Galois-fixed Points in the Bruhat-tits Building of a Reductive Group

— We give a new proof of a useful result of Guy Rousseau on Galois-fixed points in the Bruhat-Tits building of a reductive group. Résumé (Points fixes de Galois dans l’immeuble de Bruhat-Tits d’un groupe réductif ) Nous donnons une nouvelle preuve d’un résultat utile de Guy Rousseau sur les points fixes de Galois dans l’immeuble de Bruhat-Tits d’un groupe réductif. Let k be a field with a nontrivial discrete valuation. We assume that k is complete and its residue field is perfect. Let p (≥ 0) be the characteristic of the residue field. Let G be an absolutely almost simple simply connected algebraic group defined over k. The Bruhat-Tits building B(G/ ) of G/ exists for any algebraic extension of k and it is functorial in (see [2, § 5] or [4]). If is a Galois extension of k, there is a natural action, by simplicial isometries, of the Galois group Gal( /k) on the building B(G/ ) (see [2, 4.2.12], or [4, Chap. II]). The convex subset consisting of points of B(G/ ) fixed under Gal( /k) will be denoted by B(G/ ) ; B(G/ ) /k) contains B(G/k). It is known (and, in fact, this result is an important component of the Bruhat-Tits theory) that if is an unramified extension of k, then B(G/ ) /k) coincides with B(G/k), see [2, 5.1.25]. However, in general, the former is larger than B(G/k) (see [8, 2.6.1]). Guy Rousseau in his unpublished thesis [4] proved that if is a Texte reçu le 4 janvier 2000, révisé le 8 mars 2000 Gopal Prasad, Department of Mathematics, University of Michigan, Ann Arbor MI 481091109 (USA) • E-mail : [email protected]