Invariant and homogeneous bundles on G/Γ

AbstractLet Γ be a cocompact lattice in a connected complex Lie group G. Given an invariant holomorphic vector bundle E on G/Γ, we show that there is a trivial holomorphic subbundle F⊂E such that any holomorphic section of E factors through holomorphic sections of F. Given two homomorphisms γ1 and γ2 from Γ to a complex linear algebraic Lie group H, with relatively compact image, we prove that any holomorphic isomorphism between the associated holomorphic principal H–bundles EH(γ1) and EH(γ2) is automatically G–equivariant