On an invariance property of the space of smooth vectors

Abstract

Let $(\pi ,\mathcal{H})$ be a continuous unitary representation of the (infinite dimensional) Lie group $G$ and $\gamma \mathbb{R}\to \mathrm{Aut}(G)$ define a continuous action of $\mathbb{R}$ on $G$. Suppose that ${\pi }^{\#}(g,t)=\pi (g)U_t$ defines a continuous unitary representation of the semidirect product group $G{⋊}_{\gamma }\mathbb{R}$. The first main theorem of the present note provides criteria for the invariance of the space ${\mathcal{H}}^{\infty }$ of smooth vectors of $\pi$ under the operators $U_f={\int }_{\mathbb{R}}f(t)U_{t}dt$ for $f\in L^1(\mathbb{R})$, resp., $f\in \mathcal{S}(\mathbb{R})$. Using this theorem we show that, for suitably defined spectral subspaces ${\mathfrak{g}}_{\mathbb{C}}(E)$, $E\subseteq \mathbb{R}$, in the complexified Lie algebra ${\mathfrak{g}}_{\mathbb{C}}$, and ${\mathcal{H}}^{\infty }(F)$, $F\subseteq \mathbb{R}$, for $U$ in ${\mathcal{H}}^{\infty }$, we have \[ \mathsf{d}π(\mathfrak g_{\mathbb C}(E)) \mathcal H^\infty(F) \subseteq \mathcal H^\infty(E + F).\]

Accepted by Kyoto Journal of Math

References in corpus (2)

• Classification of positive energy representations of the Virasoro group
• Positive definite superfunctions and unitary representations of Lie supergroups

Cited by in corpus (2)

• Classification of positive energy representations of the Virasoro group
• Ground state representations of topological groups