 
Summary: CATEGORIES OF MODULES OVER AN AFFINE KAC{MOODY
ALGEBRA AND FINITENESS OF THE KAZHDAN{LUSZTIG
TENSOR PRODUCT
MILEN YAKIMOV
Abstract. To each category C of modules of nite length over a complex
simple Lie algebra g; closed under tensoring with nite dimensional modules,
we associate and study a category AFF(C) of smooth modules (in the sense
of Kazhdan and Lusztig [13]) of nite length over the corresponding aĆne
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{Lusztig [13] and Lian{Zuckerman [18, 19]. In the main part
of this paper we establish a niteness result for the Kazhdan{Lusztig tensor
product which can be considered as an aĆne version of a theorem of Kostant
[17]. It contains as special cases the niteness results of Kazhdan, Lusztig
[13] and Finkelberg [7], and states that for any subalgebra f of g which is
reductive in g the \aĆnization" of the category of nite length admissible (g; f)
modules is stable under Kazhdan{Lusztig's tensoring with the \aĆnization"
of the category of nite dimensional g modules (which is O in the notation
of [13, 14, 15]).
1. Introduction
