Classical Nilpotent Orbits as Hyperkähler Quotients

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 ℝ 4N ≅ℍ N . 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.