4. Cartan subgroups

All unramified reductive groups are classified by their root data. This includes the classification of unramified tori $T$ as a special case (in this case, the set of roots and the set of coroots are empty):

$$(X^*(T),X_*(T),\emptyset ,\emptyset ,\sigma ).$$

We can extend this classification to ramified tori. If $T$ is any torus over $F$, it is classified by

$$(X^*(T),X_*(T),\rho ),$$

where $\rho$ is now allowed to be any homomorphism

$$\rho :\operatorname {Gal}(\bar F/F)\to \operatorname {Aut}(X^*(T))$$

with finite image.

A basic fact is that $T$ embeds over $F$ as a Cartan subgroup in a given unramified reductive group $G$ if and only if the following two conditions hold.

• The image of $\rho$ in $\operatorname {Aut}(X^*(T))$ is contained in $W(\Phi _G)\rtimes \langle \sigma _G\rangle$.

• There is a commutative diagram:

$$\begin{equation*} \begin{CD} \operatorname {Gal}(\bar F/F)@>>>\operatorname {Gal}(F^{un}/F)\\@V\rho VV @VV{\operatorname {Frob}\mapsto \sigma _G}V\\W(\Phi _G)\rtimes \langle \sigma _G\rangle @>>w\rtimes \tau \mapsto \tau > \langle \sigma _G\rangle . \end{CD} \end{equation*}$$

It follows that every Cartan subgroup $T_H$ of $H$ is isomorphic over $F$ with a Cartan subgroup $T_G$ of $G$. (To check this, simply observe that these two conditions are more restrictive for $H$ than the corresponding conditions for $G$.) The isomorphism can be chosen to induce an isomorphism of Galois modules between the character group (and cocharacter group) of $T_H$ and that of $T_G$.

We say that a semisimple element in a reductive group is strongly regular, if its centralizer is a Cartan subgroup. If $\gamma \in H(F)$ is strongly regular semisimple, then its centralizer $T_H$ is isomorphic to some $T_G\subset G$. Let $\gamma _0\in T_G(F)\subset G(F)$ be the element in $G(F)$ corresponding to $\gamma \in T_H(F)\subset H(F)$, under this isomorphism.