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...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2022 |
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Інститут математики НАН України
2022
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| Цитувати: | Dirac Operators for the Dunkl Angular Momentum Algebra. Kieran Calvert and Marcelo De Martino. SIGMA 18 (2022), 040, 18 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859619106645344256 |
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| author | Calvert, Kieran De Martino, Marcelo |
| author_facet | Calvert, Kieran De Martino, Marcelo |
| citation_txt | Dirac Operators for the Dunkl Angular Momentum Algebra. Kieran Calvert and Marcelo De Martino. SIGMA 18 (2022), 040, 18 pages |
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| description | 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.
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| first_indexed | 2026-03-14T06:49:40Z |
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| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 040, 18 pages
Dirac Operators for the Dunkl Angular Momentum
Algebra
Kieran CALVERT a and Marcelo DE MARTINO b
a) Department of Mathematics, University of Manchester, UK
E-mail: kieran.calvert@manchester.ac.uk
b) Department of Electronics and Information Systems, University of Ghent, Belgium
E-mail: marcelo.goncalvesdemartino@ugent.be
Received November 10, 2021, in final form May 24, 2022; Published online June 01, 2022
https://doi.org/10.3842/SIGMA.2022.040
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.
Key words: Dirac operators; Calogero–Moser angular momentum; rational Cherednik alge-
bras
2020 Mathematics Subject Classification: 16S37; 17B99; 20F55; 81R12
1 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
mailto:kieran.calvert@manchester.ac.uk
mailto:marcelo.goncalvesdemartino@ugent.be
https://doi.org/10.3842/SIGMA.2022.040
2 K. Calvert and M. De Martino
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 for W̃ ,
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
D0 = 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 DC depending on certain central elements C of CW̃
(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 = Sn.
2 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.
Remark 2.1. Under the isometry B : E → E∗, we have α = 2B(α∨)|α∨|−2 and 2 = |α||α∨|.
Further, if {y1, . . . , yn} ⊂ E is an orthornormal basis then {x1, . . . , xn} ⊂ E∗ is an orthonormal
basis, where xi = B(yi) for all i, and the pairings are related via
⟨xi, α∨⟩ = B(yi, α
∨) =
2
|α|2
B∗(xi, α) =
2
|α|2
⟨α, yi⟩ =
|α∨|2
2
⟨α, yi⟩, (2.1)
for all 1 ≤ i ≤ n.
Fix, once and for all, a positive system R+ ⊆ R and let c : R+ → C be a parameter function,
that is, an assignment α 7→ cα ∈ C such that cα = cwα for all w ∈ W . Let ∆ be the simple
roots determined by R+. Denote by h = EC and h∗ = E∗
C. For any α ∈ R, the element sα is the
reflection in W acting by
sα(y) = y − ⟨α, y⟩α∨,
for all y ∈ E.
Dirac Operators for the Dunkl Angular Momentum Algebra 3
Definition 2.2 ([11]). The rational Cherednik algebra H = H(h,W, c) is the quotient of the
smash product algebra T(h∗ ⊕ h)#W modulo the relations [x, x′] = 0 = [y, y′] and
[y, x] = ⟨x, y⟩+
∑
α>0
cα⟨α, y⟩⟨x, α∨⟩sα,
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 {y1, . . . , yn} ⊂ E and let {x1, . . . , xn} ⊂ E∗ be the dual basis, i.e.,
with xi = B(yi) for all i. Consider the vector notation x := (x1, . . . , xn) and y := (y1, . . . , yn)
with the usual dot product of vectors. As customary, we shall write x2 for x · x and similarly
for y2. It is well-known (see [15]) that the elements H := 1
2(x · y + y · x), X := −1
2x
2 and
Y := 1
2y
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 Mij := xiyj − xjyi 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
Sij := [yi, xj ] = δij +
∑
α>0
cα⟨α, yj⟩⟨xi, α∨⟩sα ∈ CW
and let Z :=
∑
α>0 cαsα. 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 Sij = Sji,
for all i, j.
Lemma 2.4. We have∑
i
Sii =
∑
i
[yi, xi] = n+ 2Z. (2.2)
Proof. Using Sii = [yi, xi] = 1 +
∑
α>0 cα⟨α, yi⟩⟨xi, α∨⟩sα and the identity
∑
i⟨α, yi⟩⟨xi, α∨⟩ =
⟨α, α∨⟩ = 2, the claim follows. ■
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
[M ij ,Mkl] = M ilSjk +M jkSil −M ikSjl −M jlSik (2.3)
and the crossing-relations
M ijMkl +M jkM il +MkiM jl = M ijSkl +M jkSil +MkiSjl (2.4)
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 {Mij | 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 ).
4 K. Calvert and M. De Martino
Let w0 be the longest element of W with respect to the simple roots ∆. Then, −w0 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 w0 = (−1)h, then w0 is in the centre of W and acts on h and h∗ by −1
and hence trivially on ∧2h.
Lemma 2.9. The only elements of W which act trivially on ∧2h are, respectively, 1W and (−1)h
if w0 = (−1)h, and only 1W if w0 ̸= (−1)h.
Proof. Note that, by Schur’s lemma, the only elements of the orthogonal group acting trivially
on ∧2h 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 Ωsl(2) := H2+2(XY +Y X).
The centre of A is given in terms of Ωsl(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[Ωsl(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, G0 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 → WG.
All that said, if Z is in the center of A, then Z commutes with every generator Mij of A and
hence Z ∈ WG0 . Further, as Z is also in the image of φ, it follows that Z = φ(Z̃) for some
Z̃ ∈ U(g)G0 = U(g)G, which then implies that Z ∈ WG. 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[Ωsl(2)] on the Casimir with
coefficients R = C[(−1)h], if w0 = (−1)h, or R = C otherwise.
Proof. The proof can be split in two cases; either w0 = (−1)h or not. In the later, the proof is
identical to [13, Theorem 5]. For the remainder of this proof we assume that w0 = (−1)h. Since
the element (−1)h is in the centre of W and acts by 1 on ∧2h it is in the centre of A. We are left
to prove that the subalgebra generated by Ωsl(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 H = C[h] ⊗ C[h∗] ⊗ CW , let F0 be the
highest degree component occurring in F , say, of degree d. We can write F0 =
∑
w∈W pww,
where pw is a homogeneous polymonomial on the basis {x1, . . . , xn, y1, . . . , yn} of degree d and
w ∈ W . We claim pw = 0 unless w = 1W or w = (−1)h. Suppose not and let w ∈ W \ {1W , w0}.
By Lemma 2.9, w does not act trivially on ∧2h and hence there exists an Mij such that w(Mij) ̸=
Mij , thus [F,Mij ] has terms of degree bigger than d. However, as F is in the centre of A we get
[F,Mij ] = 0. This contradiction proves that the only group elements occurring in F0 are 1W
or (−1)h.
Furthermore, the top degree elements in [F0,Mij ] agrees with the top degree elements of
[F,Mij ]. Therefore, modulo lower order terms F0 is in Z(A). Write F0 = p−1(−1)h + p11W . We
claim that p1 and p−1 are polynomials over C in the variable Ωsl(2). Note that the top degree
elements of [p−1,Mij ] and [p1,Mij ] agree with the classical commutators (at c = 0). The classical
Dirac Operators for the Dunkl Angular Momentum Algebra 5
center (c = 0) is generated by Ωsl(2) and we can write p−1 and p1 as the corresponding elements
in the classical center, modulo lower degree terms. We have thus proved that, modulo lower
degree terms, F0 is in the algebra C[(−1)h][Ωsl(2)] and F −F0 has lower degree. By induction F
is in R[Ωsl(2)] and we are done. ■
Remark 2.12. The above result is not novel. In [13], it was shown that for W = Sn, the centre
of A is equal to the univariate polynomial ring on the angular Calogero–Moser Hamiltonian,
which coincide with Ωsl(2), modulo lower degree terms (see Remark 2.15, below). In [14, Re-
mark 3.3], the above theorem was stated, without proof, for general W . We decided to present
the argument here for completeness.
Now let M2 :=
∑
i<j M
2
ij ∈ A be the Dunkl angular momentum square. In what comes next,
we shall compute the precise relationship between M2 and the Casimir Ωsl(2). Recall the central
element Z =
∑
α>0 cαsα of CW .
Proposition 2.13. The Dunkl angular momentum square satisfy the identity
M2 = x2y2 − (x · y)2 − (x · y)(2Z + n− 2).
Proof. This is equation (2.14) in [13], we add further details for completeness. Define Q :=∑
i,j xixjyiyj and Σ :=
∑
α>0 cααα
∨sα. Here, we see α ∈ h∗ and α∨ ∈ h as elements of H.
Explictly, α =
∑
i⟨α, yi⟩xi and similarly for α∨. We note the identities∑
i,j
xi[yj , xi]yj = (x · y)− Σ (2.5)
(where we used Sij = [yi, xj ] = [yj , xi] = Sji and αsαα
∨ = −αα∨sα) and∑
i,j
xi[yj , xj ]yi = n(x · y) + 2(x · y)Z − 2Σ. (2.6)
That said, we compute
(x · y)2 =
∑
i,j
xiyixjyj = Q+ (x · y)− Σ. (2.7)
Further, using (2.5), (2.6) and (2.7), we get
M2 =
∑
i<j
M2
ij
=
∑
i,j
x2i y
2
j − (xixjyiyj) + xi[yj , xi]yj − xi[yj , xj ]yi
= x2y2 −Q+ ((x · y)− Σ)− (n(x · y) + 2(x · y)Z − 2Σ)
= x2y2 − (x · y)2 − (x · y)(2Z + n− 2),
where we used −Q+ (x · y)− Σ = −(x · y)2 + 2(x · y)− 2Σ. This finishes the proof. ■
Proposition 2.14. The Dunkl angular momentum square and the Casimir are related via the
identity
Ωsl(2) = −M2 + Z(Z + n− 2) + n(n−4)
4 = −M2 +
(
Z + n−2
2
)2 − 1.
6 K. Calvert and M. De Martino
Proof. We start by noting that the elementH = 1
2(x·y+y·x) can be written asH = x·y+n
2+Z.
