Extending real-valued characters of finite general linear and unitary groups on elements related to regular unipotents

Let GL(n,F-q)<tau > and U(n,F-q2)<tau > denote the finite general linear and unitary groups extended by the transpose inverse automorphism, respectively, where q is a power of the prime p. Let n be odd, and let X be an irreducible character of either of these groups which is an extension of a real-valued character of GL (n, F-q) or U(n, F-q2). Let y tau be an element of GL(n, F-q) <tau > or U(n, F-q2)<tau > such that (y tau)(2) is regular unipotent in GL(n, F-q) or U(n, F-q2), respectively. We show that chi(y tau) = +/- 1 if chi(1) is prime to p and chi(y tau) = 0 otherwise. Several intermediate results on real conjugacy classes and real-valued characters of these groups are obtained along the way.