## Galois-fixed points in the Bruhat-Tits building of a reductive group.(English)Zbl 0992.20032

Let $$k$$ be a field with a non-trivial valuation. Assume $$k$$ is complete and its residue field is perfect. Let $$G$$ be an absolutely almost simple, simply connected algebraic group defined over $$k$$. For any algebraic extension $$\ell$$ of $$k$$, the Bruhat-Tits building $${\mathcal B}(G/\ell)$$ of $$G/\ell$$ exists and it is functorial in $$\ell$$. If $$\ell$$ is a Galois extension of $$k$$, then there is a natural action of the Galois group $$\text{Gal}(\ell/k)$$ on the building $${\mathcal B}(G/\ell)$$. Let $${\mathcal B}(G/\ell)^{\text{Gal}(\ell/k)}$$ denote the convex subset consisting of fixed points, which contains $${\mathcal B}(G/k)$$. G. Rousseau in his unpublished thesis proved that if $$\ell$$ is a tamely ramified finite Galois extension of $$k$$, then $${\mathcal B}(G/\ell)^{\text{Gal}(\ell/k)}$$ coincides with $${\mathcal B}(G/k)$$. The author of this paper gives a short and easy proof of this result.
Reviewer: Li Fu-an (Beijing)

### MSC:

 20G15 Linear algebraic groups over arbitrary fields 20E42 Groups with a $$BN$$-pair; buildings
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