On the variety of a highest weight module. (English) Zbl 0539.17006

Let \({\mathfrak g}\) be a complex semisimple Lie algebra. Let I be a primitive ideal in \(U({\mathfrak g})\). One of the goals of primitive ideal theory for \(U({\mathfrak g})\) is to give some algebro-geometric invariants of I which determine it completely. (This goal is made precise in the Jantzen conjecture, as formulated in the author’s paper in Invent. Math. 56, 191- 213 (1980; Zbl 0446.17006).) One candidate for one of these invariants is the associated variety \({\mathcal V}(I)\), which is a Zariski closed cone contained in the nilpotent cone in \({\mathfrak g}^*\). Work of D. Barbasch and the reviewer [J. Algebra 80, 350-382 (1983; Zbl 0513.22009)] replaced this invariant by an analytically defined one, and proved some of the desired classification results for it.
The goal of this paper is to find a replacement for those analytic ideas, adequate for studying \({\mathcal V}(I)\) itself. Here is the basic idea. Let \({\mathfrak b}\) be a Borel subalgebra of \({\mathfrak g}\). Write \({\mathfrak n}^-\) for the space of linear functionals on \({\mathfrak g}\) vanishing on \({\mathfrak b}\). The closure of the intersection of any nilpotent orbit \({\mathcal O}\) of G with \({\mathfrak n}^-\) has finitely many components \({\mathcal V}_ 1,...,{\mathcal V}_ t\). To each such component, the author attaches a homogeneous polynomial \(p_ i\) in the symmetric algebra of a Cartan subalgebra \({\mathfrak h}\) of \({\mathfrak g}\). This is done by studying the action of \({\mathfrak h}\) on functions on \({\mathcal V}_ i\). He proves that the span of the various \(p_ i\) is stable under the action of the Weyl group and conjectures that the resulting representation is (up to a factor of the sign representation) the one attached to \({\mathcal O}\) by Springer. This fact was subsequently proved by R. Hotta [Tôhoku Math. J., II. Ser. 36, 49-74 (1984)].
On the other hand, suppose M is an irreducible \({\mathfrak b}\)-highest weight module with annihilator I. Then the associated variety \({\mathcal V}(M)\) is a union of several \({\mathcal V}_ i\). (To be precise, we should at this point allow several different \({\mathcal O}'s\) as well; but it turns out that only one is needed.) The author proves a relationship between his polynomials \(p_ i\), and the Goldie rank polynomial for I. (Here again I have been a little imprecise.) All of this sets the stage for the subsequent proof of the Jantzen conjecture, using Hotta’s resuls already mentioned, an equidimensionality result of Gabber, and the Borho-Brylinski theorem that \({\mathcal V}(Ann M)=G\cdot {\mathcal V}(M)\) for highest weight modules.
Reviewer: D.Vogan


17B35 Universal enveloping (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
22E46 Semisimple Lie groups and their representations
Full Text: DOI


