Peterzil-Steinhorn subgroups and $\mu$-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 $S_G^{\mu }(k)$, consisting of $\mu$-types on $G$. We showed that for each $p_{\mu }\in S_G^{\mu }(k)$, ${\text{Stab}}^{\mu }(p)=\text{Stab}(p_{\mu })$ is a solvable infinite algebraic group when $p_{\mu }$ is centered at infinity and residually algebraic. Moreover we give a description of the dimension $\text{Stab}(p_{\mu })$ in terms of dimension of $p$.