The McKay Conjecture on character degrees

We prove that for any prime $\mathrm{\ell }$, any finite group has as many irreducible complex characters of degree prime to $\mathrm{\ell }$ as the normalizers of its Sylow $\mathrm{\ell }$-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 $\mathbf{S}$ ($d\ge 3$) in a simply-connected algebraic group $\mathbf{G}$ of type D${}_l$ ($l\ge 4$) for which $F$ is a Frobenius endomorphism. We also introduce a certain class of $F$-stable reductive subgroups $\mathbf{M}\le \mathbf{G}$ 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 $\mathrm{\ell }$-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.