A primitive derivation and logarithmic differential forms of Coxeter arrangements

TL;DR

This paper constructs an explicit basis for logarithmic differential forms associated with Coxeter arrangements using primitive derivations, extending the Hodge filtration to all integers and connecting it with pole order.

Contribution

It provides a new explicit basis for these forms and extends the Hodge filtration to a broader index set, linking it to pole order.

Findings

Explicit basis for logarithmic differential forms constructed

Hodge filtration extended to all integers

Filtration coincides with pole order

Abstract

Let $W$ be a finite irreducible real reflection group, which is a Coxeter group. We explicitly construct a basis for the module of differential 1-forms with logarithmic poles along the Coxeter arrangement by using a primitive derivation. As a consequence, we extend the Hodge filtration, indexed by nonnegative integers, into a filtration indexed by all integers. This filtration coincides with the filtration by the order of poles. The results are translated into the derivation case.