Representations of Cherednik Algebras Associated to Symmetric and Dihedral Groups in Positive Characteristic

TL;DR

This paper investigates the structure of irreducible lowest-weight representations of Cherednik algebras linked to symmetric and dihedral groups over fields of positive characteristic, focusing on maximal graded submodules and singular polynomials.

Contribution

It introduces new results and conjectures on generators of maximal submodules in Verma modules for Cherednik algebras in positive characteristic, using Dunkl operators.

Findings

Identification of generators of maximal submodules

Computation of singular polynomials for Dunkl operators

Formulation of conjectures on submodule structures

Abstract

We consider irreducible lowest-weight representations of Cherednik algebras associated to certain classes of complex reflection groups in characteristic p. In particular, we study maximal graded submodules of Verma modules associated to these algebras. Various results and conjectures are presented concerning generators of these maximal submodules, which are found by computing singular polynomials of Dunkl operators.