The McKay Conjecture on character degrees

• 成果类型: Article
• 署名作者: Cabanes, MARc; Spaeth, Britta
• 署名单位: Sorbonne Universite; Universite Paris Cite; University of Wuppertal
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X; 1939-8980
• DOI: 10.4007/annals.2026.203.3.5
• 发表日期: 2026-05
• 页码: 933-1032
• 关键词: McKay conjecture; Clifford theory; finite groups of Lie type; Sylow tori; relative Weyl groups; SYLOW D-TORI; reduction theorem; classical-groups; Finite; blocks; REPRESENTATION

摘要

We prove that for any prime P, any finite group has as many irreducible complex characters of degree prime to P as the normalizers of its Sylow P-subgroups. This equality was conjectured by John McKay in 1971. The conjecture was reduced by Isaacs-Malle-Navarro (2007) to a conjecture on representations, linear and projective, of finite simple groups that we finish proving here using the classification of those groups. We study mainly characters of normalizers N-G(S)(F )of Sylow d-tori S (d >= 3) in a simply-connected algebraic group G of type D-l (l >= 4) for which F is a Frobenius endomorphism. We also introduce a certain class of F-stable reductive subgroups M <= G of maximal rank where M-degrees is of type D-k & times; Dl-k. The finite groups M-F are an efficient substitute for N-G(S)(F) or the P-local subgroups of G(F) relevant to McKay's abstract statement. For a general class of those subgroups M-F, we describe their characters and the action of Aut(G(F))MF on them, showing in particular that Irr(M-F) and Irr(G(F)) share some key features in that regard.