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

Abstract

Let $\mathbf{G}$ be a quasi-split reductive group and $\mathbb{K}$ be a Henselian field equipped with a valuation $\omega :{\mathbb{K}}^{\times }\to \Lambda$, where $\Lambda$ is a non-zero 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.

137 pages