PETERZIL–STEINHORN SUBGROUPS AND -STABILIZERS IN ACF

We consider G , a linear algebraic group defined over $\Bbbk $ , an algebraically closed field (ACF). By considering $\Bbbk $ as an embedded residue field of an algebraically closed valued field K , we can associate to it a compact G -space $S^\mu _G(\Bbbk )$ consisting of $\mu $ -types on G . We show that for each $p_\mu \in S^\mu _G(\Bbbk )$ , $\mathrm {Stab}^\mu (p)=\mathrm {Stab}\left (p_\mu \right )$ 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 $\mathrm {Stab}\left (p_\mu \right )$ in terms of the dimension of p .