J. reine angew. Math. 433 (1992), 69—100 Journal f r die reine und angewandte Mathematik © Walter de Gruyter Berlin · New York 1992 Symmetrie spaces associated to split algebraic groups over a local field By M. van der Put and H. Voskuil at Groningen Introduction Let k be a non-archimedean local field, i.e. a finite extension of Q p or F p ((t)). Let k° be its valuation ring, ð an uniformizing element and K the residue field of k. We consider a simple linear algebraic group G over fc°, which is simply connected and split over k°. For a parabolic subgroup P c G defined over k° we consider the homo- geneous space X = G/P over k°. The purpose of this paper is to construct an open (rigid) analytic subspace Õ of X® k with the following properties: 1. Õ is invariant under the action of G (k). 2. The complement of Õ in X® k is the union of a compact family of Zariski-closed subsets. 3. For every discrete, co-compact subgroup à of G (k) the quotient Õ/ à exists and is a separated and proper analytic space over k. Such a space Õwill be called a Symmetrie space over k. The space Õ is first defined s a subset of points of X(K) where K denotes the completion of the algebraic closure of k. Then an admissible covering of Yby affinoid subspaces of X® k is constructed. In 3.7 the connection with formal schemes over k° is explained. Independently from the analytic construction above a formal scheme 3E over k° with 'generic fibre' 7is constructed. Here we follow closely [Ku]. The formal scheme £ has the properties: 1. 3E is flat and locally of finite type over k°. There is an action of G (k) on £. Brought to you by | University of Iowa Libraries Authenticated Download Date | 6/9/15 7:28 AM