Discrete Series for Loop Groups.I. An algebraic Realization of Standard Modules

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.