Abstract
McKay’s conjecture (1971) on character degrees was reduced by Isaacs–Malle–Navarro (2007) to a so-called inductive condition on characters of finite quasisimple groups [IMN07], thus opening the way to a proof of McKay’s conjecture using the classification of finite simple groups. After [MA07], [MA08], [S12], [CS13], [KS16], [MS16], [CS17a], [CS17b], [CS19], [S23a], [S23b], we complete here the last step of a proof with an analysis of the representations of certain normalizers $\mathrm{N}_G(\mathbf{S})$ in $G = \mathbf{G}^F$ of maximal $d$-tori $\mathbf{S} (d\ge 3)$ of the ambient simple simply-connected algebraic group $\mathbf{G}$ of type $D_l(l \ge 4)$ for which $F$ is a Frobeneius endomorphism. To establish the so-called local conditions $\mathbf{A}(d)$ and $\mathbf{B}(d)$, we introduce a certain class of $F$-stable reductive subgroups $M\le \mathbf{G}$ of maximal rank where $\mathbf{M}^\circ$ is of type $\mathrm{D}_{l_1}\times \mathrm{D}_{l-l_1}$ with $\mathbf{M}/\mathbf{M}^\circ$ of order $2$. The finite groups $\mathbf{M}^F$ are an efficient substitute for $\mathrm{N}_G(\mathbf{S})$ or the local subgroups in non-defining characteristic relevant to McKay’s abstract statement. For a general class of those subgroups $\mathbf{M}^F$ we describe their characters and the action of $\mathrm{Aut}(\mathbf{G}^F)_{\mathbf{M}^F}$ on them, showing in particular that $\mathrm{Irr}(\mathbf{M}^F)$ and $\mathrm{Irr}(\mathbf{G}^F)$ share some key features in that regard.
With this established, McKay’s conjecture is now a theorem stating McKay’s equality: For any prime $\ell$, any finite group has as many irreducible complex characters of degree prime to $\ell$ as the normalizers of its Sylow $\ell$-subgroups.
