  We consider the conjugation-action of an arbitrary upper-block parabolic
subgroup of the general linear group on the variety of nilpotent matrices in
its Lie algebra. Lie-theoretically, it is natural to wonder about the number of
orbits of this action. We translate the setup to a representation-theoretic one
and obtain a finiteness criterion which classifies all actions with only a
finite number of orbits over an arbitrary infinite field. These results are
applied to commuting varieties and nested punctual Hilbert schemes.
