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

Abstract

Fix a simple complex Lie algebra g , not of type G 2, F 4, or E 8. Let Ō min denote the Zariski closure of the minimal non-zero nilpotent orbit in g , and let g = n + ⊕ h ⊕ n − be a triangular decomposition. We prove THEOREM. (1) If g is not of type A n then there exists an irreducible component X̄ of Ō min ∩ n + such that U(g)/J o = D(X̄), where J o is the Joseph ideal and D(X̄) denotes the ring of differential operators on X̄. (2) If g is of type A n then for n − 2 of the n irreducible components X̄ i of Ō min∩ n + there exist (distinct) maximal ideals J i of U(g) such that U(g)/J i= D(X̄ i) .