ABSTRACT:

We consider finite reflection groups acting over fields of arbitrary characteristic. Many arguments of classical invariant theory break down in the modular setting, when the characteristic of the field divides the order of the group. Here, we examine the action on differential derivations, which arise in Catalan combinatorics. We develop an analogue of Saito's freeness criterion that distinguishes the characteristic 2 case and obtain an explicit description of the invariant differential derivations for special linear groups and general linear groups over finite fields.