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We define and analyze various generalizations of the punctual Hilbert schemeof the plane, associated to complex or real Lie algebras. Out of these, weconstruct new geometric structures on surfaces whose moduli spaces sharemultiple properties with Hitchin components, and which are conjecturallyhomeomorphic to them. For simple complex Lie algebras, this generalizes thehigher complex structure. For real Lie algebras, this should give analternative description of the Hitchin-Kostant-Rallis section.

Generalized Punctual Hilbert Schemes and $\mathfrak{g}$-complex structures