$\Gamma$-action on maximal tori in Borel-Tits

Question

This is about section 6.2 in Borel-Tits' Groupes réductifs where they define a certain $\Gamma$-action on maximal split tori, denoted as $_\Delta \gamma$, distinct from the "usual" one. (If I am not completely wrong, the new action is the one visible in Satake-Tits diagrams, i.e. the *-action in a Tits index (Tits' classification article in the Boulder proceedings), and the $[\; \cdot \; ]$-action in Satake's Classification theory of semi-simple algebraic groups).)

They say that one has the "evident relation" (equation (3))

$\gamma (_K P_\theta) = \, _K P_{\theta'}$ where $\theta' = \, _\Delta \gamma(\theta)$.

Question: How to see this?

Namely, the $_K P_\theta$ are standard parabolic subgroups (defined in 5.12), and I do not see that the $\Gamma$-action stabilises these. With the notation of 5.23, I would rather have expected the weaker $\gamma (_K P_\theta) \in \, _K \mathscr{P}_{\theta'}$, i.e. $\gamma (_K P_\theta)$ is a parabolic subgroup conjugate to $_K P_{\theta'}$.

Answer 1

EDIT: I didn't comment at first on your actual question, since I wasn't familiar enough with that passage in Borel-Tits. Their (3) strikes me as wrongly stated. Moreover, it doesn't seem to come up later on. Maybe they intended to refer to the parabolics defined over $k$ including the minimal $k$-parabolic, but I'm not sure what this passage is really about. Instead, my remarks below deal with some other aspects of the Galois action.

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Maybe it's useful to deal a little more fully with this than I did in my two deleted comments. In the foundational paper by Borel and Tits, emphasis is placed on the structure theory of a reductive $k$-group over an arbitrary field $k$, whereas in his Boulder notes Tits emphasizes more the classification. There are differences in notation as well as emphasis in these sources (and others). So the literature is tricky to compare, though the explicit examples in these sources and in textbooks on algebraic groups are a useful guide.

Here $G$ is split over some separable extension $K$ of $k$, and $\Gamma$ is the associated Galois group. In the Borel-Tits structure theory one tends to work with a fixed $K$-split maximal torus $T$ defined over $k$ and a maximal $k$-split subtorus $S$, along with absolute and relative root systems in which there are compatibly chosen simple systems. (To get some uniformity in the classification, one assumes $\dim S > 0$. Otherwise $G$ is $k$-anisotropic.)

One also needs a minimal $k$-parabolic subgroup containing $T$ (not usually a Borel subgroup). For these groups and the tori there is a conjugacy theorem. In this framework, the standard $k$-parabolic subgroups are in natural bijection with subsets of the relative simple roots.

Now $\Gamma$ acts on these data, where the usual action is modified by combining with the conjugacy theorems to ensure that the tori and root systems are kept stable by the action. In particular, $\Gamma$ now permutes the simple roots which have nontrivial restriction to $S$. The $\Gamma$-orbits (distinguished in the diagrams of Tits and Satake) may consist of multiple roots under a graph automorphism of the root system (if it has components of type $A,D,E_6$), or may consist of single roots.

Note that the outline of the classification given by Tits in his Boulder lectures is not completely precise (the main theorem needs to be more carefully stated and proved). His student M. Selbach in Bonn worked out more complete details in his 1976 thesis, published in Bonner Math. Schriften 83.