Weight polytopes and saturation of Demazure characters

For G a reductive group and T ⊂ B a maximal torus and Borel subgroup, Demazure modules are certain B -submodules, indexed by elements of the Weyl group, of the finite irreducible representations of G . In order to describe the T -weight spaces that appear in a Demazure module, we study the convex hull of these weights—the Demazure polytope. We characterize these polytopes both by vertices and by inequalities, and we use these results to prove that Demazure characters are saturated, in the case that G is simple of classical Lie type. Specializing to G = G L n , we recover results of Fink, Mészáros, and St. Dizier, and separately Fan and Guo, on key polynomials, originally conjectured by Monical, Tokcan, and Yong.