Differential operators on some nilpotent orbits

In recent work, Astashkevich and Brylinski construct some differential operators of Euler degree − 1 -1 (thus, they lower the degree of polynomials by one) on the coordinate ring O ( O min ( g ) ) \mathcal {O}(\mathbb {O}_{\min }(\mathfrak {g})) of the minimal nilpotent orbit O min ( g ) \mathbb O_{\min }(\mathfrak {g}) for any classical, complex simple Lie algebra g \mathfrak {g} . They term these operators “exotic” since there is “(apparently) no geometric or algebraic theory that explains them”. In this paper, we provide just such an algebraic theory for s l ( n ) {\mathfrak {sl}}(n) by giving a complete description of the ring of differential operators on O min ( s l ( n ) ) . \mathbb O_{\min }({\mathfrak {sl}}(n)). The method of proof also works for various related varieties, notably for the Lagrangian submanifolds of the minimal orbit of classical Lie algebras for which Kostant and Brylinski have constructed exotic differential operators.