Iscrete S Eries for L Oop G Roups

In this paper we consider the category C(̃k, H̃) of the (̃k, H̃)-modules, including all the Verma modules, where k is some compact Lie algebra and H some Cartan subgroup, k̃ and H̃ are the corresponding affine Lie algebra and the affine Cartan group, respectively. To this category we apply the Zuckermann functor and its derivatives. By using the determinant bundle structure, we prove the natural duality of the Zuckermann derived functors, deduce a Borel-Weil-Bott type theorem on decomposition of the nilpotent part cohomology. Date: September 20, 1998, A revised version of 92-015(1992), pp IV.1-IV.16, Sonderforschungbereich 343, Universität Bielefeld The author was supported in part by the Alexander von Humboldt Foundation, the Abdus Salam ICTP and the Vietnam National Foundation for Reserach in Fundamental Sciences Forwarded Email: [email protected]