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Transactions of the American Mathematical Society

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Highest weight modules over graded Lie algebras: resolutions, filtrations and character formulas


Authors: Alvany Rocha-Caridi and Nolan R. Wallach
Journal: Trans. Amer. Math. Soc. 277 (1983), 133-162
MSC: Primary 17B10; Secondary 17B65, 17B70
DOI: https://doi.org/10.1090/S0002-9947-1983-0690045-3
MathSciNet review: 690045
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Abstract: In this paper the study of multiplicities in Verma modules for Kac-Moody algebras is initiated. Our analysis comprises the case when the integral root system is Euclidean of rank two. Complete results are given in the case of rank two, Kac-Moody algebras, affirming the Kazhdan-Lusztig conjectures for the case of infinite dihedral Coxeter groups.

The main tools in this paper are the resolutions of standard modules given in [21] and a generalization to the case of Kac-Moody Lie algebras of Jantzen's character sum formula for a quotient of two Verma modules (one of the main results of this article).

Finally, a precise analogy is drawn between the rank two, Kac-Moody algebras and the Witt algebra (the Lie algebra of vector fields on the circle).


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  • [1] I. N. Bernstein, I. M. Gelfand and S. I. Gelfand, Structure of representations generated by vectors of highest weight, Funkcional. Anal. i Priložen. (1) 5 (1971), 1-9. MR 0291204 (45:298)
  • [2] N. Bourbaki, Groupes et algèbres de Lie, Eléments de Mathématique, Chap. IV-VI, Hermann, Paris, 1968. MR 0240238 (39:1590)
  • [3] J. Dixmier, Enveloping algebras, North-Holland Math. Library, no. 14, North-Holland, New York, 1977. MR 0498740 (58:16803b)
  • [4] H. Garland and J. Lepowsky, Lie algebra homology and the Macdonald-Kac formulas, Invent. Math. 34 (1976), 37-76. MR 0414645 (54:2744)
  • [5] L. V. Goncharova, The cohomologies of Lie algebras of formal vector fields on the line, Funkcional. Anal. i Priložen. (2) 7 (1973), 6-14. MR 0339298 (49:4058a)
  • [6] -, Cohomologies of Lie algebras of formal vector fields on the straight line, Funkcional. Anal. i Priložen. (3) 7 (1973), 33-44. MR 0339299 (49:4058b)
  • [7] R. Goodman and N. R. Wallach, Whittaker vectors and conical vectors, J. Funct. Anal. (2) 39 (1980), 199-277. MR 597811 (82i:22018)
  • [8] P. J. Hilton and U. Stammbach, A course in homological algebra, Springer-Verlag, Berlin and New York, 1971. MR 0346025 (49:10751)
  • [9] J. C. Jantzen, Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren, Math. Ann. 226 (1977), 53-65. MR 0439902 (55:12783)
  • [10] -, Moduln mit einem höchsten Gewicht, Lecture Notes in Math., vol. 750, Springer-Verlag, Berlin and New York, 1979. MR 552943 (81m:17011)
  • [11] V. G. Kac, Contravariant form for Lie algebras and superalgebras, Lecture Notes in Physics, vol. 94, Springer-Verlag, Berlin and New York, 1979, pp. 441-445.
  • [12] -, Simple irreducible graded Lie algebras of finite growth, Math. USSR Izv. 2 (1968), 1271, 1311.
  • [13] -, Some problems on infinite dimensional Lie algebras and their representations (for AMS meeting at Amherst, 1981), preprint.
  • [14] V. G. Kac and D. A. Kazhdan, Structure of representations with highest weight of infinite dimensional Lie algebras, Adv. in Math. 34 (1979), 97-108. MR 547842 (81d:17004)
  • [15] D. Kazhdan and G. Lusztig, Representations of coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. MR 560412 (81j:20066)
  • [16] J. Lepowsky, Lectures on Kac-Moody Lie algebras, Université de Paris VI (1978) (mimeographed notes).
  • [17] R. V. Moody, A new class of Lie algebras, J. Algebra 10 (1968), 211-230. MR 0229687 (37:5261)
  • [18] -, Euclidean Lie algebras, Canad. J. Math. 21 (1969), 1432-1454. MR 0255627 (41:287)
  • [19] -, Root systems of hyperbolic type, Adv. in Math. 33 (1979), 144-160. MR 544847 (81g:17006)
  • [20] A. Rocha-Caridi, Resolutions of irreducible highest weight modules over infinite dimensional graded Lie algebras, Proc. 1981 Conf. on Lie Algebras and Related Topics, Lecture Notes in Math., vol. 933, Springer-Verlag, Berlin and New York, 1982.
  • [21] A. Rocha-Caridi and N. R. Wallach, Projective modules over graded Lie algebras. I, Math. Z. 180 (1982), 151-177. MR 661694 (83h:17018)
  • [22] -, Characters of irreducible representations of the Lie algebra of vector fields on the circle, Invent. Math. (to appear). MR 696690 (85a:17010)
  • [23] N. N. Shapovalov, On a bilinear form on the universal enveloping algebra of a complex semi-simple Lie algebra, Funct. Anal. Appl. 6 (1972), 307-312.

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DOI: https://doi.org/10.1090/S0002-9947-1983-0690045-3
Article copyright: © Copyright 1983 American Mathematical Society

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