Dunkl symplectic algebra in generalized Calogero models

TL;DR

This work develops a Dunkl-deformed sp(2N) symmetry framework for generalized Calogero models, revealing a deformed unitary subalgebra together with a nondeformed sl(2,R) conformal sector that together generate the spectrum. A key result is the explicit relation between the Casimir operators of the conformal and Dunkl angular momentum algebras, enabling a description of eigenfunctions in terms of deformed spherical harmonics organized into infinite-dimensional lowest-weight SL(2,R) multiplets. The paper then constructs conformal multiplets for the commuting integrals via rotated SL(2,R) basing, and derives Weyl-ordered descendants that encode higher-order integrals, with additional integrals arising from product multiplets. These findings illuminate the superintegrability of Calogero-Moser systems and provide a concrete, algebraic route to classify and generate all conserved quantities through the Dunkl-symplectic/conformal structure, with potential implications for related integrable models and quantum algebras.

Abstract

We study the properties of the symplectic sp(2N) algebra deformed using Dunkl operators, which describe the dynamical symmetry of the generalized N-particle quantum Calogero model. It contains a symmetry subalgebra formed by the deformed unitary generators as well as the (nondeformed) sl(2,R) conformal subalgebra. An explicit relation among the deformed symplectic generators is derived. Based on the matching between the Casimir elements of the conformal spin and Dunkl angular momentum algebras, the independent wavefunctions of the both the standard and generalized Calogero models, expressed in terms of the deformed spherical harmonics, are classified according to infinite-dimensional lowest-state sl(2,R) multiplets. Meanwhile, any polynomial integral of motion of the (generalized) Calogero-Moser model generates a finite-dimensional highest-state conformal multiplet with descendants expressed via the Weyl-ordered product in quantum field theory.