Almost cyclic regular semisimple elements in irreducible representations of simple algebraic groups

Let G be a simple linear algebraic group defined over an algebraically closed field of characteristic p ≥ 0 and let φ be a nontrivial p-restricted irreducible representation of G. Let T be a maximal torus of G and s ∈ T . We say that s is Ad-regular if α(s) is different from β(s) for all distinct T -roots α and β of G. Our main result states that if all but one of the eigenvalues of φ(s) are of multiplicity 1 then, with a few specified exceptions, s is Ad-regular. This can be viewed as an extension of our earlier work, in which we show that, under the same hypotheses, either s is regular or G is a classical group and φ is “essentially” (a twist of) the natural representation of G.