Field of Definition for Subgroups of F-Groups
Let \(G\) be a linear algebraic group defined over a field \(F\), with separable closure \(F_s\).
- The identity component \(G^\circ\) is an \(F\)-subgroup.
- If \(G\) acts on an \(F\)-variety \(X\) and \(x\in X(F)\), then the orbit \(Gx\) is defined over \(F\). The stabilizer \(G_x\) is defined over \(F\) if \(F\) is perfect, or if the tangent map \(T_eG\to T_x(Gx)\) is surjective.
- If \(\varphi:G\to G'\) is a homomorphism of \(F\)-groups, then \(\ker(\varphi)\) is defined over \(F\) under the same perfectness or tangent-surjectivity hypotheses.
- If \(G\) is connected and \(x,y\in G(F)\) are semisimple, then every nonempty transporter \[ N_G(x,y)=\{g\in G:gxg^{-1}=y\} \] is defined over \(F\). In particular, centralizers of semisimple \(F\)-points are \(F\)-subgroups.
- If \(H,K\subseteq G\) are \(F\)-subgroups, then \(H\cap K\) is defined over \(F\) if \(F\) is perfect, or if \[ \operatorname{Lie}(H\cap K)=\operatorname{Lie}(H)\cap\operatorname{Lie}(K). \]
Proof
The identity component is stable under \(\operatorname{Gal}(F_s/F)\), because it is the unique component containing the \(F\)-rational identity element. Galois descent for closed subvarieties then shows it is defined over \(F\).
For actions, the orbit morphism \(G\to X\) is defined over \(F\). Its image closure is therefore Galois-stable, and the orbit is open in its closure. Descent gives the field of definition of the orbit. The stabilizer is the fiber over \(x\); perfectness, or equivalently the stated separability condition for the orbit morphism, ensures that this fiber descends as an \(F\)-subgroup.
Kernels are stabilizers for the action of \(G\) on \(G'\) by \(\varphi(g)x\). Transporters and centralizers of semisimple elements are fibers of separable conjugacy orbit maps. Intersections descend when the scheme-theoretic intersection has the expected tangent space; over a perfect field this condition is automatic for the subgroup situations above.