# Convex Hulls of Demazure Modules

Let $$G$$ be a semisimple algebraic group over $$\mathbb{C}$$ and for a highest weight $$\lambda$$, denote by $$V_{\lambda}^w$$ the Demazure module associated with $$\lambda$$ and $$w$$. More precisely, $$V_{\lambda}^w=U(\mathfrak{b}).v_{w \lambda}$$. There are many nice descriptions of these modules (sections of line bundles restricted to Schubert varieties, and more concretely the Demazure character formula).

If $$P(V_{\lambda}^w)$$ denotes the set of weights which appear in the Demazure module, I can consider the convex hull in $$X^*(T) \otimes_{\mathbb{Z}} \mathbb{R}$$. The extremal vertices of the resulting polytope will be $$w'(\lambda)$$ for $$w' \leq w$$ via an argument of Bernstein, Gelfand and Gelfand.

My questions generally run as follows: Is there a nice description of the facets of the resulting polytope? Is it true that if $$F$$ is a facet of this polytope that there exists a coweight $$\check{\Lambda_i}$$ such that for all $$v \in F$$ we have $$\langle v, \check{\Lambda_i} \rangle =n$$ for $$n \in \mathbb{Z}$$? Are they pseudo-Weyl polytopes in the sense of Kamnitzer (MV Cycles and Polytopes)?

I would be grateful for any knowledge in this direction.