Lattice Polytopes Associated to Certain Demazure Modules of sl n + 1

Let w be an element of the Weyl group of sl_{n + 1}. We prove that for a certain class of elements w (which includes the longest element w₀ of the Weyl group), there exist a lattice polytope R^l(w), for each fundamental weight _i of sl_{n + 1}, such that for any dominant weight = _{i = 1}ⁿ a_ii, the number of lattice points in the Minkowski sum _w = _{i = 1}ⁿa_ii^w is equal to the dimension of the Demazure module E_w(). We also define a linear map A^w : R^l(w) P _Z R where P denotes the weight lattice, such that char E_w() = e e^–A(x) where the sum runs through the lattice points x of ^w.