Spaltenstein Varieties Associated with Pseudo-Polarizations

TL;DR

This paper introduces pseudo-polarizations for nilpotent orbits in classical Lie algebras, classifies minimal Richardson orbits, and proves smoothness and fibration structures of associated Spaltenstein varieties, extending duality properties.

Contribution

It defines pseudo-polarizations for nilpotent orbits and classifies minimal Richardson orbits, establishing geometric properties of Spaltenstein varieties in classical Lie types.

Findings

Spaltenstein varieties are smooth and pure dimensional.

They have iterated orthogonal or isotropic Grassmannian fibrations.

Extended duality and seesaw properties to all special orbits in types B and C.

Abstract

We introduce minimal Richardson orbits and pseudo-polarizations for nilpotent orbits in classical Lie algebras of types B, C, and D. For any nilpotent orbit, we classify all minimal Richardson orbits containing it and thereby determine the associated pseudo-polarizations. We prove that the corresponding Spaltenstein varieties are smooth and pure dimensional, with iterated orthogonal/isotropic Grassmannian fibrations. As an application, we extend the seesaw property and duality of Fu-Ruan-Wen from Richardson orbits to all special orbits in types B and C.