TY - JOUR

T1 - Dirac Operators for the Dunkl Angular Momentum Algebra

AU - Calvert, Kieran

AU - De Martino, Marcelo

PY - 2022/6/1

Y1 - 2022/6/1

N2 - 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.

AB - 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.

U2 - 10.3842/sigma.2022.040

DO - 10.3842/sigma.2022.040

M3 - Journal article

VL - 18

JO - SIGMA (Symmetry, Integrability and Geometry: Methods and Applications)

JF - SIGMA (Symmetry, Integrability and Geometry: Methods and Applications)

SN - 1815-0659

M1 - 040

ER -