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How tempered representations of a semisimple Lie group contract to its Cartan motion group

hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
hal.structure.identifierInstitut de Mathématiques de Jussieu - Paris Rive Gauche [IMJ-PRG (UMR_7586)]
dc.contributor.authorAfgoustidis, Alexandre
HAL ID: 747228
ORCID: 0000-0002-6580-7951
dc.date.accessioned2023-03-06T14:00:58Z
dc.date.available2023-03-06T14:00:58Z
dc.date.issued2015
dc.identifier.urihttps://basepub.dauphine.psl.eu/handle/123456789/24540
dc.language.isoenen
dc.subjectRepresentations of semisimple Lie groupsen
dc.subjectLie group contractionsen
dc.subjectMackey analogyen
dc.subjectTempered dualen
dc.subject.ddc519en
dc.titleHow tempered representations of a semisimple Lie group contract to its Cartan motion groupen
dc.typeDocument de travail / Working paper
dc.description.abstractenGeorge W. Mackey suggested in 1975 that there should be analogies between the irreducible unitary representations of a noncompact semisimple Lie group G and those of its Cartan motion group − the semidirect product G 0 of a maximal compact subgroup of G and a vector space. In these notes, I focus on the carrier spaces for these representations and try to give a precise meaning to some of Mackey's remarks. I first describe a bijection, based on Mackey's suggestions, between the tempered dual of G − the set of equivalence classes of irreducible unitary representations which are weakly contained in L 2 (G) − and the unitary dual of G 0. I then examine the relationship between the individual representations paired by this bijection : there is a natural continuous family of groups interpolating between G and G 0 , and starting from the Hilbert space H for an irreducible representation of G, I prove that there is an essentially unique way of following a vector through the contraction from G to G 0 within a fixed Fréchet space that contains H. It then turns out that there is a limit to this contraction process on vectors, and that the subspace of our Fréchet space thus obtained naturally carries an irreducible representation of G 0 whose equivalence class is that predicted by Mackey's analogy.en
dc.publisher.cityParisen
dc.identifier.citationpages51en
dc.relation.ispartofseriestitleCahier de recherche CEREMADE, Université Paris Dauphine-PSLen
dc.subject.ddclabelProbabilités et mathématiques appliquéesen
dc.identifier.citationdate2015
dc.description.ssrncandidatenon
dc.description.halcandidatenonen
dc.description.readershiprechercheen
dc.description.audienceInternationalen
dc.date.updated2023-03-06T13:57:39Z
hal.author.functionaut


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