Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type
We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skew-polynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W and two polynomial-Clifford subalgebras. There is y...
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Інститут математики НАН України
2009
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| Цитувати: | Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type / T. Khongsap, W. Wang // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 17 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860167109982552064 |
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| author | Khongsap, T. Wang, W. |
| author_facet | Khongsap, T. Wang, W. |
| citation_txt | Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type / T. Khongsap, W. Wang // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 17 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skew-polynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W and two polynomial-Clifford subalgebras. There is yet a third algebra containing a spin Weyl group algebra which is Morita (super)equivalent to the above two algebras. We establish the PBW properties and construct Verma-type representations via Dunkl operators for these algebras.
|
| first_indexed | 2025-12-07T17:57:07Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 012, 27 pages
Hecke–Clifford Algebras and Spin Hecke Algebras IV:
Odd Double Affine Type?
Ta KHONGSAP and Weiqiang WANG
Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA
E-mail: tk7p@virginia.edu, ww9c@virginia.edu
URL: http://people.virginia.edu/∼tk7p/, http://www.math.virginia.edu/∼ww9c/
Received October 15, 2008, in final form January 22, 2009; Published online January 28, 2009
doi:10.3842/SIGMA.2009.012
Abstract. We introduce an odd double affine Hecke algebra (DaHa) generated by a classical
Weyl group W and two skew-polynomial subalgebras of anticommuting generators. This
algebra is shown to be Morita equivalent to another new DaHa which are generated by W
and two polynomial-Clifford subalgebras. There is yet a third algebra containing a spin Weyl
group algebra which is Morita (super)equivalent to the above two algebras. We establish
the PBW properties and construct Verma-type representations via Dunkl operators for these
algebras.
Key words: spin Hecke algebras; Hecke–Clifford algebras; Dunkl operators
2000 Mathematics Subject Classification: 20C08
1 Introduction
1.1. The Dunkl operator [3], which is an ingenious mixture of differential and reflection ope-
rators, has found numerous applications to orthogonal polynomials, representation theory, non-
commutative geometry, and so on in the past twenty years. To a large extent, the Dunkl
operators helped to motivate the definition of double affine Hecke algebras of Cherednik, which
have played important roles in several areas of mathematics. In recent years, the representa-
tion theory of a degenerate version of the double affine Hecke algebra (known as the rational
Cherednik algebra or Cherednik–Dunkl algebra) has been studied extensively ([5, 4]; see the
review paper of Rouquier [14] for extensive references).
In [16], the second author initiated a program of constructing the so-called spin Hecke alge-
bras associated to Weyl groups with nontrivial 2-cocycles, by introducing the spin affine Hecke
algebra as well as the rational and trignometric double affine Hecke algebras associated to the
spin symmetric group of I. Schur [15]. Subsequently, in a series of papers [8, 9, 10, 17], the
authors have extended the constructions of [16] in several different directions.
The construction of [16, 10] provided two (super)algebras Ḧc
W and Ḧ−
W associated to any
classical Weyl group W which are Morita super-equivalent in the sense of [17]. These algebras
admit the following PBW type properties:
Ḧc
W
∼= C[h∗]⊗ Ch∗ ⊗ CW ⊗ C[h], Ḧ−
W
∼= C{h∗} ⊗ CW− ⊗ C[h].
Here we denote by h the reflection representation of W , by C[h∗] the polynomial algebra on h∗,
by Ch∗ a Clifford algebra, by C{h∗} a skew-polynomial algebra with anti-commuting generators,
and by CW− the spin Weyl group algebra associated to the element −1 in the Schur multiplier
H2(W, C∗).
