On the annihilator variety of a highest weight module for classical Lie algebras

Abstract

Let $g$ be a classical complex simple Lie algebra, and let $L\left(\lambda \right)$ be the irreducible highest weight module of $g$ with the highest weight $\lambda -\rho$, where $\rho$ is half the sum of positive roots. The associated variety of the annihilator ideal of $L\left(\lambda \right)$ is known as the annihilator variety of $L\left(\lambda \right)$. It is established by Joseph that the annihilator variety of a highest weight module is the Zariski closure of a nilpotent orbit in $g^{\ast }$. However, describing this nilpotent orbit for a given highest weight module $L\left(\lambda \right)$ can be quite challenging. In this paper, we present some efficient algorithms based on the Robinson–Schensted insertion algorithm to compute these orbits for classical Lie algebras. Our formulae are given by introducing two algorithms, that is, bipartition algorithm and partition algorithm. To get a special or metaplectic special partition from a domino type partition, we define the H-algorithm based on the Robinson–Schensted insertion algorithm. By using this H-algorithm, we can easily determine this nilpotent orbit from the information of $\lambda$.