Character formulas for tilting modules over Kac-Moody algebras

Soergel, Wolfgang

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

Character formulas for tilting modules over Kac-Moody algebras
HTML articles powered by AMS MathViewer

by Wolfgang Soergel

Represent. Theory 2 (1998), 432-448

DOI: https://doi.org/10.1090/S1088-4165-98-00057-0

Published electronically: December 28, 1998

PDF

Original Article: Represent. Theory 1 (1997)

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.

References

M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
S. M. Arkhipov, Semi-infinite cohomology of associative algebras and bar duality, Internat. Math. Res. Notices 17 (1997), 833–863. MR 1474841, DOI 10.1155/S1073792897000548
I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Differential operators on the base affine space and a study of ${\mathfrak {g}}$-modules, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 21–64. MR 0578996
David H. Collingwood and Ronald S. Irving, A decomposition theorem for certain self-dual modules in the category ${\scr O}$, Duke Math. J. 58 (1989), no. 1, 89–102. MR 1016415, DOI 10.1215/S0012-7094-89-05806-7
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, DOI 10.1016/0021-8693(87)90232-8
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
S. Donkin, Finite resolutions of modules for reductive algebraic groups, J. Algebra 101 (1986), no. 2, 473–488. MR 847172, DOI 10.1016/0021-8693(86)90206-1
Peter John Hilton and Urs Stammbach, A course in homological algebra, Graduate Texts in Mathematics, Vol. 4, Springer-Verlag, New York-Berlin, 1971. MR 0346025, DOI 10.1007/978-1-4684-9936-0
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
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, DOI 10.1007/978-0-8176-4575-5_{1}0
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, DOI 10.1090/S0894-0347-1993-99999-X
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, DOI 10.1090/S0894-0347-1994-1239506-X
Patrick Polo, Projective versus injective modules over graded Lie algebras and a particular parabolic category $\mathcal {O}$ for affine Kac-Moody algebras, Preprint, 1991.
Olga Taussky, An algebraic property of Laplace’s differential equation, Quart. J. Math. Oxford Ser. 10 (1939), 99–103. MR 83, DOI 10.1093/qmath/os-10.1.99
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, DOI 10.1007/BF02571521
Wolfgang Soergel, Kazhdan-Lusztig-Polynome and eine Kombinatorik für Kipp-Moduln, Representation Theory (An electronic Journal of the AMS) (1997).
Alexander A. Voronov, Semi-infinite homological algebra, Invent. Math. 113 (1993), no. 1, 103–146. MR 1223226, DOI 10.1007/BF01244304