Nilpotent orbits, normality and Hamiltonian group actions

Abstract:

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