Fusion Product Structure of Demazure Modules

Let g be a finite-dimensional complex simple Lie algebra. Fix a non-negative integer l, we consider the set of dominant weights {\lambda} of g such that l{\Lambda}_0+{\lambda} is a dominant weight for the corresponding untwisted affine Kac-Moody algebra. For these special family of dominant weights, we show that the fusion product of an irreducible g-module V({\lambda}) and a finite number of special family of g-stable Demazure modules of level l (considered in [15] and [16]), for the current algebra g[t] associated to g, again turns out to be a Demazure module. This fact is closely related with several important conjectures. We use this result to construct the g[t]-module structure of the irreducible module V(l{\Lambda}_0 + {\lambda}) as a semi-infinite fusion product of finite dimensional g[t]-modules as conjectured in [16]. As a second application we give further evidence to the conjecture on the generalization of Schur positivity (see [7]).