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Let $S$ be a maximal split torus of a connected, reductive group $G$. Let $P_0$ be a minimal $k$-parabolic containing $S$, $T$ a maximal torus of $P_0$ which is defined over $k$ and contains $S$, and $B$ a Borel subgroup contained in $P_0$ and containing $T$.

The choice of $P_0$ and $B$ determine simple roots $_k\Delta$ and $\Delta$ for $_k\Phi = \Phi(G,S)$ and $\Phi = \Phi(G,T)$.

For each $a \in \space _k\Delta$, the set of $\alpha \in \Delta$ which restrict to $a$ form an orbit under the $\ast$-action of $\operatorname{Gal}(k_s/k)$. If $G$ is quasisplit, then the $\ast$-action is just the usual Galois action on characters. This is explained in section 12 of Brian Conrad's notes on reductive groups over fields.

What if we take an arbitrary $a \in \space _k\Phi$? Do the set of roots in $\Phi$ which restrict to $a$ also form a Galois orbit?

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To answer your question, it's enough to point out that every root (in a reduced root system such as $\Phi$) plays the role of simple root in some basis $\Delta$. Note however that some absolute roots may be trivial (i.e., "zero") on restriction to $S$ and thus not ropts in the (possibly non-reduced) root system $_k\Phi$.

On the other hand, limiting the discussion of examples to the quasisplit case is artificial though convenient for exposition as in Conrad's notes. It's most helpful to go back to the more detailed treatment in the 1965 paper by Borel and Tits here: see especially $\S3-\S5$, followed by discussion in $\S6$ of the action by the Galois group $\Gamma$ of the separable closure over $k$ in an algebraic closure of an arbitrary field $k$. (The extreme case of an anisotropic $G$ over $k$ is left aside, since the classification of possibilities depends very much on $k$. The Borel-Tits structure theory for a connected reductive $G$ over a field is only uniform in the $k$-isotropic situation, i.e., when $S$ is nontrivial. And of course the case when $G$ is nontrivial and semisimple is the essential one.)

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    $\begingroup$ What are some good examples to keep in mind when reading these sections? $\endgroup$
    – Not a grad student
    Commented Aug 22, 2018 at 17:02
  • $\begingroup$ @CoffeeBliss: Probably the best examples are found in the classification survey by Tits (in English) in the 1965 Boulder AMS proceedings volume. Though his theoretical formulation needed some tweaking later on (in the work of his Bonn student), his case-by-case study is quite useful. He also indicates what is true for various familiar types of fields. $\endgroup$
    – Jim Humphreys
    Commented Aug 22, 2018 at 17:34
  • $\begingroup$ Thanks. For example, Borel-Tits write in section 6.2 that if K/k is a field extension, and T and T' are maximal K-split tori both invariant under Aut(K/k), then the usual Aut(K/k)-modules X^*(T) and X^*(T') need not be isomorphic. $\endgroup$
    – Not a grad student
    Commented Aug 22, 2018 at 17:58
  • $\begingroup$ @JimHumphreys, who is Tits's Bonn student? $\endgroup$
    – LSpice
    Commented Nov 14, 2018 at 15:43
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    $\begingroup$ @LSpice I think he was referring to Selbach. $\endgroup$
    – Not a grad student
    Commented Nov 15, 2018 at 16:07

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