Representations Of Quantized Function Algebras Of Kac- Moody Algebras

Let C be a symmetrizable generalized Cartan Matrix, and q an indeterminate. g (C) is the Kac-Moody Lie algebra and U = Uq( g (C)) the associated quantum enveloping algebra over k = Q (q). The quantum function algebra Cq [G] is defined as a suitable U-bisubalgebra of the dual space Homk(U,k) which can be described using matrix elements of the integrable U-modules of non-vanishing level. The highest weight modules of Cq[ G] are constructed and their (unitarizable) irreducible quotients are shown to be in a 1-1 correspondence with the reduced elements of the Weyl group of g (C). It is further established that these simple modules are described in terms of the Cq[SL 2]- modules obtained by restriction, and that they satisfy a Tensor Product Theorem, similar to the finite type case