On the T-equivariant cohomology of Hessenberg varieties

For an endomorphism $s:V\to V$ of a finite dimensional complex vector space and an action of a torus $T$ on the full flag variety ${\text{GL}}_n(\mathbb{C})\mathrm{/}B$, 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$foranyHessenbergfunction$h$. For the action of the one dimensional torus $S$ and a regular nilpotent endomorphism $N:V\to V$, we give a new computation of the equivariant cohomology of the Hessenberg variety $\text{Hes}(N,h)$ for any Hessenberg function using determinantal conditions.