Schur orthogonality relations for certain non square integrable representations of real semisimple Lie groups
type: Item
instance of: scholarly article
title: Schur orthogonality relations for certain non square integrable representations of real semisimple Lie groups
zbMATH Open document ID: 0609.22007
author: Hisaichi Midorikawa
published in: Tokyo Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Let G be a noncompact connected real semisimple Lie group with finite center and K its maximal compact subgroup. Denote by d(x) the distance from x to the origin in the Riemannian symmetric space G/K. Let (\(\pi\),H) be an irreducible unitary representation of G. Assume that there exists a K-finite vector \(\phi_ 0\) in H such that \[ 0<\lim_{\epsilon \to +0}\epsilon \int_{G}| (\pi (x)\phi_ 0,\phi_ 0)|^ 2 e^{- \epsilon d(x)} dx<+\infty \quad. \] Then there exists a positive constant \(d_{\pi}\) such that \[ \lim_{\epsilon \to +0}\epsilon \int_{G}(\pi (x)\phi,\psi) \overline{(\pi (x)\phi ',\psi ')} e^{-\epsilon d(x)} dx=d_{\pi}^{-1} (\phi,\phi ') \overline{(\psi,\psi ')} \] for arbitrary K-finite vectors \(\phi\), \(\phi\) ', \(\psi\) and \(\psi\) '. The character \(\theta_{\pi}\) of such representation is a tempered distribution on G.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 22D10
Mathematics Subject Classification ID: 46F10
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 3985531
zbMATH Keywords: real semisimple Lie group
zbMATH Keywords: Riemannian symmetric space
zbMATH Keywords: unitary representation
zbMATH Keywords: tempered distribution
Wikidata QID: Q115219783
reviewed by: Dragan Milicic
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270151217
OpenAlex ID: W2007229114
DOI: 10.3836/TJM/1270151217
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description: scientific article; zbMATH DE number 3985531