Homogeneous principal bundles over the upper half-plane

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 g₀∈K such that χ'=Ad(g₀)°χ and A'=Ad(g₀)(A).