For an endomorphism s : V → V of a finite dimensional complex vector space and an action of a torus T on the full flag variety GL n (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 Hess(s, h) ⊆ GL n (C)/B for any Hessenberg function h. For the action of the one dimensional torus S and a regular nilpotent endomorphism N : V → V, we give a new computation of the equivariant cohomology of the Hessenberg variety Hess(N, h) for any Hessenberg function using determinantal conditions.