Borel-Weil Theory for Groups over Commutative Banach-Algebras

Neeb KH, Seppänen H (2011)

Publication Type: Journal article, Original article

Publication year: 2011

Journal

Journal für die reine und angewandte Mathematik Walter de Gruyter

Publisher: Walter de Gruyter

Book Volume: 655

Pages Range: 165-187

Abstract

Let $\cA$ be a commutative unital Banach algebra, $\g$ be a semisimple complex Lie algebra and $G(\cA)$ be the 1-connected Banach--Lie group with Lie algebra $\g \otimes \cA$. Then there is a natural concept of a parabolic subgroup $P(\cA)$ of $G(\cA)$ and we obtain generalizations $X(\cA) := G(\cA)/P(\cA)$ of the generalized flag manifolds. In this note we provide an explicit description of all homogeneous holomorphic line bundles over $X(\cA)$ with non-zero holomorphic sections. In particular, we show that all these line bundles are tensor products of pullbacks of line bundles over $X(\C)$ by evaluation maps.
For the special case where $\cA$ is a C∗-algebra, our results lead to a complete classification of all irreducible involutive holomorphic representations of $G(\cA)$ on Hilbert spaces.

Authors with CRIS profile

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

How to cite

APA:

Neeb, K.H., & Seppänen, H. (2011). Borel-Weil Theory for Groups over Commutative Banach-Algebras. Journal für die reine und angewandte Mathematik, 655, 165-187.

MLA:

Neeb, Karl Hermann, and Henrik Seppänen. "Borel-Weil Theory for Groups over Commutative Banach-Algebras." Journal für die reine und angewandte Mathematik 655 (2011): 165-187.

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