The minimal nilpotent orbit, the Joseph ideal, and differential operators

AbstractFix a simple complex Lie algebra g, not of type G2, F4, or E8. Let Ōmin denote the Zariski closure of the minimal non-zero nilpotent orbit in g, and let g = n+ ⊕ h ⊕ n− be a triangular decomposition. We proveTHEOREM. (1) If g is not of type An then there exists an irreducible component X̄ of Ōmin ∩ n+ such that U(g)/Jo = D(X̄), where Jo is the Joseph ideal and D(X̄) denotes the ring of differential operators on X̄. (2) If g is of type An then for n − 2 of the n irreducible components X̄i of Ōmin∩ n+ there exist (distinct) maximal ideals Ji of U(g) such that U(g)/Ji= D(X̄i)