A new proof for the partition algorithm of the annihilator varieties of highest weight modules

Abstract

Let $L(\lambda )$ be a simple highest weight module of a classical Lie algebra $\mathfrak{g}$ with highest weight $\lambda -\rho$, where $\rho$ is half the sum of positive roots. Joseph proved that the associated variety of the annihilator ideal of $L(\lambda )$ (also called the annihilator variety) is the Zariski closure of a nilpotent orbit in ${\mathfrak{g}}^{\ast }$. Recently, Bai--Ma--Wang introduced a partition algorithm to describe this corresponding nilpotent orbit for a given highest weight module $L(\lambda )$. In this paper, we present a new direct proof of Bai--Ma--Wang's partition algorithm using Sommers duality.