On the T-equivariant cohomology of Hessenberg varieties

TL;DR

This paper investigates the T-equivariant cohomology of Hessenberg varieties, providing descriptions of fixed points, one-dimensional orbits, and new computational methods for the cohomology in the case of regular nilpotent endomorphisms.

Contribution

It offers a new description of fixed points and orbits under torus actions and introduces a novel computation method for the equivariant cohomology of Hessenberg varieties with regular nilpotent endomorphisms.

Findings

Describes fixed point sets for semisimple and regular nilpotent endomorphisms.

Computes one-dimensional orbits on Hessenberg subvarieties.

Provides a new determinantal approach to equivariant cohomology for regular nilpotent cases.

Abstract

For an endomorphism $s:V\rightarrow V$ of a finite dimensional complex vector space and an action of a torus $T$ on the full flag variety $\text{GL}_n({\mathbb C})/B$, we give a description of its fixed point set when $s$ is semisimple or regular nilpotent. We also compute the one dimensional orbits of this action on the Hessenberg subvariety $\text{Hes}(s,h)\subseteq \text{GL}_n({\mathbb C})/B$ for any Hessenberg function $h$. For the action of the one dimensional torus $S$ and a regular nilpotent endomorphism $N:V\rightarrow V$, we give a new computation of the equivariant cohomology of the Hessenberg variety $\text{Hes}(N,h)$ for any Hessenberg function using determinantal conditions.