Acyclic coefficient systems on buildings

Abstract

For cohomological (resp. homological) coefficient systems $\mathcal{F}$ (resp. $\mathcal{V}$) on affine buildings $X$ with Coxeter data of type ${\tilde{A}}_d$ we give for any $k\ge 1$ a sufficient local criterion which implies $H^k(X,\mathcal{F})=0$ (resp. $H_k(X,\mathcal{V})=0)$. Using this criterion we prove a conjecture of de Shalit on the acyclicity of coefficient systems attached to hyperplane arrangements on the Bruhat-Tits building of the general linear group over a local field. We also generalize an acyclicity theorem of Schneider and Stuhler on coefficient systems attached to representations.