Alcove Walk Models for Parabolic Mirković-Vilonen Intersections and Branching to Levi Subgroups

Abstract This article proves an alcove walk description of intersections of Schubert cells and partially semi-infinite orbits (depending on a choice of parabolic subgroup) in the affine Grassmannian of a split connected reductive group (we call these intersections parabolic Mirković-Vilonen intersections ). We deduce from this a parameterization of the irreducible components of the maximal possible dimension in these intersections by the alcove walks of maximal possible dimension. As a consequence we present 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.