Since x · y commutes with Z we have that H2 = (x · y)2 + (2Z + n)(x · y) +
(
Z + n
2
)2
. Next,
note that similarly to (2.5), using [yj , xi] = [yi, xj ], we have the identities∑
i,j
xiyj [yj , xi] =
∑
i,j
xiyj
(
δij +
∑
α>0
cα⟨α, yi⟩⟨xj , α∨⟩sα
)
= (x · y) + Σ
and ∑
i,j
[yj , xi]yjxi =
∑
i,j
(
δij +
∑
α>0
cα⟨α, yi⟩⟨xj , α∨⟩sα
)
yjxi = (y · x) + Σ′,
where Σ′ =
∑
α>0 cαα
∨αsα. Similarly we have
∑
i,j yj [yj , xi]xi = (y ·x)−Σ′. Recall, X := −1
2x
2
and Y := 1
2y
2. Using
[
y2, x2
]
= [y, x]yx+ y[y, x]x+ xy[y, x] + x[y, x]y, we get
(−4)(XY + Y X) =
∑
i,j
x2i y
2
j + y2jx
2
i
= 2
(
x2y2
)
+ 2(x · y + y · x)
= 2
(
x2y2
)
+ 4x · y + 2n+ 4Z,
from which
Ωsl(2) = H2 + 2(XY + Y X)
= (x · y)2 + (2Z + n)(x · y) +
(
Z + n
2
)2 − (
x2y2
)
− 2(x · y)− n− 2Z
= −M2 +
(
Z + n
2
)2 − n− 2Z
= −M2 +
(
Z + n−2
2
)2 − 1,
as required. ■
Remark 2.15. Comparing the computations above for Ωsl(2) and the computations in [13] for
the angular Calogero–Moser Hamiltonian HΩ, we get
Ωsl(2) = 2HΩ + 1
4n(n− 4).
3 Clifford algebra and AMA-Dirac elements
Let CR = CR(E,B) denote the Clifford algebra associated to the pair (E,B). The Clifford
algebra CR is the quotient of the tensor algebra TR(E) = ⊕i≥0T
i(E) on E modulo the ideal
generated by the expressions
y ⊗ y′ + y′ ⊗ y − 2B(y, y′)
for all y, y′ ∈ E (see [20] for more details). Furthermore, with respect to the canonical map
ι : E → CR, the pair (CR, 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 homomorphism φ̃ : CR → A such that φ̃ι = φ. For each 1 ≤ j ≤ n, let cj := ι(yj), where
{y1, . . . , yn} is our fixed orthonormal basis of E. Then, CR is generated by {c1, . . . , cn}, with
Clifford relations
{ci, cj} := (cicj + cjci) = 2B(yi, yj) = 2δij , (3.1)
for all 1 ≤ i, j ≤ n.
Dirac Operators for the Dunkl Angular Momentum Algebra 7
3.1 Pin cover of W
The reference for this part is [20]. We have the Z2-grading CR = C0
R ⊕C1
R, where C0
R is the image
of ⊕i≥0T
2i(E) ⊂ T (E) while C1
R is the image of the odd powers in the tensor algebra. We let
ε : CR → CR denote the automorphism which acts as the identity on C0
R and minus the identity
on C1
R. The anti-automorphism (·)t of TR(E) that sends η = η1⊗· · ·⊗ηp to ηt = ηp⊗· · ·⊗η1, for
all η1, . . . , ηp ∈ E, descends to an anti-automorphism of CR, called the transpose. Furthermore,
let ∗ denote the anti-automorphism η∗ = ε(ηt), for all η ∈ CR and let N(η) = η∗η, for η ∈ CR,
denote the spinorial norm. Recall that the group Γ = Γ(E,B) defined by
Γ =
{
η ∈ C×
R | ε(η)yη−1 ∈ E for all y ∈ E
}
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
1 −→ R× −→ Γ
p−→ O −→ 1 (3.2)
is a short exact sequence. The pinorial group Pin = Pin(E,B) is given by
Pin =
{
η ∈ Γ | N(η)2 = 1
}
⊂ Γ
and the sequence (3.2) restricts to a short exact sequence
1 −→ {±1} −→ Pin
p−→ O −→ 1. (3.3)
The Pin-cover of W ⊂ O is defined as W̃ := p−1(W ) ⊂ O. Given a coroot α∨ ∈ R∨ ⊂ E, recall
that we can write α∨ =
∑
i⟨xi, α∨⟩yi. Using (2.1), note that
1
|α∨| ι(α
∨) = 1
|α∨|
∑
i
B(yi, α
∨)ci =
1
|α|
∑
i
B∗(xi, α)ci =
1
|α| ι
(
B−1(α)
)
.
We are thus justified to abuse the notation and define, for any α ∈ R,
s̃α := |α∨|−1α∨ ∈ CR.
One can show (see [20, Proposition 2.6]) that p(s̃α) = sα. Further, p
−1(sα) = {±s̃α}. Then, with
respect to generators and relations, we have that (see [20, Theorem 4.2]), on the one hand W
has presentations
W = ⟨sα, α ∈ R | s2α = 1, sαsβsα = sγ , γ = sα(β)⟩,
W = ⟨sα, α ∈ ∆ | (sαsβ)mα,β = 1⟩
while the double-cover has presentations
W̃ = ⟨θ, s̃α, α ∈ R | s̃2α = 1 = θ2, s̃αs̃β s̃α = σs̃γ , γ = sα(β), θ central⟩, (3.4)
W̃ = ⟨θ, s̃α, α ∈ ∆ | (s̃αs̃β)mα,β = (θ)mα,β−1, θ central⟩. (3.5)
We let C = CR ⊗ C be the complexification. Letting θ = −1 ∈ C the group W̃ is a subgroup
of Pin ⊂ C. However, the group algebra CW̃ does not inject into C. Decomposing the identity
as two idempotents 1 = 1
2(1+ θ)+ 1
2(1− θ), the group algebra CW̃ splits as a direct sum of two
algebras
CW̃ = CW̃+ ⊕ CW̃−, (3.6)
8 K. Calvert and M. De Martino
where the central element θ is specialised to either +1 or −1 in CW̃+ and CW̃− respectively.
The algebra CW̃+ is isomorphic to CW . Following [18], we refer to the algebra CW̃− as the
twisted group algebra.
