Reduction rules for Demazure modules

TL;DR

This paper introduces a reduction rule for calculating weight multiplicities in Demazure modules of complex reductive groups, simplifying the problem by relating it to Levi subgroups.

Contribution

It provides a novel reduction technique for weight multiplicity calculations in Demazure modules using Levi subgroups.

Findings

Reduction rule simplifies weight multiplicity computations.

Applicable to weights on faces of the weight polytope.

Facilitates calculations in representation theory of reductive groups.

Abstract

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 $G$.