  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 f is a non-trivial
irreducible representation of G in some GL(V) for which there exists a
non-regular non-central semisimple element s in G such that f(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.
