# $G^F$ conjugacy class of $F$-stable maximal tori, in an algebraic group $G$ defined over $\mathbb{F}_{q}$

Let $$G$$ be an affine algebraic group over $$k=\bar{\mathbb{F}_{p}}$$. Let $$q$$ be a power of $$p$$, and assume that $$G$$ is defined over $$\mathbb{F}_q$$. Let $$\mathcal{T}$$, be the collection of all maximal tori of $$G$$. Let $$F$$ be the Frobenius map from $$G$$ onto $$G$$.

Clearly $$G$$ acts on $$\mathcal{T}$$, by conjugation and this action is transitive. Further if we assume that $$G$$ is connected, by standard theory in "Finite groups of Lie type", $$G^{F}$$, acts on $$\mathcal{T}^{F}$$ by conjugation,where $$G^{F}$$ denote the set of $$F$$-rational points of $$G$$, which is the finite algebraic group associated to $$G$$, and $$\mathcal{T}^{F}$$, denote the collection of $$F-$$ stable maximal tori of $$G$$,that is,

$$\mathcal{T}^{F}=\{ T\in \mathcal{T} | F(T)=T \}.$$

We know, $$G^F$$ need not act transitively on $$\mathcal{T}^F$$. For each $$T\in \mathcal{T}^F$$, Let $$T^F$$, denote the $$F-$$ rational points in $$T$$. Now, let $$\mathcal{T}^{[F]}= \{ T^F | T\in \mathcal{T}^{F} \}.$$

It is clear that $$G^F$$ acts on $$\mathcal{T}^{[F]}$$, by conjugation. Also, if $$T_1, T_2 \in \mathcal{T}^F$$ are in same orbit under $$G^F$$ action then $$T_1^{F},T_2^{F}$$ are also in same orbit under $$G^F$$ action.

Is it true other way around, that is, if $$T_1^{F}, T_2^{F}$$, are $$G^F$$-conjugate, then $$T_1,T_2$$ are $$G^F$$ conjugate. It seems to be that it is true. But, I couldn't come up with any proof.

I would appreciate any kind of help. Thank you!

• Have you looked at the long article by Springer and Steinberg in Lecture Notes in Mathematics 131 (or at the related work of Steinberg)? May 28, 2019 at 20:59
• Yeah I know the lecture notes you are talking about. I will have a look at it!
– Riju
May 28, 2019 at 21:40
• I got an answer to my question in the book “ Finite Groups of Lie type” by R. Carter. The answer is in Chapter 3, Pg 96-97. Maybe I’ll write a short answer to my question.
– Riju
May 30, 2019 at 15:40
• As you realize, the discussion in Chapter 3 of Carter's sprawling book answers your question affirmatively (and more). But most of this information comes from the long paper of Borel-Tits (1965) on reductive groups over am aritrary field, and the article by Springer-Steinberg in Lecture Notes 131 treats carefully the case of a finite field. where the group $G$ is either split or quasi-split. May 30, 2019 at 23:22
• Sorry if my comments have been too superficial; but I wanted to emphasize that the history is compllicated, while your insistence on regular semisimple elements in the other question seems misleading and undermotivated (so I've been inclined to ignore that condition). Most of the theory comes originally from Borel-Tits and then Springer and Steinberg papers, whereas Carter's book is a serious attempt to write a thorough summary. Jun 1, 2019 at 21:44