Generalization of the Richardson-orbit characterization

Generalize the characterization of Richardson nilpotent orbits given by the number of Legendrian components of the tangent-direction variety, or by the number of isotropy-group orbits of such components, from square-zero nilpotent orbits to a suitable larger class of nilpotent orbits.

Background

The paper characterizes Richardson orbits among square-zero nilpotent orbits using the cardinality of the set of Legendrian components of the variety of tangent directions of lines, and also using the cardinality of its quotient by the stabilizer of a general point. The authors ask whether an analogous criterion can hold for a broader class of nilpotent orbits.

The paper cautions that no such characterization may exist for all nilpotent orbits, giving as an example a Richardson orbit closure associated with a product of projective lines whose tangent-direction variety is itself a single linear Legendrian space.

References

Can Theorem~\ref{main thm: sq zero Richardson} be generalized to a suitable larger class of nilpotent orbits?

— Legendrian families of lines on nilpotent orbit closures  (2609.05003 - Kwon, 4 Sep 2026) in Problem 1, Section 1, subsection “Further questions” (label: prob:number of Leg)

Which Legendrian subvarieties of the odd dimensional projective spaces can be Legendrian components of $F(\overline{Z},\,z)$?

— Legendrian families of lines on nilpotent orbit closures  (2609.05003 - Kwon, 4 Sep 2026) in Problem 3, Section 1, subsection “Further questions” (label: prob: characterization)