On the irreducible characters of a Sylow 2-subgroup of the finite symplectic group in characteristic 2

We show that if P-m is a Sylow 2-subgroup of the finite symplectic group Sp (2m)(q), where q is a power of 2, then P-m has irreducible complex characte rs of degree 2(-t)q(m(m-1)/2), where t is any integer satisfying 0 less tha n or equal to t less than or equal to [m/2], and that q(m(m)-1)/2 is the la rgest possible degree of an irreducible complex character of P-m. We includ e related results on the degrees of the irreducible characters of a Sylow 2 -subgroup of the split special orthogonal group SO2m+(q) when g is a power of 2. (C) 2001 Academic Press.