A characterization of the stable invariant integral
type: Item
instance of: scholarly article
title: A characterization of the stable invariant integral
zbMATH Open document ID: 0607.22005
author: Garth Warner
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1986-01-01T00:00:00Z
review text: The study of the invariant integral and its Fourier transform is a central problem in harmonic analysis on a reductive Lie group G, e.g., in Harish-Chandra's work on the Plancherel formula and in the theory of the Selberg trace formula. The stabilized invariant integral of a function f relative to a \(\theta\)-stable Cartan A is the sum over w in \(W\setminus W_ C\) of the sign \(\epsilon\) (w) times the conjugate of the invariant integral \(F^ A_ f\) by w. Discrete series representations of G are partitioned into ''stable'' subsets as follows. Let T be a \(\theta\)-stable compact Cartan subgroup of G. For each regular integral \(\chi\), let \(\hat G_ d(\chi)\) be the set of cardinality \(W\setminus W_ C\) consisting of those representations \(\omega\) with infinitesimal character equal to \(\chi\). The sum \(\Theta_{\Omega}\) of the characters of the representations in such a set defines a central eigendistribution on G, which is stable in the sense of \textit{D. Shelstad} [Compos. Math. 39, 11-45 (1979; Zbl 0431.22011)]. Stabilized sums of the functions \(\Delta_ T \Theta_{\omega}\) are \(W_ C\)-skew and allow Fourier expansion of \(C^{\infty}\) functions on T which are \(W_ C\)-skew. Shelstad determines a ''pointwise'' characterization of the invariant integrals for functions f in the Schwartz space of G. The main result of the author is a characterization in terms of Fourier transforms of stabilized invariant integrals for f in \(C_ c^{\infty}(G)\). His proof uses explicit inversion formulae of Herb for the invariant integrals and a Paley-Wiener type theorem due to Clozel and Delorme. Their results are reviewed in the present paper.
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 43A80
zbMATH DE Number: 3981452
zbMATH Keywords: invariant integral
zbMATH Keywords: reductive Lie group
zbMATH Keywords: Discrete series representations
zbMATH Keywords: central eigendistribution
zbMATH Keywords: stable
zbMATH Keywords: Schwartz space
zbMATH Keywords: Fourier transforms
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228487
OpenAlex ID: W1968615659
cites work: Q3861916
cites work: Le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs. (The invariant Paley-Wiener theorem for reductive Lie groups)
cites work: Q3722748
cites work: Invariant Eigendistributions on a Semisimple Lie Group
cites work: Discrete series for semisimple Lie groups. I: Construction of invariant eigendistributions
cites work: Discrete series for semisimple Lie groups. II: Explicit determination of the characters
cites work: Harmonic analysis on real reductive groups. I: The theory of the constant term
cites work: Harmonic analysis on real reductive groups. III: The Maass-Selberg relations and the plancherel formula
cites work: Q4750881
cites work: Fourier Inversion of Invariant Integrals of Semisimple Real Lie Groups
cites work: Fourier Inversion and the Plancherel Theorem for Semisimple Real Lie Groups
cites work: Discrete Series Characters and Fourier Inversion on Semisimple Real Lie Groups
cites work: The characters of some induced representations of semi-simple Lie groups
cites work: A note on automorphisms of real semisimple Lie algebras
cites work: Q4108530
cites work: Q3949226
cites work: The Fourier transform on semisimple Lie groups of real rank one
cites work: Q3868000
cites work: Invariant harmonic analysis on split rank one groups with applications
cites work: Harmonic analysis on real reductive groups
cites work: Q5682517
cites work: Q5682518
cites work: The action of a real semisimple Lie group on a complex flag manifold. II. Unitary representations on partially holomorphic cohomology spaces
DOI: 10.2748/TMJ/1178228487
Recommended article: Integral representation of invariant functionals
Recommended article: Integral representation of invariant functions
Recommended article: Q4207052
Recommended article: Q4876636
Recommended article: A stability theory for integral equations
Recommended article: Q3641539
Recommended article: Invariant functionals and Zamolodchikovs' integral
Recommended article: A Note on the Stability of an Integral Equation
description: scientific article; zbMATH DE number 3981452