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Let $G$ be the base change of a connected reductive group over ${\mathbb{F}}_q$ to $\overline{\mathbb{F}}_q$, and let $F$ be the associated geometric Frobenius on $G$. Let $W$ be the Weyl group of a quasisplit $F$-stable maximal torus of $G$.

By definition, $F$ acts on $W$. We know that, the $G^F$-conjugacy classes of $F$-stable maximal tori in $G$ are one to one corresponding to the $F$-conjugacy classes of $W$, and the quasisplit ones form a single class corresponding to $1\in W$.

Question: Under this picture, is it true that, the $F$-stable maximal tori which are NOT contained in any $F$-stable parabolic subgroup of $G$ form a single class?

I checked standard references like Carter's book and Digne--Michel's book but couldn't find the answer; any reference containing a positive proof or counterexamples would be appreciated!

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No. For example, in (split, simply connected) type $\mathrm C_2$, there are the tori whose rational points are isomorphic to $\ker \mathrm N_{\mathbb F_{q^4}/\mathbb F_{q^2}}$, and those whose rational points are isomorphic to the product of $\ker \mathrm N_{\mathbb F_{q^2}/\mathbb F_q}$ with itself.

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  • $\begingroup$ Thank you. Could you provide a reference discussing these examples? $\endgroup$
    – user148212
    Commented May 21, 2017 at 14:07
  • $\begingroup$ @user148212, I'm not sure what you mean. I hope it's obvious that these tori are elliptic but not conjugate. What would you want the reference to discuss? $\endgroup$
    – LSpice
    Commented May 21, 2017 at 14:13
  • $\begingroup$ I would like a reference discussing the possible $F$-stable maximal tori which are not contained in any $F$-stable proper parabolic subgroup. (A bit shame to say, but I don't familiar with the term ``elliptic torus'', does an $F$-stable elliptic torus never contained in $F$-stable proper parabolic?) $\endgroup$
    – user148212
    Commented May 21, 2017 at 14:25
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    $\begingroup$ @user148212, hmm, I don't know a reference off the top of my head. I looked in Carter, but didn't see it there. "Elliptic" is just another word for the kind of torus you want; it is often equivalently defined as not containing any non-central, split torus. In terms of the Weyl-group parameterisation, it means that the element there corresponding to the torus lies in no proper parabolic subgroup (of the Weyl group). $\endgroup$
    – LSpice
    Commented May 22, 2017 at 2:46
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    $\begingroup$ @user148212, the two tori that I've mentioned correspond, respectively, to the Coxeter element of $\mathrm C_2$, and to the Coxeter element of the $\mathrm A_1 + \mathrm A_1$ sub-system. $\endgroup$
    – LSpice
    Commented May 22, 2017 at 2:51

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