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

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 6= 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 [33] related to the Delta Conjecture of Haglund–Remmel–Wilson [16]. Finally, we examine analogues of a classic theorem of Steinberg [30] and the Operator Theorem of Haiman [18, 19]. Our arguments build on the type-independent classification of semi-invariant harmonic differential forms carried out in the first part of this series [31]. In this paper we use concrete constructions including Gröbner 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.