Dominant weights classify the simple rational representations of a split reductive group

Existence: by [F1] there is a rational representation $W$ of $G$ containing a primitive vector $v$ of weight $\lambda$. Let $W_0\subseteq W$ be the $G$-submodule generated by $v$; by [F2] applied to $W_0$ and $v$, one has $W_0=kv\oplus {⨁}_{\mu <\lambda }(W_0{)}_{\mu }$ with $(W_0{)}_{\lambda }=kv$ one-dimensional, and the quotient $V(\lambda )=W_0\mathrm{/}{W}_0^{\mathrm{\prime }}$ by the largest proper submodule is a simple $G$-module generated by the image of $v$. The image of $v$ is nonzero of weight $\lambda$, and the weight spaces of the quotient are the images of the weight spaces of $W_0$, so $V(\lambda {)}_{\lambda }$ is the line generated by the image of $v$ and every other weight $\mu$ of $V(\lambda )$ satisfies $\mu <\lambda$. By [F5] $V(\lambda )$ is finite-dimensional.

Uniqueness and exhaustiveness: let $V$ be any simple rational representation of $G$. By [F3] $V$ contains a primitive vector $v$, unique up to scalar, whose weight ${\lambda }_V$ is dominant and is the highest weight of $V$; the weight ${\lambda }_V$ is therefore uniquely determined by $V$. If $V_1,V_2$ are simple with ${\lambda }_{V_1}={\lambda }_{V_2}$, then $V_1\cong V_2$ by [F4]. Hence the map $V\mapsto {\lambda }_V$ induces a bijection from the set of isomorphism classes of simple rational representations of $G$ to the set of dominant characters realized as highest weights.

Combining steps 1.1 and 1.2: for every $\lambda \in X(T{)}^{+}$ the module $V(\lambda )$ of step 1.1 is simple with $V(\lambda )=V(\lambda {)}_{\lambda }\oplus {⨁}_{\mu <\lambda }V(\lambda {)}_{\mu }$ and ${\dim }_{k}V(\lambda {)}_{\lambda }=1$, and any simple rational representation is isomorphic to $V(\lambda )$ for the unique $\lambda \in X(T{)}^{+}$ given by its highest weight. No step used a hypothesis on the characteristic of $k$, so the classification holds in every characteristic.