Affine Demazure Weight Polytopes and Twisted Bruhat Orders

For an untwisted affine Kac-Moody Lie algebra $\mathfrak{g}$ with Cartan and Borel subalgebras $\mathfrak{h}\subset \mathfrak{b}\subset \mathfrak{g}$, affine Demazure modules are certain $U(\mathfrak{b})$-submodules of the irreducible highest-weight representations of $\mathfrak{g}$. We introduce here the associated affine Demazure weight polytopes, given by the convex hull of the $\mathfrak{h}$-weights of such a module. Using methods of geometric invariant theory, we determine inequalities which define these polytopes; these inequalities come in three distinct flavors, specified by the standard, opposite, or semi-infinite Bruhat orders. We also give a combinatorial characterization of the vertices of these polytopes lying on an arbitrary face, utilizing the more general class of twisted Bruhat orders.