Note that C has the same presentation by generators and relations as in (3.1). As is well-
known, if n = dimR(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(ηt) defined above. Finally, we let ρ : CW̃ → A⊗C
denote the homomorphism obtained from the diagonal embedding of W̃ defined by
ρ(w̃) = p(w̃)⊗ w̃ (3.7)
for all w̃ ∈ W̃ and extended linearly, where p : W̃ → W is the double-cover projection map
and w̃ is considered as an element in Pin ⊂ C.
3.2 AMA-Dirac elements
Both algebras H and C contain a copy of the vector space ∧2h with basis {M ij | 1 ≤ i < j ≤ n}.
In H, these are realised by the elements Mij = xiyj − xjyi for 1 ≤ i < j ≤ n that forms part
of the generating set of A and in C they are realised by quadratic elements cicj ∈ 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 Mij or w ∈ W .
Definition 3.1. The Dirac element of the angular momentum algebra is defined by
D =
∑
i<j
Mij ⊗ cicj ∈ A⊗ C.
For brevity, we shall refer to this element as the AMA-Dirac element.
Proposition 3.2. The AMA-Dirac element is independent of the choice of orthonormal basis
{y1, . . . , yn} made. In particular, it is ρ
(
W̃
)
-invariant.
Proof. The ρ
(
W̃
)
-invariance follows from the independence of the basis since conjugating
D by ρ(s̃α) = sα ⊗ s̃α causes us to write the expression for D with respect to the bases
{sα(y1), . . . , sα(yn)} and {sα(x1), . . . , sα(xn)}.
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 Qjkyk and x′j = B(y′j) =
∑
k Qjkxk
where the collection {Qjk | 1 ≤ j, k ≤ n} satisfy
∑
k QikQjk = δij . It is then straightforward to
check that
2D′ =
∑
i,j
M ′
ij ⊗ c′ic
′
j =
∑
k,l
Mkl ⊗ ckcl = 2D,
where M ′
ij = x′iy
′
j − x′jy
′
i ∈ A and c′i = ι(y′i) ∈ C. ■
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 Ωsl(2), modulo a constant. Before we compute D2, we shall need some preliminary
computations.
Let Π =
{
(i, j) ∈ Z2; 1 ≤ i < j ≤ n
}
. Note that we can write the Cartesian product as the
disjoint union
Π2 = Π0 ∪Π1 ∪Π2 (3.8)
Dirac Operators for the Dunkl Angular Momentum Algebra 9
where Πq :=
{
((i, j), (k, l)) ∈ Π2; |{i, j} ∩ {k, l}| = q
}
, for q ∈ {0, 1, 2}. If π = (i, j) ∈ Π, we
shall write cπ = cicj in the Clifford algebra and Mπ = Mij in A. Then,
D2 =
∑
(π,σ)∈Π2
MπMσ ⊗ cπcσ = Σ0 +Σ1 +Σ2, (3.9)
where Σq is the sum over Πq, in the decomposition (3.8).
Lemma 3.3. With notations as in (3.9), we have Σ2 = −M2 and Σ0 = 0.
Proof. As (cicj)
2 = −1 when i ̸= j, it immediately follows that Σ2 = −M2. As for Σ0, to each
pair ((i, j), (k, l)) ∈ Π0, noting that [cicj , ckcl] = 0, after ordering the 4-tuple i < j < k < l, and
fixing the Clifford element cicjckcl to the right-hand side of the tensor product, the contribution
on the left-hand side becomes
(MijMkl +MklMij −MikMjl −MjlMik +MilMjk +MjkMil)⊗ cicjckcl,
from which we obtain
Σ0 =
∑
1≤i<j<k<l≤n
2(MijMkl +MjkMil +MkiMjl)⊗ cicjckcl
+ ([Mkl,Mij ] + [Mil,Mjk] + [Mjl,Mki])⊗ cicjckcl.
Using the relation (2.3) of A and the symmetry Sab = Sba for any indices a, b, we obtain
[Mkl,Mij ] + [Mil,Mjk] + [Mjl,Mki] = −2(MijSkl +MjkSil +MkiSjl),
from which, using now (2.4), we obtain Σ0 = 0. ■
Lemma 3.4. With notations as in (3.9), we have Σ1 = (n− 2)D + {D, Z}.
Proof. Each pair ((i, j), (k, l)) ∈ Π1 has exactly three distinct entries. Using the Clifford
relations, each product cπcσ with (π, σ) ∈ Π1 reduces to a product of the type cicj , for distinct
indices i, j. For example, cickcjck = −cicj and so on. Moreover, we can label the sum Σ1 in
terms of ordered triples (i < j < k) and we obtain
Σ1 =
∑
i<j<k
[Mik,Mij ]⊗ cjck + [Mij ,Mjk]⊗ cick + [Mjk,Mik]⊗ cicj ,
which, after applying the relations of A and the symmetry Sab = Sba for the indices, yields
Σ1 =
∑
i<j<k
{
(MjkSii −MjiSik −MikSji)⊗ cjck + (MikSjj −MijSjk −MjkSij)⊗ cick
+ (MijSkk −MikSkj −MkjSik)⊗ cicj
}
.
Thus, each Clifford element cicj , contributes to the sum Σ1 with the quantity C(i, j) ∈ A given
by
C(i, j) =
∑
k/∈{i,j}
(MijSkk −MikSkj −MkjSik)
= Mij(n+ 2Z)−
n∑
k=1
(MikSkj +MkjSik).
