On the Irreducible Characters of a Sylow 2-Subgroup of the Finite Symplectic Group in Characteristic 2

We show that if Pm is a Sylow 2-subgroup of the finite symplectic group Sp2m(q), where q is a power of 2, then Pm has irreducible complex characters of degree 2− tqm(m − 1)/2, where t is any integer satisfying 0 ≤ t ≤ [m/2], and that qm(m − 1)/2 is the largest possible degree of an irreducible complex character of Pm. We include related results on the degrees of the irreducible characters of a Sylow 2-subgroup of the split special orthogonal group SO+ 2m(q) when q is a power of 2.