Classification of irreducible weight modules
Annales de l'Institut Fourier, Volume 50 (2000) no. 2, pp. 537-592.

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= 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.

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= 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.

@article{AIF_2000__50_2_537_0,
     author = {Mathieu, Olivier},
     title = {Classification of irreducible weight modules},
     journal = {Annales de l'Institut Fourier},
     pages = {537--592},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {50},
     number = {2},
     year = {2000},
     doi = {10.5802/aif.1765},
     zbl = {0962.17002},
     mrnumber = {2001h:17017},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1765/}
}
TY  - JOUR
AU  - Mathieu, Olivier
TI  - Classification of irreducible weight modules
JO  - Annales de l'Institut Fourier
PY  - 2000
DA  - 2000///
SP  - 537
EP  - 592
VL  - 50
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1765/
UR  - https://zbmath.org/?q=an%3A0962.17002
UR  - https://www.ams.org/mathscinet-getitem?mr=2001h:17017
UR  - https://doi.org/10.5802/aif.1765
DO  - 10.5802/aif.1765
LA  - en
ID  - AIF_2000__50_2_537_0
ER  - 
%0 Journal Article
%A Mathieu, Olivier
%T Classification of irreducible weight modules
%J Annales de l'Institut Fourier
%D 2000
%P 537-592
%V 50
%N 2
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.1765
%R 10.5802/aif.1765
%G en
%F AIF_2000__50_2_537_0
Mathieu, Olivier. Classification of irreducible weight modules. Annales de l'Institut Fourier, Volume 50 (2000) no. 2, pp. 537-592. doi : 10.5802/aif.1765. http://www.numdam.org/articles/10.5802/aif.1765/

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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

[D] J. Dixmier, Algèbres enveloppantes, Gauthier-Villars, Paris, 1974. | MR | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

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

Cited by Sources: