Matrices of simple spectrum in irreducible representations of cyclic extensions of simple algebraic groups (Q6077850)

Let \(G\) be a simple algebraic group over an algebraically closed field \(F\) of characteristic \(p \geq 0\). The irreducible rational representations of \(G\) whose weights have multiplicity at most \(1\) are classified in the case \(F = \mathbb{C}\) by Howe and in the case of positive characteristic by the author of the paper under review and \textit{I. D. Suprunenko} [Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk 1987, No. 6, 9--15 (1987; Zbl 0708.20011)]. In the paper under review, the author considers the following: Let \(\sigma\) be a non-trivial graph automorphism of \(G\) and let \(H = \left\langle G, \sigma \right\rangle\). Then, the author describes the irreducible \(H\)-modules \(V\) such that \(V\) is rational and irreducible as a \(G\)-module and there exists an element \(h \in H\) that has a simple spectrum on \(V\) (Theorem 1.1). Related results in the case \(p=0\) are proven independently by Katz and Tiep. In addition, the author also considers a similar question for the group \(G^{\mathrm{Fr}}\), where \(G\) is a simple algebraic group over a field of positive characteristic, \(\mathrm{Fr}\) is a Frobenius endomorphism of \(G\) and \(G^{\mathrm{Fr}}\) denotes the subgroup of elements of \(G\) that are invariant under \(\mathrm{Fr}\). Then in the case when the graph automorphism \(\sigma\) of \(G\) stabilizes \(G^{\mathrm{Fr}}\), the author obtains partial results about irreducible representations of \(\left\langle G^{\mathrm{Fr}}, \sigma \right\rangle\) whose image contains a matrix with a simple spectrum.