Invariant derivations and differential forms for reflection groups

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.