Nilpotent orbits, normality, and Hamiltonian group actions

Abstract

Let $M$ be a $G$-covering of a nilpotent orbit in $\g$ where $G$ is a complex semisimple Lie group and $\g=\text{Lie}(G)$. 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 $\g$. The action of $\g'\simeq R[2]$ exponentiates to an action of the corresponding Lie group $G^{\prime }$ on a $G^{\prime }$-cover $M^{\prime }$ of a nilpotent orbit in $\g'$ such that $M$ is open dense in $M^{\prime }$. We determine all such pairs $(\g\subset\g')$.

7 pages