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

Let GL and U 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 χ be an irreducible character of either of these groups which is an extension of a real-valued character of GL or U. Let yτ be an element of GL or U such that ( yτ)2 is regular unipotent in GL or U, respectively. We show that is prime to p and otherwise. Several intermediate results on real conjugacy classes and real-valued characters of these groups are obtained along the way. [ABSTRACT FROM AUTHOR]