Closures of locally divergent orbits of maximal tori and values of homogeneous forms

Let G be a semisimple algebraic group over a number field K, S a finite set of places of K, K-S the direct product of the completions K-v, v is an element of S, and O the ring of S-integers of K. Let G = G(K-S), Gamma = G(O) and pi : G -> G/ Gamma the quotient map. We describe the closures of the locally diverergent orbits T pi(g) where T is a maximal K-S-split torus in G. If #S = 2 then the closure <(T pi(g))over bar> is a finite union of T-orbits stratified in terms of parabolic subgroups of G x G and, consequently, <(T pi(g))over bar> is homogeneous (i.e. <(T pi(g))over bar> = H pi (g) for a subgroup H of G) if and only if <(T pi(g))over bar> is closed. On the other hand, if #S > 2 and K is not a CM-field then T TC(g) is homogeneous for G = SLn and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple K-ranks for G not equal SLn. As an application, we prove that <(f (O-n))over bar> = K-S for the class of non-rational locally K-decomposable homogeneous forms f is an element of K-S[x(1), ..., x(n)].