10 K. Calvert and M. De Martino
Furthermore, denoting ϵ(i, j) =
∑n
k=1(MikSkj +MkjSik), we obtain
ϵ(i, j) = 2Mij +
∑
α>0
cα(α(⟨xi, α∨⟩yj − ⟨xj , α∨⟩yi)− (⟨α, yi⟩xj − ⟨α, yj⟩xi)α∨)sα
= 2Mij +
∑
α>0
cα(Mijsα − sαMij).
We conclude, therefore, that
Σ1 =
∑
i<j
C(i, j)⊗ cicj
= (n− 2)D + 2DZ + [Z,D],
and the claim follows from {Z,D} = 2DZ + [Z,D]. ■
Theorem 3.5. We have
D2 = −M2 + (n− 2)D + {D, Z}.
Proof. Follows directly from the previous lemmas and the identity (3.9). ■
Corollary 3.6. Let ϕ := 1
2(2Z+n−2). The element D0 = (D−ϕ) is a square root of a Casimir
element of sl(2).
Proof. We compute directly to get
D2
0 = D2 + ϕ2 − {D, ϕ}
= −M2 + (n− 2)D + {D, Z} −
{
D, Z + n−2
2
}
+
(
Z + n−2
2
)2
= Ωsl(2) + 1,
as required, where use was made of Proposition 2.14 in the last equality. ■
4 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
D =
∑
i
yi ⊗ ci, x =
∑
i
xi ⊗ ci,
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).
Proposition 4.1. We have the following identity:
−2D = [D,x]− (n+ 2Z) = [D,x]− 2(ϕ+ 1).
Proof. Using that, for all i ̸= j, we have yixj−yjxi = xjyi+Sij−xiyj−Sji = xjyi−xiyj = −Mij
in H, it is straightforward to compute
[D,x] =
∑
i,j
(yixj ⊗ cicj − xjyi ⊗ cjci)
Dirac Operators for the Dunkl Angular Momentum Algebra 11
=
∑
i,j
((yixj + xjyi)⊗ cicj − xjyi ⊗ 2δij
=
∑
i
[yi, xi]⊗ 1 +
∑
i<j
(yixj + xjyi − yjxi − xiyj)⊗ cicj
= (n+ 2Z)⊗ 1− 2D,
where, in the last equation, we used (2.2). The claim now follows immediately. ■
Corollary 4.2. As elements of H ⊗ C, the AMA-Dirac element and the SCasimir of osp(1|2)
satisfy D + S = 1
2 + ϕ.
Proof. With our notational conventions, the SCasimir of osp(1|2) is given by S = 1
2([D,x]− 1)
(see [9, equation (3.3)]). The claim follows from the previous proposition. ■
5 Vogan’s conjecture and Dirac cohomology
Inspired by [2] we prove an analogue of Vogan’s conjecture in the context of the angular mo-
mentum 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 a K̃-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.
5.1 An analogue of Vogan’s conjecture
Denote by ZW̃ the centre of CW̃ . Denote also by ∗ the anti-linear involution of C defined in
Section 3.1 restricted to W̃ and extended anti-linearly to a star operation on CW̃ .
Definition 5.1. Let W̃ be the double cover of W . If w0 ̸= (−1)h, we define Ŵ = W̃ . Else, we
define Ŵ = W̃ × C2, where, abusing notation, (−1)h generates C2. In this case, we extend the
homomophism ρ of (3.7) to Ŵ → A⊗C by ρ((−1)h) = (−1)h ⊗ 1. Furthermore, extend ∗ to Ŵ
by (−1)∗h = (−1)−1
h = (−1)h.
When w0 = (−1)h, the algebra CŴ is a central extension of CW̃ . In this case, there is
a 2-to-1 map from ZŴ to ZW̃ .
Definition 5.2. An element C ∈ CŴ is called admissible if C is central and C∗ = C. For any
admissible C ∈ ZŴ , define
DC := (D − ϕ) + ρ(C), (5.1)
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.
Theorem 5.4. Given an admissible C ∈ CŴ , there is an algebra homomorphism
ζC : Z(A) → ZŴ
such that, for all z ∈ Z(A) there exists a ∈ A⊗C satisfying
z ⊗ 1 = ρ(ζC(z)) +DCa+ aDC .
12 K. Calvert and M. De Martino
Proof. In this proof, we abbreviate Ω = Ωsl(2). Because Z(A) = R[Ω] has a very simple
algebraic structure, we can give a straightforward proof, without having to use the more sophis-
ticated ideas from [17]. Let γ := ρ
(
C2
)
− 1 ∈ ρ
(
ZŴ
)
. We show, by induction, that for every
m ∈ Z≥1, there is am ∈ A⊗C such that
Ωm ⊗ 1 = γm + {DC , am}.
Indeed, since C ∈ CW̃ we have that ρ(C) commutes with D0 and DC . Thus
D2
C = D2
0 + ρ(C)2 + 2D0ρ(C)
= Ω + 1 + ρ(C)(ρ(C) + 2D0)
= Ω + 1− ρ(C)2 + 2ρ(C)DC ,
from which we conclude that upon defining a1 :=
1
2DC − ρ(C), we have
Ω⊗ 1 = Ω = γ + {DC , a1}.
Note that a1 commutes withDC . Now assume we have Ωm = γm+{DC , am} for some am ∈ A⊗C
that commutes with DC . It is then straightforward to compute that
Ωm+1 = (γm + {DC , am})(γ + {DC , a1})
= γm+1 + {DC , am+1},
with am+1 := amγ+a1γ
m+2DCama1. Therefore, the homomorphism ζC is defined by ζC(Ω) =(
C2 − 1
)
and ζC((−1)h) = (−1)h, if w0 = (−1)h. ■
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 self-
adjoint.
