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Invariant connections and invariant holomorphic bundles on homogeneous manifolds

Let $X$ be a differentiable manifold endowed with a transitive action $α:A\times X\longrightarrow X$ 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: {enumerate} 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 $K^\C$-bundle $Q\longrightarrow X$ and a $K$-reduction $P$ of $Q$ (when $X$ has an $α$-invariant complex structure). {enumerate}