Dirac Operators for the Dunkl Angular Momentum Algebra

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.


Introduction
Let (E, B) be a Euclidean space and consider the action of partial differential operators with polynomial coefficients in the space of polynomial functions C[E]. This framework is very fruitful and yields many applications most importantly to physics. Angular momentum, for instance, is a fundamental property of particle dynamics and the quantum angular momentum operators are realized within this setup. We consider the situation in which the partial differential operators are deformed to differential-difference operators, the so-called Dunkl operators. For this, we also need a real reflection group W inside the orthogonal group O(E, B) and a parameter function c on the conjugacy classes of reflections of W . Together, the pair (W, c), the Dunkl operators and the multiplication operators generate the so-called rational Cherednik algebra (see Definition 2.2) inside the endomorphism space of the polynomial ring C [E].
The subalgebra of the Cherednik algebra generated by W and the Dunkl angular momentum operators is called the Dunkl angular momentum algebra (see Definition 2.5). In [13], Feigin and Hakobyan obtained important structural results about this algebra. In particular they obtained all the defining relations and showed that its centre is, essentially, a univariate polynomial ring on the angular part of the Calogero-Moser Hamiltonian (see also [14,Remark 3.3]). Later in [7], it was shown that this algebra naturally arises in the context of deformed Howe dualities as the centralizer algebra of the Dunkl-Cherednik version of the polynomial sl(2)-triple obtained from the Laplacian and the norm-squared operator. It is then clear that the angular Calogero-Moser Hamiltonian is, up to scalars, the Casimir operator of sl(2) (see Remark 2.15,below).
In this paper, inspired by the successful theory of Dirac operators for Lie theory [1,17,19,21,22] and Drinfeld algebras [2,4,5,6,8], we propose to define a theory of Dirac operators for the Dunkl angular momentum algebra. In slightly more details, we work with the Clifford algebra associated to (E, B) and we define the Dirac element D inside the tensor product of the angular momentum algebra and the Clifford algebra. We then show that this element is invariant forW , the Pin-cover of the Weyl group W , and that by a suitable modification φ (see Definition 5.1), akin to the one made by Kostant [19] in the context of cubic Dirac operators, the element D 0 = D − φ is essentially a square-root of the Casimir of sl(2) (see Corollary 3.6). Furthermore, we introduce a family of Dirac operators D C depending on certain central elements C of CW (see Definition 5.2) with respect to which we prove an analogue of Vogan's conjecture (see Theorem 5.4) and, using the celebrated notion of Dirac cohomology (see Definition 5.8), we show that the Dirac cohomology, when non-zero, determines the central character of representations of the angular momentum algebra (see Theorem 5.12). We expect that such results can aid in a systematic study of the representation theory of the angular momentum algebra, since its representation theory, just like for the rational Cherednik algebra, is highly dependant on the parameter function c.
Finally, we give a break-down of the contents of the paper. In Section 2, we recall the definition of the rational Cherednik algebra, introduce the angular momentum algebra and obtain a linear relation between the Casimir of sl(2) and the angular Calogero-Moser Hamiltonian. Next, in Section 3 we recall the definitions of the Clifford algebra, the Pin-cover of the Weyl group and we introduce the Dirac elements of the angular momentum algebra. The highlight of this section is the computation of the square. Afterwards, in Section 4 we relate our Dirac element D with the SCasimir of the closely related algebra osp(1|2) (see De Bie et al. [9,10] for the explicit realization) while in Section 5, we prove the main results on Vogan's conjecture and Dirac cohomology. In the last section, we describe and study a non-trivial example of an admissible central element that yields a Dirac operator and relate such element with the Dirac operator obtained by [2], in the context of a graded affine Hecke algebra. We also discuss the set of admssible elements in the case when W = S n .

Preliminaries
Let (E, B) be a Euclidean space affording the reflection representation of a finite reflection group W ⊂ O(E, B). Put n = dim(E). Let R ⊆ E * denote the root system of W and R ∨ ⊆ E its dual root system normalized by the condition α, α ∨ = 2, for all α in R, where −, − : E * × E → R denotes the natural pairing. We shall identify E and E * isometrically using the Euclidean structure B and we denote by B * the inherited Euclidean structure on E * . This identification B : E → E * is defined by B(y), η = B(y, η) for all y, η ∈ E.
