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

Abstract

Given a holomorphic principal bundle $Q⟶X$, the universal space of holomorphic connections is a torsor $C_1(Q)$ for $adQ\otimes T^{\ast }X$ 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.