Dirac Operators for the Dunkl Angular Momentum Algebra

TL;DR

This paper introduces a new family of Dirac operators for the Dunkl angular momentum algebra, proving an analogue of Vogan's conjecture and linking Dirac cohomology to representation central characters, with connections to Calogero-Moser Hamiltonian.

Contribution

It defines Dirac operators for the Dunkl angular momentum algebra and establishes their properties, including an analogue of Vogan's conjecture and a relation to the Calogero-Moser Hamiltonian.

Findings

Dirac cohomology determines the central character of representations.

The Dirac operator's square relates to the angular Calogero-Moser Hamiltonian.

An analogue of Vogan's conjecture is proved for this family of operators.

Abstract

We define a family of Dirac operators for the Dunkl angular momentum algebra depending on certain central elements of the group algebra of the Pin cover of the Weyl group inherent to the rational Cherednik algebra. We prove an analogue of Vogan's conjecture for this family of operators and use this to show that the Dirac cohomology, when non-zero, determines the central character of representations of the angular momentum algebra. Furthermore, interpreting this algebra in the framework of (deformed) Howe dualities, we show that the natural Dirac element we define yields, up to scalars, a square root of the angular part of the Calogero-Moser Hamiltonian.