McKay correspondence for linearly reductive finite group schemes in positive characteristic

Abstract

We obtain a motivic and a cohomological McKay correspondence for finite linearly reductive group schemes in arbitrary characteristic. In particular, we prove that if $V$ is a finite dimensional vector space and $G$ is a finite linearly reductive subgroup scheme of $\mathrm{SL}(V)$, then the Euler number of any crepant resolution of $V/G$ is equal to the number of irreducible algebraic representations of $G$. We obtain these McKay correspondences as a consequence of a motivic change of variables formula applied to $[V/G]\to V/G$. If $G$ is non-reduced, as can happen in positive characteristic, the stack quotient $[V/G]$ is not Deligne-Mumford. Therefore in order to prove this change of variables formula and the resulting McKay correspondences, we generalize the authors' theory of motivic integration for Artin stacks to arbitrary characteristic, which may be of independent interest.