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

Joseph, Anthony

J. Algebra 88, 238-278 (1984).

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

MSC:

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

Keywords:

enveloping algebra; Weil group representations; semisimple Lie algebra; primitive ideal; invariants; Jantzen conjecture; variety; highest weight modules

Citations:

Zbl 0446.17006; Zbl 0513.22009

References:

[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: Gauthier-Villars Paris · Zbl 0308.17007
[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: 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) · Zbl 0157.07201
[14]	Jantzen, J. C., Moduln mit einem höchsten Gewicht, (Lecture Notes in Mathematics No. 750 (1979), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York) · 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, (Lecture Notes in Mathematics No. 728 (1979), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York), 116-135 · Zbl 0422.17004
[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
[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, (Lecture Notes in Mathematics, No. 946 (1982), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York) · 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, (Proc. Konin. Nederl. Akad., 79 (1976)), 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, (Lecture Notes in Mathematics No. 366 (1974), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York) · 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