The irreducible representations of rational Cherednik algebras associated to the symmetric groups S2 and S3 in positive characteristic

Abstract

Let k be an algebraically closed field of positive characteristic p. Given parameters t, c ∈ k we form the rational Cherednik algebra Ht,c(Sn, h) for n = 2, 3 and consider its representation theory over k. For each irreducible representation τ of Sn there is a Verma module Mt,c(τ ) which is a module for Ht,c(Sn, h) and this module has a unique maximal proper graded submodule Jt,c(τ ). The quotient Mt,c(τ )/Jt,c(τ ) is a graded, irreducible representation Lt,c(τ ) of Ht,c(Sn, h) in category O and all simple objects in O are of this form. Our goal is to describe the representations Lt,c(τ ) for all possible values of p, t, c, and τ , by calculating their characters, Hilbert polynomials, and specifying the generators of Jt,c(τ ). We achieve this goal, filling gaps in the literature and describing these modules completely and explicitly, in all cases except one where we provide a conjecture.