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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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Borel–Weil theory for groups over commutative Banach algebras

1Department of Mathematics, FAU Erlangen-Nürnberg, Bismarckstr. 1½, 91054 Erlangen, Germany

2Universität Paderborn, Fakultät für Elektrotechnik, Informatik und Mathematik, Institut für Mathematik, Warburger Str. 100, 33098 Paderborn, Germany

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2011, Issue 655, Pages 165–187, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crelle.2011.040, March 2011

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Let be a commutative unital Banach algebra, 𝔤 be a semisimple complex Lie algebra and be the 1-connected Banach–Lie group with Lie algebra . Then there is a natural concept of a parabolic subgroup of and we obtain generalizations of the generalized flag manifolds. In this note we provide an explicit description of all homogeneous holomorphic line bundles over 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 is a C*-algebra, our results lead to a complete classification of all irreducible involutive holomorphic representations of on Hilbert spaces.

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