The universal connection for principal bundles over homogeneous spaces and twistor space of coadjoint orbits (Q2636384)

Summary: Given a holomorphic principal bundle \(Q\longrightarrow X\), the universal space of holomorphic connections is a torsor \(C_1(Q)\) for \(\text{ad}\, Q\otimes T^\ast X\) such that the pullback of \(Q\) to \(C_1(Q)\) has a tautological holomorphic connection. When \(X= G/P\), where \(P\) is a parabolic subgroup of a complex simple group \(G\), and \(Q\) is the frame bundle of an ample line bundle, we show that \(C_1(Q)\) may be identified with \(G/L\), where \(L \subset P\) is a Levi factor. We use this identification to construct the twistor space associated to a natural hyper-Kähler metric on \(T^\ast(G/P)\), recovering Biquard's description of this twistor space, but employing only finite-dimensional, Lie-theoretic means.