The minimal nilpotent orbit, the Joseph ideal, and differential operators (Q1109864)

Let \({\mathfrak g}\) be a finite-dimensional simple Lie algebra over \({\mathbb{C}}\) and let \(\bar O_{\min}\) denote the Zariski closure of the minimal (nonzero) nilpotent orbit in \({\mathfrak g}\). If \({\mathfrak g}\) is not of type \(A_ n\), then the Joseph ideal \(J_ 0\) is the unique completely prime primitive ideal of U(\({\mathfrak g})\) with associated variety \(\bar O_{\min}\). \textit{A. Joseph} [Ann. Sci. Ec. Norm. Supér., IV. Sér. 9, 1-29 (1976; Zbl 0346.17008)] introduced \(J_ 0\) as the kernel of a \({\mathbb{C}}\)-algebra homomorphism \(\psi\) : U(\({\mathfrak g})\to {\mathcal D}({\mathbb{A}}^ n\setminus H)\), for a certain hyperplane \(H\subset {\mathbb{A}}^ n\), where \({\mathcal D}(Y)\) denotes the ring of differential operators on an algebraic variety Y. This homomorphism \(\psi\) is never surjective. If \({\mathfrak g}\) is of type \(B_ n\), \(C_ n\), \(D_ n\), \(E_ 6\) or \(E_ 7\) there exists a homomorphism \(\psi\) : U(\({\mathfrak g})\to {\mathcal D}(\bar X)\) with kernel \(J_ 0\), where \({\mathfrak g}={\mathfrak n}^+\oplus {\mathfrak h}\oplus {\mathfrak n}^-\) is a triangular decomposition and \(\bar X\) a suitable irreducible component of \(\bar O_{\min}\cap {\mathfrak n}^+\). This construction is due to \textit{A. B. Goncharov} [Funct. Anal. Appl. 16, 133-135 (1982); translation from Funkts. Anal. Prilozh. 16, No.2, 70-71 (1982; Zbl 0504.17002)]. The first main result is that \(\psi\) is surjective. The proof reduces to proving \(GK\dim (Rd+R/R)\leq GK\dim (R)- 2\) for all \(d\in {\mathcal D}(\bar X)\), where \(R=\psi (U({\mathfrak g}))\). By passing to the associated graded ring of R this can be reduced to the problem of showing that \(\dim (\bar O_{\min}\cap {\mathfrak p}^-)\leq \dim (\bar O_{\min})-2\) for a certain parabolic subalgebra \({\mathfrak p}\) of \({\mathfrak g}\) having an Abelian nilpotent radical. The second main result shows that for \({\mathfrak g}\) of type \(A_ n\), certain completely prime primitive ideals of U(\({\mathfrak g})\) associated to the minimal nilpotent orbit may be obtained as the kernel of a surjective map from U(\({\mathfrak g})\) to the ring of differential operators on an irreducible component of \(\bar O_{\min}\cap n^+.\) An interesting consequence is the existence of two non-isomorphic singular irreducible affine varieties \(\bar X_ 1\) and \(\bar X_ n\) with \({\mathcal D}(\bar X_ 1)\) isomorphic to \({\mathcal D}(\bar X_ n)\).