Invariant connections and invariant holomorphic bundles on homogeneous manifolds

Let X be a differentiable manifold endowed with a transitive action alpha: AxX -> X of a Lie group A. Let K be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects: equivalence classes of alpha-invariant K-connections on X alpha-invariant gauge classes of K-connections on X, and alpha-invariant isomorphism classes of pairs (Q,P) consisting of a holomorphic K-C-bundle Q -> X and a K-reduction P of Q (when X has an alpha-invariant complex structure).