On an invariance property of the space of smooth vectors

Abstract

Let $(\pi ,\mathcal{H})$ be a continuous unitary representation of the (infinite-dimen- sional) Lie group $G$, and let $\gamma :\mathbb{R}\to Aut\left(G\right)$ be a group homomorphism which defines a continuous action of $\mathbb{R}$ on $G$ by Lie group automorphisms. Let ${\pi }^{\#}(g,t)=\pi \left(g\right)U_t$ be a continuous unitary representation of the semidirect product group $G{⋊}_{\gamma }\mathbb{R}$ on $\mathcal{H}$. 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 $U_f={\int }_{\mathbb{R}}f\left(t\right)U_{t}dt$ for $f\in L^1\left(\mathbb{R}\right)$ and $f\in \mathcal{S}\left(\mathbb{R}\right)$, respectively. When $\mathfrak{g}$ is complete and the actions of $\mathbb{R}$ on $G$ and $\mathfrak{g}$ are continuous, we use the above theorem to show that, for suitably defined spectral subspaces ${\mathfrak{g}}_{\mathbb{C}}\left(E\right)$, $E\subseteq \mathbb{R}$, in the complexified Lie algebra ${\mathfrak{g}}_{\mathbb{C}}$ and ${\mathcal{H}}^{\infty }\left(F\right)$, $F\subseteq \mathbb{R}$, for $U_t$ in ${\mathcal{H}}^{\infty }$, we have

$$\mathrm{d}\pi \left({\mathfrak{g}}_{\mathbb{C}}\right(E\left)\right){\mathcal{H}}^{\infty }\left(F\right)\subseteq {\mathcal{H}}^{\infty }(E+F).$$