TY - JOUR

T1 - Dirac Operators for the Dunkl Angular Momentum Algebra

AU - Calvert, Kieran

AU - De Martino, Marcelo

N1 - Funding Information:
This research was supported by Heilbronn Institute for Mathematical Research and the special research fund (BOF) from Ghent University [BOF20/PDO/058]. We would also like to thank Roy Oste for the many discussions while preparing this manuscript and the anonymous referees for their comments and corrections, which greatly improved the manuscript. In particular, we would like to thank them for inspiring us to add Proposition 6.8 which guarantees that the theory of Dirac operators for the AMA is not a vacuous theory.
Publisher Copyright:
© 2022, Institute of Mathematics. All rights reserved.

PY - 2022

Y1 - 2022

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.

KW - Calogero–Moser angular momentum

KW - Dirac operators

KW - rational Cherednik algebras

UR - http://www.scopus.com/inward/record.url?scp=85133846588&partnerID=8YFLogxK

U2 - 10.3842/SIGMA.2022.040

DO - 10.3842/SIGMA.2022.040

M3 - Article

AN - SCOPUS:85133846588

SN - 1815-0659

VL - 18

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

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

M1 - 040

ER -