This talk is on joint work with Bhama Srinivasan. Let G = U(n,q) denote the unitary group defined over the finite field F_q, of characteristic p. In this talk, we consider the irreducible complex characters of G which have degree prime to p, that is, which are semisimple, and which are real-valued, but are not the character of a real representation, i.e. symplectic characters. In the case n=2m is even and p is odd, we prove that the number of such characters is q^{m-1}, while there are 0 such characters otherwise. In fact, we find a bijection between these characters and the self-dual irreducible polynomials of degree 2m over F_q with constant term -1. In the process of this enumeration, we obtain a general result for a large class of finite reductive groups which gives a correspondence between the real-valued semisimple characters of the group and the real semisimple classes of the dual group.