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

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.

19 pages, 1 figure. Minor changes in wording made in introduction and in Proposition 3.2. Additional references added. Statement of Theorem A corrected, to account for a mistake in the previous version of Proposition 5.5. To appear, Transformation Groups; this arXiv version is close to but not the same as the Version of Record: https://doi.org/10.1007/s00031-026-09956-0