2023/01/12 by Milan Niestijl, Niestijl, Milan
Mathematics · Physics and Astronomy · #22E45 #22E66 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #FOS: Mathematics #Noncommutative and Quantum Gravity Theories #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2301.05129
openalex publication_date 2023/01/12 · openalex created_date 2023/01/14 · openalex updated_date 2026/07/28
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) Fréchet-Lie group. Given a smooth ℝ-action α 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 ρ is a unitary ground-state representation of G \rtimesαℝ for which the energy-zero subspace Hρ(0) admits a dense set of G-analytic vectors, then ρ|G is holomorphically induced from the representation of the connected subgroup H := (Gα)0 of α-fixed points on Hρ(0). As a consequence, we obtain an isomorphism B(Hρ)G ≅ B(Hρ(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 Hρ(0).