Equations for some nilpotent varieties

TL;DR

This paper describes minimal generators for the ideals of certain nilpotent orbit closures in simple Lie algebras, extending previous results and providing new proofs of normality and invariant properties.

Contribution

It provides a minimal generating set for the defining ideals of Richardson nilpotent orbit closures induced from orthogonal short roots, extending Broer's results and connecting invariant properties.

Findings

Ideal generated by at most two copies of a specific representation and invariants

Extended Broer's result on normality of orbit closures

Connected invariant properties to ideal generation conditions

Abstract

Let $\mathcal{O}$ be a Richardson nilpotent orbit in a simple Lie algebra $\mathfrak{g}$ over $\mathbb C$, induced from a Levi subalgebra whose 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 $S \mathfrak{g}^*$. In such cases, the ideal is generated by bases of at most two copies of the representation whose highest weight is the dominant short root, along with some fundamental invariants. This extends Broer's result for the subregular nilpotent orbit. Along the way we give another proof of Broer's result that $\overline{ \mathcal{O}}$ is normal. We also prove a result connecting a property of invariants related to flat bases to the question of when one copy of the adjoint representation is in the ideal in $S \mathfrak{g}^*$ generated by another copy of the adjoint representation and the fundamental invariants.