Holomorphic Induction Beyond the Norm-Continuous Setting, With Applications to Positive Energy Representations

Abstract

We extend the theory of holomorphic induction of unitary representations of a possibly infinite-dimensional Lie group $G$ beyond the setting where the representation being induced is required to be norm-continuous. We allow the group $G$ to be a connected regular BCH(Baker-Campbell-Hausdorff) Fr\'echet-Lie group. Given a smooth $\mathbb{R}$-action $\alpha$ on $G$, we proceed to show that the corresponding class of so-called positive energy representations is intimately related with holomorphic induction. Assuming that $G$ is regular, we in particular show that if $\rho$ is a unitary ground-state representation of $G \rtimes_\alpha \mathbb{R}$ for which the energy-zero subspace $\mathcal{H}_\rho(0)$ admits a dense set of $G$-analytic vectors, then $\rho\big|_G$ is holomorphically induced from the representation of the connected subgroup $H := (G^\alpha)_0$ of $\alpha$-fixed points on $\mathcal{H}_\rho(0)$. As a consequence, we obtain an isomorphism $\mathcal{B}(\mathcal{H}_\rho)^G \cong \mathcal{B}(\mathcal{H}_\rho(0))^H$ between the corresponding commutants. We also find that any two such ground-state representations are necessarily unitary equivalent if their energy-zero subspaces are unitarily equivalent as $H$-representations. These results were previously only available under the assumption of norm-continuity of the $H$-representation on $\mathcal{H}_\rho(0)$.