Tensor product structure of affine Demazure modules and limit constructions

Let g be a simple complex Lie algebra, we denote by g the affine Kac-Moody algebra associated to the extended Dynkin diagram of g. Let Lambda(0) be the fundamental weight of p corresponding to the additional node of the extended Dynkin diagram. For a dominant integral g-coweight lambda(V), the Demazure submodule V-lambda(V) (m Lambda(0)) is a g-module. We provide a description of the g-module structure as a tensor product of "smaller" Demazure modules. More precisely; for any partition of lambda(V) = Sigma(j) lambda j(V) as a sum of dominant integral g-coweights, the Demazure module is (as g-module) isomorphic to circle times j V-lambda(V)(j) (m Lambda(0)). For the "smallest" case, lambda(V) = w(V) a fundamental coweight, we provide for g of classical type a decomposition of V-w(v) (m Lambda(0)) into irreducible g-modules, so this can be viewed as a natural generalization of the decomposition formulas in [13] and [16]. A comparison with the Uq(g)-characters of certain finite dimensional Uq'((q) over bar)-modules (Kirillov-Reshetikhin-modules) suggests furthermore that all quantized Demazure modules V-lambda(V)(,q),Q(m Lambda(0)) can be naturally endowed with the structure of a U-q'((g) over bar)-module. We prove, in the classical case (and for a lot of non-classical cases), a conjecture by Kashiwara. [10], that the "smallest" Demazure modules are, when viewed as g-modules, isomorphic to some KR-modules. For an integral dominant g-weight A let V(A) be the corresponding irreducible g-representation. Using the tensor product decomposition for Demazure modules, we give a description of the g-module structure of V (A) as a semi-infinite tensor product of finite dimensional g-modules. The case of twisted affine Kac-Moody algebras can be treated in the same way, some details are worked out in the last section.