Restriction of Odd Degree Characters and Natural Correspondences

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) × GL₁ (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. Next, we construct a canonical bijection between the odd-degree irreducible characters of G = S_n, 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.