ON THE ACTION OF THE UNITARY-GROUP ON THE PROJECTIVE FLAME OVER A LOCAL-FIELD

Let G be a unitary group of rank one over a non-archimedean local fiel d K (whose residue field has a characteristic not equal 2). We conside r the action of G on the projective plane. A G(K) equivariant map from the set of points in the projective plane that are semistable for eve ry maximal K split torus in G to the set of convex subsets of the buil ding of G(K) is constructed. This map gives rise to an equivariant map from the set of points that are stable for every maximal K split toru s to the building. Using these maps one describes a G(K) invariant pur e affinoid covering of the set of stable points. The reduction of the affinoid covering is given.