5.2 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 = yi,
y•i = xi 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(w̃)⋆ = ρ(w̃∗) for any w̃ ∈ 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
(σ(η)s1, s2)S = (s1, σ(η
∗)s2)S ,
for all η ∈ C and s1, s2 ∈ 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
DC = (π⊗ σ)(DC), for admissible C ∈ ZŴ , are self-adjoint. Furthermore, if X ⊗S is unitary,
then (
D2
C(x⊗ s), x⊗ s
)
X⊗S
≥ 0
for all x ∈ X and all s ∈ S.
Dirac Operators for the Dunkl Angular Momentum Algebra 13
Proof. It is straightforward to check that M•
ij = −Mij and (cicj)
∗ = −(cicj), 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 Mc(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 Mc(τ) is a unitary H-module. For such
parameters, the modules Xc(τ)m = ker(∆c) ∩ Mc(τ)m are irreducible unitary A-modules (see
[7, Theorem B]), where ∆c is the Dunkl–Laplacian and Mc(τ)m are the homogeneous elements
of degree m of Mc(τ). Let λ(c, τ,m) = m+ n
2 +Nc(τ), where Nc(τ) is the scalar on which the
central element Z =
∑
α>0 cαsα acts on τ . Then, the Casimir Ω = Ωsl(2) acts on Xc(τ)mby 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 Mc(τ) satisfy χ ≥ −1.
5.3 Dirac cohomology
In the proof of Theorem 5.4, we computed the square
D2
C = Ωsl(2) −
(
ρ(C)2 − 1
)
+ 2ρ(C)DC , (5.2)
for any admissible C ∈ ZŴ . Thus, in the kernel of a Dirac operator DC = (π ⊗ σ)(DC) ∈
End(X ⊗ S), where (π,X) is an A-module, we get the equation
(π ⊗ σ)(Ωsl(2)) = (π ⊗ σ)
(
ρ(C)2 − 1
)
.
Suppose w0 = (−1)h. Then w0 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 DC , it acts on the kernel of a Dirac operator DC 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 DC .
Definition 5.8. Let (π,X) be an A-module and C be an admissible element in ZW̃ . The Dirac
cohomology of C is defined by
H(X,C) =
ker(DC)
ker(DC) ∩ im(DC)
,
where DC = (π ⊗ σ)(DC) ∈ End(X ⊗ S).
Proposition 5.9. The Dirac cohomology of C is a Ŵ -module. Moreover, if X is a •-Hermitian
A-module, then H(X,C) = ker(DC).
Proof. Clear, as DC is ρ(Ŵ )-invariant and DC 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.
14 K. Calvert and M. De Martino
Definition 5.11. Let C ∈ ZŴ be an admissible element and ζC : Z(A) → ZŴ be the homo-
morphism of Theorem 5.4. For any irreducible Ŵ representation τ̂ , define the homomorphism
χτ̂ : Z(A) → C via
χτ̂ (z) =
1
dim τ̂
Tr
(
τ̂(ζC(z))
)
,
for any z in Z(A).
Theorem 5.12. Let C ∈ ZŴ be an admissible element, τ̂ be an irreducible Ŵ representation
and (π,X) be an A-module with central character χ. Suppose that
HomŴ
(
τ̂ , H(X,C)
)
̸= 0.
Then, χ = χτ̂ .
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 the τ̂ -isotypic component of X⊗S which is in ker(DC) but not in im(DC). For any
z ∈ Z(A), since both z ⊗ 1 and ρ(ζC(z)) act by a scalar on ξ, we get, using Theorem 5.4 that
(π ⊗ σ)(z ⊗ 1− ρ(ζC(z)))ξ = (π ⊗ σ)(DCa+ aDC)ξ
= (π ⊗ σ)(DCa)ξ
= 0,
since otherwise ξ would be in the image of DC , 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 DC , so DCa + aDC = 2aDC . Thus, the last bit of the proof of the previous theorem can
be simplified, in our context.
6 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
C2 :=
1
4
∑
α,β>0
cαcβ s̃αs̃β ∈ CW̃ ⊂ CŴ . (6.1)
Proposition 6.1. The element C2 of (6.1) is admissible.
Proof. Note that the element Z̃ = 1
2
∑
α>0 cαs̃α is such that C2 = Z̃2. Since s̃∗α = θs̃α for any
α ∈ R, we get Z̃∗ = θZ̃ and hence C∗
2 =
(
Z̃2
)∗
= C2. Recall θ is the central involution in W̃ (see
presentation (3.4)). Equation (3.6) shows that CW̃ splits into CW ⊕ CW̃− and Z̃ decomposes
as
Z̃ = 1
2(1 + θ)Z̃ + 1
2(1− θ)Z̃,
where the element 1
2(1 + θ)Z̃ is equal to the central element Z = 1
2
∑
α>0 cαsα of CW and we
denote T := 1
2(1− θ)Z̃. Hence, we are left to prove that the element T 2 = 1
2(1− θ)Z̃2 is central
in CW̃−. Let τα = 1
2(1− θ)s̃α for any α ∈ R. Presentation (3.5) shows that the simple ‘pseudo’
reflections τα, α ∈ ∆ generate CW̃−. Hence, it is sufficient to prove that T 2 commutes with
Dirac Operators for the Dunkl Angular Momentum Algebra 15
every simple pseudo reflection. To that end, note that we can express T in terms of the pseudo
reflections as
1
2(1− θ)Z̃ = T = 1
2
∑
α>0
cατα.
