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

TL;DR

This paper provides explicit formulas and algorithms to determine the nilpotent orbit associated with the annihilator variety of highest weight modules for classical Lie algebras, simplifying their classification.

Contribution

It introduces bipartition and partition algorithms, including the H-algorithm, to explicitly characterize the nilpotent orbit from the highest weight.

Findings

Provides simple formulas for nilpotent orbit classification.

Introduces bipartition and partition algorithms for analysis.

Uses the H-algorithm based on Robinson-Schensted insertion.

Abstract

Let $\mathfrak{g}$ be a classical complex simple Lie algebra. Let $L(\lambda)$ be a highest weight module of $\mathfrak{g}$ with highest weight $\lambda-\rho$, where $\rho$ is half the sum of positive roots. The associated variety of the annihilator ideal of $L(\lambda)$ is called the annihilator variety of $L(\lambda)$.It is known that the annihilator variety of any highest weight module $L(\lambda)$ is the Zariski closure of a nilpotent orbit in $\mathfrak{g}^*$. But in general, this nilpotent orbit is not easy to describe for a given highest weight module $L(\lambda)$. In this paper, we will give some simple formulas to characterize this unique nilpotent orbit appearing in the annihilator variety of a highest weight module for classical Lie algebras. Our formulas are given by introducing two algorithms, i.e., 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 $\lambda$.