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Equivariant sheaves for classical groups acting on Grassmannians

Pramod N. Achar, Tamanna Chatterjee

TL;DR

The paper develops a comprehensive parity-sheaf framework for Grassmannians equipped with stratifications from products of classical groups, proving the existence and parity-vanishing of parity sheaves across Q_B-orbits and all indecomposable local systems when the coefficient ring has 2 inverted or char κ ≠ 2. It classifies Q_B orbits, analyzes their equivariant fundamental groups, constructs even resolutions and π_1-injective fibrations to realize parity sheaves for nontrivial local systems, and extends these results to direct sums of bilinear forms. The main technical contributions include explicit tangent-slice descriptions, smooth resolutions of orbit closures, and compactified fibrations that realize the regular representation in the equivariant setting, all aligning with aims in Springer theory and Mautner’s cleanness conjecture. Overall, the results establish parity-structure, semisimplicity of local systems, and parity-cohomology vanishing in a broad isotropic-Grassmannian context, with potential applications to the modular Springer correspondence for classical groups.

Abstract

Let $V$ be a finite-dimensional complex vector space. Assume that $V$ is a direct sum of subspaces each of which is equipped with a nondegenerate symmetric or skew-symmetric bilinear form. In this paper, we introduce a stratification of the Grassmannian $\mathrm{Gr}_k(V)$ related to the action of the appropriate product of orthogonal and symplectic groups, and we study the topology of this stratification. The main results involve sheaves with coefficients in a field of characteristic other than $2$. We prove that there are "enough" parity sheaves, and that the hypercohomology of each parity sheaf also satisfies a parity-vanishing property. This situation arises in the following context: let $x$ be a nilpotent element in the Lie algebra of either $G = \mathrm{Sp}_N(\mathbb{C})$ or $G = \mathrm{SO}_N(\mathbb{C})$, and let $V = \ker x \subset \mathbb{C}^N$. Our stratification of $\mathrm{Gr}_k(V)$ is preserved by the centralizer $G^x$, and we expect our results to have applications in Springer theory for classical groups.

Equivariant sheaves for classical groups acting on Grassmannians

TL;DR

The paper develops a comprehensive parity-sheaf framework for Grassmannians equipped with stratifications from products of classical groups, proving the existence and parity-vanishing of parity sheaves across Q_B-orbits and all indecomposable local systems when the coefficient ring has 2 inverted or char κ ≠ 2. It classifies Q_B orbits, analyzes their equivariant fundamental groups, constructs even resolutions and π_1-injective fibrations to realize parity sheaves for nontrivial local systems, and extends these results to direct sums of bilinear forms. The main technical contributions include explicit tangent-slice descriptions, smooth resolutions of orbit closures, and compactified fibrations that realize the regular representation in the equivariant setting, all aligning with aims in Springer theory and Mautner’s cleanness conjecture. Overall, the results establish parity-structure, semisimplicity of local systems, and parity-cohomology vanishing in a broad isotropic-Grassmannian context, with potential applications to the modular Springer correspondence for classical groups.

Abstract

Let be a finite-dimensional complex vector space. Assume that is a direct sum of subspaces each of which is equipped with a nondegenerate symmetric or skew-symmetric bilinear form. In this paper, we introduce a stratification of the Grassmannian related to the action of the appropriate product of orthogonal and symplectic groups, and we study the topology of this stratification. The main results involve sheaves with coefficients in a field of characteristic other than . We prove that there are "enough" parity sheaves, and that the hypercohomology of each parity sheaf also satisfies a parity-vanishing property. This situation arises in the following context: let be a nilpotent element in the Lie algebra of either or , and let . Our stratification of is preserved by the centralizer , and we expect our results to have applications in Springer theory for classical groups.

Paper Structure

This paper contains 17 sections, 29 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Overview and statement of the main results
  3. Motivation
  4. Organization of the paper
  5. General conventions
  6. Preliminaries on isotropic Grassmannians
  7. Symplectic and orthogonal group orbits on Grassmannians
  8. Stratification by rank
  9. Tangent spaces
  10. The symplectic case
  11. The orthogonal case
  12. Some results on orbit closures
  13. Direct sums of bilinear forms
  14. Tangent spaces and slices
  15. Resolution of singularities for orbit closures
  16. ...and 2 more sections

Key Result

Lemma 2.1

Let $V$ be a finite-dimensional complex vector space equipped with a (possibly degenerate) symmetric or skew-symmetric bilinear form $\langle{-},{-}\rangle$. Let $M_1 \subset M_2 \subset \cdots \subset M_c$ be a sequence of isotropic subspaces of $V$. For $0 \le k \le \dim V$, the variety $\mathrm{G such that for each piece $S_\alpha$ and each $i$ with $1 \le i \le c$, the function is constant.

Theorems & Definitions (58)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 48 more