A McKay Correspondence in Positive Characteristic

We establish a McKay correspondence for finite and linearly reductive subgroup schemes of $\mathrm{SL}_2$ in positive characteristic. As an application, we obtain a McKay correspondence for all rational double point singularities in characteristic $p\geq7$. We discuss linearly reductive quotient singularities and canonical lifts over the ring of Witt vectors. In dimension 2, we establish simultaneous resolutions of singularities of these canonical lifts via $G$-Hilbert schemes. In the appendix, we discuss several approaches towards the notion of conjugacy classes for finite group schemes: This is an ingredient in McKay correspondences, but also of independent interest.