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

  • Indranil Biswas School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
  • Michael Lennox Wong Universit\"at Duisburg-Essen, Fakult\"at f\"ur Mathematik, Thea-Leymann-Str.~9, 45127 Essen, Germany
Keywords: $\lambda$-connection, rational homogeneous space, twistor space, complexification, Levi subgroup


Given a holomorphic principal bundle $Q\longrightarrow X$, the universal space of holomorphic connections is a torsor
$C_1(Q)$ for $\ad Q\otimes T^*X$ 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-K\"ahler metric on $T^*(G/P)$, recovering Biquard's description of
this twistor space, but employing only finite-dimensional, Lie-theoretic means.


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