×

Orbits and primitive ideals of solvable Lie algebras. (English) Zbl 0788.17008

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 [J. Dixmier, J. Math. Pures Appl., IX. Sér. 45, 1-66 (1966; Zbl 0136.306); M. Duflo, Ann. Sci. Éc. Norm. Supér. 5, 71-120 (1972; Zbl 0241.22030); 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 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. 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.

MSC:

17B35 Universal enveloping (super)algebras
22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
17B30 Solvable, nilpotent (super)algebras
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Borho, W., Gabriel, P., Rentschler, R.: Primideale in Einh?llenden aufl?sbarer Lie-Algebren. Berlin Heidelberg New York: Springer 1973 · Zbl 0293.17005
[2] Chevalley, C.: Th?orie des groupes de Lie. Paris: Hermann 1968 · Zbl 0186.33104
[3] Conze, N., Duflo, M.: Sur l’alg?bre enveloppante d’une alg?bre de Lie r?soluble. Bull. Sci. Math.94, 201-208 (1970) · Zbl 0202.04101
[4] Corwin, L., Greenleaf, F.: Commutativity of invariant differential operators on nilpotent homogeneous spaces K/G with finite multiplicity. Preprint (1989) · Zbl 0812.43004
[5] Dixmier, J.: Repr?sentations irr?ductibles des alg?bres de Lie r?solubles. J. Math. Pures Appl.45, 1-66 (1966) · Zbl 0136.30603
[6] Dixmier, J.: Alg?bres enveloppantes. Paris: Gauthier-Villars 1974 · Zbl 0308.17007
[7] Dixmier, J.: Id?aux primitifs dans les alg?bres enveloppantes. J. Algebra48, 96-112 (1977) · Zbl 0366.17007
[8] Duflo, M.: Sur les extensions des repr?sentations irr?cdutibles des groupes de Lie nilpotents. Ann. Sci. ?c. Norm. Sup?r., IV S?r.5, 71-120 (1972) · Zbl 0241.22030
[9] Felix, R.: Zerlegung von Distributionen, die unter einer unipotenten Gruppenoperation invariant sind. Comment. Math. Helv.55, 528-546 (1980) · Zbl 0454.22005
[10] Godfrey, C.: Ideals of coadjoint orbits of nilpotent Lie algebras. Trans. Am. Math. Soc.233, 295-307 (1977) · Zbl 0316.17006
[11] Kirillov, A.A.: Unitary representations of nilpotent Lie groups. Usp. Mat. Nauk.17, 57-110 (1962) · Zbl 0106.25001
[12] Kostant, B.: Quantization and unitary representations. In: Lecture notes in modern analysis and applications. III. Berlin Heidelberg New York: Springer 1970 · Zbl 0223.53028
[13] Mathieu, O.: Bicontinuity of the Dixmier map. J. Am. Math. Soc.4, 837-863 (1991) · Zbl 0743.17013
[14] Matsushima, Y.: On algebraic Lie groups and algebras. J. Am. Math. Soc. Japan1, 46-57 (1948) · Zbl 0038.02201
[15] Pedersen, N.V.: On the characters of exponential solvable Lie groups. Ann. Sci. ?c. Norm. Sup?r., IV S?r.17, 1-29 (1984)
[16] Pedersen, N.V.: On the infinitesimal kernel of irreducible representations of nilpotent Lie groups. Bull. Soc. Math. Fr.112, 423-467 (1984) · Zbl 0589.22009
[17] Pedersen, N.V.: Composition series ofC*(G) andC C ? (G), whereG is a solvable Lie group. Invent. Math.82, 191-206 (1985) · Zbl 0589.22010
[18] Pedersen, N.V.: On the symplectic structure of coadjoint orbits of (solvable) Lie groups and applications. I. Math. Ann.281, 633-669 (1988) · Zbl 0629.22004
[19] Pedersen, N.V.: Geometric quantization and the universal enveloping algebra of a nilpotent Lie group. Trans. Am. Math. Soc.315, 511-563 (1989) · Zbl 0684.22004
[20] Pukanszky, L.: Le?ons sur les r?presentations des groupes. Paris: Dunod 1967 · Zbl 0152.01201
[21] Pukanszky, L.: On the characters and the Plancherel formula of nilpotent groups. J. Funct. Anal.1, 255-280 (1967) · Zbl 0165.48603
[22] Pukanszky, L.: Characters of algebraic solvable groups. J. Funct. Anal.3, 435-491 (1969) · Zbl 0186.20004
[23] Pukanszky, L.: Unitary representations of solvable Lie groups. Ann. Sci. ?c. Norm. Sup?r., IV S?r.4, 457-608 (1971) · Zbl 0238.22010
[24] Rentschler, R.: Sur le centre du quotient de l’alg?bre enveloppante d’une alg?bre de Lie nilpotente par un id?al premier. C.R. Acad. Sci. Paris S?r. A268, 689-692 (1969) · Zbl 0248.17009
[25] Rentschler, R.: L’injectivit? de l’application de Dixmier pour les alg?bres de Lie r?solubles Invent. Math.23, 49-71 (1974) · Zbl 0299.17003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.