Combinatorics in affine flag varieties

TL;DR

This paper develops a refined combinatorial model using alcove walks to encode points in affine flag varieties, linking geometric structures with affine Hecke algebra combinatorics.

Contribution

It introduces a new alcove walk model that encodes affine flag variety points and relates to Mirkovic-Vilonen intersections.

Findings

Refined alcove walk model encodes affine flag points

Model indexes cells in Mirkovic-Vilonen intersections

Connects combinatorics with geometric structures in affine flag varieties

Abstract

The Littelmann path moel gives a realization of the crystals of integrable representations of symmetrizable Kac-Moody Lie algebras. Recent work of Gaussent-Littelmann and others has demonstrated a connection between this model and the geometry of the loop Grassmannian. The alcove walk model is a version of the path model which is intimately connected to the combinatorics of the affine Hecke algebra. In this paper we define a refined alcove walk model which encodes the points of the affine flag variety. We show that this combinatorial indexing naturally indexes the "cells" in generalized Mirkovic-Vilonen intersections.