Invariant derivations and differential forms for reflection groups

Abstract

Classical invariant theory of a complex reflection group W beautifully describes the W-invariant polynomials, the W-invariant differential forms, and the relative invariants of any W-representation. 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 do not rely on any reflection group classification.