Kyoto Journal of Mathematics

On an invariance property of the space of smooth vectors

Karl-Hermann Neeb, Hadi Salmasian, and Christoph Zellner

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Abstract

Let (π,H) be a continuous unitary representation of the (infinite-dimen- sional) Lie group G, and let γ:RAut(G) be a group homomorphism which defines a continuous action of R on G by Lie group automorphisms. Let π#(g,t)=π(g)Ut be a continuous unitary representation of the semidirect product group GγR on H. The first main theorem of the present note provides criteria for the invariance of the space H of smooth vectors of π under the operators Uf=Rf(t)Utdt for fL1(R) and fS(R), respectively. When g is complete and the actions of R on G and g are continuous, we use the above theorem to show that, for suitably defined spectral subspaces gC(E), ER, in the complexified Lie algebra gC and H(F), FR, for Ut in H, we have

dπ(gC(E))H(F)H(E+F).

Article information

Source
Kyoto J. Math., Volume 55, Number 3 (2015), 501-515.

Dates
Received: 6 January 2014
Revised: 24 April 2014
Accepted: 25 April 2014
First available in Project Euclid: 9 September 2015

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1441824031

Digital Object Identifier
doi:10.1215/21562261-3089019

Mathematical Reviews number (MathSciNet)
MR3395973

Zentralblatt MATH identifier
1325.22012

Subjects
Primary: 22E65: Infinite-dimensional Lie groups and their Lie algebras: general properties [See also 17B65, 58B25, 58H05] 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05} 17B65: Infinite-dimensional Lie (super)algebras [See also 22E65]

Keywords
Infinite-dimensional Lie groups unitary representation integrated representations smooth vectors Arveson spectral theory

Citation

Neeb, Karl-Hermann; Salmasian, Hadi; Zellner, Christoph. On an invariance property of the space of smooth vectors. Kyoto J. Math. 55 (2015), no. 3, 501--515. doi:10.1215/21562261-3089019. https://projecteuclid.org/euclid.kjm/1441824031


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