3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients

Using 3-Sasakian reduction techniques we obtain infinite familiesof new 3-Sasakian manifolds M (p1,p2, p3) andM (p1,p2, p3, p4) in dimension 11 and 15 respectively. The metric cone on (p1,p2, p3) is a generalization ofthe Kronheimer hyperkähler metric on the regular maximalnilpotent orbit of sl (3, C)whereas the cone on M (p1,p2, p3, p4)generalizes the hyperkähler metric onthe 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(p1,p2, p3).These yield examples of nontoric 3-Sasakian orbifold metrics in dimensions 7. As a result we obtain explicit families O(Θ) of compact self-dual positivescalar curvature Einstein metrics with orbifoldsingularities and with only one Killing vector field.