?This paper is a contribution to the Special Issue on Dunkl Operators and Related Topics. The full collection
is available at http://www.emis.de/journals/SIGMA/Dunkl operators.html
mailto:tk7p@virginia.edu
mailto:ww9c@virginia.edu
http://people.virginia.edu/~tk7p/
http://www.math.virginia.edu/~ww9c/
http://dx.doi.org/10.3842/SIGMA.2009.012
http://www.emis.de/journals/SIGMA/Dunkl_operators.html
2 T. Khongsap and W. Wang
In contrast to the rational Cherednik algebra (cf. [5, 14]) which admits a nontrivial automor-
phism group, the construction of the algebras Ḧc
W is asymmetric as Ḧc
W contains as subalgebras
one polynomial algebra and one polynomial-Clifford subalgebras C[h∗] ⊗ Ch∗ (the polynomial-
Clifford algebra also appeared in the affine Hecke–Clifford algebra of type A introduced by
Nazarov [13]). Moreover, in type A case, Ḧc
W contains Nazarov’s algebra as a subalgebra,
see [16].
1.2. In the present paper, we introduce three new algebras Hcc
W , H−c
W and HW , which are shown
to be Morita (super)equivalent to each other and to have PBW properties as follows:
Hcc
W
∼= C[h]⊗ Ch ⊗ CW ⊗ Ch∗ ⊗ C[h∗],
H−c
W
∼= C{h} ⊗ CW− ⊗ Ch∗ ⊗ C[h∗],
HW
∼= C{h} ⊗ CW ⊗ C{h∗}.
A novel feature here is that the algebra Hcc
W contains two isomorphic copies of the polynomial-
Clifford subalgebra and there is an automorphism of Hcc
W which switches these two copies.
Similar remark applies to the algebra HW . We further show that the odd DaHa HW of type
A contains the degenerate affine algebra of Drinfeld and Lusztig as a subalgebra (see [5] for
a similar phenomenon).
It turns out that the number of parameters in the algebras Hcc
W , H−c
W and HW is equal to one
plus the number of conjugacy classes of reflections in W , which is the same as for the corre-
sponding rational Cherednik algebras and differs by one from the algebras introduced in [16, 10].
However, in contrast to the usual Cherednik algebras, we show that each of the algebras Hcc
W ,
H−c
W and HW contain large centers and are indeed module-finite over their respective centers.
1.3. In Section 2 we present a finite dimensional version of the Morita (super) equivalence
of the DaHa mentioned above, and introduce the necessary concepts such as spin Weyl group
algebras and Clifford algebras associated to the reflection representation h.
The Schur multipliers H2(W, C∗) for finite Weyl groups W were computed by Ihara and
Yokonuma [6] (cf. Karpilovsky [7, Theorem 7.2.2]). For example, H2(WBn , C∗) = Z2 × Z2 × Z2
for n ≥ 4. Given any finite Weyl group W (not necessarily classical) and any 2-cocycle α ∈
H2(W, C∗), we establish a superalgebra isomorphism (in two versions +, −)
Φ̇α
± : Ch∗ o± CW−α '−→ Ch∗ ⊗ CWα.
For the purpose of the rest of this paper, only the case when W is classical and α = ±1 is
needed. The special case when α = −1 was established in [9], and this special case was in turn
a generalization of a theorem of Sergeev and Yamaguchi for symmetric group.
We construct and study the algebras Hcc
W , H−c
W and HW in the next three sections, i.e., in
Sections 3, 4, and 5, respectively. Among other results, we establish the PBW properties as men-
tioned earlier and construct Verma-like representations of the three algebras via Dunkl operators.
Note in particular that a representation for HW (see Theorems 5.10, 5.13, and 5.14) is realized
on the skew-polynomial algebra with anti-commuting Dunkl operators. Anti-commuting Dunkl
operators first appeared in [16], also cf. [10]. In a very recent work [1], Bazlov and Berenstein
introduced a notion of braided Cherednik algebra where anti-commuting Dunkl operators also
make a natural appearance. After the second author communicated to them our construction
of HW for type A, they have also produced a similar algebra in their second version (cf. [1,
Corollary 3.7]).
Finally, in the Appendix A, we collect the proofs of several lemmas stated in Section 3 and
Section 5.
