  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.
