  Let $q$ be an odd prime power, $n > 1$, and let $P$ denote a maximal
parabolic subgroup of $GL_n(q)$ with Levi subgroup $GL_{n-1}(q) \times
GL_1(q)$. We restrict the odd-degree irreducible characters of $GL_n(q)$ to $P$
to discover a natural correspondence of characters, both for $GL_n(q)$ and
$SL_n(q)$. A similar result is established for certain finite groups with
self-normalizing Sylow $p$-subgroups. We also construct a canonical bijection
between the odd-degree irreducible characters of $S_n$ and those of $M$, where
$M$ is any maximal subgroup of $S_n$ of odd index; as well as between the
odd-degree irreducible characters of $G = GL_n(q)$ or $GU_n(q)$ with $q$ odd
and those of $N_{G}(P)$, where $P$ is a Sylow $2$-subgroup of $G$. Since our
bijections commute with the action of the absolute Galois group over the
rationals, we conclude that the fields of values of character correspondents
are the same. We use this to answer some questions of R. Gow.
