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Cylindrical representations of some infinite dimensional nuclear Lie groups

Published online by Cambridge University Press:  17 April 2009

Jean Marion
Jean Marion
Affiliation:
Département de Mathématiques Faculté des Sciences, Case 901 163, Avenue de Luminy 13288 Marseille, Cedex 9, France
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Abstract

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Let Γ.𝒜 be the semi-direct product group of a nuclear Lie group Γ with the additive group 𝒜 of a real nuclear vector space. We give an explicit description of all the continuous representations of Γ.𝒜 the restriction of which to 𝒜 is a cyclic unitary representation, and a necessary and sufficient condition for the unitarity of such cylindrical representations is stated. This general result is successfully used to obtain irreducible unitary representations of the nuclear Lie groups of Riemannian motions, and, in the setting of the theory of multiplicative distributions initiated by I.M. Gelfand, it is proved that for any connected real finite dimensional Lie group G and for any strictly positive integer k there exist non located and non trivially decomposable representations of order k of the nuclear Lie group (M;G) of all the G-valued test-functions on a given paracompact manifold M.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 46 , Issue 2 , October 1992 , pp. 295 - 310
Copyright
Copyright © Australian Mathematical Society 1992

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