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Restriction of odd degree characters and natural correspondences. (English) Zbl 1404.20006
Summary: Let $$q$$ be an odd prime power, $$n>1$$, and let $$P$$ denote a maximal parabolic subgroup of $$\mathrm{GL}_n(q)$$ with Levi subgroup $$\mathrm{GL}_{n-1}(q) \times \mathrm{GL}_1(q)$$. We restrict the odd-degree irreducible characters of $$\mathrm{GL}_n(q)$$ to $$P$$ to discover a natural correspondence of characters, both for $$\mathrm{GL}_n(q)$$ and $$\mathrm{SL}_n(q)$$. 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 $$G=\mathsf{S}_n$$, $$\mathrm{GL}_n(q)$$ or $$\mathrm{GU}_n(q)$$ with $$q$$ odd, and those of $$\mathbf{N}_G(P)$$, 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.

##### MSC:
 20C15 Ordinary representations and characters 20G05 Representation theory for linear algebraic groups 20G40 Linear algebraic groups over finite fields
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