Spaltenstein Varieties Associated with Pseudo-Polarizations

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.

💡 Research Summary

The paper addresses a long‑standing gap in the geometry of nilpotent orbits for the classical Lie algebras of types B, C, and D. While every nilpotent orbit in type A is Richardson and thus admits a polarization, many special nilpotent orbits in the orthogonal and symplectic series are not Richardson and consequently lack a conventional polarization. To overcome this, the authors introduce two new notions: a “minimal Richardson orbit” for a given nilpotent orbit 𝒪ₑ and a “pseudo‑polarization”, defined as any polarization of such a minimal Richardson orbit.

The first part of the work (Sections 2–3) translates nilpotent orbits into partitions and then decomposes each partition uniquely into elementary blocks (B1, B1*, B2, B3 for type B, analogous families for types C and D). Using these blocks the authors give a complete classification of all minimal Richardson orbits containing a prescribed orbit. The classification is constructive: one modifies blocks by elementary operations (e.g., turning a B1 block