Hecke–Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type 3
2 Schur multipliers of Weyl groups and Clifford algebras
2.1 A distinguished double cover
As in [9, 10], we shall be concerned about a distinguished double covering W̃ of W :
1 −→ Z2 −→ W̃ −→ W −→ 1.
We denote by Z2 = {1, z}, and by t̃i a fixed preimage of the generators si of W for each i. The
group W̃ is generated by z, t̃1, . . . , t̃n with relations
z2 = 1, (t̃it̃j)mij =
{
1, if mij = 1, 3,
z, if mij = 2, 4, 6.
The quotient algebra CW− := CW̃/〈z + 1〉 of CW̃ by the ideal generated by z + 1 is called
the spin Weyl group algebra associated to W . Denote by ti ∈ CW− the image of t̃i. It follows
that CW− is isomorphic to the algebra generated by ti, 1 ≤ i ≤ n, subject to the relations
(titj)mij = (−1)mij+1 ≡
{
1, if mij = 1, 3,
−1, if mij = 2, 4, 6.
The algebra CW− has a natural superalgebra (i.e. Z2-graded) structure by letting each ti be
odd.
Example 2.1. Let W be the Weyl group of type An, Bn, or Dn, which will be considered
extensively in later sections. Then the spin Weyl group algebra CW− is generated by t1, . . . , tn
with relations listed in Table 2.1.
Table 2.1. The defining relations of CW−.
Type of W Defining Relations for CW−
An t2i = 1, titi+1ti = ti+1titi+1,
(titj)2 = −1 if |i− j| > 1
t1, . . . , tn−1 satisfy the relations for CW−
An−1
,
Bn t2n = 1, (titn)2 = −1 if i 6= n− 1, n,
(tn−1tn)4 = −1
t1, . . . , tn−1 satisfy the relations for CW−
An−1
,
Dn t2n = 1, (titn)2 = −1 if i 6= n− 2, n,
tn−2tntn−2 = tntn−2tn
2.2 Clifford algebra
Denote by h the reflection representation of the Weyl group W (i.e. a Cartan subalgebra of the
corresponding complex Lie algebra g). In the case of type An−1, we will always choose to work
with the Cartan subalgebra h = Cn of gln instead of sln in this paper.
Note that h carries a W -invariant nondegenerate bilinear form (−,−), which gives rise to an
identification h∗ ∼= h and also a bilinear form on h∗ which will be again denoted by (−,−). We
identify h∗ with a suitable subspace of CN in a standard fashion (cf. e.g. [9, Table in 2.3]). Then
describe the simple roots {αi} for g using a standard orthonormal basis {ei} of CN . It follows
that (αi, αj) = −2 cos(π/mij).
4 T. Khongsap and W. Wang
Denote by Ch∗ the Clifford algebra associated to (h∗, (−,−)), which is regarded as a subalgebra
of the Clifford algebra CN associated to (CN , (−,−)). We shall denote by ci the generator in CN
corresponding to
√
2ei and denote by βi the generator of Ch∗ corresponding to the simple root αi
normalized with β2
i = 1. In particular, CN is generated by c1, . . . , cN subject to the relations
c2
i = 1, cicj = −cjci if i 6= j.
For example, we have
βi =
1√
2
(ci − ci+1), 1 ≤ i ≤ n− 1
and an additional one
βn =
{
cn if W = WBn ,
1√
2
(cn−1 + cn) if W = WDn .
Note that N = n in the above three cases. For a complete list of βi for each Weyl group W , we
refer to [9, Section 2] for details.
The action of W on h and h∗ preserves the bilinear form (−,−) and thus W acts as auto-
morphisms of the algebra Ch∗ . This gives rise to a semi-direct product Ch∗ o CW . Moreover,
the algebra Ch∗ o CW naturally inherits the superalgebra structure by letting elements in W be
even and each βi be odd.
2.3 A superalgebra isomorphism
We recall the following result of Morris (the type A case goes back to Schur).
Proposition 2.2 ([12, 15]). Let W be a f inite Weyl group. Then, there exists a surjective
superalgebra homomorphism Ω : CW−−→Ch∗ which sends ti to βi for each i.
