Tilings defined by affine Weyl groups

TL;DR

This paper extends known tiling properties of Weyl groups to affine Weyl groups, demonstrating that their images of the Weyl alcove form a disjoint tiling of the entire Euclidean space.

Contribution

It proves that affine Weyl group images of the Weyl alcove are disjoint and cover the whole Euclidean space, generalizing Waldspurger's results for finite Weyl groups.

Findings

Affine Weyl group images of the alcove are disjoint.

Their union covers the entire Euclidean space.

Generalizes tiling properties from finite to affine Weyl groups.

Abstract

Let W be a Weyl group, presented as a crystallographic reflection group on a Euclidean vector space V, and C an open Weyl chamber. In a recent paper, Waldspurger proved that the images (id-w)(C), for Weyl group elements w, are all disjoint, and their union is the closed cone spanned by the positive roots. We show that similarly, if A is the Weyl alcove, the images (id-w)(A), for affine Weyl group elements w, are all disjoint, and their union is V.