$\Lambda$-buildings associated to quasi-split groups over $\Lambda$-valued fields

Auguste Hébert (1, 2) , Diego Izquierdo (3, 4) , Benoit Loisel (1, 2)

1 ENS de Lyon - École normale supérieure de Lyon
2 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
3 CMLS - Centre de Mathématiques Laurent Schwartz
4 X - École polytechnique

Let $\mathbf{G}$ be a quasi-split reductive group and $\mathbb{K}$ be a Henselian field equipped with a valuation $\omega:\mathbb{K}^{\times}\rightarrow \Lambda$, where $\Lambda$ is a totally ordered abelian group. In 1972, Bruhat and Tits constructed a building on which the group $\mathbf{G}(\mathbb{K})$ acts provided that $\Lambda$ is a subgroup of $\mathbb{R}$. In this paper, we deal with the general case where there are no assumptions on $\Lambda$ and we construct a set on which $\mathbf{G}(\mathbb{K})$ acts. We then prove that it is a $\Lambda$-building, in the sense of Bennett.

Domaines: Théorie des représentations [math.RT]; Théorie des groupes [math.GR]

Auguste Hébert, Diego Izquierdo, Benoit Loisel. $\Lambda$-buildings associated to quasi-split groups over $\Lambda$-valued fields. 2020. ⟨hal-02430546v1⟩