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Preprints, Working Papers, ... (Preprint) Year : 2023

Arithmetic subgroups of Chevalley group schemes over function fields II: Conjugacy classes of maximal unipotent subgroups

Abstract

Let $\mathcal{C}$ be a smooth, projective, geometrically integral curve defined over a perfect field $\mathbb{F}$. Let $k=\mathbb{F}(\mathcal{C})$ be the function field of $\mathcal{C}$. Let $\mathbf{G}$ be a split simply connected semisimple $\mathbb{Z}$-group scheme. Let $\mathcal{S}$ be a finite set of places of $\mathcal{C}$. In this paper, we investigate on the conjugacy classes of maximal unipotents subgroups of $\mathcal{S}$-arithmetic subgroups. These are parameterized thanks to the Picard group of $\mathcal{O}_{\mathcal{S}}$ and the rank of $\mathbf{G}$. Furthermore, these maximal unipotent subgroups can be realized as the unipotent part of natural stabilizer, that are the stabilizers of sectors of the associated Bruhat-Tits building. We decompose these natural stabilizers in terms of their diagonalisable part and unipotent part, and we precise the group structure of the diagonalisable part.
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hal-04196262 , version 1 (05-09-2023)

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Claudio Bravo, Benoit Loisel. Arithmetic subgroups of Chevalley group schemes over function fields II: Conjugacy classes of maximal unipotent subgroups. 2023. ⟨hal-04196262⟩
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