[1] Barbasch, D; Vogan, D, Primitive ideals and orbital integrals in complex classical groups, Math. ann., 259, 153-199, (1982) · Zbl 0489.22010
[2] Barbasch, D; Vogan, D, Primitive ideals and orbital integrals in complex exceptional groups, preprint, J. alg., 80, 350-382, (1983) · Zbl 0513.22009
[3] Beilinson, A; Bernstein, J, Localisation de g modules, C. R. acad. sci. Paris, 292A, 15-18, (1981) · Zbl 0476.14019
[4] Bernstein, I.N, Modules over a ring of differential operators, Func. anal. appl., 5, 89-101, (1971) · Zbl 0233.47031
[5] Borho, W; Brylinski, J.-L, Differential operators on homogeneous spaces. I, Invent. math., 69, 437-476, (1982) · Zbl 0504.22015
[6] Brylinski, J.-L; Kashiwara, M, Kazhdan-Lusztig conjecture and holonomic systems, Invent. math., 64, 387-410, (1981) · Zbl 0473.22009
[7] Demazure, M, Désingularisation des variétés de Schubert généralisées, Ann. école norm. sup., 7, 53-88, (1974) · Zbl 0312.14009
[8] Dixmier, J, Algébres enveloppantes, cahiers scientifiques XXXVII, (1974), Gauthier-Villars Paris
[9] {\scT. J. Enright}, to appear.
[10] Gabber, O, The integrability of the characteristic variety, Amer. J. math., 103, 445-468, (1981) · Zbl 0492.16002
[11] Gabber, O, Equidimensionalité de la variété caractéristique, exposé de O, (1982), Gabber rédigé par T. Levasseur Paris 6
[12] Gabber, O; Joseph, A, On the Bernstein-Gelfand-Gelfand resolution and the Duflo sum formula, Compositio math., 43, 107-131, (1981) · Zbl 0461.17004
[13] Hadziev, Dz, Some questions in the theory of vector invariants, Mat. sb., 72, 420-435, (1967)
[14] Jantzen, J.C, Moduln mit einem höchsten gewicht, () · Zbl 0426.17001
[15] Joseph, A, Gelfand-Kirillov dimension for the annihilators of simple quotients of Verma modules, J. London math. soc., 18, 50-60, (1978) · Zbl 0401.17007
[16] Joseph, A, Sur la classification des idéaux primitifs dans l’algèbre enveloppante de sl(n + 1, \(C\)), C. R. acad. sci. Paris, 287A, 303-306, (1978) · Zbl 0399.17002
[17] Joseph, A, W-module structure in the primitive spectrum of the enveloping algebra of a semisimple Lie algebra, (), 116-135
[18] Joseph, A, Kostant’s problem, Goldie rank and the Gelfand-Kirillov conjecture, Invent. math., 56, 191-213, (1980) · Zbl 0446.17006
[19] Joseph, A, Goldie rank in the enveloping algebra of a semisimple Lie algebra, I, J. alg., 65, 269-283, (1980) · Zbl 0441.17004
[20] Joseph, A, Goldie rank in the enveloping algebra of a semisimple Lie algebra, II, J. alg., 65, 284-306, (1980) · Zbl 0441.17004
[21] Joseph, A, Towards the jantzen conjecture III, Compositio math., 41, 23-30, (1981) · Zbl 0446.17005
[22] Joseph, A, Goldie rank in the enveloping algebra of a semisimple Lie algebra, III, J. alg., 73, 295-326, (1981) · Zbl 0482.17002
[23] Joseph, A, The enright functor on the Bernstein-Gelfand-Gelfand category \(O\), Invent. math., 67, 423-445, (1982) · Zbl 0502.17006
[24] Joseph, A, Application de la théorie des anneaux aux algèbres enveloppantes, cours de troisième cycle, Paris VI, (1981)
[25] Joseph, A; Small, L.W, An additivity principle for Goldie rank, Israel J. math., 31, 105-114, (1978) · Zbl 0395.17010
[26] Kazhdan, D; Lusztig, G, A topological approach to Springer’s representations, Advan. math., 38, 222-228, (1980) · Zbl 0458.20035
[27] King, D.R, The character polynomial of the annihilator of an irreducible harish-chandra module, Amer. J. math., 103, 1195-1240, (1981) · Zbl 0486.17003
[28] {\scG. Lusztig}, to appear.
[29] Lusztig, G; Spaltenstein, N, Induced unipotent classes, J. London math. soc., 19, 41-52, (1979) · Zbl 0407.20035
[30] Spaltenstein, N, Classes unipotentes de sous-groupes de Borel, () · Zbl 0486.20025
[31] Spaltenstein, N, On the fixed point set of a unipotent element on the variety of Borel subgroups, Topology, 16, 203-204, (1977) · Zbl 0445.20021
[32] Spaltenstein, N, The fixed point set of a unipotent transformation on the flag manifold, (), 452-456 · Zbl 0343.20029
[33] Springer, T.A, A construction of representations of Weyl groups, Invent. math., 44, 279-293, (1978) · Zbl 0376.17002
[34] Steinberg, R, Conjugacy classes in algebraic groups, () · Zbl 0192.36202
[35] Steinberg, R, On the desingularization of the unipotent variety, Invent. math., 36, 209-224, (1976) · Zbl 0352.20035
[36] Vogan, D, Gelfand-Kirillov dimension for harish-chandra modules, Invent. math., 48, 75-98, (1978) · Zbl 0389.17002
[37] Vogan, D, A generalized τ-invariant for the primitive spectrum of a semisimple Lie algebra, Math. ann., 242, 209-224, (1979) · Zbl 0387.17007
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