Weights of infinitesimally irreducible representations of Chevalley groupsover a field of prime characteristic

Let $K$ be an algebraically closed field of characteristic $p>0$, $G$ a universal Chevalley group over $K$ with an irreducible root system $R$, $B$ a basis of $R$, $Q_+$ the set of radical weights that are nonnegative with respect to the natural ordering associated with $B$, $P_{++}$ the set of dominant weights, and $e(R)$ the maximum of the squares of the ratios of the lengths of the roots in $R$. It is well known that $e(R)=1$ if $R$ is of type $A_n$, $D_n$, $E_6$, $E_7$, or $E_8$, $e(R)=2$ if $R$ is of type $B_n$, $C_n$, or $F_4$, and $e(R)=3$ if $R$ is of type $G_2$. A rational representation $\pi\colon G\to\mathrm{GL}(V)$ is called infinitesimally irreducible if its differential $d\pi$ defines an irreducible representation of the Lie algebra $\mathfrak g$ of the group $G$. Let $\mathfrak g_{\mathbf C}$ be a simple complex Lie algebra with the same root system as $G$.
In this paper it is proved that for $p>e(R)$ the system of weights of an infinitesimally irreducible representation $\pi$ of a group $G$ with highest weight $\lambda$ coincides with the system of weights of an irreducible complex representation $\pi_{\mathbf C}$ of a Lie algebra $\mathfrak g_{\mathbf C}$ with the same highest weight. In particular, the set of dominant weights of the representation is $(\lambda-Q_+)\cap P_{++}$.
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