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 -