  We consider Hessenberg varieties in the flag variety of $GL_n(\mathbb{C})$
with the property that the corresponding Hessenberg function defines an
ad-nilpotent ideal. Each such Hessenberg variety is contained in a Springer
fiber. We extend a theorem of Tymoczko to this setting, showing that these
varieties have an affine paving obtained by intersecting with Schubert cells.
Our method of proof constructs an an affine paving for each Springer fiber that
restricts to an affine paving of the Hessenberg variety. We use the
combinatorial properties of this paving to prove that Hessenberg varieties of
this kind are connected.
