Stiefel-Whitney Classes for Finite Symplectic Groups

Let q be an odd prime power, and G = Sp(2n, q) the finite symplectic group. We give an expression for the total Stiefel-Whitney Classes (SWCs) for orthogonal representations π of G, in terms of character values of π at elements of order 2. We give “universal formulas” for the fourth and eighth SWCs. For n = 2, we compute the subring of the mod 2 cohomology generated by the SWCs w k (π).