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We extend the theory of holomorphic induction of unitary representations of apossibly infinite-dimensional Lie group $G$ beyond the setting where therepresentation being induced is required to be norm-continuous. We allow thegroup $G$ to be a connected regular BCH(Baker-Campbell-Hausdorff) Fr\'echet-Liegroup. Given a smooth $\mathbb{R}$-action $\alpha$ on $G$, we proceed to showthat the corresponding class of so-called positive energy representations isintimately related with holomorphic induction. Assuming that $G$ is regular, wein 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 theconnected 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 suchground-state representations are necessarily unitary equivalent if theirenergy-zero subspaces are unitarily equivalent as $H$-representations. Theseresults were previously only available under the assumption of norm-continuityof the $H$-representation on $\mathcal{H}_\rho(0)$.

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