Low-Dimensional Representations of Special Unitary Groups

We prove gap results for the low-dimensional representation of unitary groups in nondefining characteristics. More precisely, we give a lower bound for the third smallest degree of a nontrivial absolutely irreducible representation of the groups mentioned above, which is almost in the order of magnitude of the square of the two smallest such representations. As a corollary we obtain the uniqueness of Weil representations in all cross characteristics for the unitary groups. Our approach uses some results on ordinary representations which can be proved using Deligne–Lusztig theory and which might be of independent interest, as well as information on decomposition matrices.