Further, if {y 1 , . . . , y n } ⊂ E is an orthornormal basis then {x 1 , . . . , x n } ⊂ E * is an orthonormal basis, where x i = B(y i ) for all i, and the pairings are related via Fix, once and for all, a positive system R + ⊆ R and let c : R + → C be a parameter function, that is, an assignment α → c α ∈ C such that c α = c wα for all w ∈ W . Let ∆ be the simple roots determined by R + . Denote by h = E C and h * = E * C . For any α ∈ R, the element s α is the reflection in W acting by s α (y) = y − α, y α ∨ , for all y ∈ E.
for all y, y ∈ h and x, x ∈ h * . Remark 2.3. More generally, rational Cherednik algebras are defined with respect to finite complex reflection groups inside the unitary group with respect to the Hermitian extension of B. However, for the existence of the sl(2)-triple and the Duality Theorem stated below, it is fundamental that W is a real reflection group.
Fix an orthonormal basis {y 1 , . . . , y n } ⊂ E and let {x 1 , . . . , x n } ⊂ E * be the dual basis, i.e., with x i = B(y i ) for all i. Consider the vector notation x := (x 1 , . . . , x n ) and y := (y 1 , . . . , y n ) with the usual dot product of vectors. As customary, we shall write x 2 for x · x and similarly for y 2 . It is well-known (see [15]) that the elements H := 1 2 (x · y + y · x), X := − 1 2 x 2 and Y := 1 2 y 2 of H satisfy the sl(2)-commutation relations and span a copy of sl(2, C) inside H. On the other hand, consider the Dunkl angular momentum elements M ij := x i y j − x j y i of H for 1 ≤ i, j ≤ n. Note that they span a vector space isomorphic to ∧ 2 (h). For each pair (i, j) with 1 ≤ i, j ≤ n define and let Z := α>0 c α s α . Note that Z is in the centre of CW since the parameter function c is uniform on conjugacy classes of reflections. Since W is a real reflection group, we get S ij = S ji , for all i, j.
Definition 2.5 ( [13]). Let {M ij | 1 ≤ i < j ≤ n} be a vector space basis of ∧ 2 (h). The Dunkl angular momentum algebra A(h, W, c) is the quotient of the smash product algebra T ∧ 2 (h) #W modulo the commutation relations and the crossing-relations for all 1 ≤ i, j, k, l ≤ n. Note that M ii = 0 for any i = 1, . . . , n.
In what follows, we shall refer to this algebra only as the angular momentum algebra, or just AMA. The relevance of this subalgebra of H is manifested by the following fact (see [13] and [7]): Theorem 2.6. The associative subalgebra A of H generated by the elements {M ij | 1 ≤ i < j ≤ n} and W is isomorphic to the angular momentum algebra A(h, W, c). Furthermore, A is the centralizer algebra in H of the sl(2)-triple (H, X, Y ).
Let w 0 be the longest element of W with respect to the simple roots ∆. Then, −w 0 acts on the root system R and it is an automorphism of the associated Dynkin diagram. Definition 2.7. We will denote by (−1) h the element −Id ∈ End(h).
Remark 2.8. If we have w 0 = (−1) h , then w 0 is in the centre of W and acts on h and h * by −1 and hence trivially on ∧ 2 h. Lemma 2.9. The only elements of W which act trivially on ∧ 2 h are, respectively, Proof . Note that, by Schur's lemma, the only elements of the orthogonal group acting trivially on ∧ 2 h are ±Id. The statement follows from this observation.
Since (H, X, Y ) span a Lie algebra isomorphic to sl(2, C), the associative algebra subalgebra of H generated by this triple contains the quadratic Casimir element Ω sl(2) := H 2 +2(XY +Y X). The centre of A is given in terms of Ω sl (2) and, possibly, (−1) h . Lemma 2.10. When c = 0 and W is the trivial group, the center of A is the univariate polynomial ring C[Ω sl (2) ].
