Nilpotent orbits, normality, and Hamiltonian group actions

Abstract

Let M be a G-covering of a nilpotent orbit in ɡ where G is a complex semisimple Lie group and ɡ = Lie(G). We prove that under Poisson bracket the space R[2] of homogeneous functions on M of degree 2 is the unique maximal semisimple Lie subalgebra of R = R(M) containing ɡ. The action of (Equation present)exponentiates to an action of the corresponding Lie group G′on a cover M′of a nilpotent orbit in ɡ′ such that M is open dense in M′. We determine all such pairs (ɡ ⊂ ɡ′). © 1992, American Mathematical Society.