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Character formulas for tilting modules over Kac-Moody algebras. (English) Zbl 0964.17018
Author’s introduction: In this article the author determines the characters of indecomposable tilting modules in the category \({\mathcal O}\) over an affine Kac-Moody Lie algebra. By an equivalence of categories due to Kazhdan and Lusztig, this leads to character formulas for tilting modules over quantum groups; in particular he proves Conjecture 7.2 from ibid. 1, 37-68 (1997; Zbl 0886.05124)] in many cases.
He found the key to the determination of these characters in [S. M. Arkhipov, Int. Math. Res. Not. 1997, 833-863 (1997; Zbl 0884.16025)]. There Arkhipov extends Feigin’s semi-infinite cohomology [B. Feigin, Usp. Mat. Nauk 39, No. 2, 195-196 (1984; Zbl 0544.17009)] and shows in particular, that the category of all modules with a Weyl filtration in positive level is contravariantly equivalent to the analogous category in negative level. Under this equivalence, projective objects have to be transformed into tilting modules; thus the Kazhdan-Lusztig conjectures in positive level lead to character formulas for tilting modules in negative level.
In Arkhipov (loc. cit.) the contravariant equivalence alluded to above appears as an illustration of a much stronger and deeper semi-infinite duality. The author shows in the subsequent sections, how one can get it directly. Then he discusses the application to tilting modules.

MSC:
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
20G05 Representation theory for linear algebraic groups
20G42 Quantum groups (quantized function algebras) and their representations
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