  In this paper we consider the category $C (\tilde k, \tilde H)$ of the
$(\tilde k, \tilde H)$-modules, including all the Verma modules, where $k$ is
some compact Lie algebra and H some Cartan subgroup, $\tilde k$ and $\tilde H$
are the corresponding affine Lie algebra and the affine Cartan group,
respectively. To this category we apply the Zuckerman functor and its
derivatives. By using the determinant bundle structure, we prove the natural
duality of the Zuckerman derived functors, and deduce a Borel-Weil-Bott type
theorem on decomposition of the nilpotent part cohomology.
