Combinatorial results on Demazure modules

In this paper, we introduce the notion of polytopes with integral structure, which generalizes the notion of usual polytopes whose vertices have integral coordinates. Then we consider a symmetrizable Kac-Moody algebra. For every dominant weight lambda and every element of the Weyl group w, we construct a polytope with integral structure such that its Ehrhart polynomial describes the dimension of the Demazure module of highest weight n lambda associated to w.