Proof . This statement is a consequence of classical invariant theory (see [16] and [23]), but we provide the argument for completeness. In the present situation, A is a subalgebra of the Weyl algebra W acting in the space of polynomial functions C[E]. Furthermore, if we denote by G the orthogonal group, G 0 its identity component (the special orthogonal group), g = Lie(G) and U(g) the universal enveloping algebra, then there is a G-equivariant homomorphism ϕ : U(g) → W, whose image coincides with A. By equivariance, it follows that ϕ maps U(g) G → W G .
All that said, if Z is in the center of A, then Z commutes with every generator M ij of A and hence Z ∈ W G 0 . Further, as Z is also in the image of ϕ, it follows that Z = ϕ(Z) for somẽ Z ∈ U(g) G 0 = U(g) G , which then implies that Z ∈ W G . By classical invariant theory, Z is thus in the associative subalgebra of W generated by the sl(2)-triple (H, X, Y ), from which the statement follows.
Theorem 2.11. The centre of A is equal to the polynomial ring R[Ω sl (2) ] on the Casimir with Proof . The proof can be split in two cases; either w 0 = (−1) h or not. In the later, the proof is identical to [13,Theorem 5]. For the remainder of this proof we assume that w 0 = (−1) h . Since the element (−1) h is in the centre of W and acts by 1 on ∧ 2 h it is in the centre of A. We are left to prove that the subalgebra generated by Ω sl(2) , (−1) h and the constants is the full centre Z(A).
Let F be an arbitrary element in Z(A). With respect to the usual filtration of H whose associated graded object gives the PBW isomorphism where p w is a homogeneous polymonomial on the basis {x 1 , . . . , x n , y 1 , . . . , y n } of degree d and w ∈ W . We claim p w = 0 unless w = 1 W or w = (−1) h . Suppose not and let w ∈ W \ {1 W , w 0 }. By Lemma 2.9, w does not act trivially on ∧ 2 h and hence there exists an M ij such that w(M ij ) = M ij , thus [F, M ij ] has terms of degree bigger than d. However, as F is in the centre of A we get [F, M ij ] = 0. This contradiction proves that the only group elements occurring in F 0 are 1 W or (−1) h . Furthermore, the top degree elements in [F 0 , M ij ] agrees with the top degree elements of [F, M ij ]. Therefore, modulo lower order terms F 0 is in Z(A). Write F 0 = p −1 (−1) h + p 1 1 W . We claim that p 1 and p −1 are polynomials over C in the variable Ω sl (2) . Note that the top degree elements of [p −1 , M ij ] and [p 1 , M ij ] agree with the classical commutators (at c = 0). The classical center (c = 0) is generated by Ω sl(2) and we can write p −1 and p 1 as the corresponding elements in the classical center, modulo lower degree terms. We have thus proved that, modulo lower degree terms, F 0 is in the algebra C[(−1) h ][Ω sl (2) ] and F − F 0 has lower degree. By induction F is in R[Ω sl (2) ] and we are done.
Remark 2.12. The above result is not novel. In [13], it was shown that for W = S n , the centre of A is equal to the univariate polynomial ring on the angular Calogero-Moser Hamiltonian, which coincide with Ω sl (2) , modulo lower degree terms (see Remark 2.15,below). In [14, Remark 3.3], the above theorem was stated, without proof, for general W . We decided to present the argument here for completeness.
ij ∈ A be the Dunkl angular momentum square. In what comes next, we shall compute the precise relationship between M 2 and the Casimir Ω sl (2) . Recall the central element Z = α>0 c α s α of CW .
Proposition 2.14. The Dunkl angular momentum square and the Casimir are related via the identity Proof . We start by noting that the element H = 1 2 (x·y+y·x) can be written as H = x·y+ n 2 +Z. Since x · y commutes with Z we have that H 2 = (x · y) 2 + (2Z + n)(x · y) + Z + n 2 2 . Next, note that similarly to (2.5), using [y j , as required.

