Holomorphic realization of unitary representations of Banach-Lie groups. (English) Zbl 1282.22012
Huckleberry, Alan (ed.) et al., Lie groups: structure, actions, and representations. In honor of Joseph A. Wolf on the occasion of his 75th birthday. New York, NY: Birkhäuser/Springer (ISBN 978-1-4614-7192-9/hbk; 978-1-4614-7193-6/ebook). Progress in Mathematics 306, 185-223 (2013).
This paper is part of a long-term research project of the author aiming at a systematic study of unitary representations of Banach-Lie groups. In this paper the author extends the classification of complex bundle structures from finite dimensional Lie groups to Banach-Lie groups. In Section 2 the author studies holomorphic Banach bundles; in Section 3 he investigates Hilbert spaces of holomorphic sections. The existence of analytic vectors is established and a realization result for unitary representations by holomorphic sections is given. In Section 4 the theory is applied to the construction of positive energy representations.
For the entire collection see [Zbl 1276.00017].
For the entire collection see [Zbl 1276.00017].
Reviewer: Walter Freyn (Augsburg)
MSC:
| 22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |
| 22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |
Keywords:
infinite-dimensional Lie group; unitary representation; smooth vector; analytic vector; holomorphic Hilbert bundle; semibounded representation; Arveson spectrumReferences:
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