  We define and analyze various generalizations of the punctual Hilbert scheme
of the plane, associated to complex or real Lie algebras. Out of these, we
construct new geometric structures on surfaces whose moduli spaces share
multiple properties with Hitchin components, and which are conjecturally
homeomorphic to them. For simple complex Lie algebras, this generalizes the
higher complex structure. For real Lie algebras, this should give an
alternative description of the Hitchin-Kostant-Rallis section.
