Let $\mathbb{G}$ be a connected reductive $\mathbb{F}\_q$ algebraic group over its algebraic closure $\bar{\mathbb{F}\_q}$, and $\mathbb{T}$ be an $\mathbb{F}\_q$-defined maximal torus. Let $\Phi$ be the root system of $\mathbb{G}$ wrt $\mathbb{T}$, and, given $g\in\mathbb{T}$, put $$\Phi(g)=\lbrace\alpha\in \Phi:\alpha(g)=1\rbrace.$$


There is a very clean criterion by Deriziotis for which closed subsystems $\Sigma\le \Phi$ can occur as $\Phi(g)$ for some $g\in \mathbb{T}$; namely, these are precisely the subsystems $\Sigma\le \Phi$ which admit a basis which is a subset of the set of affine simple roots of $\Phi$, and all such subsystems occur for some $g\in \mathbb{T}$. These are often called *pseudo-Levi* subsystems. Furthermore, if $g\in\mathbb{T}(\mathbb{F}\_q)$, then $\Sigma$ is stable under the action of the Frobenius map associated with the $\mathbb{F}\_q$-structure on $\mathbb{G}$.



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> **Question** Given a subsystem $\Sigma\le \Phi$ as in the last paragraph, stable under the Frobenius map, does there exist $g\in \mathbb{T}(\mathbb{F}\_q)$ for which $\Sigma=\Phi(g)$? Do there exist counterexamples for this?
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**What I know by now:** If $\Sigma=\Phi(g)$ is a Levi subsystem, meaning that it has a basis of simple elements of $\Phi$ then one can always take $g$ to be $\mathbb{F}\_q$-rational. To show this, one can compute the dimension of the subgroups $$\mathbb{T}\_{\Sigma'}=\bigcap\_{\alpha\in \Sigma'}\ker(\alpha)\le \mathbb{T}$$
for all $\Sigma\le \Sigma'\le \Phi$, and verify that
$$\dim\mathbb{T}\_{\Sigma'}\le \mathrm{rk}(\Phi)-\mathrm{rk}(\Sigma),$$
with equality iff $\Sigma'=\Sigma$, and, consequently, deduce that $\mathbb{T}\_{\Sigma}^\circ\setminus(\bigcup\_{\Sigma<\Sigma'}\mathbb{T}\_{\Sigma'})$ is irreducible of dimension $\mathrm{rk}(\Phi)-\mathrm{rk}(\Sigma)$, and therefore admits an $\mathbb{F}\_q$-rational point.


In the more general case, where $\Sigma$ is merely pseudo-Levi, this argument fails more-or-less completely. However, in all cases I have computed thus far it seems that one can find elements $g\in\mathbb{T}$ with $\Phi(g)=\Sigma$ whose representing matrices only have the entries $0,1$ and $-1$... I wonder if maybe there is a simpler argument that my dimension computation above overshoots.



Centralizers of $\mathbb{F}_q$-rational semisimple elements of a finite group of Lie type
As @LSpice already pointed out, you need $q$ to be sufficiently large even in the case of a Levi subgroup. Just take $G = \operatorname{GL}\_n(\overline{\mathbb{F}}\_q)$ and $G^F = \operatorname{GL}\_n(\mathbb{F}\_q)$ under the usual Frobenius endomorphism. If $T \leqslant G$ is the maximal torus of diagonal matrices then $(\mathsf{C}\_{q-1})^n \cong T^F = C\_G(s)$ for some semisimple element $s \in G^F$ if and only if $q-1 \geqslant n$. That is because $s$ needs $n$ distinct eigenvalues to be regular.


Deciding the exact conditions for your subgroup to be realisable as the centraliser of a rational semisimple element involves a detailed case by case analysis. For the exceptional groups Frank Lübeck's GAP calculations [Centralizers and numbers of semisimple classes in exceptional groups of Lie type](http://www.math.rwth-aachen.de/%7EFrank.Luebeck/chev/CentSSClasses/index.html) are invaluable here. But if you're just interested in a "$q$ sufficiently large" statement then this was obtained by Deriziotis's advisor R. W. Carter in Corollary 20 of the following paper:


* "[Centralizers of semisimple elements in finite groups of Lie type](https://doi.org/10.1112/plms/s3-37.3.491)", Proc. London Math. Soc. (3), vol. 37, (1978), 491–507.


It is essentially a counting argument. It's pointed out explicitly in Theorem 21 that a Levi subgroup is always the centraliser of a rational semisimple element assuming $q$ is sufficiently large.


Another paper that streamlines things here is the following paper of Bonnafé:


* "[Quasi-isolated elements in reductive groups](https://doi.org/10.1081/AGB-200063602)", Comm. Algebra (7), vol. 33, (2005), 2315–2337.


This paper makes parts of Carter's and Deriziotis' work much clearer. In particular, Bonnafé gives a clean construction for elements whose centraliser is not contained in any proper parabolic subgroup.


