# $\Lambda$-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 $\omega:\mathbb{K}^{\times}\rightarrow \Lambda$, 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.
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Cited literature [74 references]

https://hal.archives-ouvertes.fr/hal-02430546
Contributor : Auguste Hébert <>
Submitted on : Tuesday, June 16, 2020 - 10:46:11 AM
Last modification on : Thursday, June 18, 2020 - 3:36:26 AM

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### Identifiers

• HAL Id : hal-02430546, version 2
• ARXIV : 2001.01542

### Citation

Auguste Hébert, Diego Izquierdo, Benoît Loisel. $\Lambda$-buildings associated to quasi-split groups over $\Lambda$-valued fields. 2020. ⟨hal-02430546v2⟩

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