Cross characteristic representations of odd characteristic symplectic groups and unitary groups

Abstract

In [LS], Landazuri and Seitz gave lower bounds for irreducible representations of Chevalley groups in non-defining characteristic (when referring to irreducible representations for quasisimple groups G, we will assume that the modules are nontrivial on F ∗(G)). See also [SZ], [GPPS], [Ho] for some improvements on these bounds. These results have proved to be useful in many applications. In particular, they have been used to classify the maximal subgroups of classical groups containing an element of prime order acting irreducibly on a subspace of large dimension (cf. [GPPS]), and to show that low dimensional modules in characteristic p for groups with no normal p-subgroup are semisimple (see [Gu]). It is also important to identify the modules which have dimension close to the smallest possible dimension and to prove that there are no irreducible modules with dimension in a certain range above it. This was done in [GPPS] and [GT] for SLn(q). Further improvements were obtained by Brundan and Kleshchev [BrK]. Hiss and Malle [HM] have obtained results similar to [GT] for unitary groups. In this paper, we consider the groups G = Sp2n(q) with n ≥ 2 and q = p odd and G = SUn(q) with n ≥ 3. Landazuri and Seitz [LS] had already shown that the minimal dimension d of any irreducible module in the nondefining characteristic has dimension exactly (q − 1)/2 for the symplectic case, and [(q − 1)/(q + 1)] for the unitary case. It was proved in [GPPS] that (aside from some small exceptions), every irreducible kG-module in a nondefining characteristic has dimension d, d+1 or at least dimension 2d. In characteristic 0, Tiep and Zalesskii [TZ1] (using Deligne-Lusztig