On the existence of regular vectors

2015/10/29 by Zellner, Christoph
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1510.08727

Abstract

Let G be a locally convex Lie group and π:G → U(H) be a continuous unitary representation. π is called smooth if the space of π-smooth vectors H^∞⊂ H is dense. In this article we show that under certain conditions, concerning in particular the structure of the Lie algebra \mathfrakg of G, a continuous unitary representation of G is automatically smooth. As an application, this yields a dense space of smooth vectors for continuous positive energy representations of oscillator groups, double extensions of loop groups and the Virasoro group. Moreover we show the existence of a dense space of analytic vectors for the class of semibounded representations of Banach-Lie groups. Here π is called semibounded, if π is smooth and there exists a non-empty open subset U⊂\mathfrakg such that the operators idπ(x) from the derived representation are uniformly bounded from above for x∈ U.

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