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Erschienen in:
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2013 | OriginalPaper | Buchkapitel

Holomorphic Realization of Unitary Representations of Banach–Lie Groups

verfasst von : Karl-Hermann Neeb

Erschienen in: Lie Groups: Structure, Actions, and Representations

Verlag: Springer New York

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Abstract

In this paper we explore the method of holomorphic induction for unitary representations of Banach–Lie groups. First we show that the classification of complex bundle structures on homogeneous Banach bundles over complex homogeneous spaces of real Banach–Lie groups formally looks as in the finite-dimensional case. We then turn to a suitable concept of holomorphic unitary induction and show that this process preserves commutants. In particular, holomorphic induction from irreducible representations leads to irreducible ones. Finally we develop criteria to identify representations as holomorphically induced and apply these to the class of so-called positive energy representations. All this is based on extensions of Arveson’s concept of spectral subspaces to representations on Fréchet spaces, in particular on spaces of smooth vectors.

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Vorheriges Kapitel Weak Harmonic Maaß Forms and the Principal Series for
Nächstes Kapitel The Segal–Bargmann Transform on Compact Symmetric Spaces and Their Direct Limits
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Metadaten
Titel
Holomorphic Realization of Unitary Representations of Banach–Lie Groups
verfasst von
Karl-Hermann Neeb
Copyright-Jahr
2013
Verlag
Springer New York
Buch
Lie Groups: Structure, Actions, and Representations

Print ISBN: 978-1-4614-7192-9

Electronic ISBN: 978-1-4614-7193-6

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-1-4614-7193-6_10