Classification of irreducible weight modules

Mathieu, Olivier

Annales de l'Institut Fourier, Tome 50 (2000), p. 537-592 / Harvested from Numdam

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Résumé
Abstract

Soit $𝔤$ une algèbre de Lie réductive et soit $𝔥$ une sous-algèbre de Cartan. Un $𝔤$ -module $M$ est dit module de poids si et seulement si il admet une décomposition $M = \oplus_{λ} M_{λ}$ , où chaque espace de poids $M_{λ}$ est de dimension finie. Notre résultat principal est la classification de tous les $𝔤$ -modules de poids simples. Également, leurs caractères sont déduits de formules des caractères des modules simples de la catégorie $𝒪$ .

Let $𝔤$ be a reductive Lie algebra and let $𝔥$ be a Cartan subalgebra. A $𝔤$ -module $M$ is called a weighted module if and only if $M = \oplus_{λ} M_{λ}$ , where each weight space $M_{λ}$ is finite dimensional. The main result of the paper is the classification of all simple weight $𝔤$ -modules. Further, we show that their characters can be deduced from characters of simple modules in category $𝒪$ .

Publié le : 2000-01-01

MR 2001h:17017 | Zbl 0962.17002 | 2 citations dans Numdam

DOI : https://doi.org/10.5802/aif.1765

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     author = {Mathieu, Olivier},
     title = {Classification of irreducible weight modules},
     journal = {Annales de l'Institut Fourier},
     volume = {50},
     year = {2000},
     pages = {537-592},
     doi = {10.5802/aif.1765},
     mrnumber = {2001h:17017},
     zbl = {0962.17002},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2000__50_2_537_0}
}

Mathieu, Olivier. Classification of irreducible weight modules. Annales de l'Institut Fourier, Tome 50 (2000) pp. 537-592. doi : 10.5802/aif.1765. http://gdmltest.u-ga.fr/item/AIF_2000__50_2_537_0/

Bibliographie

[B1] N. Bourbaki, Groupes et algèbres de Lie, Ch 4-6, Herman, Paris, 1968.

[B2] N. Bourbaki, Groupes et algèbres de Lie, Ch 7-8, Herman, Paris, 1975.

[BLL] G. Benkart, D. Britten and F.W. Lemire, Modules with bounded multiplicities for simple Lie algebras, Math. Z., 225 (1997), 333-353. | MR 98h:17004 | Zbl 0884.17004

[BHL] D. J. Britten, J. Hooper and F.W. Lemire, Simple Cn-modules with multiplicities 1 and applications, Canad. J. Phys., 72 (1994), 326-335. | MR 96d:17004 | Zbl 0991.17501

[BFL] D. J. Britten, V.M. Futorny and F.W. Lemire, Simple A2-modules with a finite-dimensional weight space, Comm. Algebra, 23 (1995), 467-510. | MR 95k:17005 | Zbl 0819.17007

[BL1] D. J. Britten and F.W. Lemire, A classification of simple Lie modules having a 1-dimensional weight space, Trans. Amer. Math. Soc., 299 (1987), 683-697. | MR 88b:17013 | Zbl 0635.17002

[BL2] D. J. Britten and F.W. Lemire, On Modules of Bounded Multiplicities For The Symplectic Algebras, Trans. amer. math. Soc., 351 (1999), 3413-3431. | MR 99m:17008 | Zbl 0930.17005

[BL3] D. J. Britten and F.W. Lemire, The torsion free Pieri formula, Canad. J. Math., 50 (1998), 266-289. | MR 99f:17005 | Zbl 0908.17005

[CFO] A. Cylke, V. Futorny and S. Ovsienko, On the support of irreducible non-dense modules for finite-dimensional Lie algebras, Preprint.

[DMP] I. Dimitrov, O. Mathieu and I. Penkov, On the structure of weight modules, to appear in Trans. Amer. Math. Soc. | Zbl 0984.17006

[D] J. Dixmier, Algèbres enveloppantes, Gauthier-Villars, Paris, 1974. | MR 58 #16803a | Zbl 0308.17007

[Fe] S. Fernando, Lie algebra modules with finite dimensional weight spaces, I, TAMS, 322 (1990), 757-781. | MR 91c:17006 | Zbl 0712.17005

[Fu] V. Futorny, The weight representations of semisimple finite dimensional Lie algebras, Ph. D. Thesis, Kiev University, 1987.

[GJ] O. Gabber, A. Joseph, Towards the Kazhdan-Lusztig conjecture, Ann. Sci. E.N.S., 14 (1981), 261-302. | Numdam | MR 83e:17009 | Zbl 0476.17005

[Gab] Gabriel, Exposé au Séminaire Godement, Paris (1959-1960), unpublished.

[Gai] P.Y. Gaillard, Formes différentielles sur l'espace projectif réel sous l'action du groupe linéaire général, Comment. Math. Helv., 70 (1995), 375-382. | MR 96d:58004 | Zbl 0852.58001

[Ja] J. C. Jantzen, Moduln mit einem hochsten Gewicht, Lect. Notes Math. 750 (1979). | MR 81m:17011 | Zbl 0426.17001

[Jo1] A. Joseph, Topics in Lie algebras, unpublished notes (1995).

[Jo2] A. Joseph, The primitive spectrum of an enveloping algebra, Astérisque, 173-174 (1989), 13-53. | MR 91b:17012 | Zbl 0714.17011

[Jo3] A. Joseph, Some ring theoretic techniques and open problems in enveloping algebras, in Non-commutative Rings, ed. S. Montgomery and L. Small, Birkhäuser (1992), 27-67. | MR 94j:16045 | Zbl 0752.17008

[K] B. Kostant, Lie algebra cohomology and the generalized Borel-Weil-Bott theorem, Ann. of Math., 74 (1961), 329-387. | MR 26 #265 | Zbl 0134.03501

[Mi] W. Miller, On Lie algebras and some special functions of mathematical physics, Mem. A.M.S., 50 (1964). | MR 30 #3246 | Zbl 0132.29602

[S] W. Soergel, Kategorie O, perverse Garben und Moduln uber den Koinvarianten zur Weylgruppe, J. A.M.S., 3 (1990), 421-445. | Zbl 0747.17008