Peterzil-steinhorn subgroups and -stabilizers in acf

We consider G, a linear algebraic group defined over k, an algebraically closed field (ACF). By considering k as an embedded residue field of an algebraically closed valued field K, we can associate to it a compact G-space S-G(mu)(k) consisting of mu-types on G. We show that for each p(mu) is an element of S-G(mu) (k), Stab(mu) (p) = Stab (p(mu)) is a solvable infinite algebraic group when p(mu) is centered at infinity and residually algebraic. Moreover, we give a description of the dimension of Stab (p(mu)) in terms of the dimension of p.