Birational geometry for the covering of a nilpotent orbit closure II

Let $O$ be a nilpotent orbit of a complex semisimple Lie algebra $\mathfrak{g}$ and let $\pi: X \to \bar{O}$ be the finite covering associated with the universal covering of $O$. In the previous article we have explicitly constructed a $\mathbf{Q}$-factorial terminalization $\tilde{X}$ of $X$ when $\mathfrak{g}$ is classical. In the present article, we count how many different $\mathbf{Q}$-factorial terminalizations $X$ has. We construct the universal Poisson deformation of $\tilde{X}$ over $H^2(\tilde{X}, \mathbf{C})$ and look at the action of the Weyl group $W(X)$ on $H^2(\tilde{X}, \mathbf{C})$. The main result is an explicit geometric description of $W(X)$.