Constructible sheaves on nilpotent cones in rather good characteristic

We study some aspects of modular generalized Springer theory for a complex reductive group $G$ with coefficients in a field $mathbb k$ under the assumption that the characteristic $ell$ of $mathbb k$ is rather good for $G$, i.e., $ell$ is good and does not divide the order of the component group of the centre of $G$. We prove a comparison theorem relating the characteristic-$ell$ generalized Springer correspondence to the characteristic-$0$ version. We also consider Mautners characteristic-$ell$ `cleanness conjecture; we prove it in some cases; and we deduce several consequences, including a classification of supercuspidal sheaves and an orthogonal decomposition of the equivariant derived category of the nilpotent cone.