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Let $G$ be a reductive group over a local non-arch field $F$. For convenience let's assume $G$ has anisotropic center (or even that $G$ is semisimple if preferred). Let $x$ be any point in the building $\mathcal{B}(G,F)$, and $G_x$ its stabilizer in $G(F)$. It also contains the parahoric, which I denote by $G_{x,0}$, attached to (the facet that contains) $x$.

For the purpose of some computation of orbital integrals, I'm interested in this quotient group $G_x/G_{x,0}$. When $x$ is a vertex or a general point of a facet, 3.5 of Tits' Corvallis article "Reductive groups over local fields" provide an algorithm: Let $G'$ be the centralizer of a maximal split torus. Then $G'(F)$ maps to the automorphism group of the affine Dynkin diagram. The kernel of this map can be computed - let's just call it $\text{ker}$. Tits explains how $G_x/G_{x,0}$ appears as a subgroup of $G'(F)/\text{ker}$ (more precisely, he explains this in the group scheme language, and the result I mention can be deduced).

I'm wondering if exactly the same result is true (and moreover, does it always come from an explicit isomorphism from a subgroup of $G'(F)/\text{ker}$ to $G_x/G_{x,0}$) when $x$ is any point or even when $x$ is replaced by any bounded subset of an apartment. Thank you!

p.s. I fail to find such a statement in the original paper of Bruhat-Tits. But I'm very unfamiliar with that paper - hope I'm not being stupid.

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  • $\begingroup$ Minor edits to clarify the language. Note that Bruhat-Tits wrote a number of papers with different amounts of detail. $\endgroup$ Apr 27, 2014 at 13:27
  • $\begingroup$ Thanks. I didn't realize about some other papers. I will go try. $\endgroup$ Apr 27, 2014 at 19:43
  • $\begingroup$ It's in a very different spirit, but is it helpful to know that the stabiliser in $N$ (the normaliser of a maximally split torus) of $x$ surjects onto the quotient in which you are interested? This is Proposition 4.6.28(ii) of BT2. $\endgroup$
    – LSpice
    Jan 12, 2015 at 21:43
  • $\begingroup$ Thanks Loren! This should help, but I forget precisely why I wanted this question. It must be for computing orbital integral using Cartan decomposition. But then I decided that for theoretical purpose it's no harm to pass to $G^{sc}$. $\endgroup$ Jan 13, 2015 at 15:47
  • $\begingroup$ It may be helpful to use generalized tits system. In short for Chevalley groups, G_{x,0} shall be generated by one Iwahori and simple reflections fixing x, while G_x additionally contains elements in fundamental groups fixing x. $\endgroup$
    – YUAN Zhiri
    Nov 9, 2018 at 13:14

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