Homogeneous Bases for Demazure Modules

We study the PBW filtration on the Demazure modules $V_{r_{\gamma }}(\lambda )$ associated to reflections $r_{\gamma }$ at positive roots in type $A_n$, long roots in type $C_n$, short roots in type $B_n$ and positive roots not involving the simple root ${\alpha }_{n-1}$ in type $D_n$. In each type, we construct a normal polytope labelling a basis for the associated graded space $\mathrm{gr}{V}_{r_{\gamma }}(\lambda )$ and prove that the annihilating ideal of $\mathrm{gr}{V}_{r_{\gamma }}(\lambda )$ is a monomial ideal.