Free reflection multiarrangements and quasi-invariants

TL;DR

This paper explores the relationship between logarithmic vector fields and quasi-invariants in complex reflection arrangements, establishing new freeness results for multiarrangements and connections to Catalan arrangements.

Contribution

It introduces a novel relation between modules of quasi-invariants and logarithmic vector fields, leading to new freeness results for complex reflection multiarrangements.

Findings

Established a close relation between modules of quasi-invariants and logarithmic vector fields.

Proved freeness of Catalan arrangements for the non-reduced root system BC_N.

Extended the theory to non-homogeneous quasi-invariants and their applications.

Abstract

To a complex reflection arrangement with an invariant multiplicity function one can relate the space of logarithmic vector fields and the space of quasi-invariants, which are both modules over invariant polynomials. We establish a close relation between these modules. Berest-Chalykh freeness results for the module of quasi-invariants lead to new free complex reflection multiarrangements. K. Saito's primitive derivative gives a linear map between certain spaces of quasi-invariants. We also establish a close relation between non-homogeneous quasi-invariants for root systems and logarithmic vector fields for the extended Catalan arrangements. As an application, we prove the freeness of Catalan arrangements corresponding to the non-reduced root system $BC_N$.