Peterzil–Steinhorn subgroups and μ-stabilizers in ACF

We consider G, a linear group defined over k, an algebraically closed field. 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^\mu_G(k), consisting of μ-types on G. We showed that for each p_μ∈S^\mu_G(k), \textStab^μ(p)=\textStab(p_μ) is a solvable infinite algebraic group when p_μis centered at infinity and residually algebraic. Moreover we give a description of the dimension \textStab(p_μ) in terms of dimension of p.