Homogeneous twistor spaces and nilpotent orbits (Q1298130)

Suppose \(M\) is a quaternionic Kähler manifold with twistor space \(Z\). We study the case when \(Z\) is homogeneous under a group \(G^\mathbb{C}\). If \(G^\mathbb{C}\) is reductive, it is shown that there is a nilpotent orbit \({\mathcal O}\subset {\mathfrak g}^\mathbb{C}\) such that \(Z\) is a discrete cover of an open set in \(\mathbb{P}({\mathcal O})\). When \(G\) is compact, we construct such a twistor space for each nilpotent orbit \({\mathcal O}\subset {\mathfrak g}^\mathbb{C}\). The corresponding quaternionic Kähler manifolds have positive scalar curvature and we describe them via equivariant Morse theory on the Grassmannian \(\widetilde{Gr}_3({\mathfrak g})\). We show that most of the metrics obtained are not locally symmetric.