On Chevalley group schemes over function fields: quotients of the Bruhat-Tits building by $\{\wp\}$-arithmetic subgroups
Résumé
Let $\mathbf{G}$ be a reductive Chevalley group scheme (defined over $\mathbb{Z}$). Let $\mathcal{C}$ be a smooth, projective, geometrically integral curve over a field $\mathbb{F}$. Let $\wp$ be a closed point on $\mathcal{C}$. Let $A$ be the ring of functions that are regular outside $\lbrace \wp \rbrace$. The fraction field $k$ of $A$ has a discrete valuation $\nu=\nu_{\wp}: k^{\times} \rightarrow \mathbb{Z}$ associated to $\wp$. In this work, we study the action of the group $ \textbf{G}(A)$ of $A$-points of $\mathbf{G}$ on the Bruhat-Tits building $\mathcal{X}=\mathcal{X}(\textbf{G},k,\nu_\wp)$ in order to describe the structure of the orbit space $ \textbf{G}(A)\backslash \mathcal{X}$. We obtain that this orbit space is the ``gluing'' of a closed connected CW-complex with some sector chambers. The latter are parametrized by a set depending on the Picard group of $\mathcal{C} \smallsetminus \{\wp\}$ and on the rank of $\mathbf{G}$. Moreover, we observe that any rational sector face whose tip is a special vertex contains a subsector face that embeds into this orbit space. We deduce, from this description, a writing of $\mathbf{G}(A)$ as a free product with amalgamation. We also obtain a counting of the $\Gamma$-conjugacy classes of maximal unipotent subgroups contained in a finite index subgroup $\Gamma \subseteq \mathbf{G}(A)$, together with a description of these maximal unipotent subgroups.
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