Summary: AFFINE JACQUET FUNCTORS AND
HARISH-CHANDRA CATEGORIES
MILEN YAKIMOV
Abstract. We define an affine Jacquet functor and use it to describe the
structure of induced affine Harish-Chandra 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 Harish-Chandra
modules of Lian and Zuckerman [13]. The latter provides an extension of the
works of Rocha-Caridi, Wallach [15] and Deodhar, Gabber, Kac [5] on block
decompositions of BGG categories for Kac-Moody 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
Harish-Chandra 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

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

Collections: Mathematics