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

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 a 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 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.