The McKay Conjecture on character degrees

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 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 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 ${\mathbf M}^\circ$ is of type some D$_{k}\times\ $D$_{l-k}$. The finite groups ${\mathbf M}^F$ are an efficient substitute for 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 Aut$({\mathbf G}^F)_{{\mathbf M}^F}$ on them, showing in particular that Irr$({\mathbf M}^F)$ and Irr$({\mathbf G}^F)$ share some key features in that regard.