Equations for Some Nilpotent Varieties

Abstract

AbstractLet ${\mathcal{O}}$ be a Richardson nilpotent orbit in a simple Lie algebra $\mathfrak{g}$ of rank $n$ over $\mathbb C$, induced from a Levi subalgebra whose $s$ simple roots are orthogonal, short roots. The main result of the paper is a description of a minimal set of generators of the ideal defining $\overline{\mathcal{O}}$ in $S \mathfrak{g}^{\ast }$. In such cases, the ideal is generated by bases of either one or two copies of the representation whose highest weight is the dominant short root, along with $n-s$ fundamental invariants of $S \mathfrak{g}^{\ast }$. This extends Broer’s result for the subregular nilpotent orbit, which is the case of $s=1$. Along the way we give another proof of Broer’s result that $\overline{\mathcal{O}}$ is normal. We also prove a result relating a property of the invariants of $S \mathfrak{g}^{\ast }$ to the following question: when does a copy of the adjoint representation in $S \mathfrak{g}^{\ast }$ belong to the ideal in $S \mathfrak{g}^{\ast }$ generated by another copy of the adjoint representation together with the invariants of $S \mathfrak{g}^{\ast }$?