Supports of simple modules in cyclotomic Cherednik categories O

TL;DR

This paper computes the supports of simple modules in categories O for rational Cherednik algebras related to complex reflection groups, using combinatorial maps and crystal structures.

Contribution

It introduces new combinatorial tools and crystal structures to determine module supports in Cherednik categories O, advancing understanding of their representation theory.

Findings

Computed supports of simple modules in Cherednik categories O

Developed combinatorial maps including wall-crossing bijections

Linked module supports to crystal structures and algebra actions

Abstract

The goal of this paper is to compute the supports of simple modules in the categories $\mathcal{O}$ for the rational Cherednik algebras associated to groups $G(\ell,1,n)$. For this we compute some combinatorial maps on the set of simples: wall-crossing bijections and a certain $\mathfrak{sl}_\infty$-crystal associated to a Heisenberg algebra action on a Fock space.