Peterzil-Steinhorn subgroups and $μ$-stabilizers in ACF

Abstract: 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 $\mu$-types on $G$. We showed that for each $p_\mu\in S^\mu_G(k)$, $\text{Stab}^\mu(p)=\text{Stab}(p_\mu)$ 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 $p$.