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).