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Group automorphisms preserving equivalence classes of unitary representations

Group automorphisms preserving equivalence classes of unitary representations Abstract. We introduce and investigate the notion of a quasi-complete group. A group G is quasi-complete if every automorphism f A AutðGÞ, with the property that p and p f are unitarily equivalent for every unitary irreducible representation p of G, is an inner automorphism of G. Our main result is that every connected linear real reductive Lie group is quasi-complete. 2000 Mathematics Subject Classification: 22D45; 22E46, 43A65. Introduction The study of the automorphisms group AutðGÞ of a topological group G and of its distinguished normal subgroups raises many interesting questions. In connection with the normal subgroup of inner automorphisms, it was shown in [DG] that any group G can be realized as the outer automorphism group of some group H, i.e. G G AutðHÞ=InnðHÞ. In another direction, an abstract characterization of inner automorphisms in terms of their extension properties was recently given in [Sc]. Other distinguished normal subgroups of AutðGÞ are the automorphisms AutðGÞC preserving the conjugacy classes of G and the automorphisms AutðGÞG preserving the ^ equivalence classes of (continuous) unitary representations of G. These definitions appear in [Bu] for finite groups, but carry over to more general settings. In general, one has the inclusions InnðGÞ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Group automorphisms preserving equivalence classes of unitary representations

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Publisher
de Gruyter
Copyright
Copyright © 2004 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2004.023
Publisher site
See Article on Publisher Site

Abstract

Abstract. We introduce and investigate the notion of a quasi-complete group. A group G is quasi-complete if every automorphism f A AutðGÞ, with the property that p and p f are unitarily equivalent for every unitary irreducible representation p of G, is an inner automorphism of G. Our main result is that every connected linear real reductive Lie group is quasi-complete. 2000 Mathematics Subject Classification: 22D45; 22E46, 43A65. Introduction The study of the automorphisms group AutðGÞ of a topological group G and of its distinguished normal subgroups raises many interesting questions. In connection with the normal subgroup of inner automorphisms, it was shown in [DG] that any group G can be realized as the outer automorphism group of some group H, i.e. G G AutðHÞ=InnðHÞ. In another direction, an abstract characterization of inner automorphisms in terms of their extension properties was recently given in [Sc]. Other distinguished normal subgroups of AutðGÞ are the automorphisms AutðGÞC preserving the conjugacy classes of G and the automorphisms AutðGÞG preserving the ^ equivalence classes of (continuous) unitary representations of G. These definitions appear in [Bu] for finite groups, but carry over to more general settings. In general, one has the inclusions InnðGÞ

Journal

Forum Mathematicumde Gruyter

Published: Mar 30, 2004

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