On elliptic Calogero-Moser systems for complex crystallographic reflection groups

TL;DR

This paper introduces new integrable systems associated with complex crystallographic reflection groups, generalizing elliptic Calogero-Moser systems to non-real cases, and provides explicit and geometric constructions of their integrals.

Contribution

It extends elliptic Calogero-Moser systems to complex reflection groups and offers two novel constructions of their integrals, including a geometric approach via elliptic Cherednik algebras.

Findings

New integrable systems for complex crystallographic reflection groups.

Explicit limits connecting to elliptic Dunkl operators.

Proof of algebraic integrability under certain conditions.

Abstract

To every irreducible finite crystallographic reflection group (i.e., an irreducible finite reflection group G acting faithfully on an abelian variety X), we attach a family of classical and quantum integrable systems on X (with meromorphic coefficients). These families are parametrized by G-invariant functions of pairs (T,s), where T is a hypertorus in X (of codimension 1), and s in G is a reflection acting trivially on T. If G is a real reflection group, these families reduce to the known generalizations of elliptic Calogero-Moser systems, but in the non-real case they appear to be new. We give two constructions of the integrals of these systems - an explicit construction as limits of classical Calogero-Moser Hamiltonians of elliptic Dunkl operators as the dynamical parameter goes to 0 (implementing an idea of arXiv:hep-th/9403178), and a geometric construction as global sections of sheaves of elliptic Cherednik algebras for the critical value of the twisting parameter. We also prove algebraic integrability of these systems for values of parameters satisfying certain integrality conditions.