Low-dimensional representations of special linear groups in cross characteristics

The low-dimensional projective irreducible representations in cross charact eristics of the projective special linear group L-n(q) are investigated. If n greater than or equal to 3 and (n,q)not equal(3,2), (3,4), (4,2), (4,3), all such representations of the first degree (which is (q(n) - q)/(q - 1) - kappa(n) With kappa(n) = 0 or 1) and the second degree (which is (q(n) - 1)/(q - 1)) come from Weil representations. We show that the gap between th e second and the third degree is roughly q(2n-4).