Let $G$ be a connected algebraic group defined over a field $E$ of characteristic $0$. Suppose $G$ reductive $E$-split and let $T \subset G$ a maximal (split) torus defined over $E$.

Set $G' = Res_{E/F} G$ where $E/F$ is a finite field extension, $T' = Res_{E/F} T$ and $S' \subset T'$ a maximal $F$-split torus. Then $T'$ is the centralizer of $S'$ in $G'$ and a maximal torus of $G'$.

Now, what is the relation between the Weyl group $W = N_G(T)/T$ of $(G,T)$ and the relative Weyl group $W' = N_{G'}(S') / T'$ of $(G',S')$ ?


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    $\begingroup$ They're equal. This can be seen in multiple ways. For example, the evident isomorphism $S'_E \rightarrow T$ induced by $G_{E'} \twoheadrightarrow G$ gives an identification ${\rm{X}}_F(S') = {\rm{X}}_F(T)$ under which $\Phi(G',S')$ is carried isomorphically onto $\Phi(G,T)$ (using the equality of $\mathfrak{g}'$ with the underlying $F$-vector space of $\mathfrak{g}$ to match root spaces), and the Weyl group of this common root system is naturally identified with the finite constant groups $W$ and $W'$. This respects the induced map $W'_E \rightarrow W$, so the latter is an isomorphism. $\endgroup$ – user28172 Apr 30 '13 at 15:35
  • $\begingroup$ [I assume when you wrote "maximal (split) torus" you mean that $T$ is a split maximal $E$-torus of $G$ that is also maximal as an $E$-torus.] $\endgroup$ – user28172 Apr 30 '13 at 15:37
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    $\begingroup$ nosr. please put this as answer. $\endgroup$ – Marc Palm Apr 30 '13 at 17:09
  • $\begingroup$ Thank you nosr. If you could post your answer so I can accept it. $\endgroup$ – Arkandias Apr 30 '13 at 22:58

While waiting for nosr to make the comments here into a full answer, I'll add some references to the literature. The early sources are quite technical and don't provide much in the way of user-friendly examples, but they do show the origins of the ideas involved in Weil restriction:

A. Weil, Adeles and Algebraic Groups (Birkhauser, PM 23, 1982): this small monograph contains a verbatim copy of Weil's 1959-60 IAS lectures on Tamagawa numbers, along with some updates by T. Ono and a bibliography. See section 1.3 for the starting formalism of Weil restriction of scalars.

A. Borel and J.-P. Serre, Theoremes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111-164: see section 2.8. [This paper and the following one are available online, the second at numdam.org]

A. Borel and J. Tits, Groupes reductifs, Publ. Math. IHES 27 (1965): see especially section 6.

M. Demazure and P. Gabriel, Groupes algebriques, tome I, Masson and North-Holland, 1970: see I, section 4, 6.4 and 6.6. (Their framework is more scheme-theoretic than the early ones.)

Much more recently, Weil restriction has played a large role in the book Pseudo-reductive Groups (Cambridge, 2010) by B. Conrad, O. Gabber, G. Prasad. In their index, see the many specific topics listed under "Weil restriction". Here I haven't tracked down an explicit comparison of the Weyl groups, but for instance Prop. A.5.15 comes close in its treatment of tori.

My only editorial comment would be that no one over the years seems to have provided an ideal intuitive discussion of what Weil restriction is all about and why it's important. Some explicit examples could be very enlightening, including the three-dimensional simple group $\mathrm{SL}_2$.

ADDED: For a typical application of Weil restriction (in language close to Weil's) which brings out implicitly the connection between the maximal split tori (and Weyl groups) in the groups over the two fields involved, see 3.1.2 in the article by J. Tits, "Classification of algebraic semisimple groups", Proc. Symp. Pure Math. 9, Amer. Math. Soc., 1966 (proceedings of 1965 Boulder summer institute). As in other sources, the emphasis is on comparing tori root systems and rather than directly on the Weyl groups.


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