On the inverse of the Dixmier map for a solvable Lie algebra (Q1974167)

Let \(\mathfrak g\) be a completely solvable Lie algebra over a field of characteristic zero and let \(\widehat A({\mathfrak g})\) be the algebra of the differential operators on \(\mathfrak g\) with formal power series coefficients. \textit{J. Dixmier} [``Non-commutative harmonic analysis, Proc., Marseille-Luminy 1978'', Lect. Notes Math. 728, 42-63 (1979; Zbl 0409.22003)] defined an embedding \(L_{\mathfrak g}:U({\mathfrak g})\to \widehat A({\mathfrak g})\) of the universal enveloping algebra \(U({\mathfrak g})\) of \(\mathfrak g\) into \(\widehat A({\mathfrak g})\). For a derivation \(x\) of \(\mathfrak g\) let \(W_{\mathfrak g}(x)\) be the adjoint vector field on \(\mathfrak g\) defined by \(-x\). For a prime ideal \(I\) in \(U({\mathfrak g})\) let \(M_{\mathfrak g}(I)\) be the quotient of the subalgebra \(\widehat P({\mathfrak g})\) of \(\widehat A({\mathfrak g})\) generated by the Weyl algebra \(A({\mathfrak g})\) and the formal power series with rational coefficients in the weights of the adjoint representation of \(\mathfrak g\) modulo the left ideal generated by \(L_{\mathfrak g}\) and \(W_{\mathfrak g}({\mathfrak n}_a)\) where \({\mathfrak n}_a\) is the biggest nilpotent ideal in the algebraic hull of the image of \(\mathfrak g\) by the adjoint representation. The main result of the paper under review is stated in the language of categories of modules and connects \(M_{\mathfrak g}(I)\) and the inverse image of the Dixmier map for \(\mathfrak g\).