Associated varieties of integral minimal highest weight modules

Let $\mathfrak{g}$ be a complex simple Lie algebra and $L(\lambda)$ be a highest weight module of $\mathfrak{g}$ with highest weight $\lambda-\rho$, where $\rho$ is half the sum of positive roots. A simple $\mathfrak{g}$-module $L_w:=L(-w\rho)$ is called integral minimal if the associated variety of its annihilator ideal equals the closure of the minimal special nilpotent orbit. In this paper, we find that the associated variety of any integral minimal module $L_w$ is irreducible and equal to the orbital variety corresponding to the minimal length element in the Kazhdan--Lusztig right cell containing $w$.