Alcove walk models for parabolic Mirković-Vilonen intersections and branching to Levi subgroups

TL;DR

The paper develops an explicit alcove-walk framework to model parabolic Mirković-Vilonen intersections in the affine Grassmannian, producing a cellular paving indexed by positively-folded alcove walks and a bijection between maximal-dimension walks and the irreducible components of these intersections. It provides a direct, elementary description of Levi-subgroup branching for highest-weight representations via these walks, yielding a new algorithm for computing characters and a PRV-type result for branching to Levi factors. The approach relies on Bruhat-Tits retractions and alcove-walk combinatorics, connecting MV-geometry with explicit combinatorics in the base apartment, and it extends to tensor-product multiplicities through a variant model. Concrete examples, including a Type A2 GL3 case, illustrate the counting and geometry, while the appendix clarifies key length-difference invariants guiding the counts. Overall, the work offers a self-contained, explicit, and computationally usable bridge between MV theory, affine Grassmannians, and representation-theoretic branching.

Abstract

This article establishes alcove walk models for intersections of Schubert varieties and partially semi-infinite orbits in the affine Grassmannian of a split reductive group (we call such intersections parabolic Mirković-Vilonen intersections). More precisely, we describe explicit cellular pavings of these intersections, indexed by certain positively-folded alcove walks. We prove a parametrization of the irreducible components of maximal possible dimension, in terms of alcove walks of maximal possible dimension. We then deduce a new combinatorial description of branching to Levi subgroups of irreducible highest weight representations, and in particular we give a new algorithm for computing the characters of such representations.

Figures (1)

• Figure 1: The base alcove ${\bf a}$ is colored in light gray, and the bolded vertices are the elements in $W_0(-\mu)$, with corresponding translation alcoves $-w(\mu) + {\bf a}$ colored in light blue. Each alcove $(t_{-w(\mu)})_{\bf 0}({\bf a})$ is labelled in magenta with the $W_{\rm aff}$-part of a chosen reduced word expression (e.g., we write $12012$ in place of $s_{12012}\tau$). Other relevant alcoves have their reduced words colored in light pink. The three vertices $-\lambda_i + {\bf 0}$ are colored: $-\lambda_1 = \hbox{\color{red} red square}$, $-\lambda_2 = \hbox{\color{green} green square}$, and $-\lambda_3 = \hbox{\color{blue} blue square}$. The alcove walks of maximum possible dimension which terminate at $-\lambda_i + {\bf 0}$ are given the same color as that vertex.

Theorems & Definitions (45)

• Theorem A: Theorem \ref{['Thm_A_body']}
• Theorem B: Theorem \ref{['main_thm_P']}
• Remark 2.1
• Definition 3.1
• Proposition 3.2
• proof
• Definition 4.1
• Definition 4.2
• Definition 4.3
• Definition 4.4