[For the entire collection see Zbl 0621.00011.]
This paper is a timely and eminently readable account of the classification of primitive ideals in the enveloping algebra U(\({\mathfrak g})\) of a finite dimensional Lie algebra \({\mathfrak g}\) over an algebraically closed field. The results are primarily due to M. Duflo, and the author in collaboration with C. Moeglin. The classification is reduced to the semisimple case, and as expected the ideas are an interesting amalgam of the techniques in the solvable and semisimple cases, although there are considerable complications. In this brief review we avoid the technicalities, and encourage the interested reader to read the paper under review.
One ingredient is the Dixmier-Duflo map J: \({\mathfrak g}^*_{sp}\to \Pr im U({\mathfrak g})\) defined on those \(f\in {\mathfrak g}^*\) having a solvable polarization; J(f) is the usual twisted induction from a solvable polarization of f. If \({\mathfrak g}\) is solvable, J is the Dixmier map. If \({\mathfrak g}\) is semisimple \({\mathfrak g}^*_{sp}\) coincides with the regular elements of \({\mathfrak g}^*\) and J(f) is a minimal primitive ideal. An arbitrary primitive ideal, I say, contains a canonical J(f), denoted \(I_{\min}\). Given J(f), there are only a finite number of primitive ideals I for which \(I_{\min}=J(f)\). The I for which \(I_{\min}=J(f)\) are then classified by the primitive ideals in a certain reductive subalgebra of \({\mathfrak g}.\)
The definition of \(I_{\min}\) requires knowing that I is itself an “induced” ideal of a certain type. The precise definition of this induction is rather technical, but the idea is to start with a “unipotent form” \(f\in {\mathfrak g}^*\) and a primitive ideal \(\xi\) in U(\({\mathfrak r})\) where \({\mathfrak r}\) is a Levi factor of \({\mathfrak g}(f)\). The “induced ideal” is denoted I(f,\(\xi)\). Every primitive ideal of U(\({\mathfrak g})\) is of this form, and there is an action of the adjoint algebraic group on such pairs (f,\(\xi)\), such that pairs in the same orbit give the same induced ideal. If \(I=I(f,\xi)\), then \(I_{\min}=I(f,\xi_{\min})\) where \(\xi_{\min}\) is the minimal primitive ideal of U(\({\mathfrak r})\) contained in \(\xi\).