Low-dimensional representations of special unitary groups

We prove gap results fur the low-dimensional representation of unitary grou ps in nondefining characteristics. More precisely, we give a lower bound fo r the third smallest degree of a nontrivial absolutely irreducible represen tation of the groups mentioned above, which is almost in the order of magni tude of the square of the two smallest such representations. As a corollary we obtain the uniqueness of Well representations in all cross characterist ics for the unitary groups. Our approach uses some results on ordinary repr esentations which can be proved using Deligne-Lusztig theory and which migh t be of independent interest, as well as information on decomposition matri ces. (C) 2001 Academic Press.