Every dominant character of a split reductive group is a highest weight

By [F1] form $H=Z(G{)}_t\times G_{\mathrm{d}\mathrm{e}\mathrm{r}}$ with maximal torus $T_H=Z(G{)}_t\times T_{\mathrm{d}\mathrm{e}\mathrm{r}}$ and central isogeny $m:H\to G$ with finite kernel $N$. Its torus restriction maps $T_H$ onto $T$ and induces an injection $m^{\ast }:X(T)\hookrightarrow X(T_H)$ with finite cokernel. Put ${\lambda }_H=m^{\ast }\lambda$, which lies in that image.

We establish the root correspondence, without assuming the total Lie differential is an isomorphism. The central kernel lies in $Z(H)\subseteq T_H$ by [F1] applied to $H$. For a root $\beta$ of $H$, $N\cap U_{\beta }$ is both unipotent and diagonalizable, so it is trivial; hence $m{\mathrm{\mid }}_{U_{\beta }}$ is an isomorphism onto a smooth closed image $I_{\beta }$ by [F6]. Centrality makes $N$ act trivially on the root tangent line, so $\beta {\mathrm{\mid }}_N=1$ and exact character duality gives a unique $\alpha \in X(T)$ with $m^{\ast }\alpha =\beta$. The image is $T$-stable, with nonzero tangent weight $\alpha$, so $\alpha$ is a root of $G$, and the root-group containment criterion gives $U_{\alpha }\subseteq I_{\beta }$. Both are smooth connected curves, hence equal. This gives an injection of root sets; it is a bijection since $\dim H=\dim G$, $\dim T_H=\dim T$, and the root decomposition with one-dimensional root spaces gives $\mathrm{\mid }\mathrm{\Phi }\mathrm{\mid }=\dim G-\dim T$ for either group.

The torus isogeny maps the maximal subtorus of $\ker \beta$ onto that of $\ker \alpha$, so it maps the source rank-one centralizer into the target one. Hence a rank-one reflection representative $n_{\beta }$ maps into that target rank-one subgroup. Its action on $T$ is nontrivial, because torus pullback is injective and intertwines conjugation, so it represents $s_{\alpha }$ by [F6]. Comparing the two reflection formulas and using $m^{\ast }\alpha =\beta$ gives $⟨m^{\ast }x,{\beta }^{\vee }⟩=⟨x,{\alpha }^{\vee }⟩$ for every $x\in X(T)$. The root bijection preserves addition, so the inverse images of ${\mathrm{\Phi }}^{+}(B)$ form a positive system in $H$. Choose its Borel $B_H$ by [F7]; it is a product Borel since the central torus has no roots. By [F6] and [F7], $B_H$ and $B$ are generated by their maximal tori and positive root groups; their identified generators give $m(B_H)=B$. All highest weights below use these compatible pairs.

For every positive root $\beta =m^{\ast }\alpha$ of $H$, step 3.1 gives $⟨{\lambda }_H,{\beta }^{\vee }⟩=⟨\lambda ,{\alpha }^{\vee }⟩\ge 0$. Hence ${\lambda }_H$ is dominant for the product Borel $B_H$, so the hypothesis of [F2] is satisfied.

By [F2] there is a rational representation $W$ of $H$ containing a primitive vector $v$ of weight ${\lambda }_H$. Let $V_H\subseteq W$ be the $H$-submodule generated by $v$; applying [F4] to $V_H$ and $v$, the quotient $Q=V_H\mathrm{/}{V}_H^{\mathrm{\prime }}$ by the largest proper $H$-submodule $V_H^{\mathrm{\prime }}$ is a simple $H$-module, and the image of $v$ is a nonzero primitive vector of weight ${\lambda }_H$ in $Q$, so ${\lambda }_H$ is the highest weight of the simple module $Q$.

By [F3] applied to the central isogeny $m:H\to G$ (with $\pi =m$) and the simple $H$-module $Q$ of highest weight ${\lambda }_H\in X(T)$, the kernel $N$ acts trivially on $Q$ and $Q$ descends along $H\mathrm{/}N=G$ to a simple $G$-module of highest weight ${\lambda }_H$, which is $\lambda$ under the identification $X(T)\subseteq X(T_H)$. This descended module is finite-dimensional by [F5], since $G$ is affine of finite type. Hence every dominant $\lambda$ is the highest weight of a simple finite-dimensional rational representation of $G$.

For the equivalence: a simple finite-dimensional $G$-module of highest weight $\lambda$ contains a primitive vector of weight $\lambda$ by [F5], while conversely a rational representation containing a primitive vector $v$ of weight $\lambda$ has, by [F4] applied to the submodule generated by $v$, a simple quotient of highest weight $\lambda$, finite-dimensional by [F5]. Thus the two formulations are equivalent.