Neeb KH (2013)
Publication Type: Book chapter / Article in edited volumes
Publication year: 2013
Publisher: Springer
Edited Volumes: Lie Groups: Structure, Actions and Representations - In Honor of Joseph A. Wolf on the Occasion of his 75th Birthday
Series: Progress in Mathematics
City/Town: New York
Book Volume: 306
Pages Range: 185 - 223
ISBN: 978-1-4614-7192-9
URI: https://arxiv.org/abs/1011.1210
DOI: 10.1007/978-1-4614-7193-6_10
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.
APA:
Neeb, K.H. (2013). Holomorphic Realization of Unitary Representations of Banach-Lie Groups. In Huckleberry, A., Penkov, I., Zuckerman, G. (Eds.), Lie Groups: Structure, Actions and Representations - In Honor of Joseph A. Wolf on the Occasion of his 75th Birthday. (pp. 185 - 223). New York: Springer.
MLA:
Neeb, Karl Hermann. "Holomorphic Realization of Unitary Representations of Banach-Lie Groups." Lie Groups: Structure, Actions and Representations - In Honor of Joseph A. Wolf on the Occasion of his 75th Birthday. Ed. Huckleberry, A., Penkov, I., Zuckerman, G., New York: Springer, 2013. 185 - 223.
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