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

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