Continuity conditions for finite-dimensional representations of connected locally compact groups
type: Item
instance of: scholarly article
title: Continuity conditions for finite-dimensional representations of connected locally compact groups
zbMATH Open document ID: 1263.22006
DOI: 10.1134/S1061920812040097
published in: Russian Journal of Mathematical Physics
publication date: 2013-04-08T00:00:00Z
review text: The discontinuity subgroup of a relatively compact homomorphism \(\pi\) between a connected locally compact group \(G\) and a topological group \(H\) is the intersection \(\bigcap\limits_U \overline{\pi(U)}\), where \(U\) runs over all the neighbourhoods of the identity of \(G\). It is a closed connected normal subgroup of the closure \(\overline{\pi(G)}\) of the image of \(\pi\), as proved in the author's paper [Izv. Math. 72, No. 1, 169--205 (2008); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 1, 183--224 (2008; Zbl 1158.22002)], and measures in some sense how discontinuous \(\pi\) is. The first result of the paper under review addresses the case of a locally bounded finite-dimensional homomorphism \(\pi\) between connected Lie groups. Using structural facts established by \textit{K. Iwasawa} in [Ann. Math. (2) 50, 507--558 (1949; Zbl 0034.01803)] and results of his own [Izv. Math. 72, No. 1, 169--205 (2008); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 1, 183--224 (2008; Zbl 1158.22002)], the author proves that the discontinuity subgroup of such a morphism is a finite-dimensional torus in the center of \(\overline{\pi(G)}\). The rest of the paper is devoted to the case where \(\pi\) is a locally bounded finite-dimensional representation of a connected locally compact group \(G\). Let \(K\) be a compact normal subgroup such that \(G/K\) is a Lie group, and denote by \(H\) the commutant of \(K\) so that \(G=HK\). It is proved that, if \(\pi\) is continuous on \(K\), then it is automatically continuous on \(H'K\), where \(H'\) is the commutator subgroup of \(H\). Building on this and results he obtained in [Trans. Mosc. Math. Soc. 2011 (2011); translation from Tr. Mosk. Mat. O.-va 2011, No. 1, 79--95 (2011; Zbl 1242.22010)] regarding the structure of irreducible finite-dimensional representations, the author finally states a continuity criterion: a locally bounded finite-dimensional representation of a connected locally compact group \(G\) …
reviewed by: Pierre Clare
Mathematics Subject Classification ID: 22D12
Mathematics Subject Classification ID: 22D05
Mathematics Subject Classification ID: 22E15
zbMATH DE Number: 6151584
zbMATH Keywords: connected locally compact group
zbMATH Keywords: connected Lie group
zbMATH Keywords: locally bounded group homomorphism
zbMATH Keywords: finite-dimensional representation
author: A. I. Shtern
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6151584