Plancherel formula for reductive symmetric spaces
type: Item
instance of: scholarly article
title: Plancherel formula for reductive symmetric spaces
zbMATH Open document ID: 0906.22015
DOI: 10.2307/121014
author: Patrick Delorme
published in: Annals of Mathematics. Second Series
publication date: 1999-02-14T00:00:00Z
review text: Let \(G\) be a reductive Lie group of Harish-Chandra class, \(\theta\) a Cartan involution of \(G\), \(K\) the subgroup of the fixed point set of \(\theta\), \(\sigma\) an involution of \(G\) commuting with \(\theta\) and let \(H\) be an open subgroup of the fixed point subgroup of \(\sigma\). Based on many previous works, the author proves a Plancherel formula for the reductive symmetric space \(G/H\). Fixing a finite-dimensional unitary representation \(\tau\) of \(K\) and using the algebra \(D(G/H)\) of \(G\)-invariant differential operators on \(G/H\) to characterize functional spaces, one defines the normalized Fourier transformation on the space of \(\tau\)-spherical or \(K\)-finite Schwartz functions on \(G/H\). The author gives the inverse Fourier transformation making use of Eisenstein integrals. He then determines the image of the Schwartz space under the Fourier transformation and shows the Fourier inversion formula. The paper demands expert knowledge on representation theory for semisimple Lie groups [cf. \textit{P. Delorme}, J. Funct. Anal. 136, 422-509 (1996; Zbl 0865.43010); Acta Math. 179, 41-77 (1997; Zbl 0896.43001); \textit{J. Carmona} and \textit{P. Delorme}, J. Funct. Anal. 122, 152-221 (1994; Zbl 0831.22004); Transformation de Fourier pour les espaces symétriques réductifs (to appear in Invent. Math.)].
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 43A90
zbMATH DE Number: 1187684
zbMATH Keywords: reductive Lie group
zbMATH Keywords: Plancherel formula
zbMATH Keywords: symmetric space
zbMATH Keywords: Fourier transformation
zbMATH Keywords: Eisenstein integrals
zbMATH Keywords: representation theory
zbMATH Keywords: semisimple Lie groups
reviewed by: Hidenori Fujiwara
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/121014
OpenAlex ID: W2023117756
description: scientific article; zbMATH DE number 1187684