Classical nilpotent orbits as hyperkahler quotients

TY - JOUR

T1 - Classical nilpotent orbits as hyperkahler quotients

AU - Kobak, Piotr Z.

AU - Swann, Andrew

PY - 1996/4/1

Y1 - 1996/4/1

N2 - We show that on an arbitrary nilpotent orbit script O sign in gℂ, 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 gℂ0 ⊂ gÇ1 such that there is a finite Gℂ0-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).

AB - We show that on an arbitrary nilpotent orbit script O sign in gℂ, 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 gℂ0 ⊂ gÇ1 such that there is a finite Gℂ0-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).

UR - http://www.scopus.com/inward/record.url?scp=0030550841&partnerID=8YFLogxK

U2 - 10.1142/S0129167X96000116

DO - 10.1142/S0129167X96000116

M3 - Journal article

AN - SCOPUS:0030550841

SN - 0129-167X

VL - 7

SP - 193

EP - 210

JO - International Journal of Mathematics

JF - International Journal of Mathematics

IS - 2

ER -