Biswas, Indranil
(2010)
*Homogeneous principal bundles over the upper half-plane*
Kyoto Journal of Mathematics, 50
(2).
pp. 325-363.
ISSN 0023-608X

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Official URL: http://projecteuclid.org/DPubS?service=UI&version=...

Related URL: http://dx.doi.org/10.1215/0023608X-2009-016

## Abstract

Let G be a connected complex reductive linear algebraic group, and let K⊂G be a maximal compact subgroup. The Lie algebra of K is denoted by k. A holomorphic Hermitian principal G-bundle is a pair of the form (E_{G},E_{K}), where E_{G} is a holomorphic principal G-bundle and E_{K}⊂E_{G} is a C^{∞}-reduction of structure group to K. Two holomorphic Hermitian principal G-bundles (E_{G},E_{K}) and (E'_{G},E'_{K}) are called holomorphically isometric if there is a holomorphic isomorphism of the principal G-bundle E_{G} with E'_{G} which takes E_{K }to E'_{K}. We consider all holomorphic Hermitian principal G-bundles (E_{G},E_{K}) over the upper half-plane H such that the pullback of (E_{G},E_{K}) by each holomorphic automorphism of H is holomorphically isometric to (E_{G},E_{K}) itself. We prove that the isomorphism classes of such pairs are parameterized by the equivalence classes of pairs of the form (χ,A), where χ:R→K is a homomorphism, and A∈k⊗RC such that [A,dχ(1)]=2√ -1^{.}A. (Here d_{X}:R→k is the homomorphism of Lie algebras associated to χ.) Two such pairs (χ,A) and (χ',A') are called equivalent if there is an element g_{0}∈K such that χ'=Ad(g_{0})°χ and A'=Ad(g_{0})(A).

Item Type: | Article |
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Source: | Copyright of this article belongs to Duke University Press. |

ID Code: | 3597 |

Deposited On: | 18 Oct 2010 10:21 |

Last Modified: | 27 Jan 2011 06:11 |

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