CLASSICAL NILPOTENT ORBITS AS HYPERKÄHLER QUOTIENTS

Abstract

We show that on an arbitrary nilpotent orbit [Formula: see text] in [Formula: see text] where [Formula: see text] is a direct sum of classical simple Lie algebras, there is a G-invariant hyperKähler structure obtainable as a hyperKäher quotient of the flat hyperKähler manifold ℝ4N≅ℍN. Coïncidences between various low-dimensional simple Lie groups lead to some nilpotent orbits being described as hyperKähler quotients (in some cases in fact finite quotients) of other nilpotent orbits. For example, from the construction we are able to read off pairs of orbits [Formula: see text] in different classical Lie algebras [Formula: see text] such that there is a finite [Formula: see text]-equivariant surjection [Formula: see text] between the orbit closures. We include a table listing examples of hyperKähler quotients between small nilpotent orbits. The above-mentioned results have consequences in quaternionic Kähler geometry: it is known that nilpotent orbits in complex semisimple Lie algebras give rise to quaternionic Kähler manifolds. Our approach gives a more direct proof of this in the classical case as these manifolds turn out to be quaternionic Kähler quotients of quaternionic projective spaces. We find that many of these manifolds can also be constructed as quaternionic Kähler quotients of complex Grassmannians [Formula: see text].