Homogeneous principal bundles over the upper half-plane

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


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 (EG,EK), where EG is a holomorphic principal G-bundle and EK⊂EG is a C-reduction of structure group to K. Two holomorphic Hermitian principal G-bundles (EG,EK) and (E'G,E'K) are called holomorphically isometric if there is a holomorphic isomorphism of the principal G-bundle EG with E'G which takes EK to E'K. We consider all holomorphic Hermitian principal G-bundles (EG,EK) over the upper half-plane H such that the pullback of (EG,EK) by each holomorphic automorphism of H is holomorphically isometric to (EG,EK) 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 dX: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 g0∈K such that χ'=Ad(g0)°χ and A'=Ad(g0)(A).

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ID Code:3597
Deposited On:18 Oct 2010 10:21
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