Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits

We give an elementary proof of a theorem of Bruhat, Tits and Rousseau, and also of a theorem of Tits. Let k be a field with a non-trivial non-archimedean valuation v. We shall assume that the valuation v has a (up to equivalence) unique extension to any finite field extension of k, or, equivalently, k is henselian for v (i. e. the HenseFs lemma holds in k with respect to v). We fix an algebraic closure Jf of k and shall denote the unique valuation on it, which extends the given valuation on k, again by v. Let K be the separable closure of k in Jf; the extended valuation on K is obviously invariant under the Galois group Ga\(K/k). Let V be a finite dimensional ^-vector space. Let G be a connected reductive /^-subgroup of SL (V}. For any extension L of k contained in JT, let G{L) be the group ofL-rational points of G endowed with the Hausdorff topology and the bomology induced by the valuation on L. Let G {k}^ be the normal subgroup o!G(k) generated by the ^-rational points of the unipotent radicals of parabolic ^-subgroups of G. G is said to be isotropic over Aif G contains a non-trivial k-sp\it torus, and k-anisotropic (or anisotropic over k) otherwise. The object of this note is to give a simple proof of the following theorem proved first by F. Bruhat and J. Tits in case A: is a discretely valuated complete field with perfect residue field and then in general by G. Rousseau in his thesis (Orsay, 1977). (*) Textere<?ule l^juin 1981. G. PRASAD, Tata Institute of Fundamental Research, Colaba, Bombay 5, India. Partially supported by the Sonderfbrschungsbereich fur Theoretische Mathematik at the University of Bonn. BULLETIN DE LA SOCIETE MATHEMAT1QUE DE FRANCE 003 7-9484/19 82/197/S 5.00 © Gauthier-Villars