On the existence of regular vectors

TL;DR

This paper establishes conditions under which continuous unitary representations of certain Lie groups are automatically smooth or analytic, enhancing understanding of their structure and properties.

Contribution

It proves that under specific conditions, continuous unitary representations are inherently smooth or analytic, with applications to oscillator groups, loop groups, and the Virasoro group.

Findings

Continuous unitary representations are automatically smooth under certain Lie algebra conditions.

Existence of dense spaces of smooth vectors for positive energy representations.

Existence of dense spaces of analytic vectors for semibounded Banach-Lie group representations.

Abstract

Let $G$ be a locally convex Lie group and $\pi:G \to \mathrm{U}(\mathcal{H})$ be a continuous unitary representation. $\pi$ is called smooth if the space of $\pi$-smooth vectors $\mathcal{H}^\infty\subset \mathcal{H}$ is dense. In this article we show that under certain conditions, concerning in particular the structure of the Lie algebra $\mathfrak{g}$ 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 $\pi$ is called semibounded, if $\pi$ is smooth and there exists a non-empty open subset $U\subset\mathfrak{g}$ such that the operators $i\mathrm{d}\pi(x)$ from the derived representation are uniformly bounded from above for $x\in U$.