Let, $G(k^{al})$ be an algebraic group, over an algebraically closed field, and $\Gamma\_{G}$ is the set of all closed subgroups of $G(k^{al})$.


Then is the map $Z\_{G}: \Gamma\_{G} \rightarrow \Gamma\_{G}$ which takes a closed subgroup to its centralizer in $G$, an involution? (probably not true)


If we now assume that $G(k^{al})$ is reductive or semisimple is there a characterization of all such closed subgroups for which $Z\_{G}$ is an involution?


More generally if $G\_{k}$ is an algebraic group scheme (now $k$ is no longer algebraically closed ) and $\Gamma\_{G}$ is the set of closed group subschemes of $G\_{k}$, do the previous two questions have a meaningful answer?



Is a double centralizer type theorem ( encountered in semisimple algebras)  true for algebraic groups ?
When $G=\mathrm{GL}\_n$, then the centraliser $C$ of a subgroup scheme $H$ form the invertible elements of the algebra $M$ of matrices commuting with $H$. The group of such elements is Zariski dense in $M$ so $C$ and $M$ determine each other. Hence, the image of $Z\_{\mathrm{GL}\_n}$ and the question of involutivity on that image is completely reduced to double centraliser results for algebras.


