Kazhdan-Lusztig conjecture for generalized Kac-Moody algebras. II. Proof of the conjecture
HTML articles powered by AMS MathViewer
- by Satoshi Naito PDF
- Trans. Amer. Math. Soc. 347 (1995), 3891-3919 Request permission
Abstract:
Generalized Kac-Moody algebras were introduced by Borcherds in the study of Conway and Norton’s moonshine conjectures for the Monster sporadic simple group. In this paper, we prove the Kazhdan-Lusztig conjecture for generalized Kac-Moody algebras under a certain mild condition, by using a generalization (to the case of generalized Kac-Moody algebras) of Jantzen’s character sum formula. Our (main) formula generalizes the celebrated result for the case of Kac-Moody algebras, and describes the characters of irreducible highest weight modules over generalized Kac-Moody algebras in terms of the "extended" Kazhdan-Lusztig polynomials.References
- Richard Borcherds, Generalized Kac-Moody algebras, J. Algebra 115 (1988), no. 2, 501–512. MR 943273, DOI 10.1016/0021-8693(88)90275-X
- Richard E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), no. 2, 405–444. MR 1172696, DOI 10.1007/BF01232032
- Luis Casian, Kazhdan-Lusztig multiplicity formulas for Kac-Moody algebras, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 6, 333–337 (English, with French summary). MR 1046507
- Vinay V. Deodhar, Ofer Gabber, and Victor Kac, Structure of some categories of representations of infinite-dimensional Lie algebras, Adv. in Math. 45 (1982), no. 1, 92–116. MR 663417, DOI 10.1016/S0001-8708(82)80014-5
- Roe Goodman and Nolan R. Wallach, Whittaker vectors and conical vectors, J. Functional Analysis 39 (1980), no. 2, 199–279. MR 597811, DOI 10.1016/0022-1236(80)90013-0
- Jens Carsten Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, vol. 750, Springer, Berlin, 1979 (German). MR 552943, DOI 10.1007/BFb0069521
- Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
- V. G. Kac and D. A. Kazhdan, Structure of representations with highest weight of infinite-dimensional Lie algebras, Adv. in Math. 34 (1979), no. 1, 97–108. MR 547842, DOI 10.1016/0001-8708(79)90066-5
- Masaki Kashiwara, Kazhdan-Lusztig conjecture for a symmetrizable Kac-Moody Lie algebra, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 407–433. MR 1106905
- Masaki Kashiwara and Toshiyuki Tanisaki, Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody Lie algebra. II. Intersection cohomologies of Schubert varieties, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 159–195. MR 1103590
- David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI 10.1007/BF01390031
- David Kazhdan and George Lusztig, Schubert varieties and Poincaré duality, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 185–203. MR 573434
- S. Naito, Kazhdan-Lusztig conjecture for generalized Kac-Moody algebras. I. Towards the conjecture, Comm. Algebra 23 (1995), no. 2, 703–736. MR 1311810, DOI 10.1080/00927879508825242
- Alvany Rocha-Caridi and Nolan R. Wallach, Highest weight modules over graded Lie algebras: resolutions, filtrations and character formulas, Trans. Amer. Math. Soc. 277 (1983), no. 1, 133–162. MR 690045, DOI 10.1090/S0002-9947-1983-0690045-3 N. N. Shapovalov, On a bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra, Functional Anal. Appl. 6 (1972), 307-312.
- Nolan R. Wallach, A class of nonstandard modules for affine Lie algebras, Math. Z. 196 (1987), no. 3, 303–313. MR 913657, DOI 10.1007/BF01200353
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 3891-3919
- MSC: Primary 17B67; Secondary 17B10
- DOI: https://doi.org/10.1090/S0002-9947-1995-1316859-2
- MathSciNet review: 1316859