Spectra of non-regular elements in irreducible representations of simple algebraic groups

Abstract

We study the spectra of non-regular semisimple elements in irreducible representations of simple algebraic groups. More precisely, we prove that if \(G\) is a simply connected simple linear algebraic group and \(\phi:G\to {\mathrm{GL}}(V)\) is a non-trivial irreducible representation for which there exists a non-regular non-central semisimple element \(s\in G\) such that \(\phi(s)\) has almost simple spectrum, then, with few exceptions, \(G\) is of classical type and \(\dim V\) is minimal possible. Here the spectrum of a diagonalizable matrix is called simple if all eigenvalues are of multiplicity 1, and almost simple if at most one eigenvalue is of multiplicity greater than 1. This yields a kind of characterization of the natural representation (up to their Frobenius twists) of classical algebraic groups in terms of the behavior of semisimple elements.