Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with single non-trivial Jordan block

In this paper we prove the following result. Let $G$ be a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed field $F$ of characteristic $p\geq 0$, and let $u\in G$ be a nonidentity unipotent element. Let $\phi$ be a non-trivial irreducible representation of $G$. Then the Jordan normal form of $\phi(u)$ contains at most one non-trivial block if and only if $G$ is of type $G_2$, $u$ is a regular unipotent element and $\dim \phi\leq 7$. Note that the irreducible representations of the simple classical algebraic groups in which a non-trivial unipotent element is represented by a matrix whose Jordan form has a single non-trivial block were determined by I.D. Suprunenko (Unipotent elements of non-prime order in representations of the classical algebraic groups: two big Jordan blocks, J. Math. Sci. 199(2014), 350 -- 374.