Unitary Highest Weight Representations in Hilbert Spaces of Holomorphic Functions on Infinite Dimensional Domains
Automorphism groups of symmetric domains in Hilbert spaces form a natural class of infinite dimensional Lie algebras and corresponding Banach Lie groups. We give a classification of the algebraic category of unitary highest weight modules for such Lie algebras and show that infinite dimensional vers...
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| Vydané v: | Journal of functional analysis Ročník 156; číslo 1; s. 263 - 300 |
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| Jazyk: | English |
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20.06.1998
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| Abstract | Automorphism groups of symmetric domains in Hilbert spaces form a natural class of infinite dimensional Lie algebras and corresponding Banach Lie groups. We give a classification of the algebraic category of unitary highest weight modules for such Lie algebras and show that infinite dimensional versions of the Lie algebras so(2,n) have no unitary highest weight representations and thus do not meet the physical requirement of having positive energy. Highest weight modules correspond to unitary representations of global Banach Lie groups realized in Hilbert spaces of vector valued holomorphic functions on the relevant domains in Hilbert spaces. The construction of such holomorphic representations of certain Banach Lie groups, followed by the application of the general framework of Harish-Chandra type groups in an appropriate Banach setting, leads to the integration of the Lie algebra representation to a group representation. The extension of this theory to infinite dimensional settings is explored. |
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| AbstractList | Automorphism groups of symmetric domains in Hilbert spaces form a natural class of infinite dimensional Lie algebras and corresponding Banach Lie groups. We give a classification of the algebraic category of unitary highest weight modules for such Lie algebras and show that infinite dimensional versions of the Lie algebras so(2,n) have no unitary highest weight representations and thus do not meet the physical requirement of having positive energy. Highest weight modules correspond to unitary representations of global Banach Lie groups realized in Hilbert spaces of vector valued holomorphic functions on the relevant domains in Hilbert spaces. The construction of such holomorphic representations of certain Banach Lie groups, followed by the application of the general framework of Harish-Chandra type groups in an appropriate Banach setting, leads to the integration of the Lie algebra representation to a group representation. The extension of this theory to infinite dimensional settings is explored. |
| Author | Neeb, Karl-Hermann Ørsted, Bent |
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| Cites_doi | 10.1007/BF01465868 10.1090/S0002-9947-1972-0296359-6 10.1006/jabr.1994.1173 10.1070/IM1975v009n02ABEH001480 10.1016/0022-1236(80)90106-8 10.1007/BF02392042 |
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| References_xml | – reference: K.-H. Neeb, B. Ørsted, Hardy spaces in an infinite dimensional setting, Clausthal Proceedings, Odense Univ. – volume: 22:4 start-page: 273 year: 1988 end-page: 285 ident: FU973233RF14 article-title: Method of holomorphic extension in the theory of unitary representations of infinite dimensional classical groups publication-title: Funct. Anal. Appl. – volume: 136 start-page: 1 year: 1976 end-page: 59 ident: FU973233RF17 article-title: Analytic continuation of the holomorphic discrete series of a semisimple Lie group publication-title: Acta Math. – reference: J. Hilgert, K.-H. Neeb, Riesz distributions associated to Euclidian Jordan algebras – reference: Y. Neretin, G. I. 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