A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups

TL;DR

This paper constructs an explicit basis for logarithmic vector fields in well-generated complex reflection groups, establishing a Hodge filtration and generalizing previous results for real reflection groups.

Contribution

It introduces a new explicit basis and Hodge filtration for logarithmic vector fields in complex reflection groups, unifying and extending prior work.

Findings

Explicit basis for vector fields with logarithmic poles

Hodge filtration of the module established

Generalization of results from real to complex reflection groups

Abstract

Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration of that module. Our approach is based on a detailed analysis of a flat connection applied to the primitive vector field. This generalizes and unifies analogous results for real reflection groups.