Abstract
We prove that 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. This equality was conjectured by John McKay.
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 $\mathrm{N}_{\mathbf{G}}(\mathbf{S})^F$ of Sylow $d$-tori $\mathbf{S} (d\geq 3)$ in a simply-connected algebraic group $\mathbf{G}$ of type $\mathrm{D}_l$ ($l\geq 4$) for which $F$ is a Frobenius endomorphism. We also introduce a certain class of $F$-stable reductive subgroups $\mathbf{M}\leq \mathbf{G}$ of maximal rank where $\mathbf{N}^\circ$ is of type $\mathrm{D}_{k}\times \mathrm{D}_{l-k}$. The finite groups $\mathbf{M}^F$ are an efficient substitute for $\mathrm{N}_{\mathbf{G}}(\mathbf{S})^F$ or the $\ell$-local subgroups of $\mathbf{G}^F$ 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.
