The Universal Connection for Principal Bundles over Homogeneous Spaces and Twistor Space of Coadjoint Orbits

Abstract

Given a holomorphic principal bundle Q\longrightarrow X , the universal space of holomorphic connections is a torsor C_1(Q) for {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.