Characterization of nonempty Legendrian-component sets

Determine precisely when the set of Legendrian components of the tangent-direction variety of lines through a general point of a nilpotent orbit closure is nonempty.

Background

The paper proves that every irreducible component of the tangent-direction variety is integral in the contact hyperplane, but not every component is necessarily Legendrian. In particular, computations for square-zero nilpotent orbits exhibit examples in which the variety of lines is nonempty while its set of Legendrian components is empty.

The unresolved problem seeks necessary and sufficient conditions for the existence of Legendrian components. The paper notes two known sufficient conditions: the orbit is Richardson, or the open nilpotent orbit itself contains a line; neither condition is necessary.