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

TL;DR

This paper classifies conjugacy classes of maximal unipotent subgroups in arithmetic groups over function fields, linking them to geometric and algebraic structures like the Picard group and Bruhat-Tits buildings.

Contribution

It provides a detailed parameterization of these conjugacy classes using the Picard group and explores their realization as stabilizers in Bruhat-Tits buildings, including their structural decomposition.

Findings

Conjugacy classes are parameterized by the Picard group and the rank of the group.

Maximal unipotent subgroups are realized as stabilizers of sectors in Bruhat-Tits buildings.

The structure of the diagonalisable part of stabilizers is explicitly described.

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 unipotent 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, which 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.