Journal Article

Borel–Weil Theory for Root Graded Banach–Lie Groups

Christoph Müller, Karl-Hermann Neeb and Henrik Seppänen

in International Mathematics Research Notices

Volume 2010, issue 5, pages 783-823
Published in print January 2010 | ISSN: 1073-7928
Published online October 2009 | e-ISSN: 1687-0247 | DOI: https://dx.doi.org/10.1093/imrn/rnp146

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In this article, we introduce (weakly) root graded Banach–Lie algebras and corresponding Lie groups as natural generalizations of group like for a Banach algebra A or groups like C(X,K) of continuous maps of a compact space X into a complex semisimple Lie group K. We study holomorphic induction from holomorphic Banach representations of so-called parabolic subgroups P to representations of G on holomorphic sections of homogeneous vector bundles over G/ P. One of our main results is an algebraic characterization of the space of sections which is used to show that this space actually carries a natural Banach structure, a result generalizing the finite dimensionality of spaces of sections of holomorphic bundles over compact complex manifolds. We also give a geometric realization of any irreducible holomorphic representation of a (weakly) root graded Banach–Lie group G and show that all holomorphic functions on the spaces G/ P are constant.

Journal Article.  11165 words.  Illustrated.

Subjects: Mathematics