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Perverse sheaves, nilpotent Hessenberg varieties, and the modular law

Martha Precup, Eric Sommers

Abstract

We consider generalizations of the Springer resolution of the nilpotent cone of a simple Lie algebra by replacing the cotangent bundle with certain other vector bundles over the flag variety. We show that the analogue of the Springer sheaf has as direct summands only intersection cohomology sheaves that arise in the Springer correspondence. The fibers of these general maps are nilpotent Hessenberg varieties, and we build on techniques established by De Concini, Lusztig, and Procesi to study their geometry. For example, we show that these fibers have vanishing cohomology in odd degrees. This leads to several implications for the dual picture, where we consider maps that generalize the Grothendieck-Springer resolution of the whole Lie algebra. In particular we are able to prove a conjecture of Brosnan. As we vary the maps, the cohomology of the corresponding nilpotent Hessenberg varieties often satisfy a relation we call the geometric modular law, which also has origins in the work on De Concini, Lusztig, and Procesi. We connect this relation in type $A$ with a combinatorial modular law defined by Guay-Paquet that is satisfied by certain symmetric functions and deduce some consequences of that connection.

Perverse sheaves, nilpotent Hessenberg varieties, and the modular law

Abstract

We consider generalizations of the Springer resolution of the nilpotent cone of a simple Lie algebra by replacing the cotangent bundle with certain other vector bundles over the flag variety. We show that the analogue of the Springer sheaf has as direct summands only intersection cohomology sheaves that arise in the Springer correspondence. The fibers of these general maps are nilpotent Hessenberg varieties, and we build on techniques established by De Concini, Lusztig, and Procesi to study their geometry. For example, we show that these fibers have vanishing cohomology in odd degrees. This leads to several implications for the dual picture, where we consider maps that generalize the Grothendieck-Springer resolution of the whole Lie algebra. In particular we are able to prove a conjecture of Brosnan. As we vary the maps, the cohomology of the corresponding nilpotent Hessenberg varieties often satisfy a relation we call the geometric modular law, which also has origins in the work on De Concini, Lusztig, and Procesi. We connect this relation in type with a combinatorial modular law defined by Guay-Paquet that is satisfied by certain symmetric functions and deduce some consequences of that connection.
Paper Structure (25 sections, 26 theorems, 128 equations, 2 tables)

This paper contains 25 sections, 26 theorems, 128 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Preliminaries
  4. Grading induced by a nilpotent orbit
  5. $P$-orbits on $\mathcal{B}$
  6. A decomposition of Hessenberg varieties
  7. Definition of Hessenberg varieties
  8. Building block varieties
  9. The decomposition
  10. Proof of Theorem \ref{['ThmA']}
  11. Generalized Grothendieck--Springer setting
  12. Brosnan's conjecture
  13. Monodromy action
  14. Nilpotent Hessenberg varieties
  15. The dot action and LLT representations
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $I \in \mathfrak Id$. If $V^I_{\mathcal{O}, \mathcal{L}} \neq 0$ in eqn.gen_Springer.sheaf2_intro, then $(\mathcal{O}, \mathcal{L}) \in \Theta_{sp}$.

Theorems & Definitions (58)

  • Theorem 1.1
  • Proposition 1.2: The geometric modular law
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • proof
  • Theorem 3.5: Theorem 3.9 in dCLP1988
  • ...and 48 more