Invariant differential operators in positive characteristic

TL;DR

This paper classifies invariant differential operators over one-dimensional manifolds in positive characteristic, revealing new operators analogous to classical ones and introducing a novel operator type.

Contribution

It extends Veblen's classification problem to positive characteristic, discovering new high-order operators and analogs of known operators in this setting.

Findings

Identified analogs of the Berezin integral in positive characteristic.

Discovered two new types of indecomposable operators, including a novel operator.

Extended classification of invariant differential operators to positive characteristic.

Abstract

We consider an analog of the problem Veblen formulated in 1928 at the IMC: classify invariant differential operators between "natural objects" (spaces of either tensor fields, or jets, in modern terms) over a real manifold of any dimension. For unary operators, the problem was solved by Rudakov (no nonscalar operators except the exterior differential); for binary ones, by Grozman (there are no operators of orders higher than 3, operators of order 2 and 3 are, bar an exception in dimension 1, compositions of order 1 operators which, up to dualization and permutation of arguments, form 8 families). In dimension one, Grozman discovered an indecomposable selfdual operator of order 3 that does not exist in higher dimensions. We solve Veblen's problem in the 1-dimensional case over the ground field of positive characteristic. In addition to analogs of the Berezin integral (strangely overlooked so far) and binary operators constructed from them, we discovered two more (up to dualization) types of indecomposable operators of however high order: analogs of the Grozman operator and a completely new type of operators.