Legendrian families of lines on nilpotent orbit closures

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, and $\overline{Z}$ a nilpotent orbit closure in the projectivization $\mathbb{P}(\mathfrak{g})$. We investigate the space of tangent directions of lines on $\overline{Z}$ passing through a general point $z$, denoted by $F(\overline{Z},\,z)$. We