A building-theoretic approach to relative Tamagawa numbers of semisimple groups over global function fields

Abstract

Let $G$ be a semisimple almost simple group defined over a global function field $K$, not anisotropic of type $A_n$. We express the (relative) Tamagawa number of $G$ in terms of local data, including the number $t_\infty(G)$ of types in one orbit of a special vertex in the Bruhat--Tits building of $G_\infty(\hat{K}_\infty)$ for some place $\infty$ and the class number $h_\infty(G)$ of $G$ at $\infty$.