The McKay conjecture for exceptional groups and odd primes.
The McKay conjecture is one of the main conjectures in the representation theory of finite groups. In the important paper [Invent. Math. 170, No. 1, 33-101 (2007; Zbl 1138.20010)], \textit{I. M. Isaacs, G. Malle}, and \textit{G. Navarro} have been able to reduce the conjecture to prove that all finite simple non-Abelian groups are ``good'' for every prime \(\ell\) in a certain sense. Now let \(S\) be a finite simple group of Lie type defined in characteristic \(p\neq\ell\) and let \(G\) be the maximal perfect central \(\ell'\)-extension of \(S\). Then it was shown by \textit{G. Malle} [in Represent. Theory 11, 192-220 (2007; Zbl 1139.20008)] that \(S\) is good for the prime \(\ell\), if the normalizers \(N_G(T)\) of Sylow tori \(T\) satisfy a set of three specific conditions. In the same paper, Malle proved that \(N_G(T)\) satisfies the first condition, and also (part of) the second condition, provided that all irreducible characters of \(C_G(T)\) extend to their inertia group in \(N_G(T)\). The main result of this paper establishes this extendibility condition in the case \(G\) is an exceptional group of Lie type. This result also allows the author to verify the McKay conjecture for the exceptional groups of Lie type \(G\) and odd primes \(\ell\neq p\).
scientific article; zbMATH DE number 5533140
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