Hessenberg varieties associated to ad-nilpotent ideals

TL;DR

This paper studies Hessenberg varieties linked to ad-nilpotent ideals within the flag variety of GL(n,C), extending Tymoczko's theorem to show they have affine pavings and are connected.

Contribution

It extends Tymoczko's theorem to Hessenberg varieties associated with ad-nilpotent ideals, demonstrating they have affine pavings and are connected.

Findings

Hessenberg varieties with ad-nilpotent ideals have affine pavings.

These varieties are contained in Springer fibers.

Hessenberg varieties of this kind are connected.

Abstract

We consider Hessenberg varieties in the flag variety of $GL_n(\mathbb{C})$ with the property that the corresponding Hessenberg function defines an ad-nilpotent ideal. Each such Hessenberg variety is contained in a Springer fiber. We extend a theorem of Tymoczko to this setting, showing that these varieties have an affine paving obtained by intersecting with Schubert cells. Our method of proof constructs an an affine paving for each Springer fiber that restricts to an affine paving of the Hessenberg variety. We use the combinatorial properties of this paving to prove that Hessenberg varieties of this kind are connected.