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Now I am reading Groupes réductifs sur un corps local : I. Données radicielles valuées written by Bruhat and Tits in 1972. Let $\Phi$ be a root system, $G$ a group with root data $(T,(U_{a},M_{a})_{a\in \Phi})$ of type $\Phi$, and $\varphi=(\varphi_{a}\colon U_{a}\longrightarrow \mathbb{R}\cup \{\infty\})$ a valuation of these root data. We denote by $A$ the equivalence class of $\varphi$. Then the bijective $$ V\longrightarrow A;v\longmapsto \varphi+v $$ provides $A$ with a structure of affine apartment. Bruhat and Tits defined an equivalence relation $\sim$ on $G\times A$ to be $$ (g,x)\sim (h,y) \iff \exists n\in N \text{ such that } y=\nu(n)(x) \text{ and } g^{-1}hn\in P_{x}, $$ where all notations are the same in Bruhat–Tits paper. They called the quotient $X=G\times A/{\sim}$ the building of $G$. My questions are that “Do the following hold?

  1. For any $x,y\in X$, there exist $g\in G$ such that $x,y\in gA$.
  2. The action of $G$ on $X$ is strongly transitive.”

Since it is so called building, perhaps these are valid, but I would like to know the proof.

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  • $\begingroup$ I think that the standard English translation of the French "données radicielles" is "root data" $\endgroup$
    – YCor
    Sep 29, 2022 at 10:49
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    $\begingroup$ @YCor And with the singular "root datum". $\endgroup$ Sep 29, 2022 at 12:13
  • $\begingroup$ $N = N_G(T)$, I guess. (1) is true, and is almost certainly in both BT1 which you are reading, and in Tits's Corvallis article which is much more accessible. Is "strong transitivity" in (2) the requirement of transitivity on finite subsets of the same cardinality? This is certainly not true; the action of $G$ is not even transitive on points. So perhaps you want transitivity on finite sets of vertices, maybe with some distance condition? That, too, can fail, again even for single vertices, even if $G$ is adjoint, in special cases. $\endgroup$
    – LSpice
    Sep 29, 2022 at 13:44
  • $\begingroup$ Sorry. I did not explain the strong transitivity. Let $\mathcal{A}$ be a system of apartments, now in this case, $\mathcal{A}=\{gA\mathrel{\vert}g\in G\}$. We say that the action of $G$ is strongly transitive if for any $A_{1},A_{2}\in \mathcal{A}$ and chambers $C_{1}\subset A_{1}, C_{2}\subset A_{2}$, there exists $g\in G$ such that $g(A_{1},C_{1})=(A_{2},C_{2})$. $\endgroup$
    – M masa
    Sep 29, 2022 at 14:49

1 Answer 1

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(1) is Théorème 7.4.18 of [BT1]: Bruhat and Tits - Groupes réductifs sur un corps local.

Now let $(A_1, C_1)$ and $(A_2, C_2)$ be two apartment–chamber pairs. Upon replacing $(A_1, C_1)$ and $(A_2, C_2)$ by $G$-conjugates, we may, and do, assume that $A_1$ and $A_2$ both equal $A$. Now the entire computation is happening with respect to a fixed affine root system, and has nothing to do with the full valuation of root datum, so we go back to the very beginning. (1.3.3) of [BT1] shows (or, rather, quotes from Bourbaki the fact) that there is some element of $W$ that conjugates $C_1$ to $C_2$. This element of $W$ may be lifted to an element of $N \subseteq G$. Per your definition, this is what is required in (2).

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