Volume growth, temperedness and integrability of matrix coefficients on a real spherical space
type: Item
instance of: scholarly article
title: Volume growth, temperedness and integrability of matrix coefficients on a real spherical space
zbMATH Open document ID: 1342.22035
author: Friedrich Knop
author: Bernhard Krötz
author: Eitan Sayag
author: Henrik Schlichtkrull
published in: Journal of Functional Analysis
publication date: 2016-05-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1407.8006
review text: First, the authors consider a real algebraic homogeneous space \(Z=G/H\), attached to an algebraic real reductive group \(G\) and assumed to carry a \(G\)-invariant measure, where they define the volume weight \(v\) and study the growth of \(v\) as a function of \(z\in Z\). Next, they study the Harish-Chandra Schwartz space on \(Z\) by means of a function space \( \mathcal C (Z)\) and show several interesting properties (e.g., \( \mathcal C (Z)\) is nuclear). Finally, they define the integrability property \((I)\) by means of unitary irreducible representations \((\pi,\mathcal H)\) of an \(H\)-invariant distribution vector \(\eta\) and of the \(L^{p}\)-integrability of the matrix coefficient \(m_{v,\eta}\). They discuss different cases where \(Z\) has the property \((I)\) (e.g. if \(Z=G/H\) is a wavefront real spherical space with \(H\) reductive). The investigation of several related phenomena is also very interesting.
reviewed by: Eugen Pascu
Mathematics Subject Classification ID: 22F30
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 6579043
zbMATH Keywords: real spherical spaces
zbMATH Keywords: harmonic analysis on reductive spaces
zbMATH Keywords: \(L^{p}\)-integrability
zbMATH Keywords: homogenous spaces
MaRDI profile type: Publication
OpenAlex ID: W2963225356
arXiv ID: 1407.8006
cites work: The principal series for a reductive symmetric space. II: Eisenstein integrals
cites work: On the support of Plancherel measure
cites work: Smooth Fréchet globalizations of Harish-Chandra modules
cites work: A comparison theorem for Lie algebra homology groups
cites work: Cohomological properties of the canonical globalizations of Harish-Chandra modules. (Consequences of theorems of Kashiwara-Schmid, Casselman, and Schneider-Stuhler)
cites work: Meromorphic basis of \(H\)-invariant distribution vectors for generalized principal series of reductive symmetric spaces: Functional equation
cites work: Analytic representation theory of Lie groups: general theory and analytic globalizations of Harish-Chandra modules
cites work: Q5183990
cites work: Analytic localization of group representations
cites work: Q4369235
cites work: Simple compactifications and polar decomposition of homogeneous real spherical spaces
cites work: The local structure theorem for real spherical varieties
cites work: The tempered spectrum of a real spherical space
cites work: Finite multiplicity theorems for induction and restriction
cites work: Decay of matrix coefficients on reductive homogeneous spaces of spherical type
cites work: The harmonic analysis of lattice counting on real spherical spaces
cites work: Multiplicity bounds and the subrepresentation theorem for real spherical spaces
cites work: Decay of eigenfunctions on semisimple symmetric spaces
cites work: Eigenspaces of the Laplacian on hyperbolic spaces: composition series and integral transforms
cites work: Q3994337
DOI: 10.1016/J.JFA.2016.04.001
description: scientific article; zbMATH DE number 6579043