Dirac Operators for the Dunkl Angular Momentum Algebra

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.

References in corpus (4)

• Algebraic and analytic Dirac induction for graded affine Hecke algebras
• The total angular momentum algebra related to the ${\mathrm{S}}_3$ Dunkl Dirac equation
• Dirac cohomology, the projective supermodules of the symmetric group and the Vogan morphism
• Dirac cohomology and simple modules of the Dunkl-Opdam subalgebra via inherited Drinfeld properties

Cited by in corpus (1)

• Supercentralizers for deformations of the Pin osp dual pair