Orbits and primitive ideals of solvable Lie algebras
Let \({\mathfrak g}\) be a complex solvable Lie algebra, and let \(U({\mathfrak g})\) be the universal enveloping algebra of \({\mathfrak g}\). The Dixmier map \(I:{\mathfrak g}^*\to \text{Prim}(U({\mathfrak g}))\) associates with each element \(g\) in the dual \({\mathfrak g}^*\) of \({\mathfrak g}\) a primitive ideal \(I(g)\) of \(U({\mathfrak g})\). This map is surjective, and the inverse image by \(I\) of \(I(g)\) is the orbit \(O\) through \(g\) of the algebraic adjoint \({\mathcal G}\) of \({\mathfrak g}\), i.e., the factorized Dixmier map \(\overline{I}: {\mathfrak g}^*/{\mathcal G} \to \text{Prim}(U({\mathfrak g}))\), where \({\mathfrak g}^*/{\mathcal G}\) is the space of \({\mathcal G}\)-orbits, is a bijection [\textit{J. Dixmier}, J. Math. Pures Appl., IX. Sér. 45, 1-66 (1966; Zbl 0136.306); \textit{M. Duflo}, Ann. Sci. Éc. Norm. Supér. 5, 71-120 (1972; Zbl 0241.22030); \textit{R. Rentschler}, Invent. Math. 23, 49- 71 (1974; Zbl 0299.17003)]. The Dixmier map is established using induction (in its Lie algebra version) of one dimensional representations of polarizations. The main purpose of this paper is to establish a direct link between orbits and primitive ideals of \({\mathfrak g}\). This is done by giving a detailed explicit description of the orbits of the algebraic adjoint of \({\mathfrak g}\), and then showing that an analogous description of the primitive ideals is possible. This shows that there is a much closer connection between orbits and primitive ideals than previously known. The author applies his results to give an explicit description of the inverse of the factorized Dixmier map. In his previous publication [ Bull. Soc. Math. Fr. 112, 423-467 (1984; Zbl 0589.22009)] the author showed how to construct a (finite) set of generators for the infinitesimal kernel of irreducible representations of a nilpotent Lie group (which is a primitive ideal of \(U({\mathfrak g})\); see also \textit{C. Godfrey}, Trans. Am. Math. Soc. 233, 295-307 (1977; Zbl 0316.17006)). This construction was based on the fact that in this case we have at our disposal Pukanszky's parametrization of the orbits of the algebraic adjoint of \({\mathfrak g}\). This parametrization exhibits the given orbit as the graph of a polynomial map. Such a description is no longer possible in the solvable case. As a substitute he gives an explicit description of the orbits of the algebraic adjoint of a complex solvable Lie algebra in terms of a finite set of polynomials vanishing on the orbit together with the condition that a certain polynomial does not vanish on the orbit (Theorem 2.6.2). He then gives a completely parallel description of the primitive ideal associated with a given orbit (Theorem 3.7.2) in this way obtaining directly the pairing between orbits and primitive ideals. The starting point of his description of the primitive ideals in terms of the orbits of the algebraic adjoint is the so-called jump index stratification of the dual \({\mathfrak g}^*\) of \({\mathfrak g}\) [Section 1.3; cf. also e.g. \textit{N. V. Pedersen}, Trans. Am. Math. Soc. 315, 511-563 (1989; Zbl 0684.22004)]. The jump index stratification gives rise to a finite family of polynomials in the symmetric algebra \(S({\mathfrak g})\) of \({\mathfrak g}\), the so-called \(Q\)-polynomials. These polynomials and their symmetrizations are well-behaved with respect to the Dixmier map (Theorem 3.2.1). They give rise to a finite filtration of the symmetric algebra \(S({\mathfrak g})\) and of the enveloping algebra \(U({\mathfrak g})\) by (ad- invariant) ideals, respectively. The (semi-prime) roots of these ideals give rise to two other finite filtrations of \(S({\mathfrak g})\) and \(U({\mathfrak g})\), respectively, in such a manner that corresponding members as well as their zero-sets correspond to each other under the Dixmier map (Propositions 1.3.2 and 3.4.2). In this way one obtains a stratification of the dual of \({\mathfrak g}^*\) and of \(\text{Prim } U({\mathfrak g})\). Finally he then carefully compares the pieces of these stratifications using explicitly constructed semi-invariant functions on both sides (Theorem 3.7.2 and Remark 3.7.3). It should be emphasized that while his description is not canonical -- it depends on the choice of appropriate bases -- it is explicit and all the objects involved can be algorithmically constructed.
scientific article; zbMATH DE number 537319
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