Kazhdan-Lusztig conjecture for generalized Kac-Moody algebras. II. Proof of the conjecture

Naito, Satoshi

doi:10.1090/S0002-9947-1995-1316859-2

Kazhdan-Lusztig conjecture for generalized Kac-Moody algebras. II. Proof of the conjecture
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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

On a bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra

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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