  We use the Springer correspondence to give a partial characterization of the
irreducible representations which appear in the Tymoczko dot-action of the Weyl
group on the cohomology ring of a regular semisimple Hessenberg variety. In
type A, we apply these techniques to prove that all irreducible summands which
appear in the pushforward of the constant sheaf on the universal Hessenberg
family have full support. We also observe that the recent results of Brosnan
and Chow, which apply the local invariant cycle theorem to the family of
regular Hessenberg varieties in type A, extend to arbitrary Lie type. We use
this extension to prove that regular Hessenberg varieties, though not
necessarily smooth, always have the "Kahler package."
