LOW-DIMENSIONAL REPRESENTATIONS OF THE SPECIAL LINEAR GROUP IN CROSS-CHARACTERISTICS

ROBERT M. GURALNICK AND PHAM HUU TIEP

Abstract. The low-dimensional projective irreducible representations in cross characteristics of the projective special linear group L_{n}(q) are investigated. If n ≥ 3 and (n, q) ≠ (3, 2), (3, 4), (4, 2), (4, 3), all such representations of the first degree (which is (q^{n} − q)/(q − 1) − κ_{n} with κ_{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 the second and the third degree is roughly q^{2n−4}.