Year: 2026
Type: preprint
Abstract: 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 first prove that every irreducible component of $F(\overline{Z},\,z)$ is an integral subvariety of the contact hyperplane in the projectivized tangent space. Next, we study Legendrian components of $F(\overline{Z},\,z)$ in two cases: the case where $\overline{Z}$ admits a Springer resolution; the case where $\overline{Z}$ is square-zero in a projectivized simple Lie algebra of classical type. As an application, two corollaries are presented: a characterization of Richardson orbits among square-zero orbits in terms of $F(\overline{Z},\,z)$; a description of $F(\overline{Z},\,z)$ for $\overline{Z}$ arising from stratified Mukai flops associated to irreducible Hermitian symmetric spaces.
Source: arXiv (Cornell University)
Author: Kwon, Minseong
Institution: Chinese Academy of Sciences
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7211867031