Given two superalgebras A and B, we view the tensor product of superalgebras A ⊗ B as
a superalgebra with multiplication defined by
(a⊗ b)(a′ ⊗ b′) = (−1)|b||a
′|(aa′ ⊗ bb′) (a, a′ ∈ A, b, b′ ∈ B),
where |b| denotes the Z2-degree of b, etc.
Now, let Cn o−CW− denote the algebra generated by the subalgebras Cn and CW− with the
following additional multiplication:
ticj = −csi
j ti ∀ i, j.
Note that Cn o− CW− has a natural superalgebra structure by setting each ci and tj to be odd
for all admissible i, j. We also endow a superalgebra structure on Ch∗ ⊗ CW by declaring all
elements of W to be even.
Theorem 2.3. We have an isomorphism of superalgebras:
Φ̇ : Ch∗ o− CW− '−→ Ch∗ ⊗ CW
which extends the identity map on Ch∗ and sends each ti to βisi. The inverse map Ψ̇ is an
extension of the identity map on Ch∗ and sends each si to βiti.
We first prepare a few lemmas.
Lemma 2.4. We have (Φ̇(ti)Φ̇(tj))mij = (−1)mij+1.
Hecke–Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type 5
Proof. Proposition 2.2 says that (titj)mij =(βiβj)mij =(−1)mij+1. Also recall that (sisj)mij =1.
Then we have
(Φ̇(ti)Φ̇(tj))mij = (βisiβjsj)mij = (βiβj)mij (sisj)mij = (−1)mij+1. �
Lemma 2.5. We have βjΦ̇(ti) = −Φ̇(ti) βsi
j for all i, j.
Proof. Note that (βi, βi) = 2β2
i = 2, and hence
βjβi = −βiβj + (βj , βi) = −βiβj +
2(βj , βi)
(βi, βi)
β2
i = −βiβ
si
j .
Thus, we have
βjΦ̇(ti) = βjβisi = −βiβ
si
j si = −βisiβ
si
j = −Φ̇(ti)βsi
j . �
Proof of Theorem 2.3. The algebra Ch∗ o− CW− is generated by βi and ti for all i. Lem-
mas 2.4 and 2.5 imply that Φ̇ is a (super) algebra homomorphism. Clearly Φ̇ is surjective, and
thus an isomorphism by a dimension counting argument.
Clearly, Ψ̇ and Φ̇ are inverses of each other. �
Let us denote by Ch∗⊕h the Clifford algebra associated to
((
h∗, (−,−)
)
⊕
(
h, (−,−)
))
, and re-
gard it as a subalgebra of the Clifford algebra C2N associated to
((
CN , (−,−)
)
⊕
(
(CN )∗, (−,−)
))
.
We shall denote by ei and νi the counterparts to ci and βi via the identification Ch∗
∼= Ch.
By [9, Theorem 2.4], there exists an isomorphism of superalgebras
Φ : Ch o CW → Ch ⊗ CW− (2.1)
which extends the identity map on Ch and sends each si to −
√
−1νiti. The isomorphism Φ was
due to Sergeev and Yamaguchi when W is the symmetric group.
Theorem 2.6. We have an isomorphism of superalgebras:
Φ̈ : Ch∗⊕h o CW
'−→ Ch∗⊕h ⊗ CW
which extends the identity map on Ch∗⊕h and sends each si to
√
−1βiνisi. The inverse map Ψ̈ is
the extension of the identity map on Ch∗⊕h which sends each si to
√
−1βiνisi.
Proof. The isomorphisms Φ̇ in Theorem 2.3 and Φ in (2.1) can be readily extended to the
following isomorphisms of superalgebras which restrict to the identity map on Ch∗⊕h:
Φ : Ch∗⊕h o CW
'−→ Ch ⊗
(
Ch∗ o− CW−) ,
Φ̇ : Ch ⊗
(
Ch∗ o− CW−) '−→ Ch∗⊕h ⊗ CW.
