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

Rocha-Caridi, Alvany; Wallach, Nolan R.

doi:10.1090/S0002-9947-1983-0690045-3

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6850(online) ISSN 0002-9947(print)

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
MathSciNet review: 690045
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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).

[1] I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Structure of representations that are generated by vectors of highest weight, Funckcional. Anal. i Priložen. 5 (1971), no. 1, 1–9 (Russian). MR 0291204 (45 #298)
[2] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 (French). MR 0240238 (39 #1590)
[3] Jacques Dixmier, Enveloping algebras, North-Holland Publishing Co., Amsterdam, 1977. North-Holland Mathematical Library, Vol. 14; Translated from the French. MR 0498740 (58 #16803b)
[4] Howard Garland and James Lepowsky, Lie algebra homology and the Macdonald-Kac formulas, Invent. Math. 34 (1976), no. 1, 37–76. MR 0414645 (54 #2744)
[5] L. V. Gončarova, Cohomology of Lie algebras of formal vector fields on the line, Funkcional. Anal. i Priložen. 7 (1973), no. 2, 6–14 (Russian). MR 0339298 (49 #4058a)
[6] L. V. Gončarova, Cohomology of Lie algebras of formal vector fields on the line, Funkcional. Anal. i Priložen. 7 (1973), no. 3, 33–44 (Russian). MR 0339299 (49 #4058b)
[7] Roe Goodman and Nolan R. Wallach, Whittaker vectors and conical vectors, J. Funct. Anal. 39 (1980), no. 2, 199–279. MR 597811 (82i:22018), http://dx.doi.org/10.1016/0022-1236(80)90013-0
[8] Peter John Hilton and Urs Stammbach, A course in homological algebra, Springer-Verlag, New York, 1971. Graduate Texts in Mathematics, Vol. 4. MR 0346025 (49 #10751)
[9] Jens C. Jantzen, Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren, Math. Ann. 226 (1977), no. 1, 53–65. MR 0439902 (55 #12783)
[10] Jens Carsten Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, vol. 750, Springer, Berlin, 1979 (German). 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), no. 1, 97–108. MR 547842 (81d:17004), http://dx.doi.org/10.1016/0001-8708(79)90066-5
[15] David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412 (81j:20066), http://dx.doi.org/10.1007/BF01390031
[16] J. Lepowsky, Lectures on Kac-Moody Lie algebras, Université de Paris VI (1978) (mimeographed notes).
[17] Robert V. Moody, A new class of Lie algebras, J. Algebra 10 (1968), 211–230. MR 0229687 (37 #5261)
[18] Robert V. Moody, Euclidean Lie algebras, Canad. J. Math. 21 (1969), 1432–1454. MR 0255627 (41 #287)
[19] Robert V. Moody, Root systems of hyperbolic type, Adv. in Math. 33 (1979), no. 2, 144–160. MR 544847 (81g:17006), http://dx.doi.org/10.1016/S0001-8708(79)80003-1
[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] Alvany Rocha-Caridi and Nolan R. Wallach, Projective modules over graded Lie algebras. I, Math. Z. 180 (1982), no. 2, 151–177. MR 661694 (83h:17018), http://dx.doi.org/10.1007/BF01318901
[22] Alvany Rocha-Caridi and Nolan R. Wallach, Characters of irreducible representations of the Lie algebra of vector fields on the circle, Invent. Math. 72 (1983), no. 1, 57–75. MR 696690 (85a:17010), http://dx.doi.org/10.1007/BF01389129
[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.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|