Computational aspects of Calogero-Moser spaces

TL;DR

This paper introduces algorithms for computing invariants of Calogero-Moser spaces and rational Cherednik algebras, enabling the verification of conjectures and analysis of geometric structures related to complex reflection groups.

Contribution

The authors developed new algorithms for invariants of Calogero-Moser spaces, implemented them in CHAMP, and applied them to verify conjectures and study birational geometry.

Findings

Confirmed open conjectures in new cases

Determined chamber decompositions of movable cones

Analyzed symplectic singularities of complex reflection groups

Abstract

We present a series of algorithms for computing geometric and representation-theoretic invariants of Calogero-Moser spaces and rational Cherednik algebras associated to complex reflection groups. Especially, we are concerned with Calogero-Moser families (which correspond to the $\mathbb{C}^\times$-fixed points of the Calogero-Moser space) and cellular characters (a proposed generalization by Rouquier and the first author of Lusztig's constructible characters based on a Galois covering of the Calogero-Moser space). To compute the former, we devised an algorithm for determining generators of the center of the rational Cherednik algebra (this algorithm has several further applications), and to compute the latter we developed an algorithmic approach to the construction of cellular characters via Gaudin operators. We have implemented all our algorithms in the Cherednik Algebra Magma Package (CHAMP) by the second author and used this to confirm open conjectures in several new cases. As an interesting application in birational geometry we are able to determine for many exceptional complex reflection groups the chamber decomposition of the movable cone of a $\mathbb{Q}$-factorial terminalization (and thus the number of non-isomorphic relative minimal models) of the associated symplectic singularity.