Invariant derivations and differential forms for reflection groups

TL;DR

This paper explores the structure of invariant differential forms and derivations associated with complex reflection groups, providing explicit descriptions and resolving conjectures, especially for duality groups, without relying on classification.

Contribution

It offers an explicit description of the isotypic component in differential forms for duality reflection groups and proves that invariant differential derivations form a finitely generated, often free, module over invariant forms.

Findings

Resolved a conjecture relating to differential forms in reflection groups.

Proved that invariant differential derivations are finitely generated modules.

Established case-free results for duality groups without classification reliance.

Abstract

Classical invariant theory of a complex reflection group $W$ highlights three beautiful structures: -- the $W$-invariant polynomials constitute a polynomial algebra, over which -- the $W$-invariant differential forms with polynomial coefficients constitute an exterior algebra, and -- the relative invariants of any $W$-representation constitute a free module. When $W$ is a duality (or well-generated) group, we give an explicit description of the isotypic component within the differential forms of the irreducible reflection representation. This resolves a conjecture of Armstrong, Rhoades and the first author, and relates to Lie-theoretic conjectures and results of Bazlov, Broer, Joseph, Reeder, and Stembridge, and also Deconcini, Papi, and Procesi. We establish this result by examining the space of $W$-invariant differential derivations; these are derivations whose coefficients are not just polynomials, but differential forms with polynomial coefficients. For every complex reflection group $W$, we show that the space of invariant differential derivations is finitely generated as a module over the invariant differential forms by the basic derivations together with their exterior derivatives. When $W$ is a duality group, we show that the space of invariant differential derivations is free as a module over the exterior subalgebra of $W$-invariant forms generated by all but the top-degree exterior generator. (The basic invariant of highest degree is omitted.) Our arguments for duality groups are case-free, i.e., they do not rely on any reflection group classification.