Invariant and homogeneous bundles on G/ΓG/Γ

Let ΓΓ be a cocompact lattice in a connected complex Lie group GG. Given an invariant holomorphic vector bundle EE on G/ΓG/Γ, we show that there is a trivial holomorphic subbundle F⊂E such that any holomorphic section of EE factors through holomorphic sections of FF. Given two homomorphisms γ1γ1 and γ2γ2 from ΓΓ to a complex linear algebraic Lie group HH, with relatively compact image, we prove that any holomorphic isomorphism between the associated holomorphic principal HH–bundles EH(γ1)EH(γ1) and EH(γ2)EH(γ2) is automatically GG–equivariant.