The Plancherel formula for homogeneous spaces with polynomial spectrum
type: Item
instance of: scholarly article
title: The Plancherel formula for homogeneous spaces with polynomial spectrum
zbMATH Open document ID: 0798.22005
author: Ronald L. Lipsman
published in: Pacific Journal of Mathematics
publication date: 1994-03-14T00:00:00Z
review text: Let \(G\) be a connected Lie group, and \(H\) a closed subgroup of \(G\). The author continues to study Penney's Plancherel formula [cf. \textit{R. Penney}: J. Funct. Anal. 18, 177-190 (1975; Zbl 0305.22016); \textit{H. Fujiwara}: Pac. J. Math. 127, 329-352 (1987; Zbl 0588.22008)], for the quasi-regular representation \(\tau = \text{ind}^ G_ H 1\). As a sequel to the author's previous one [Pac. J. Math. 151, 265-295 (1991; Zbl 0759.22012)], where almost all of the irreducible representations that appear in the spectrum of \(\tau\) were monomial, this paper treats the case that the spectrum of \(\tau\) consists of irreducible representations induced from a finite dimensional representation. Attention is focused on Strichartz homogeneous spaces wherein \(G\) and \(H\) are semidirect products of normal abelian subgroups by reductive Lie groups [cf. \textit{R. Strichartz}: Trans. Am. Math. Soc. 296, 387-409 (1986; Zbl 0618.43005)], especially on Strichartz symmetric spaces. It also finds place to discuss criteria for finite multiplicity and for multiplicity-free, and finally a generalization of Frobenius reciprocity.
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 22D30
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 509121
zbMATH Keywords: connected Lie group
zbMATH Keywords: Plancherel formula
zbMATH Keywords: irreducible representations
zbMATH Keywords: Strichartz homogeneous spaces
zbMATH Keywords: semidirect products
zbMATH Keywords: reductive Lie groups
zbMATH Keywords: Strichartz symmetric spaces
zbMATH Keywords: finite multiplicity
zbMATH Keywords: Frobenius reciprocity
reviewed by: Hidenori Fujiwara
MaRDI profile type: Publication
full work available at URL: pjm.1993.159.351
OpenAlex ID: W2058580236
DOI: 10.2140/PJM.1993.159.351
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description: scientific article; zbMATH DE number 509121