Summary: CATEGORIES OF MODULES OVER AN AFFINE KACMOODY
ALGEBRA AND FINITENESS OF THE KAZHDANLUSZTIG
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) of smooth modules (in the sense
of Kazhdan and Lusztig ) of finite length over the corresponding affine
KacMoody 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 KazhdanLusztig  and LianZuckerman [18, 19]. In the main part
of this paper we establish a finiteness result for the KazhdanLusztig tensor
product which can be considered as an affine version of a theorem of Kostant
. It contains as special cases the finiteness results of Kazhdan, Lusztig
 and Finkelberg , 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 KazhdanLusztig's tensoring with the "affinization"
of the category of finite dimensional g modules (which is O in the notation
of [13, 14, 15]).