Representations of rational Cherednik algebras of G(m,r,n) in positive characteristic

TL;DR

This paper investigates the structure of irreducible representations of rational Cherednik algebras associated with complex reflection groups G(m,r,n) over fields of positive characteristic, using commutative algebra techniques to derive formulas and conjectures.

Contribution

It introduces a new approach to study these representations via the kernel of the contravariant form, including formulas for Hilbert series and the concept of matrix regular sequences.

Findings

Formulas for Hilbert series in several cases

Conjectures relating Hilbert series to n modulo p

Identification of kernel generators as matrix regular sequences in small cases

Abstract

We study lowest-weight irreducible representations of rational Cherednik algebras attached to the complex reflection groups G(m,r,n) in characteristic p. Our approach is mostly from the perspective of commutative algebra. By studying the kernel of the contravariant bilinear form on Verma modules, we obtain formulas for Hilbert series of irreducible representations in a number of cases, and present conjectures in other cases. We observe that the form of the Hilbert series of the irreducible representations and the generators of the kernel tend to be determined by the value of n modulo p, and are related to special classes of subspace arrangements. Perhaps the most novel (conjectural) discovery from the commutative algebra perspective is that the generators of the kernel can be given the structure of a "matrix regular sequence" in some instances, which we prove in some small cases.