Towards a classification of finite-dimensional representations of rational Cherednik algebras of type D

TL;DR

This paper classifies finite-dimensional representations of rational Cherednik algebras of type D, showing they are all restrictions of type B representations, and demonstrates that most irreducible representations are infinite dimensional.

Contribution

It provides a combinatorial classification of finite-dimensional irreducible representations of type D rational Cherednik algebras, linking them to type B representations.

Findings

All finite-dimensional irreducible representations of type D are restrictions of type B representations.

Irreducible representations $L_c(\lambda^\pm)$ are infinite dimensional for all parameters.

The combinatorial wall crossing bijections are key to the classification.

Abstract

Using a combinatorial description due to Jacon and Lecouvey of the wall crossing bijections for cyclotomic rational Cherednik algebras, we show that the irreducible representations $L_c(\lambda^\pm)$ of the rational Cherednik algebra $H_c(D_n, \mathbb{C}^n)$ of type $D$ for symmetric bipartitions $\lambda$ are infinite dimensional for all parameters $c$. In particular, all finite-dimensional irreducible representations of rational Cherednik algebras of type $D$ arise as restrictions of finite-dimensional irreducible representations of rational Cherednik algebras of type $B$.