3-Sasakian geometry, nilpotent orbits, and exceptional quotients

Using 3-Sasakian reduction techniques we obtain infinite families of new 3-Sasakian manifolds M (p(1), p(2), p(3)) and M (p(1), p(2), p(3), p(4)) in dimension 11 and 15 respectively. The metric cone on (p(1), p(2), p(3)) is a generalization of the Kronheimer hyperkahler metric on the regular maximal nilpotent orbit of sl (3, C) whereas the cone on M (p(1), p(2), p(3), p(4)) generalizes the hyperkahler metric on the 16-dimensional orbit of so(6, C). These are the first examples of 3-Sasakian metrics which are neither homogeneous nor toric. In addition we consider some further U(1)-reductions of M(p(1), p(2), p(3)). These yield examples of nontoric 3-Sasakian orbifold metrics in dimensions 7. As a result we obtain explicit families O(Theta} of compact self-dual positive scalar curvature Einstein metrics with orbifold singularities and with only one Killing vector field.