Character formulas for tilting modules over Kac-Moody algebras

Soergel, Wolfgang

doi:10.1090/S1088-4165-98-00057-0

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-4165

Previous volume | This volume | Most recent volume | All volumes | Next volume
Previous article

Character formulas for tilting modules
over Kac-Moody algebras

Author: Wolfgang Soergel
Journal: Represent. Theory 2 (1998), 432-448
MSC (1991): Primary 17B70, 17B67, 17B37
Posted: December 28, 1998
Original Article: Represent. Theory 1 (1997)
MathSciNet review: 1663141
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show how to express the characters of tilting modules in a (possibly parabolic) category ${\mathcal O}$ over a Kac-Moody algebra in terms of the characters of simple highest weight modules. This settles, in lots of cases, Conjecture 7.2 of Kazhdan-Lusztig-Polynome and eine Kombinatorik für Kipp-Moduln, Representation Theory (An electronic Journal of the AMS) (1997), by the author, describing the character of tilting modules for quantum groups at roots of unity.

[AM69] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802 (39 #4129)
[Ark96] S. M. Arkhipov, Semi-infinite cohomology of associative algebras and bar duality, Internat. Math. Res. Notices 17 (1997), 833–863. MR 1474841 (98j:16006), http://dx.doi.org/10.1155/S1073792897000548
[BGG75] I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Differential operators on the base affine space and a study of 𝔤-modules, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), Halsted, New York, 1975, pp. 21–64. MR 0578996 (58 #28285)
[CI89] David H. Collingwood and Ronald S. Irving, A decomposition theorem for certain self-dual modules in the category 𝒪, Duke Math. J. 58 (1989), no. 1, 89–102. MR 1016415 (90k:17010), http://dx.doi.org/10.1215/S0012-7094-89-05806-7
[Deo87] Vinay V. Deodhar, On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan-Lusztig polynomials, J. Algebra 111 (1987), no. 2, 483–506. MR 916182 (89a:20054), http://dx.doi.org/10.1016/0021-8693(87)90232-8
[DGK82] 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 (83i:17012), http://dx.doi.org/10.1016/S0001-8708(82)80014-5
[Don86] S. Donkin, Finite resolutions of modules for reductive algebraic groups, J. Algebra 101 (1986), no. 2, 473–488. MR 847172 (87h:20067), http://dx.doi.org/10.1016/0021-8693(86)90206-1
[HS71] 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)
[Kac90] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219 (92k:17038)
[Kas90] 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 (93a:17026)
[KL93] D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. I, II, J. Amer. Math. Soc. 6 (1993), no. 4, 905–947, 949–1011. MR 1186962 (93m:17014), http://dx.doi.org/10.2307/2152745
[KL94] D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. III, J. Amer. Math. Soc. 7 (1994), no. 2, 335–381. MR 1239506 (94g:17048), http://dx.doi.org/10.1090/S0894-0347-1994-1239506-X
D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. IV, J. Amer. Math. Soc. 7 (1994), no. 2, 383–453. MR 1239507 (94g:17049), http://dx.doi.org/10.1090/S0894-0347-1994-1239507-1
[Pol91] Patrick Polo, Projective versus injective modules over graded Lie algebras and a particular parabolic category ${\mathcal O}$ for affine Kac-Moody algebras, Preprint, 1991.
[RCW82] Alvany Rocha-Caridi and Nolan R. Wallach, Projective modules over graded Lie algebras, Mathematische Zeitschrift 180 (1982), 151-177. MR 83h:1701
[Rin91] Claus Michael Ringel, The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991), no. 2, 209–223. MR 1128706 (93c:16010), http://dx.doi.org/10.1007/BF02571521
[Soe97] Wolfgang Soergel, Kazhdan-Lusztig-Polynome and eine Kombinatorik für Kipp-Moduln, Representation Theory (An electronic Journal of the AMS) (1997). CMP 97:11
[Vor93] Alexander A. Voronov, Semi-infinite homological algebra, Invent. Math. 113 (1993), no. 1, 103–146. MR 1223226 (94f:17021), http://dx.doi.org/10.1007/BF01244304

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