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.

The centre of the Dunkl total angular momentum algebra

“…Our main result (Theorem 4.8) is a full description of the centre of O t,c for any real reflection group W acting on V and for any parameter function c. As an application to the determination of the centre, and inspired by the successful Dirac-theories for Drinfeld algebras [9] we prove in Theorem 5.10 and 5.17 results analagous to the celebrated ideas of Vogan on Dirac cohomologies (see [35], [26], [9]). Our results build on Dirac theories for subalgebras of the Cherednik algebra (see [4] and [5]). The theory in this paper shares similarities with the Dunkl angular momentum algebra [5], in the sense that we consider a family of operators depending on certain "admissible" elements (see Definition 5.7).…”

“…Our results build on Dirac theories for subalgebras of the Cherednik algebra (see [4] and [5]). The theory in this paper shares similarities with the Dunkl angular momentum algebra [5], in the sense that we consider a family of operators depending on certain "admissible" elements (see Definition 5.7). We remark that in the present case, instead of enlarging the algebra in question with a suitably defined Clifford algebra and define a theory using a Dirac element, we do not tensor O t,c with another Clifford algebra and we use a natural element inside the algebra O t,c itself (see Definition 5.1 -classically when c = 0, the eigenvalues of this element are precisely the square root of the total angular momentum quantum numbers), this reflects the theory for Hecke-Clifford algebras as in [6].…”

“…The proof of this theorem, given Theorem 5.10, is identical to [5,Theorem 5.11] and [1,Theorem 4.5]. We refer the reader to [5] if they wish to see details.…”

“…Proof. The proof here employs identical ideas to the proof in [5,Theorem 5.4]. By Theorem 4.8 the centre of O t,c is polynomial in the element…”

“…Recall that w 0 denotes the longest element of W . In [5,20] (see also [21,Remark 3.3]) it was proved that Z(A t,c ) is the univariate polynomial ring…”