Clifford algebra and AMA-Dirac elements
for all y, y ∈ E (see [20] for more details). Furthermore, with respect to the canonical map ι : E → C R , the pair (C R , ι) satisfies the universal property, that, for any unital R-algebra A and any linear map ϕ : E → A satisfying ϕ(y)ϕ(y ) + ϕ(y )ϕ(y) = 2B(y, y ), there is a unique algebra homomorphismφ : C R → A such thatφι = ϕ. For each 1 ≤ j ≤ n, let c j := ι(y j ), where {y 1 , . . . , y n } is our fixed orthonormal basis of E. Then, C R is generated by {c 1 , . . . , c n }, with Clifford relations for all 1 ≤ i, j ≤ n.

Pin cover of W
The reference for this part is [20]. We have the Z 2 -grading R is the image of the odd powers in the tensor algebra. We let ε : C R → C R denote the automorphism which acts as the identity on C 0 R and minus the identity on C 1 R . The anti-automorphism (·) t of T R (E) that sends η = η 1 ⊗ · · · ⊗ η p to η t = η p ⊗ · · · ⊗ η 1 , for all η 1 , . . . , η p ∈ E, descends to an anti-automorphism of C R , called the transpose. Furthermore, let * denote the anti-automorphism η * = ε(η t ), for all η ∈ C R and let N (η) = η * η, for η ∈ C R , denote the spinorial norm. Recall that the group Γ = Γ(E, B) defined by is the so-called twisted Clifford group and the homomorphism p : Γ → O = O(E, B), defined via p(η)y = ε(η)yη −1 , for all η ∈ Γ and y ∈ h, is such that the sequence is a short exact sequence. The pinorial group Pin = Pin(E, B) is given by and the sequence (3.2) restricts to a short exact sequence We are thus justified to abuse the notation and define, for any α ∈ R, One can show (see [20,Proposition 2.6]) that p(s α ) = s α . Further, p −1 (s α ) = {±s α }. Then, with respect to generators and relations, we have that (see [20,Theorem 4.2]), on the one hand W has presentations We let C = C R ⊗ C be the complexification. Letting θ = −1 ∈ C the groupW is a subgroup of Pin ⊂ C. However, the group algebra CW does not inject into C. Decomposing the identity as two idempotents 1 = 1 2 (1 + θ) + 1 2 (1 − θ), the group algebra CW splits as a direct sum of two algebras where the central element θ is specialised to either +1 or −1 in CW + and CW − respectively. The algebra CW + is isomorphic to CW . Following [18], we refer to the algebra CW − as the twisted group algebra. Note that C has the same presentation by generators and relations as in (3.1). As is wellknown, if n = dim R (E), then C has one (resp. two) equivalence classes of complex irreducible representations of dimension 2 n/2 for n even (resp. n odd). Let also * denote the anti-linear extension to C of the anti-involution η * = ε(η t ) defined above. Finally, we let ρ : CW → A ⊗C denote the homomorphism obtained from the diagonal embedding ofW defined by for allw ∈W and extended linearly, where p :W → W is the double-cover projection map andw is considered as an element in Pin ⊂ C.

AMA-Dirac elements
Both algebras H and C contain a copy of the vector space ∧ 2 h with basis {M ij | 1 ≤ i < j ≤ n}.
In H, these are realised by the elements M ij = x i y j − x j y i for 1 ≤ i < j ≤ n that forms part of the generating set of A and in C they are realised by quadratic elements c i c j ∈ C. In what follows, we may use the short hand notation Y to denote Y ⊗ 1 ∈ A ⊗C for any Y ∈ A. For example, Y may be M ij or w ∈ W .
Definition 3.1. The Dirac element of the angular momentum algebra is defined by For brevity, we shall refer to this element as the AMA-Dirac element. Proof . The ρ W -invariance follows from the independence of the basis since conjugating D by ρ(s α ) = s α ⊗s α causes us to write the expression for D with respect to the bases {s α (y 1 ), . . . , s α (y n )} and {s α (x 1 ), . . . , s α (x n )}. The proof for the independence of the choice of basis is standard, and we briefly recall the steps. If {y 1 , . . . , y n } is another choice, we have y j = k Q jk y k and x j = B(y j ) = k Q jk x k where the collection {Q jk | 1 ≤ j, k ≤ n} satisfy k Q ik Q jk = δ ij . It is then straightforward to check that As in every Dirac theory, we now compute the square of the AMA-Dirac element. We will show that upon subtracting a correction term this element yields a square-root of the Casimir Ω sl(2) , modulo a constant. Before we compute D 2 , we shall need some preliminary computations.
