Maximal Orders in Central Simple Algebras and Bruhat–Tits Buildings

Abstract

We study the affine building forSLnover a local field and give a characterization of distance involving Hecke operators. Forn=3 we give an explicitly computable distance formula. We use this local information to show that the class number of a maximal order in a central simple algebra of dimensionn2over a number fieldKis equal to the number of orbits of a group of isometries (related to the unit group of the maximal order) acting on a Bruhat-Tits building forSLn(K). This generalizes results of Serre and Vignéras who considered the quaternion case in which the Bruhat-Tits building is a tree.