Harmonic differential forms for pseudo-reflection groups II. Bi-degree bounds

Abstract: This paper studies three results that describe the structure of the super-coinvariant algebra of pseudo-reflection groups over a field of characteristic $0$. Our most general result determines the top component in total degree, which we prove for all Shephard--Todd groups $G(m, p, n)$ with $m \neq p$ or $m=1$. Our strongest result gives tight bi-degree bounds and is proven for all $G(m, 1, n)$, which includes the Weyl groups of types $A$ and $B$/$C$. For symmetric groups (i.e. type $A$), this provides new evidence for a recent conjecture of Zabrocki related to the Delta Conjecture of Haglund--Remmel--Wilson. Finally, we examine analogues of a classic theorem of Steinberg and the Operator Theorem of Haiman. Our arguments build on the type-independent classification of semi-invariant harmonic differential forms carried out in the first part of this series. In this paper we use concrete constructions including Gr\"{o}bner and Artin bases for the classical coinvariant algebras of the pseudo-reflection groups $G(m, p, n)$, which we describe in detail. We also prove that exterior differentiation is exact on the super-coinvariant algebra of a general pseudo-reflection group. Finally, we discuss related conjectures and enumerative consequences.