Dirac Operators for the Dunkl Angular Momentum Algebra

Kieran Calvert^a, Marcelo De Martino<sup>b

a</sup> Department of Mathematics, University of Manchester, UK
^b Department of Electronics and Information Systems, University of Ghent, Belgium

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.

Keywords: Dirac operators, Calogero–Moser angular momentum, rational Cherednik algebras.

MSC: 16S37, 17B99, 20F55, 81R12

Received: November 10, 2021; in final form May 24, 2022; Published online June 1, 2022

Language: English

DOI: 10.3842/SIGMA.2022.040

Bibliographic databases:

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ArXiv: 2110.01353