Legendrian families of lines on nilpotent orbit closures
Abstract
Let g be a complex semisimple Lie algebra, and Z̅ a nilpotent orbit closure in the projectivization ℙ(g). We investigate the space of tangent directions of lines on Z̅ passing through a general point z, denoted by F(Z̅, z). We first prove that every irreducible component of F(Z̅, z) is an integral subvariety of the contact hyperplane in the projectivized tangent space. Next, we study Legendrian components of F(Z̅, z) in two cases: the case where Z̅ admits a Springer resolution; the case where Z̅ 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(Z̅, z); a description of F(Z̅, z) for Z̅ arising from stratified Mukai flops associated to irreducible Hermitian symmetric spaces.
Related
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- A short note on Arthur parameters with singular orbit closures
BibTeX
@misc{kwon2026legendrian,
title = {{Legendrian families of lines on nilpotent orbit closures}},
author = {Minseong Kwon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05003},
eprint = {2609.05003},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05003},
}
RIS
TY - UNPB
TI - Legendrian families of lines on nilpotent orbit closures
AU - Minseong Kwon
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05003
UR - https://arxiv.org/abs/2609.05003
AB - Let g be a complex semisimple Lie algebra, and Z̅ a nilpotent orbit closure in the projectivization ℙ(g). We investigate the space of tangent directions of lines on Z̅ passing through a general point z, denoted by F(Z̅, z). We first prove that every irreducible component of F(Z̅, z) is an integral subvariety of the contact hyperplane in the projectivized tangent space. Next, we study Legendrian components of F(Z̅, z) in two cases: the case where Z̅ admits a Springer resolution; the case where Z̅ 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(Z̅, z); a description of F(Z̅, z) for Z̅ arising from stratified Mukai flops associated to irreducible Hermitian symmetric spaces.
ER -