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

Abstract

To each category C of modules of finite length over a complex simple Lie algebra g , closed under tensoring with finite dimensional modules, we associate and study a category AFF ( C ) κ 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 [D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, I, 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, 1991, 1992, pp. 849–865; B.H. Lian, G.J. Zuckerman, An application of infinite dimensional Lie theory to semisimple 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, I, 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 κ in the notation of [D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, I, 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]).