Homogeneous principal bundles over the upper half-plane

Let G be a connected complex reductive linear algebraic group, and let K subset of G be a maximal compact subgroup. The Lie algebra of K is denoted by P. 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 subset of E-G is a C-infinity-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 (chi, A), where chi : R -> K is a homomorphism, and A is an element of t circle times(R) C such that [A, d chi(1)] = 2 root-/l.A. (Here d chi : R -> t is the homomorphism of Lie algebras associated to chi.) Two such pairs (chi, A) and (chi', A') are called equivalent if there is an element g(0) is an element of K such that chi' = Ad(g(0)) o chi and A' = Ad(g(0)) (A).