Quaternionic Kähler Reductions of Wolf Spaces

The main purpose of the following article is to introduce a Lie theoretical approach to the problem of classifying pseudo quaternionic-Kähler (QK) reductions of the pseudo QK symmetric spaces, otherwise called generalized Wolf spaces. The history of QK geometry starts with the celebrated Berger’s Theorem [Ber55] which classifies all the irreducible holonomy groups for not locally symmetric pseudoriemannian manifolds. In fact, a pseudo QK manifold (M, g) of dimension 4n (n > 1) is traditionally defined by the reduction of the holonomy group to a subgroup of Sp(k, l)Sp(1) (k + l = n). Alekseevsky proved [Ale75] that any pseudo QK manifold is Einstein and satisfies some additional curvature condition, so it is possible to extend in a natural way the definition of QK manifolds to 4-manifolds: an oriented 4-manifold is said to be QK if it is Einstein and self-dual. Furthermore, the whole definition can be naturally extended to orbifolds [GL88]. Examples of pseudo QK manifolds are not too many, and most of them are homogeneous spaces. In particular, Alekseevsky proved that all homogeneous, Riemannian QK manifolds with positive scalar curvature are (compact and) symmetric [Ale75]. These spaces were classified by Wolf [Wol65] and have been called Wolf spaces. The Wolf spaces together with their duals can be characterized as all the non-scalar flat QK manifolds admitting a transitive unimodular group of isometries [AC97]. The duals of Wolf spaces don’t classify all the homogeneous, Riemannian QK manifolds with negative scalar curvature: in fact, there are more such spaces, like the so-called Alekseevskian spaces [Ale75]. Most examples of non-homogeneous QK orbifolds emerge via the so-called symmetry reduction. This process can be seen as a variation of a well-known construction of Marsden and Weinstein developed in the context of Poisson and symplectic geometry (see [MW74] or [AM78]). The Marsden-Weinstein quotient considers a symplectic manifold with some symmetries. A new symplectic manifold of lower dimension and fewer symmetries is then obtained by “dividing out” some symmetries in a “symplectic fashion”. This simple idea has been more recently applied in many different geometric situations. Various generalizations of the symplectic reductions include Kähler quotients, hyperkähler and hypercomplex quotients, quaternionic Kähler and quaternionic quotients, 3-Sasakian, Sasakian and contact quotients to mention just a few. The so-called QK reduction has been introduced by Galicki and Lawson (see [Gal86, GL88]). Here one starts with a QK space M with some symmetry H . A new QK space of dimension 4n − 4dim(H) is constructed out of the quaternionic Kähler moment map. In this paper we specialize the QK-reduction to pseudo QK symmetric spaces, that we call generalized Wolf spaces (or sometimes just Wolf spaces). Recently, this spaces have been classified by Alekseevsky and Cortés [AC05].