Let Π = (i, j) ∈ Z 2 ; 1 ≤ i < j ≤ n . Note that we can write the Cartesian product as the disjoint union where Π q := ((i, j), (k, l)) ∈ Π 2 ; |{i, j} ∩ {k, l}| = q , for q ∈ {0, 1, 2}. If π = (i, j) ∈ Π, we shall write c π = c i c j in the Clifford algebra and M π = M ij in A. Then, where Σ q is the sum over Π q , in the decomposition (3.8). Proof . As (c i c j ) 2 = −1 when i = j, it immediately follows that Σ 2 = −M 2 . As for Σ 0 , to each pair ((i, j), (k, l)) ∈ Π 0 , noting that [c i c j , c k c l ] = 0, after ordering the 4-tuple i < j < k < l, and fixing the Clifford element c i c j c k c l to the right-hand side of the tensor product, the contribution on the left-hand side becomes Using the relation (2.3) of A and the symmetry S ab = S ba for any indices a, b, we obtain from which, using now (2.4), we obtain Σ 0 = 0. Proof . Each pair ((i, j), (k, l)) ∈ Π 1 has exactly three distinct entries. Using the Clifford relations, each product c π c σ with (π, σ) ∈ Π 1 reduces to a product of the type c i c j , for distinct indices i, j. For example, c i c k c j c k = −c i c j and so on. Moreover, we can label the sum Σ 1 in terms of ordered triples (i < j < k) and we obtain which, after applying the relations of A and the symmetry S ab = S ba for the indices, yields Thus, each Clifford element c i c j , contributes to the sum Σ 1 with the quantity C(i, j) ∈ A given by Furthermore, denoting (i, j) = n k=1 (M ik S kj + M kj S ik ), we obtain We conclude, therefore, that  Proof . Follows directly from the previous lemmas and the identity (3.9). Proof . We compute directly to get = Ω sl(2) + 1, as required, where use was made of Proposition 2.14 in the last equality.

AMA-Dirac and the SCasimir of osp(1|2)
Now recall (see for example [9] and [10]) that the algebra H ⊗ C contains a copy of the Lie superalgebra osp(1|2) spanned by the Lie triple (H, X, Y ) ⊂ H together with the elements of H ⊗ C. The element D is often referred to as the Dunkl-Dirac operator, as it squares to the Dunkl-Laplace operator when viewed as an operator on the polynomial space. Next, we relate the AMA-Dirac element with the SCasimir S of osp(1|2). Proof . Using that, for all i = j, we have where, in the last equation, we used (2.2). The claim now follows immediately.

Vogan's conjecture and Dirac cohomology
Inspired by [2] we prove an analogue of Vogan's conjecture in the context of the angular momentum algebra A. Vogan's original conjecture states that if the Dirac cohomology for a (g, K) module X is non-zero then the infinitesimal character of X can be described in terms of the highest weight of aK-type in the cohomology. This conjecture was proved in [17]. However, in our context, instead of a single Dirac operator relating the center of the algebra in question and the centre of (the double-cover of) the Weyl group, we shall construct a family of operators depending on central elements.

An analogue of Vogan's conjecture
Denote by ZW the centre of CW . Denote also by * the anti-linear involution of C defined in Section 3.1 restricted toW and extended anti-linearly to a star operation on CW .
When w 0 = (−1) h , the algebra CŴ is a central extension of CW . In this case, there is a 2-to-1 map from ZŴ to ZW .
Definition 5.2. An element C ∈ CŴ is called admissible if C is central and C * = C. For any admissible C ∈ ZŴ , define where D is the AMA-Dirac element and φ = 1 2 (2Z + n − 2). Remark 5.3. The set of admissible elements has the structure of a real vector space, and it is not empty as C = 0 is admissible. In the next section we shall exhibit and study a more interesting admissible element.
