Smooth Models Associated to Concave Functions in Bruhat-tits Theory

Let O be a henselian discrete valuation ring with perfect residue field, G a connected reductive group over k, the quotient field of O. Let S be a maximal k-split torus of G, = (G, S) the corresponding root system. For simplicity, in this introduction we assume that (G, S) is reduced. Fix a point x on the apartment A(G, S) attached to S. According to Bruhat-Tits theory, x determines a filtration {Ua(k)x,r}r∈ for each root subgroup Ua of G. Let f : ∪ {0} → R be a concave function (see 8.2 for the definition) with f (0) = 0, and let G(k)x, f be the subgroup generated by Ua(k)x, f (a) and the maximal bounded subgroup of S(k). Bruhat-Tits proved the following fundamental result: