G\) a closed subgroup, and \(\tau= \text{Ind}^G_H1\) the quasi regular representation supposed to be type I. Then we have the abstract Plancherel theorem \(\tau= \int^\oplus_{\widehat G}n_\pi \pi d\mu(\pi)\) for the homogeneous space \(G/H\). According to this formula, we can decompose the Dirac measure in the distribution sense [cf. \textit{R. Penney}, J. Funct. Anal. 18, 177-190 (1975; Zbl 0305.22016); \textit{P. Bonnet}, ibid. 55, 220-246 (1984; Zbl 0531.43001)]. Furthermore we can write down in many cases this decomposition more explicitly to get a concrete Plancherel formula. The search for such a formula has been carried out in two mutually exclusive cases according as the multiplicity function \(n_\pi\) is finite or infinite \(\mu\) -- a.e. [cf. a series of the author's works: Progr. Math. 82, 135-145 (1990; Zbl 0758.22005); Pac. J. Math. 151, 265-295 (1991; Zbl 0759.22012); ibid. 159, 351-377 (1993; Zbl 0798.22005); the reviewer's two works: Pac. J. Math. 127, 329-352 (1987; Zbl 0588.22008); Representation theory of Lie groups and Lie algebras, World Scientific, Singapore, 140-150 (1992), \textit{G. Grélaud}, J. Reine Angew. Math. 398, 92-100 (1989; Zbl 0666.43004)]. In this paper, the author considers Strichartz homogeneous spaces \(G=V\times R\), a semidirect product of a real algebraic group \(R\) by a normal vector subgroup \(V\) with a subgroup \(H=U\times S\) of the same type satisfying \(S\subset R\), \(U\subset V\), and shows that the two cases separated by the property of multiplicities should be treated in a unified way, namely that the distinction between the two cases is not really a matter of Plancherel theory but of orbital constructs or of other invariants associated to the homogeneous space \(G/H\). Several interesting remarks are given relating to Penney's and Bonnet's Plancherel formula. Unfortunately there are some typographical errors.
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 1135212
zbMATH Keywords: quasi regular representation
zbMATH Keywords: Plancherel theorem
zbMATH Keywords: homogeneous space
zbMATH Keywords: Penney's and Bonnet's Plancherel formula
reviewed by: Hidenori Fujiwara
MaRDI profile type: Publication
cites work: Multiplicité un pour les espaces symétriques exponentiels
cites work: A geometric criterion for Gelfand pairs associated with the Heisenberg group
cites work: Transformation de Fourier des distributions de type positif sur un groupe de Lie unimodulaire
cites work: Q5531996
cites work: Représentations monomiales des groupes de Lie nilpotents. (Monomial representations of nilpotent Lie groups)
cites work: Q3487656
cites work: Q3973970
cites work: Q3408327
cites work: Complex Fourier Analysis on a Nilpotent Lie Group
cites work: La formule de Plancherel pour les espaces homogènes des groupes de Heisenberg.
cites work: The plancherel formula for group extensions
cites work: Q3973973
cites work: Q3408326
cites work: The Penney-Fujiwara Plancherel formula for abelian symmetric spaces and completely solvable homogeneous spaces
cites work: The Plancherel formula for homogeneous spaces with polynomial spectrum
cites work: The multiplicity function on exponential and completely solvable homogeneous spaces
cites work: Q4317813
cites work: The Penney-Fujiwara Plancherel formula for Gelfand pairs
cites work: Q4342767
cites work: Abstract Plancherel theorems and a Frobenius reciprocity theorem
cites work: Group representations on spaces of distributions
OpenAlex ID: W4238311918
description: scientific article; zbMATH DE number 1135212