Quaternionic K Ahler Metrics and Nilpotent Orbits

Given a geometric structure on a manifold it is natural to ask what are the symmetries of that structure and what are the examples of such structures with a big group of symmetries. In this talk I wish to start by discussing such questions for quaternionic KK ahler manifolds. Let M be a quaternionic KK ahler manifold, that is a manifold whose holonomy is contained in Sp(n) Sp(1), the subgroup of SO(4n) acting on R 4n = H n via (A; q) = AAq. I will assume that the scalar curvature of M is non-zero, thus ruling out the case when M is locally hyperKK ahler, but will not necessarily assume that M is compact. Fibred over M, is the twistor space Z 36, 5]. This is a complex-contact manifold with a real structure and a (possibly indeenite) KK ahler metric. The bres of Z ! M are rational curves C P(1) = S 2 called twistor lines which are preserved by and have normal bundle = 2nO(1). I will regard the complex-contact structure as a holomorphic one-form taking values in a holomorphic line bundle L ! Z such that ^ (dd) n vanishes nowhere and = .