Further, for β ∈ ∆, write R+ = Rβ∪Γβ where Rβ = R+∩sβ(R−) = {β} and Γβ = R+∩sβ(R+) =
R+ \ {β} is the complement. Write also
Γ̃β = 1
2
∑
α∈Γβ
cατα (6.2)
so that T = 1
2cβτβ + Γ̃β ∈ CW̃−. Since β is simple then sβ permutes the elements of Γβ, so, in
view of the presentation (3.4), and using the invariance of the parameter function c, we conclude
that, the anti-commutator
{
τβ, Γ̃β
}
= 0 in CW̃− and hence
T 2 = 1
4cβ
2τ2β + Γ̃2
β +
{
1
2cβτβ, Γ̃β
}
= 1
4cβ
2τ2β + Γ̃2
β −
{
1
2cβτβ, Γ̃β
}
= sβ(T )
2.
It then follows that τβZ̃
2 = sβ
(
Z̃
)2
τβ = Z̃2τβ in CW̃−, and we are done. ■
Definition 6.2. Define elements
Ti =
1
2
∑
α∈R+
cα
⟨xi, α∨⟩
|α∨|
sα and T •
i = 1
2
∑
α∈R+
cα
⟨α, yi⟩
|α|
sα ∈ CW.
Using equation (2.1) then ⟨xi,α
∨⟩
|α∨| = |α∨|2⟨α,yi⟩
2|α∨| = ⟨α,yi⟩
|α| . Hence T •
i = Ti. Furthermore,
ρ(Z̃) = 1
2
∑
α>0
cαsα ⊗ s̃α = 1
2
n∑
α>0,i=1
cαsα ⊗ B(yi, α
∨)
|α∨|
ci
= 1
2
n∑
α>0,i=1
cαsα ⊗ ⟨xi, α∨⟩
|α∨|
ci =
n∑
i=1
Ti ⊗ ci.
Proposition 6.3. The element C2 is such that
ρ(C2) =
∑
i<j
(TiT
•
j − TjT
•
i )⊗ cicj + Z3,
where Z3 =
1
4
∑
α,β>0 cα|α∨|−1cβ|β|−1⟨β, α∨⟩sαsβ is a central element in CW .
Proof. Note that ρ(C2) = ρ
(
Z̃2
)
=
(∑n
i=1 Ti ⊗ ci
)2
. Using the super-commutation rela-
tions (3.1) of ci, we obtain that ρ(C2) =
∑
i<j [Ti, Tj ] ⊗ cicj +
∑n
i=1 T
2
i . Now T 2
i = TiT
•
i =
1
4
∑
α,β>0 cαcβ
⟨xi,α
∨⟩
|α∨|
⟨β,yi⟩
|β| sαsβ. Therefore,
n∑
i=1
T 2
i = 1
4
n∑
i=1
∑
α,β>0
cα
|α∨|
cβ
|β|
⟨xi, α∨⟩⟨β, yi⟩sαsβ = 1
4
∑
α,β>0
cα
|α∨|
cβ
|β|
⟨β, α∨⟩sαsβ,
as required. ■
Corollary 6.4. Let DC2 be the Dirac operator as defined in Definition 5.2. Then DC2 can be
expressed as:
DC2 =
∑
i<j
(xiyj − xjyi)⊗ cicj +
∑
i<j
(TiT
•
j − TjT
•
i )⊗ cicj + Z3 − ϕ.
16 K. Calvert and M. De Martino
Remark 6.5. Alternatively, we could slightly modify the generators of A, writing M̃ij = Mij +
TiT
•
j − TjT
•
i . Then we can write the Dirac operator as
DC2 =
∑
i<j
M̃ij ⊗ cicj + Z3 − ϕ.
The expression for DC2 above reflects an equivalent definition for the Dirac operator of the
degenerate affine Hecke algebra [2]. There
DH =
n∑
i=1
xi ⊗ ci +
n∑
i=1
Ti ⊗ ci,
where the elements xi are the commuting generators in the Luzstig presentation of H and
Ti = −1
2
∑
α>0 cαB
∗(xi, α)sα.
However, in the AMA-Dirac we must modify by the central element Z3 − ϕ ∈ Z(W ). This
can be interpreted as an analogue of the modification Kostant makes (see [19]) when defining
the cubic Dirac operator.
We will finish by exploring the set of admissible sets in the case when W = Sn. Recall that
the algebra CW̃ is a Z2-graded algebra, where the generators s̃α are given odd degree and θ even
degree. The idempotent 1
2(1− θ) is central and Z2-homogenous, and hence CW̃− = 1
2(1− θ)CW̃
is also Z2-graded. Recall also that the simple ‘pseudo’ reflections τα = 1
2(1 − θ)s̃α, α ∈ ∆
generate CW̃−. The decomposition of the group algebra CW̃ = RW̃ ⊕ iRW̃ is preserved under
the multiplication by the idempotent 1
2(1− θ). Therefore, CW̃− = RW̃− ⊕ iRW̃−.
Proposition 6.6. Let W be the symmetric group Sn. The set of admissible elements in RW̃−
is the even centre of RW̃−.
