  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.
