Invariants of symbols of the linear differential operators

TL;DR

This paper classifies the invariants of symbols of linear differential operators acting on sections of vector bundles, using algebraic methods to identify criteria for their equivalence.

Contribution

It introduces a classification of symbols' invariants for differential operators on vector bundles, extending algebraic invariant theory to this geometric context.

Findings

Identified generators for the field of rational invariants.

Provided a criterion for the equivalence of non-degenerated symbols.

Reduced the classification problem to linear algebra of operator tuples.

Abstract

In this paper we classify the symbols of the linear differential operators of order $k$, which act from the module $C^\infty(\xi)$ to the module $C^\infty(\xi^t)$, where $\xi\colon E(\xi)\to M$ is vector bundle over the smooth manifold $M$, bundle $\xi^t$ is either $\xi^*$ with fiber $E^*:=\mathrm{Hom}(E,\mathbb{C})$ or $\xi^\flat$ with fiber $E^\flat:=\mathrm{Hom}(E, \Lambda^n T^*)$ and $C^\infty(\xi)$, $C^\infty(\xi^t)$ are the modules of their smooth sections. To find invariants of the symbols we associate with every non-degenerated symbol the tuple of linear operators acting on space $E$ and reduce our problem to the classification of such tuples with respect to some orthogonal transformations. Using the results of C. Procesi, we find generators for the field of rational invariants of the symbols and in terms of these invariants provide a criterion of equivalence of non-degenerated symbols.