Linear semisimple Lie algebras containing an operator with small number of eigenvalues (Q1075417)

Let V be a finite-dimensional vector space over an algebraically closed field of characteristic zero, let \({\mathfrak g}\) be an irreducible Lie algebra contained in End(V), and let f be a semisimple non-zero endomorphism of V lying in \({\mathfrak g}\). The author investigates what can be concluded about \({\mathfrak g}\) from the knowledge of the eigenvalues of f and their multiplicities. In particular, he shows that if \({\mathfrak g}\) is simple and f has exactly \((r+1)\) different eigenvalues, the highest weight of the \({\mathfrak g}\)-module V is the sum of no more than r fundamental weights.