Reduction rules for Demazure modules

1. Problem Statement

The paper addresses a specific subclass of branching problems in the representation theory of complex reductive algebraic groups $G$. While classical branching problems study how representations of $G$ restrict to subgroups $G'$, this work focuses on reduction rules: equalities that reduce the computation of a specific quantity (here, weight multiplicities) for a representation of $G$ to a similar quantity for a representation of a smaller, predictable subgroup (specifically, a Levi subgroup).

The specific object of study is the Demazure module $V^w_\lambda$. Given a dominant weight $\lambda$ and an element $w$ of the Weyl group $W$, $V^w_\lambda$ is a $B$-submodule of the irreducible highest weight module $V_\lambda$. The authors seek a condition on a weight $\mu$ such that the dimension of the weight space $\dim V^w_\lambda(\mu)$ can be computed as the dimension of a weight space in a Demazure module associated with a Levi subgroup $L \subset G$.

2. Methodology

The authors employ a multi-faceted approach combining Geometric Invariant Theory (GIT), convex geometry, and combinatorial representation theory.

A. Geometric Invariant Theory and Ample Cones

The authors utilize the connection between representation theory and the geometry of flag varieties.

• They consider the Demazure weight polytope $P^w_\lambda$, defined as the convex hull of the weights of $V^w_\lambda$.
• Drawing on previous work (citing [3, 4]), they establish that the faces of $P^w_\lambda$ correspond to Levi subgroups of $G$.
• The reduction rule is triggered when a weight $\mu$ lies on a specific face $F(v, \eta)$ of the polytope, determined by a dominant coweight $\eta$ and a minimum-length coset representative $v \in W^\eta$.

B. Convex Geometry and Polytope Structure

• The paper relies on the characterization of $P^w_\lambda$ via inequalities derived from the Hilbert–Mumford criterion.
• A key structural insight is that a face $F(v, \eta)$ of $P^w_\lambda$ is itself isomorphic to a Demazure weight polytope $P^{w_L}_{\lambda_L}$ for a Levi subgroup $L$. This establishes the geometric correspondence between the "large" module and the "reduced" module.

C. Two-Stage Proof Strategy

The proof of the main reduction rule is divided into two distinct stages:

1. Algebro-Geometric Reduction (Stage 1):

   • The authors adapt Roth's proof for Littlewood–Richardson coefficients.
   • They construct a closed immersion of a Schubert variety in the flag variety of the Levi subgroup $L$ into a Schubert variety in the flag variety of $G$.
   • By analyzing the restriction of line bundles and the resulting maps on global sections (cohomology), they establish an isomorphism between specific weight spaces of the Levi Demazure module $V^{w_L}_{\lambda_L}$ and an "intermediate" Demazure module $V^q_\lambda$ (where $q$ is a specific element in the Bruhat interval $[e, w]$).

2. Convex-Geometric and Combinatorial Reduction (Stage 2):

   • The authors compare the intermediate module $V^q_\lambda$ with the original module $V^w_\lambda$.
   • They utilize Demazure operators ($D_i$) and the properties of the Demazure product ($*$) on the Weyl group.
   • Using induction on the length of the Weyl group element $v$, they show that for weights $\mu$ lying on the face $F(v, \eta)$, the weight multiplicity is invariant under the action of the Weyl group elements that define the face. This proves that the multiplicity in $V^q_\lambda$ equals the multiplicity in $V^w_\lambda$.

3. Key Contributions and Results

Main Theorem (Theorem 1.1 / Theorem 3.8)

The central result provides a precise reduction rule. Let $\mu$ be a weight of the Demazure module $V^w_\lambda$ lying on the face $F(v, \eta)$ of the associated polytope $P^w_\lambda$. Then:
$$\dim V^w_\lambda(\mu) = \dim V^{w_L}_{\lambda_L}(\mu)$$
Where:

• $L = v L_\eta v^{-1}$ is the conjugate of the Levi subgroup determined by $\eta$.
• $w_L = v (\pi_\eta(w^{-1} * v))^{-1} v^{-1}$ is the Weyl group element for the Levi.
• $\lambda_L = v (\pi_\eta(w^{-1} * v))^{-1} \lambda$ is the transformed highest weight.
• $V^{w_L}_{\lambda_L}$ is the Demazure module for the Borel subgroup $B_L \subset L$.

Technical Lemmas and Propositions

• Proposition 3.6: Establishes the commutative diagram of inclusions relating the irreducible module $V_\lambda$, the intermediate module $V^q_\lambda$, and the Levi module $V^{w_L}_{\lambda_L}$.
• Theorem 4.11: Proves the first stage of the reduction using GIT techniques, showing the isomorphism of weight spaces between the Levi module and the intermediate module.
• Theorem 5.6: Proves the second stage using induction and Demazure operators, showing that the multiplicity is preserved when moving from the intermediate module $V^q_\lambda$ to the full module $V^w_\lambda$ for weights on the specific face.

4. Significance

• Computational Efficiency: The rule allows for the computation of weight multiplicities in large, complex Demazure modules by reducing them to calculations in smaller rank groups (Levi subgroups). This is particularly useful for high-rank groups where direct computation is intractable.
• Unification of Concepts: The paper bridges the gap between the combinatorics of Weyl groups (Demazure products), the geometry of flag varieties (Schubert varieties and ample cones), and convex geometry (polytope faces).
• Generalization: It extends known reduction rules (such as those for Kostka numbers and Littlewood–Richardson coefficients) to the broader context of Demazure modules, which are fundamental in the study of Schubert calculus and the representation theory of Kac–Moody algebras.
• Geometric Insight: It reinforces the philosophy that the combinatorial structure of representation theory (weight multiplicities) is deeply encoded in the geometry of the associated polytopes and their faces.

In summary, the paper provides a rigorous framework for "cutting down" the complexity of Demazure module calculations by identifying when a weight lies on a face of the weight polytope, thereby allowing the problem to be solved within a smaller, associated Levi subgroup.