Affine Jacquet Functors and Harish-chandra Categories

We define an affine Jacquet functor and use it to describe the structure of induced affine Harish-Chandra modules at noncritical levels, extending the theorem of Kac and Kazhdan [10] on the structure of Verma modules in the Bernstein–Gelfand–Gelfand categories O for Kac–Moody algebras. 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 Kac–Moody algebras, based on the fusion tensor product.