arithmetic group over function fields and its fundamental domain

Question

Let $G$ be a semi-simple algebraic group defined over a global function field $K$. Let $S$ be a finite set of places of $K$. For a place $v$ of $K$ let $K_v$ be the completion under $v$. We take $K_S=\prod_{v\in S}K_v$ and $O=\{x\in K: x \mbox{ is integral in }K_v \mbox{ if } v\not \in S \}$. It is well known that $G(O)$ is a lattice in $G(K_S)$. Are there any fundamental domain of $G(O)$ expressed in terms of Ziegel sets?

More precisely we want the following type result. If $H$ is semi-simple $\mathbb Q$-group, then it is proved in Theorem 15.5 of Borel's book "Introduction aux groupes arithmétic" that there exists a Ziegel set $F$ and a finite subset $C\subset G(Q)$ such that $FCG(\mathbb Z)=G(\mathbb R)$. To my understanding this result is not contain in the paper of Borel and Harish-Chandra 1962 on arithmetic groups.

It will be great if one can suggest some references (in English) about it.

Answer 1

There are quite a few substantial research papers, most in English but some in German. One fairly recent book and its references would probably clarify for you what is out there: Lizhen Ji, Arithmetic groups and their generalizations. What, why, and how. AMS/IP Studies in Advanced Mathematics, 43. American Mathematical Society, Providence, RI; International Press, Cambridge, MA, 2008.

The research literature ranges over quite a bit of territory, not all directly connected with questions about fundamental domains but all somewhat connected in the case of semisimple groups. (There's also literature on solvable groups.) Typical names include Ulrich Stuhler, Jean-Pierre Serre, Gopal Prasad, and numerous others.

A key player in the study of $S$-arithmetic groups over function fields has been the Bruhat-Tits building, which substitutes for the traditional symmetric space in Lie group theory.