Differential operators on classical invariant rings do not lift modulo $p$

TL;DR

This paper demonstrates that differential operators on classical invariant rings in positive characteristic generally do not lift to characteristic zero, highlighting limitations in the transfer of algebraic structures across characteristics.

Contribution

It proves that differential operators on determinantal, Pfaffian, and symmetric determinantal hypersurfaces do not lift modulo p, and that these rings lack mod p^2 Frobenius lifts.

Findings

Differential operators on certain invariant rings do not lift to characteristic zero.

Most classical invariant rings do not admit a mod p^2 Frobenius lift.

The results extend to determinantal, Pfaffian, and symmetric hypersurfaces.

Abstract

Levasseur and Stafford described the rings of differential operators on various classical invariant rings of characteristic zero; in each of the cases that they considered, the differential operators form a simple ring. Towards an attack on the simplicity of rings of differential operators on invariant rings of linearly reductive groups over the complex numbers, Smith and Van den Bergh asked if differential operators on the corresponding rings of positive prime characteristic lift to characteristic zero differential operators. We prove that, in general, this is not the case for determinantal hypersurfaces, as well as for Pfaffian and symmetric determinantal hypersurfaces. We also prove that, with very few exceptions, these hypersurfaces -- and, more generally, classical invariant rings -- do not admit a mod $p^2$ lift of the Frobenius endomorphism.