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Classification of Harish-Chandra Modules for Current Algebras

Author: Michael Lau
Journal: Proc. Amer. Math. Soc. 146 (2018), 1015-1029
MSC (2010): Primary 17B10; Secondary 17B65, 17B67, 17B22
Published electronically: October 5, 2017
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Abstract: For any reductive Lie algebra $ \mathfrak{g}$ and commutative, associative, unital algebra $ S$, we give a complete classification of the simple weight modules of $ \mathfrak{g}\otimes S $ with finite weight multiplicities. In particular, any such module is parabolically induced from a simple admissible module for a Levi subalgebra. Conversely, all modules obtained in this way have finite weight multiplicities. These modules are isomorphic to tensor products of evaluation modules at distinct maximal ideals of $ S$. Our results also classify simple Harish-Chandra modules up to isomorphism for all central extensions of current algebras.

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  • [BeBe] Alexandre Beĭlinson and Joseph Bernstein, Localisation de $ g$-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15-18 (French, with English summary). MR 610137
  • [BeBi] Stephen Berman and Yuly Billig, Irreducible representations for toroidal Lie algebras, J. Algebra 221 (1999), no. 1, 188-231. MR 1722910,
  • [BZ] Y. Billig and K. Zhao, Weight modules over exp-polynomial Lie algebras, J. Pure Appl. Algebra 191 (2004), 23-42.
  • [Bo] N. Bourbaki, Éléments de mathématique: Groupes et algèbres de Lie, Chapitres 4,5 et 6, Masson, Paris, 1981.
  • [BLL] D. Britten, M. Lau, and F. Lemire, Weight modules for current algebras, J. Algebra 440 (2015), 245-263.
  • [BK] J.L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), 387-410.
  • [DG] I. Dimitrov and D. Grantcharov, Classification of simple weight modules over affine Lie algebras, arXiv preprint 2009: 0910.0688v1.
  • [DMP] Ivan Dimitrov, Olivier Mathieu, and Ivan Penkov, On the structure of weight modules, Trans. Amer. Math. Soc. 352 (2000), no. 6, 2857-2869. MR 1624174,
  • [Fe] S. L. Fernando, Lie algebra modules with finite dimensional weight spaces. I, Trans. Amer. Math. Soc. 322 (1990), no. 2, 757-781. MR 1013330,
  • [FT] V. Futorny and A. Tsylke, Classification of irreducible nonzero level modules with finite-dimensional weight spaces for affine Lie algebras, J. Algebra 238 (2001), no. 2, 426-441. MR 1823767,
  • [GLZ] X. Guo, R. Lu, and K. Zhao, Simple Harish-Chandra modules, intermediate series modules, and Verma modules over the loop-Virasoro algebra, Forum Math. 23 (2011), 1029-1052.
  • [Hu] J. Humphreys, Introduction to Lie algebras and representation theory, Grad. Texts in Math., vol. 9, Springer, New York, 6th corrected printing, 1994.
  • [Ma92] Olivier Mathieu, Classification of Harish-Chandra modules over the Virasoro Lie algebra, Invent. Math. 107 (1992), no. 2, 225-234. MR 1144422,
  • [Ma00] O. Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier (Grenoble) 50 (2000), 537-592.
  • [Sa] A. Savage, Classification of irreducible quasifinite modules over map Virasoro algebras, Transform. Groups 17 (2012), 547-570.

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Additional Information

Michael Lau
Affiliation: Département de mathématiques et de statistique, Université Laval Québec, QC, Canada G1V 0A6

Received by editor(s): June 29, 2016
Received by editor(s) in revised form: May 10, 2017
Published electronically: October 5, 2017
Additional Notes: Funding from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
Communicated by: Kailash C. Misra
Article copyright: © Copyright 2017 American Mathematical Society

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