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Spaltenstein Varieties Associated with Pseudo-Polarizations

Xueqing Wen, Yaoxiong Wen

TL;DR

The paper extends the study of polarizations from Richardson orbits to all special nilpotent orbits in types B, C, and D by introducing minimal Richardson orbits and pseudo-polarizations. It provides a complete classification of minimal Richardson orbits for any given nilpotent orbit and describes the associated Spaltenstein fibers, showing they are smooth and pure dimensional with an iterated orthogonal/isotropic Grassmannian fibration structure. This leads to a generalized seesaw identity and E-polynomial duality for Springer dual special orbits in types B and C, thereby connecting Springer duality with Langlands duality beyond the Richardson setting. The results offer a robust geometric framework for studying polarizations and dualities in the special-orbit regime, with implications for the structure of Spaltenstein varieties and related representation-theoretic phenomena.

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.

Spaltenstein Varieties Associated with Pseudo-Polarizations

TL;DR

The paper extends the study of polarizations from Richardson orbits to all special nilpotent orbits in types B, C, and D by introducing minimal Richardson orbits and pseudo-polarizations. It provides a complete classification of minimal Richardson orbits for any given nilpotent orbit and describes the associated Spaltenstein fibers, showing they are smooth and pure dimensional with an iterated orthogonal/isotropic Grassmannian fibration structure. This leads to a generalized seesaw identity and E-polynomial duality for Springer dual special orbits in types B and C, thereby connecting Springer duality with Langlands duality beyond the Richardson setting. The results offer a robust geometric framework for studying polarizations and dualities in the special-orbit regime, with implications for the structure of Spaltenstein varieties and related representation-theoretic phenomena.

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.
Paper Structure (5 sections, 14 theorems, 66 equations)

This paper contains 5 sections, 14 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on nilpotent orbits
  3. Minimal Richardson and pseudo-polarizations
  4. The geometry of Spaltenstein varieties
  5. E-polynomials of Spaltenstein varieties and duality

Key Result

Theorem 1.2

For any nilpotent orbit $\bf{O}_e$ of type B, C, or D and any pseudo-polarization $P$, the reduced Spaltenstein variety $\mu_P^{-1}(e)^{\mathrm{red}}$ is an iterated fibration over a point, where each fiber is isomorphic to either an orthogonal Grassmannian or an isotropic Grassmannian. In particula

Theorems & Definitions (29)

  • Definition 1.1
  • Theorem 1.2: Theorems \ref{['Thm:Spal_fiber B']}, \ref{['Thm:Spal_fiber C']}, \ref{['Thm:Spal_fiber D']}
  • Theorem 1.3: Theorem \ref{['thm.duality']}
  • Lemma 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 3.1: Minimal Richardson orbit
  • Definition 3.2: pseudo-polarization
  • ...and 19 more