Proof . In this proof, we abbreviate Ω = Ω sl (2) . Because Z(A) = R[Ω] has a very simple algebraic structure, we can give a straightforward proof, without having to use the more sophisticated ideas from [17]. Let γ := ρ C 2 − 1 ∈ ρ ZŴ . We show, by induction, that for every m ∈ Z ≥1 , there is a m ∈ A ⊗C such that Indeed, since C ∈ CW we have that ρ(C) commutes with D 0 and D C . Thus from which we conclude that upon defining a 1 : Note that a 1 commutes with D C . Now assume we have Ω m = γ m +{D C , a m } for some a m ∈ A ⊗C that commutes with D C . It is then straightforward to compute that with a m+1 := a m γ + a 1 γ m + 2D C a m a 1 . Therefore, the homomorphism ζ C is defined by ζ C (Ω) = Remark 5.5. In the proof of the previous theorem we only used that C was in CŴ . The conditions on admissibility are needed below to ensure that the operators we obtain are selfadjoint.

Unitary structures
Let * denote the anti-linear anti-involution of C defined in Section 3.1. Let also • be the restriction to A of the anti-linear anti-involution of H characterized on the generators by x • i = y i , y • i = x i and w • = w −1 , for all 1 ≤ i ≤ n and w ∈ W , where we recall that we have fixed orthonormal bases of E and E * . We then define an anti-linear anti-involution on A ⊗C by taking the tensor product of these two anti-involutions. It is straightforward to check that ρ(w) = ρ(w * ) for anyw ∈Ŵ , where ρ : CŴ → A ⊗C is the homomorphism of Definition 5.1. Now fix, once and for all, (σ, S) an irreducible module for C. Endow S with a unitary structure (−, −) S , i.e., a complex inner product on S that is also * -Hermitian for all η ∈ C and s 1 , s 2 ∈ S. For any •-Hermitian module (π, X) of A we endow X ⊗ S with a -Hermitian structure (x ⊗ s, x ⊗ s ) X⊗S = (x, x ) X (s, s ) S for all x, x ∈ X and s, s ∈ S. If the -Hermitian form on X ⊗ S is also positive definite, then we say X ⊗ S is unitary. We define operators in End(X ⊗ S) by taking the image of the AMA-Dirac elements under π ⊗ σ. Proposition 5.6. If (π, X) is a •-Hermitian A-module, then the operators D = (π ⊗ σ)(D) and D C = (π ⊗ σ)(D C ), for admissible C ∈ ZŴ , are self-adjoint. Furthermore, if X ⊗ S is unitary, then for all x ∈ X and all s ∈ S.
Proof . It is straightforward to check that M • ij = −M ij and (c i c j ) * = −(c i c j ), from which we get that the AMA-Dirac element is invariant for the -involution and thus D is indeed Hermitian. Also, it is straightforward to check that φ • = φ and the claims follow since C is admissible.
Example 5.7. Fix τ an irreducible representation of W and let M c (τ ) be the standard module at τ for the rational Cherednik algebra H with dim(E) ≥ 2. For real parameter functions c close enough to c = 0, it is known (see [12]) that M c (τ ) is a unitary H-module. For such parameters, the modules X c (τ ) m = ker(∆ c ) ∩ M c (τ ) m are irreducible unitary A-modules (see [7,Theorem B]), where ∆ c is the Dunkl-Laplacian and M c (τ ) m are the homogeneous elements of degree m of M c (τ ). Let λ(c, τ, m) = m + n 2 + N c (τ ), where N c (τ ) is the scalar on which the central element Z = α>0 c α s α acts on τ . Then, the Casimir Ω = Ω sl(2) acts on X c (τ ) m by the scalar χ = λ(c, τ, m)(λ(c, τ, m) − 2). From the previous proposition, with C = 0, we get that the parameter function c for unitary M c (τ ) satisfy χ ≥ −1.

Dirac cohomology
In the proof of Theorem 5.4, we computed the square for any admissible C ∈ ZŴ . Thus, in the kernel of a Dirac operator D C = (π ⊗ σ)(D C ) ∈ End(X ⊗ S), where (π, X) is an A-module, we get the equation Suppose w 0 = (−1) h . Then w 0 is central in A. Hence (−1) h acts by a scalar on X and X ⊗ S. The element (−1) h squares to 1 and therefore this scalar is 1 or −1. Because (−1) h commutes with D C , it acts on the kernel of a Dirac operator D C by the same scalar. We can relate the action of the whole centre Z(A) with the isotypic component of irreducible C Ŵ -representations occuring in the kernel of D C .
