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 G. Rousseau in his unpublished thesis proved that if \(\ell\) is a tamely ramified finite Galois extension of \(k\), then \({\mathcal B}(G/\ell)^{\text{Gal}(\ell/k)}\) coincides with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)


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