Categories of modules over an affine Kac-Moody algebra and finiteness of the Kazhdan-Lusztig tensor product

摘要

To each category C of modules of finite length over a complex simple Lie algebra 0, closed under tensoring with finite dimensional modules, we associate and study a category AFF(C)(K) of smooth modules (in the sense of Kazhdan and Lusztig [D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, I, J. Amer. Math. Soc. 6 (1993) 905-947]) of finite length over the corresponding affine Kac-Moody algebra in the case of central charge less than the critical level. Equivalent characterizations of these categories are obtained in the spirit of the works of Kazhdan and Lusztig I-D. Kazhdan, G. Luszfig, Tensor structures arising from affine Lie algebras, 1, J. Amer. Math. Soc. 6 (1993) 905-947] and Lian and Zuckerman [B.H. Lian, G.J. Zuckerman, BRST cohomology and noncompact coset models, in: Proceedings of the XXth International Conference on Differential Geometric methods in Theoretical Physics, New York, 199 1, 1992, pp. 849-865; B.H. Lian, G.J. Zuckerman, An application of infinite dimensional Lie theory to semi-simple Lie groups, in: Representation Theory of Groups and Algebras, in: Contemp. Math., vol. 145, 1993, pp. 249-257]. In the main part of this paper we establish a finiteness result for the Kazhdan-Lusztig tensor product which can be considered as an affine version of a theorem of Kostant [B. Kostant, On the tensor product of a finite and an infinite dimensional representation, J. Funct. Anal. 20 (1975) 257-285]. It contains as special cases the finiteness results Of Kazhdan, Lusztig [D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, 1, J. Amer. Math. Soc. 6 (1993) 905-947] and Finkelberg [M. Finkelberg, PhD thesis, Harvard University, 1993], and states that for any subalgebra f of g which is reductive in g the "affinization" of the category of finite length admissible (g, f) modules is stable under Kazhdan-Lusztig's tensoring with the "affinization" of the category of finite dimensional g modules (which is O-k in the notation of [D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, 1, J. Amer. Math. Soc. 6 (1993) 905-947; D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, II, J. Amer. Math. Soc. 6 (1994) 949-1011; D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, IV, J. Amer. Math. Soc. 7 (1994) 383-453]). (C) 2007 Elsevier Inc. All rights reserved.