Rational Cherednik algebras and categorification

TL;DR

This survey reviews how Kac-Moody and Heisenberg algebra actions on categories of rational Cherednik algebras facilitate solving key representation theory problems like classifying finite-dimensional irreducibles and computing their characters.

Contribution

It introduces a categorification approach using algebra actions to address fundamental representation theoretic questions for rational Cherednik algebras.

Findings

Classification of finite-dimensional irreducible representations

Explicit computation of characters for irreducibles

Application of algebra actions to solve representation problems

Abstract

In this survey article we review Kac-Moody and Heisenberg algebra actions on the categories $\mathcal{O}$ of the rational Cherednik algebras associated to groups $G(\ell,1,n)$. Using these actions we solve basic representation theoretic problems for these categories such as the classification of finite dimensional irreducible representations and computation of characters of the irreducibles.