Mathieu: Classification of irreducible weight modules

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Olivier Mathieu
Classification of irreducible weight modules
Annales de l'institut Fourier, 50 no. 2 (2000), p. 537-592, doi: 10.5802/aif.1765
Article PDF | Reviews MR 2001h:17017 | Zbl 0962.17002 | 2 citations in Cedram

Résumé - Abstract

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

Bibliography

[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