Classical nilpotent orbits as hyperkahler quotients

Piotr Z. Kobak*, Andrew Swann

*Corresponding author af dette arbejde

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Abstract

We show that on an arbitrary nilpotent orbit script O sign in g, where g 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 (script O sign 0, script O sign1) in different classical Lie algebras g0 ⊂ gÇ1 such that there is a finite G0-equivariant surjection script O sign1 → script O sign0 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 Gr2(ℂn).

OriginalsprogEngelsk
TidsskriftInternational Journal of Mathematics
Vol/bind7
Nummer2
Sider (fra-til)193-210
Antal sider18
ISSN0129-167X
DOI
StatusUdgivet - 1 apr. 1996
Udgivet eksterntJa

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