A note on $\mu$-stabilizers in ACVF

We study $\mu$-stabilizers for groups definable in ACVF in the valued field sort. We prove that ${\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}}^{\mu }(p)$ is an infinite unbounded definable subgroup of $G$ when $p$ is standard and unbounded. In the particular case when $G$ is linear algebraic, we show that ${\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}}^{\mu }(p)$ is a solvable algebraic subgroup of $G$, with $\mathrm{d}\mathrm{i}\mathrm{m}({\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}}^{\mu }(p))=\mathrm{d}\mathrm{i}\mathrm{m}(p)$ when $p$ is $\mu$-reduced and unbounded.