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I'm struggling to understand the definition of $\mathbb{F}_q$-rank of a torus, while reading Michel's book Representation Theory of Finite Groups of Lie Type. Essentially,

Let $T$ be a torus, $F$ an endomorphism of $T$ corresponding to an $\mathbb{F}_q$-structure. The map $\tau=q^{-1}F$ is a linear endomorphism of $X(T)\otimes\mathbb{R}$ which has finite order. The $\mathbb{F}_q$-rank of a torus $T$ is $\dim(X(T)\otimes\mathbb{R})^\tau$.

Does this mean $T$ is being viewed as an $\overline{\mathbb{F}_q}$-variety, and $T\simeq T(\mathbb{F}_q)\otimes_{\mathbb{F}_q}\overline{\mathbb{F}_q}$ for an $\mathbb{F}_q$-structure $T(\mathbb{F}_q)$, and then $F$ is the endomorphism defined on simple tensors by $x\otimes\lambda\mapsto x^q\otimes\lambda$?

If so, $\tau=q^{-1}F$ seems to be a scalar multiple of $F$, so how is this an endormorphism of $X(T)\otimes \mathbb{R}$? Does is send a character $\varphi\mapsto \varphi\circ q^{-1}F$?

Lastly, what does $\dim(X(T)\otimes\mathbb{R})^\tau$ mean? Is $\tau$ acting on $X(T)\otimes\mathbb{R}$ somehow to give an $\mathbb{R}$-vector space, and we take the dimension of that as an $\mathbb{R}$-vector space to be the $\mathbb{F}_q$-rank of $T$?

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  1. Yes, that sounds about right.

  2. Yes.

  3. The superscript means taking the invariant subspace under the action of $\tau$. In general, if $V$ is a vector space and $G$ is a group acting on it, then $V^G = \{ v \in V \mid gv = v \forall g \in G \}$.

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