Proof. Let A = A
(
RW̃−
)
be the set of admissible elements in RW̃−. If C ∈ A then C is central
and fixed by the involution ∗. Since τ∗α = −τα in RW̃− then any odd central element is not
admissible. Hence, A is contained in Z0
(
RW̃−
)
, the even centre of RW̃−. Now, the even centre
Z0(RW̃−) =
{
σ
(
M2
1 , . . . ,M
2
n
)
| σ real symmetric polynomial
}
,
where Mi are the Jucys–Murphy elements (see [18, Lemma 13.1.2, Remark 13.1.3] and references
within). The elements Mi are odd so M∗
i = −Mi and
(
M2
i
)∗
= M2
i ∈ Z0
(
RW̃−
)
. Therefore,
any expression in the squares of the Jucys–Murphy elements is ∗-invariant and thus, Z0
(
RW̃−
)
is contained in A. ■
Remark 6.7. When W = Sn, we have shown that the set of admissible elements in RW̃−
is equal to the even centre Z0
(
RW̃−
)
. From [18], Z0
(
RW̃−
)
is equal to
{
σ
(
M2
1 , . . . ,M
2
n
)
|
σ real symmetric polynomial
}
. The element σ
(
M2
1 , . . . ,M
2
n
)
acts on an irreducible representa-
tion by evaluating σ at specific real values [3, Corollary 6.3]. In particular, we can conclude that
for every CW̃ -module there is an admissible element that does not act by zero.
Proposition 6.8. Let A be a Dunkl angular momentum algebra with Weyl group W = Sn,
and let (π,X) be a •-unitary module for A, such that the spectrum of π(Ωsl(2)) is contained in
[−1,+∞). Then, there exists an admissible C ∈ CW̃− such that H(X,C) ̸= 0.
Proof. Firstly, it follows from Proposition 5.9 that, with our assumptions, H(X,C) = kerDC ,
where DC = π ⊗ σ(DC) and C is an admissible element.
Secondly, as operators on the space X ⊗ S, the element Ωsl(2) commutes with ρ(C) for
every admissible element C. Therefore, there exist simultaneous eigenspaces for Ωsl(2) and
Dirac Operators for the Dunkl Angular Momentum Algebra 17
{ρ(C) | C admissible}. Let U be a nonzero eigenspace where Ωsl(2) acts by the scalar u(Ωsl(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.
To that end, note that upon changing C to
(
u(C)−1
√
u(Ωsl(2)) + 1
)
C, we may assume that,
as operators on U , we have
Ωsl(2) = ρ(C)2 − 1.
Equation (5.2) states that D2
C = Ωsl(2) −
(
ρ(C)2 − 1
)
+ 2ρ(C)DC . Hence, on the eigenspace U
we have D2
C − 2u(C)DC = 0 from which we conclude that
{0} ≠ U ⊆ ker((DC − 2u(C)) ◦DC).
Using the fact that the composition of injective maps is injective, it is not possible that both
kerDC and ker(DC − 2u(C)) are equal to zero. Now using that
D(−C) = D0 − ρ(C) = D0 + ρ(C)− 2ρ(C) = DC − 2ρ(C),
when restricted to U , we obtain D(−C) = DC − 2u(C). Therefore, ker(DC′) ̸= 0 for some choice
of C ′ ∈ {C,−C}. ■
Example 6.9. Let Xc(τ)m ⊂ Mc(τ) be the harmonic polynomials of degree m as defined in
Example 5.7. Then the spectrum of Ωsl(2) on Xc(τ)m is contained in [−1,+∞) for every m.
Therefore, when W = Sn (with n ≥ 3), for every Xc(τ)m there exists an admissible C such that
H(Xc(τ)m, C) ̸= 0.
Acknowledgements
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.
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1 Introduction
2 Preliminaries
3 Clifford algebra and AMA-Dirac elements
3.1 Pin cover of W
3.2 AMA-Dirac elements
4 AMA-Dirac and the SCasimir of osp(1|2)
5 Vogan's conjecture and Dirac cohomology
5.1 An analogue of Vogan's conjecture
5.2 Unitary structures
5.3 Dirac cohomology
6 Examples of non-trivial admissible elements
References
|
| id | nasplib_isofts_kiev_ua-123456789-211628 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T06:49:40Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Calvert, Kieran De Martino, Marcelo 2026-01-07T13:41:04Z 2022 Dirac Operators for the Dunkl Angular Momentum Algebra. Kieran Calvert and Marcelo De Martino. SIGMA 18 (2022), 040, 18 pages 1815-0659 2020 Mathematics Subject Classification: 16S37; 17B99; 20F55; 81R12 arXiv:2110.01353 https://nasplib.isofts.kiev.ua/handle/123456789/211628 https://doi.org/10.3842/SIGMA.2022.040 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. This research was supported by the 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 vacuous. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Dirac Operators for the Dunkl Angular Momentum Algebra Article published earlier |
| spellingShingle | Dirac Operators for the Dunkl Angular Momentum Algebra Calvert, Kieran De Martino, Marcelo |
| title | Dirac Operators for the Dunkl Angular Momentum Algebra |
| title_full | Dirac Operators for the Dunkl Angular Momentum Algebra |
| title_fullStr | Dirac Operators for the Dunkl Angular Momentum Algebra |
| title_full_unstemmed | Dirac Operators for the Dunkl Angular Momentum Algebra |
| title_short | Dirac Operators for the Dunkl Angular Momentum Algebra |
| title_sort | dirac operators for the dunkl angular momentum algebra |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211628 |
| work_keys_str_mv | AT calvertkieran diracoperatorsforthedunklangularmomentumalgebra AT demartinomarcelo diracoperatorsforthedunklangularmomentumalgebra |