Reduction rules for Demazure modules

For $G$ a complex reductive group and $B\subseteq G$ a Borel subgroup, we provide a reduction rule for certain weight multiplicities in Demazure modules $V_{\lambda }^w$: given a weight $\mu$ on a face of the associated weight polytope $P_{\lambda }^w$, we reduce the computation of the dimension of the weight space $V_{\lambda }^w(\mu )$ to a similar problem of computing the weight space dimension for a Demazure module of a Levi subgroup of $G$.