Primitive filtrations of the modules of invariant logarithmic forms of Coxeter arrangements

TL;DR

This paper introduces primitive filtrations for modules of invariant logarithmic forms in Coxeter arrangements, extending previous work on irreducible cases and generalizing the Hodge filtration.

Contribution

It defines primitive derivations for possibly reducible Coxeter arrangements and constructs primitive filtrations for their invariant logarithmic forms, broadening the scope of prior theories.

Findings

Primitive derivations are defined for non-irreducible arrangements.

Primitive filtrations are constructed for arbitrary Coxeter arrangements with any multiplicity.

The work generalizes the Hodge filtration for irreducible arrangements.

Abstract

We define {\bf primitive derivations} for Coxeter arrangements which may not be irreducible. Using those derivations, we introduce the {\bf primitive filtrations} of the module of invariant logarithmic differential forms for an arbitrary Coxeter arrangement with an arbitrary multiplicity. In particular, when the Coxeter arrangement is irreducible with a constant multiplicity, the primitive filtration has already been studied, which generalizes the Hodge filtration introduced by K. Saito.