Let $\mathbf{G}$ be a connected reductive group over $\mathbb{F}\_q$, let $G=\mathbf{G}(\mathbb{F}\_q)$, and let $k$ be a large enough finite extension of $\mathbb{F}\_q$. 


If $\mathbf{G}=\mathrm{SL}\_2$, then in page-109 of Bonnafe's [book](https://www.springer.com/us/book/9780857291561) it is said that "It turns out that the simple $kG$-modules are the restrictions of simple "rational representations" of the algebraic group $\mathbf{G}\_{\overline{k}}$."


Question 1: Where can I find a proof of this statement?


Question 2: Is this true for a general $\mathbf{G}$?


Any help would be appreciated.



The defining characteristic representations of Lie type groups
This result is false for a general connected reductive algebraic group $\mathbf{G}$ but is true if $\mathbf{G}$ is simple and simply connected. This was proved by Steinberg in Theorem 1.3 of the following paper


* R. Steinberg, *[Representations of algebraic groups](https://projecteuclid.org/euclid.nmj/1118801156)*, Nagoya Math. J. **22** (1963), 33--56.


It's also summarised in Theorem 13.1 of Steinberg's monograph "Endomorphisms of linear algebraic groups".


A parameterisation of the simple modules for general $\mathbf{G}$ was recently obtained by Brunat and Lübeck:


* O. Brunat and F. Lübeck, *[On defining characteristic representations of finite reductive groups](https://doi.org/10.1016/j.jalgebra.2013.07.022)*, J. Algebra **395** (2013), 121--141.


A preprint version of this article is also available [here](https://arxiv.org/abs/1211.3692).


