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Affine Pavings of Hessenberg Ideal Fibers

Ke Xue

Abstract

We define certain closed subvarieties of the flag variety, Hessenberg ideal fibers, and prove that they are paved by affines. Hessenberg ideal fibers are a natural generalization of Springer fibers. In type $G_2$, we give explicit descriptions of all Hessenberg ideal fibers, study some of their geometric properties and use them to completely classify Tymoczko's dot actions of the Weyl group on the cohomology of regular semisimple Hessenberg varieties.

Affine Pavings of Hessenberg Ideal Fibers

Abstract

We define certain closed subvarieties of the flag variety, Hessenberg ideal fibers, and prove that they are paved by affines. Hessenberg ideal fibers are a natural generalization of Springer fibers. In type , we give explicit descriptions of all Hessenberg ideal fibers, study some of their geometric properties and use them to completely classify Tymoczko's dot actions of the Weyl group on the cohomology of regular semisimple Hessenberg varieties.

Paper Structure

This paper contains 27 sections, 43 theorems, 85 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Hessenberg ideals
  5. Affine pavings
  6. A brief roadmap
  7. A paving of the flag variety
  8. Associated parabolics
  9. Prehomogeneous vector spaces
  10. Undistinguished nilpotent elements
  11. Proof of the Main Theorem
  12. Type
  13. Some structures of $G_2$
  14. Outline of the algorithm
  15. Computation
  16. ...and 12 more sections

Key Result

Theorem 1

Let $G$ be a connected reductive algebraic group over $\mathbb{C}$ whose Lie algebra has no simple component of type $E_7$ or $E_8$. For any Hessenberg ideal $I \subset \mathfrak{u}$ and any nilpotent element $N \in \mathfrak{g}$, the Hessenberg ideal fiber $\pi_{I}^{-1}(N)$ is paved by affines when

Figures (1)

  • Figure 1: Root system of $G_2$

Theorems & Definitions (76)

  • Theorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • Theorem \oldthetheorem: bass1985linearizing, Theorem 9.1
  • Definition \oldthetheorem
  • ...and 66 more