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2013 | OriginalPaper | Chapter

Holomorphic Realization of Unitary Representations of Banach–Lie Groups

Author : Karl-Hermann Neeb

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

Publisher: 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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previous chapter Weak Harmonic Maaß Forms and the Principal Series for

next chapter The Segal–Bargmann Transform on Compact Symmetric Spaces and Their Direct Limits

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Title: Holomorphic Realization of Unitary Representations of Banach–Lie Groups
Author: Karl-Hermann Neeb
Publisher: Springer New York
Book: Lie Groups: Structure, Actions, and Representations
Print ISBN: 978-1-4614-7192-9

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

Copyright Year: 2013
DOI: https://doi.org/10.1007/978-1-4614-7193-6_10

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