Tessellations of an affine apartment by affine weight polytopes

Abstract

Let $\mathbb{A}$ be a finite-dimensional vector space and $\Phi$ be a finite root system in $\mathbb{A}$. To these data are associated an affine poly-simplicial complex. Motivated by a forthcoming construction of connectified higher buildings, we study “affine weight polytopes” associated with these data. We prove that these polytopes tessellate $\mathbb{A}$. We also prove a kind of “mixed” tessellation, involving the affine weight polytopes and the poly-simplicial structure on $\mathbb{A}$.