 
Summary: AFFINE JACQUET FUNCTORS AND
HARISHCHANDRA CATEGORIES
MILEN YAKIMOV
Abstract. We define an affine Jacquet functor and use it to describe the
structure of induced affine HarishChandra modules at noncritical levels, ex
tending the theorem of Kac and Kazhdan [10] on the structure of Verma
modules in the BernsteinGelfandGelfand categories O for KacMoody al
gebras. This is combined with a vanishing result for certain extension groups
to construct a block decomposition of the categories of affine 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 affine Jacquet functor and the Kazhdan
Lusztig tensor product. A modification of this is used to prove that the affine
HarishChandra category is stable under fusion tensoring with the Kazhdan
Lusztig category (a case of our finiteness result [17]) and will be further applied
in studying translation functors for KacMoody algebras, based on the fusion
tensor product.
1. Introduction
Let g be a complex semisimple Lie algebra and ~g be the corresponding affine
