Dunkl and Cherednik operators

TL;DR

This survey explores Dunkl and Cherednik operators, highlighting their role in integrable systems and their connections to algebraic structures, providing a comprehensive overview without detailed proofs.

Contribution

It offers a broad overview of Dunkl and Cherednik operators, emphasizing their applications in integrable systems and their algebraic relationships, with historical context.

Findings

Dunkl and Cherednik operators are key in studying integrable many-body systems.

These operators relate to rational Cherednik algebras and double affine Hecke algebras.

The survey summarizes their significance without detailed proofs.

Abstract

This survey article, written for the Encyclopedia of Mathematical Physics, 2nd edition, is devoted to the remarkable family of operators introduced by Charles Dunkl and to their $q$-analogues discovered by Ivan Cherednik. The main focus is on the r\^ole of these operators in studying integrable many-body systems such as the Calogero-Moser and the Ruijsenaars systems. To put these constructions into a wider context, we indicate their relationship with the theory of the rational Cherednik algebras and double affine Hecke algebras. While we do not include proofs, references to the original research articles are provided, accompanied by brief historical comments.