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$Λ$-buildings associated to quasi-split groups over $Λ$-valued fields

Auguste Hébert, Diego Izquierdo, Benoit Loisel

Abstract

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

$Λ$-buildings associated to quasi-split groups over $Λ$-valued fields

Abstract

Let be a quasi-split reductive group and be a Henselian field equipped with a valuation , where is a non-zero totally ordered abelian group. In 1972, Bruhat and Tits constructed a building on which the group acts provided that is a subgroup of . In this paper, we deal with the general case where there are no assumptions on and we construct a set on which acts. We then prove that it is a -building, in the sense of Bennett.

Paper Structure

This paper contains 103 sections, 151 theorems, 407 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Bennett's Lambda
  3. Examples of $\Lambda$-buildings
  4. The building I(K,w,G)
  5. Affine structure of the standard apartment
  6. Main results
  7. Structure of the paper
  8. Acknowledgements
  9. Funding
  10. The group SL(l+1)
  11. The projection pi
  12. The Z2-tree of SL(2)
  13. Abstract definition of RS-buildings and statement of the main theorem
  14. Definition of the standard apartment
  15. Root system, vectorial apartment over R and Weyl group
  16. ...and 88 more sections

Key Result

Lemma 2.2

The ring $\mathcal{O}_1$ is a valuation ring in $\mathcal{K}_1$ with valuation group:

Figures (5)

  • Figure 2.1: The building of $\mathrm{SL}(V)$ when $V$ is a $2$-dimensional vector space over a field $\mathbb{K}$ endowed with a valuation $\omega: \mathbb{K}^{\times} \rightarrow \mathbb{Z}^2$.
  • Figure 2.3: Three apartments in the $\mathbb{Z}^2$-building of $\mathrm{SL}_2$ over a $\mathbb{Z}^2$-valued field.
  • Figure 2.4: The action of the extended affine Weyl group on the standard apartment of the building of $\mathrm{SL}(V)$ when $V$ is a $2$-dimensional vector space over a field $\mathbb{K}$ endowed with a valuation $\omega: \mathbb{K}^{\times} \rightarrow \mathbb{Z}^2$.
  • Figure 11.1: Suppose for simplicity that $\pi(\Lambda)$ is discrete. Let $\mathbb{A}_\Lambda=\{x\in \mathbb{A}|\alpha(x)\in \Lambda\}$ and $\mathcal{I}_\Lambda=G.\mathbb{A}_\Lambda$. Let $[L],[L']\in \pi(\mathcal{I}_\Lambda)$. Then by Theorem \ref{['thmFibers_buildings']}, $\mathcal{I}_{[L]}=\pi^{-1}([L])$ and $\mathcal{I}_{[L']}=\pi^{-1}([L'])$ are $\mathbb{R}$-trees. By Proposition \ref{['propBouts_aretes']} and Corollary \ref{['corBijection_ends_edges']}, the set of ends of $\mathcal{I}_{[L]}$ is in bijection with the set of edges of $\pi(\mathcal{I})$ containing $\pi(x)$ and the edge $\pi([L])-\pi([L'])$ corresponds to a unique pair of ends $\mathcal{E}_{[L]}$, $\mathcal{E}_{[L']}$ of $\mathcal{I}_{[L]}$ and $\mathcal{I}_{[L']}$.
  • Figure 11.2: We work with $\mathbb{K}=\mathbb{F}_q(\!(t)\!)(\!(u)\!)$. Let $(b_1,b_2)$ be a basis of $\mathbb{K}^2$. We can choose $x_{(1,0)}=u$ and $x_{(0,1)}=t$.

Theorems & Definitions (356)

  • Example 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Proposition 2.7
  • Theorem 2.8
  • ...and 346 more