  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]).
