Invariant connections and invariant holomorphic bundles on homogeneous manifolds

Let X be a differentiable manifold endowed with a transitive action α: A×X→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 α-invariant K-connections on X
• α-invariant gauge classes of K-connections on X, and
• α-invariant isomorphism classes of pairs (Q,P) consisting of a holomorphic K ^?-bundle Q → X and a K-reduction P of Q (when X has an α-invariant complex structure).