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Let $G$ be a reductive group over a $p$-adic field $k$, $S \subset G$ a maximal $k$-split torus, $\Phi(G,S)$ the relative root system and $\Delta$ a basis of $\Phi$.

There is a group homomorphism : $$S(k) \to \mathbb{Z}^{\Delta}, s \mapsto (\mathrm{ord_p} \alpha(s))_{\alpha \in \Delta}$$ with finite cokernel and kernel equal to $S_0 (Z(k) \cap S(k))$ where $S_0 \subset S(k)$ is the maximal compact subgroup and $Z$ the center of $G$.

This is used in many papers (e.g. Tit's "Reductive groups over local fields") but I could not find a reference for the proof.

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  • $\begingroup$ I'm not sure where in the literature this first appears (can you document your own sources further?). But it may help to start with the case of relative rank 1, where the assertion looks more familiar. Since $S$ is just a product of multiplicative groups, maybe it's easy to generalize to higher rank. In any case, Tits himself had some early announcements and expositions before the more technical Bruhat-Tits papers appeared. I'll take a closer look. $\endgroup$
    – Jim Humphreys
    Commented May 30, 2013 at 22:45
  • $\begingroup$ In Tit's "Reductive groups over local fields", it is implicitly used on page 32 (see this explanation by JK Yu). $\endgroup$
    – Arkandias
    Commented May 31, 2013 at 8:50
  • $\begingroup$ Yes sorry for the mistakes, I wrote in a hurry. It's been corrected. I can find the proofs of the facts you mentioned. How do you deduce the result from these ? $\endgroup$
    – Arkandias
    Commented May 31, 2013 at 12:34
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    $\begingroup$ I found the answer in Landvogt's "A compactification of the Bruhat-Tits building". Thank you for your time ! $\endgroup$
    – Arkandias
    Commented May 31, 2013 at 14:13

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