 
Summary: AFFINE JACQUET FUNCTORS AND
HARISHCHANDRA CATEGORIES
MILEN YAKIMOV
Abstract. We dene an aĆne Jacquet functor and use it to describe the
structure of induced aĆne HarishChandra modules at noncritical levels, ex
tending the theorem of Kac and Kazhdan [10] on the structure of Verma
modules in the Bernstein{Gelfand{Gelfand categories O for Kac{Moody al
gebras. This is combined with a vanishing result for certain extension groups
to construct a block decomposition of the categories of aĆne HarishChandra
modules of Lian and Zuckerman [13]. The latter provides an extension of the
works of RochaCaridi, Wallach [15] and Deodhar, Gabber, Kac [5] on block
decompositions of BGG categories for KacMoody algebras. We also prove a
compatibility relation between the aĆne Jacquet functor and the Kazhdan{
Lusztig tensor product. A modication of this is used to prove that the aĆne
HarishChandra category is stable under fusion tensoring with the Kazhdan{
Lusztig category (a case of our niteness result [17]) and will be further applied
in studying translation functors for Kac{Moody algebras, based on the fusion
tensor product.
1. Introduction
Let g be a complex semisimple Lie algebra and ~ g be the corresponding aĆne
