Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with a single non-trivial Jordan block (Q1693062)

Let \(G\) be a simply connected simple linear algebraic group of exceptional type over an algebraically closed field \(F\). The authors look for representations \(\phi\) of \(G\) and unipotent elements \(u\) that act with a matrix in whose Jordan decomposition there is just one Jordan block whose size is not one. The main result is that this happens only if \(G\) is of type \(G_2\), \(u\) is a regular unipotent element and \(\dim \phi\leq7\). Recall that the case of classical groups has been treated in [\textit{I. D. Suprunenko}, J. Math. Sci., New York 199, No. 3, 350--374 (2014; Zbl 1315.20045)]. The main result can be easily transferred to representations of finite groups of Lie type in defining characteristic. Its proof is long and indirect, with many cases. One starts with studying representations of \(\mathrm{SL}_2\). Next, one studies weight multiplicities of \(\phi\) under the extra assumption \(u^p=1\). After an induction on the rank of \(G\), one ends up studying \(G_2\).