Eigenvalues of unipotent elements in cross-characteristic representations of finite classical groups

Let H be a finite classical group, g be a unipotent element of H of order s and θ be an irreducible representation of H with dimθ>1 over an algebraically closed field of characteristic coprime to s. We show that almost always all the s-roots of unity occur as eigenvalues of θ(g), and classify all the triples (H,g,θ) for which this does not hold. In particular, we list the triples for which 1 is not an eigenvalue of θ(g). We also give estimates of the asymptotic behavior of eigenvalue multiplicities when the rank of H grows and s is fixed.