Definition 5.8. Let (π, X) be an A-module and C be an admissible element in ZW . The Dirac cohomology of C is defined by , Proposition 5.9. The Dirac cohomology of C is aŴ -module. Moreover, if X is a •-Hermitian A-module, then H(X, C) = ker(D C ).
Proof . Clear, as D C is ρ(Ŵ )-invariant and D C is self-adjoint when X is •-Hermitian.
We finish this section by showing that the Dirac cohomology of C determines the central character of an A-module. To make this statement precise, we need some definitions. First, we say that an A-module (π, X) has central character χ : Z(A) → C if the centre z ∈ Z(A) acts by the scalar χ(z) on X.
Remark 5.10. Every irreducible A-module has a central character. However, to the best of the authors' knowledge, the representation theory of A is currently unknown and since A is the deformation of the image of the universal enveloping algebra of the Lie algebra so(n) into a smash-product of W and a Weyl algebra, there might be non-irreducible A-modules with central character resembling Verma modules.
Definition 5.11. Let C ∈ ZŴ be an admissible element and ζ C : Z(A) → ZŴ be the homomorphism of Theorem 5.4. For any irreducibleŴ representationτ , define the homomorphism χτ : Z(A) → C via Tr τ (ζ C (z)) , for any z in Z(A).
Proof . The proof is mutatis mutandis of the one in [2, Theorem 4.5], but we add the short proof here, for convenience. The assumption in the statement implies the existence of a non-zero element ξ in theτ -isotypic component of X ⊗ S which is in ker(D C ) but not in im(D C ). For any z ∈ Z(A), since both z ⊗ 1 and ρ(ζ C (z)) act by a scalar on ξ, we get, using Theorem 5.4 that since otherwise ξ would be in the image of D C , which it is not. The claim follows.
Remark 5.13. In the proof of Theorem 5.4 we actually proved that the element "a" commuted with D C , so D C a + aD C = 2aD C . Thus, the last bit of the proof of the previous theorem can be simplified, in our context.

Examples of non-trivial admissible elements
In this last section we explore the set of admissible elements. Throughout this section, we assume the parameter function c is real valued. Let c α c βsαsβ ∈ CW ⊂ CŴ . (6.1) Proposition 6.1. The element C 2 of (6.1) is admissible.
Proof . Note that ρ(C 2 ) = ρ Z 2 = n i=1 T i ⊗ c i 2 . Using the super-commutation relations (3.1) of c i , we obtain that ρ( as required.
Corollary 6.4. Let D C 2 be the Dirac operator as defined in Definition 5.2. Then D C 2 can be expressed as: {ρ(C) | C admissible}. Let U be a nonzero eigenspace where Ω sl(2) acts by the scalar u(Ω sl(2) ) and ρ(C) act by u(C) for C admissible. By Remark 6.7, there exists an admissible C such that u(C) = 0. We will show that there is λ ∈ R \ {0} such that H(X, C ) = 0 for C = λC.
Equation (5.2) states that D 2 C = Ω sl(2) − ρ(C) 2 − 1 + 2ρ(C)D C . Hence, on the eigenspace U we have D 2 C − 2u(C)D C = 0 from which we conclude that Using the fact that the composition of injective maps is injective, it is not possible that both ker D C and ker(D C − 2u(C)) are equal to zero. Now using that when restricted to U , we obtain D (−C) = D C − 2u(C). Therefore, ker(D C ) = 0 for some choice of C ∈ {C, −C}.
Example 6.9. Let X c (τ ) m ⊂ M c (τ ) be the harmonic polynomials of degree m as defined in Example 5.7. Then the spectrum of Ω sl(2) on X c (τ ) m is contained in [−1, +∞) for every m. Therefore, when W = S n (with n ≥ 3), for every X c (τ ) m there exists an admissible C such that H(X c (τ ) m , C) = 0.