Some irreducible unitary representations of G(K) for a simple algebraic group \(G\) over an algebraic number field \(K\)
type: Item
instance of: scholarly article
title: Some irreducible unitary representations of G(K) for a simple algebraic group \(G\) over an algebraic number field \(K\)
zbMATH Open document ID: 1022.22018
published in: Mathematische Zeitschrift
publication date: 2003-06-02T00:00:00Z
review text: Let \(G\) be a simply connected simple linear algebraic group defined over a number field \(K\). The authors investigate the behaviour of irreducible unitary representations of the adelic group \(G({\mathbf A}_K)\) upon restriction to the global points \(G(K)\). They prove that the restriction to \(G(K)\) stays irreducible and that inequivalent representations have inequivalent restrictions. If all local groups \(G(K_v)\) have Kazhdan's property \((T)\), it is proved for inequivalent (irreducible unitary) representations \(\pi, \pi'\) which are not both weakly contained in the regular representation that \(\pi |G(K), \pi' |G(K)\) are not even weakly equivalent. A similar result is stated for the restrictions of representations of \(\prod_{v \in S} G(K_v)\) to \(G({\mathcal O}_S)\), where \(S\) is a finite set of places containing the archimedean places.
reviewed by: Rainer Schulze-Pillot
Mathematics Subject Classification ID: 22E55
Mathematics Subject Classification ID: 20G30
zbMATH DE Number: 1922565
zbMATH Keywords: unitary representations
zbMATH Keywords: linear algebraic groups
zbMATH Keywords: restriction of representations
zbMATH Keywords: Kazhdan's property (T)
author: Bachir Bekka
author: Michael G. Cowling
MaRDI profile type: Publication
full work available at URL: s00209-002-0442-6
OpenAlex ID: W2025345542
DOI: 10.1007/S00209-002-0442-6
description: scientific article; zbMATH DE number 1922565