Fourier inversion on a reductive symmetric space
type: Item
instance of: scholarly article
title: Fourier inversion on a reductive symmetric space
zbMATH Open document ID: 0993.22006
DOI: 10.1007/BF02392823
author: Erik P. van den Ban
author: Henrik Schlichtkrull
published in: Acta Mathematica
publication date: 2000-12-06T00:00:00Z
review text: Let \(X=G/H\) be a reductive symmetric space, that is, \(G\) is a connected reductive Lie group of Harish-Chandra's class with an involution \(\sigma\), and \(H\) an open subgroup of the group of \(\sigma\)-fixed elements in \(G\). As an important step towards a complete Plancherel theorem (i.e. an explicit direct integral decomposition) for the regular representation of \(G\) in \(L^2(X)\), the authors define in the article at hand a ''pseudo-wave packet'' transform which they show to be an inverse of a suitably normalized Fourier transform. It is part of a series of papers by the authors on topics circling around this issue; in particular, it should be viewed as a direct continuation of \textit{E.P. van den Ban} and \textit{H. Schlichtkrull} [Ann. Math. (2) 145, No. 2, 267--364 (1997; Zbl 0878.43018)] and \textit{E.P. van den Ban} and \textit{H. Schlichtkrull} [Invent. Math. 130, No. 3, 517--574 (1997; Zbl 0936.22008)], which treat the ''most continuous part'' of the spectrum and the semisimple case, respectively. The article is structured as follows. Let \(K\) be a maximal compact subgroup of \(G\) invariant under \(\sigma\), let \((\tau,V_{\tau})\) be a f.d. unitary representation of \(K\), \(\lambda\in{\mathfrak a}^*_{q\mathbb C}\) and \(x\in X\). In Sections 1 and 2, the \(\tau\)-spherical Fourier transform \(\mathcal{F}\) is defined using normalized Eisenstein integrals \(E^0(\lambda:x)\) from one of the papers mentioned previously. In Section 3 (and part of the appendix), it is shown that \(E^0(\lambda:x)\) is singular only along real root hyperplanes in \({\mathfrak a}^*_{q\mathbb C}\). Pseudo-wave packets are defined in Section 4 on a suitable dense subset \(X_+\) of \(X\) by \[ \mathcal{T}_{\eta}\varphi(x)= |W|\int_{\eta+i{\mathfrak a}^*_q} E_+(\lambda:x)\varphi(x) d\lambda, \] where \(E_+\) is a linear operator into \(V_{\tau}\) appearing in the expansion of \(E^0\) and defined by its asymptotic behaviour and a vanishing property relative to some invaria…
reviewed by: Ilka Agricola
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 43A85
zbMATH DE Number: 1541201
zbMATH Keywords: reductive symmetric space
zbMATH Keywords: Fourier transformation
zbMATH Keywords: Eisenstein series
zbMATH Keywords: Plancherel theorem
zbMATH Keywords: residue operators
MaRDI profile type: Publication
cites work: A Paley-Wiener theorem for real reductive groups
cites work: Q3766269
cites work: Invariant differential operators on a semisimple symmetric space and finite multiplicities in a Plancherel formula
cites work: The principal series for a reductive symmetric space. I. $H$-fixed distribution vectors
cites work: The principal series for a reductive symmetric space. II: Eisenstein integrals
cites work: The action of intertwining operators on spherical vectors in the minimal principal series of a reductive symmetric space
cites work: Wave packets in the Schwartz space of a reductive symmetric space
cites work: Fourier transforms on a semisimple symmetric space
cites work: The most continuous part of the Plancherel decomposition for a reductive symmetric space
cites work: Expansions for Eisenstein integrals on semisimple symmetric spaces
cites work: Q4519092
cites work: Meromorphic basis of \(H\)-invariant distribution vectors for generalized principal series of reductive symmetric spaces: Functional equation
cites work: Fourier transform on the Schwartz space of a reductive symmetric space
cites work: Canonical Extensions of Harish-Chandra Modules to Representations of G
cites work: Eisenstein integrals for reductive symmetric spaces: Temperedness, majorations. Small \(B\)-matrix
cites work: Truncation for reductive symmetric spaces
cites work: Plancherel formula for reductive symmetric spaces
cites work: Towards a Paley-Wiener theorem for semisimple symmetric spaces
cites work: Q3331943
cites work: Q4301588
cites work: Intertwining operators for semisimple groups. II
cites work: Q3699041
cites work: Harmonic analysis on real reductive groups
cites work: Elementary spherical functions on symmetric spaces.
cites work: Q3994337
cites work: Q4004200
full work available at URL: https://doi.org/10.1007/bf02392823
OpenAlex ID: W2149754515
description: scientific article; zbMATH DE number 1541201