Analytic extension techniques for unitary representations of Banach-Lie groups

Merigon S, Neeb KH (2012)

Publication Type: Journal article, Original article

Publication year: 2012

Journal

International Mathematics Research Notices Oxford University Press (OUP): Policy H - Oxford Open Option A

Publisher: Oxford University Press (OUP): Policy H - Oxford Open Option A

Book Volume: 18

Pages Range: 4260 - 4300

DOI: 10.1093/imrn/rnr174

Abstract

Let (G,θ) be a Banach–Lie group with involutive automorphism θ, Graphic be the θ-eigenspaces in the Lie algebra Graphic of G, and H=(Gθ)0 be the identity component of its group of fixed points. An Olshanski semigroup is a semigroup S⊆G of the form Graphic, where W is an open Ad(H)-invariant convex cone in Graphic and the polar map Graphic is a diffeomorphism. Any such semigroup carries an involution * satisfying Graphic. Our central result, generalizing the Lüscher–Mack theorem for finite-dimensional groups, asserts that any locally bounded *-representation Graphic with a dense set of smooth vectors defines by “analytic continuation” a unitary representation of the simply connected Lie group Gc with Lie algebra Graphic. We also characterize those unitary representations of Gc obtained by this construction. With similar methods, we further show that semibounded unitary representations extend to holomorphic representations of complex Olshanski semigroups.

Authors with CRIS profile

Stephane Merigon Department Mathematik Karl Hermann Neeb Lehrstuhl für Mathematik (Lie-Gruppen und Darstellungstheorie)

How to cite

APA:

Merigon, S., & Neeb, K.H. (2012). Analytic extension techniques for unitary representations of Banach-Lie groups. International Mathematics Research Notices, 18, 4260 - 4300. https://doi.org/10.1093/imrn/rnr174

MLA:

Merigon, Stephane, and Karl Hermann Neeb. "Analytic extension techniques for unitary representations of Banach-Lie groups." International Mathematics Research Notices 18 (2012): 4260 - 4300.

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