Affine Pavings of Hessenberg Ideal Fibers

TL;DR

This paper introduces Hessenberg ideal fibers, a new class of subvarieties in the flag variety, proves they are paved by affines, and explores their properties, including explicit descriptions in type G2 and applications to Weyl group actions.

Contribution

It defines Hessenberg ideal fibers, proves their affine paving, and applies these results to classify Weyl group actions on Hessenberg variety cohomology.

Findings

Hessenberg ideal fibers are paved by affines.

Explicit descriptions of fibers in type G2.

Classification of Weyl group actions on cohomology.

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.