Observe that Φ̈ = Φ̇ ◦ Φ, and so Φ̈ is an isomorphism. �
2.4 The case of general 2-cocycles
The materials of this subsection generalize the Section 2.3 above and [9, Section 2]; however,
they will not be used in subsequent sections.
The Schur multipliers H2(W, C∗) for finite Weyl groups W were computed by Ihara and
Yokonuma [6] (also cf. Karpilovsky [7, Theorem 7.2.2]). In all cases, we have H2(W, C∗) ∼=
k∏
j=1
Z2
for suitable k = 0, 1, 2, 3.
6 T. Khongsap and W. Wang
Consider the following central extension of W by H2(W, C∗):
1 −→ H2(W, C∗) −→ W̃ −→ W −→ 1.
We denote by zi the generator of the ith copy of Z2 in H2(W, C∗) ∼=
k∏
j=1
Z2 and by ti a fixed
preimage of the generator si of W for each i. The group W̃ is generated by z1, . . . , zk, t1, . . . , tn
subject to that zi is central of order 2 for all i, and the additional relations shown in Table 2.2
below (cf. [7, Table 7.1]). In particular, the values of k can be read off from Table 2.2.
Table 2.2. Central extensions W̃ of Weyl groups.
Type of W Generators/Relations for W̃
t2i = 1, 1 ≤ i ≤ n,
An (n ≥ 3) titi+1ti = ti+1titi+1, 1 ≤ i ≤ n− 1
titj = z1tjti if mij = 2
B2 t21 = t22 = 1, (t1t2)2 = z1(t2t1)2
B3 t21 = t22 = t23 = 1, t1t2t1 = t2t1t2,
t1t3 = z1t3t1, (t2t3)2 = z2(t3t2)2
t2i = 1, 1 ≤ i ≤ n, titi+1ti = ti+1titi+1, 1 ≤ i ≤ n− 2
Bn (n ≥ 4) titj = z1tjti, 1 ≤ i < j ≤ n− 1, mij = 2
titn = z2tnti, 1 ≤ i ≤ n− 2
(tn−1tn)2 = z3(tntn−1)2
D4 t2i = 1, 1 ≤ i ≤ 4, titjti = tjtitj if mij = 3
t1t3 = z1t3t1, t1t4 = z2t4t1, t3t4 = z3t4t3
t2i = 1, 1 ≤ i ≤ n, titjti = tjtitj if mij = 3
Dn (n ≥ 5) titj = z1tjti, 1 ≤ i < j ≤ n, mij = 2, i 6= n− 1
tn−1tn = z2tntn−1
En=6,7,8 t2i = 1, 1 ≤ i ≤ n, titjti = tjtitj if mij = 3
titj = z1tjti, if mij = 2
t2i = 1, 1 ≤ i ≤ 4, titi+1ti = ti+1titi+1 (i = 1, 3)
F4 titj = z1tjti, 1 ≤ i < j ≤ 4, mij = 2,
(t2t3)2 = z2(t3t2)2
G2 t21 = t22 = 1, (t1t2)3 = z1(t2t1)3
For α = (αi)i=1,...,k ∈ H2(W, C∗), the quotient CWα := CW̃/〈zi − αi,∀ i〉 can be identified
as the algebra generated by t1, . . . , tn subject to the relations:
(titj)mij =
{
1, if mij = 1, 3,
αij , if mij = 2, 4, 6,
where αij ∈ {±1} is specified by α ∈ H2(W, C∗) as in Table 2.2.
Let Ch∗ o− CW−α denote the algebra generated by subalgebras Ch∗ and CW−α with the
following additional multiplication:
t−i βj = −βsi
j t−i ∀ i, j,
Hecke–Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type 7
where we have denoted by t−i the generators of the subalgebra CW−α of Ch∗ o− CW−α, in order
to distinguish from the generators ti of CWα below. We impose superalgebra structures on the
algebras Ch∗ o− CW−α and on Ch∗ ⊗ CWα by letting t−i be odd, ti be even, and βi be odd for
all i.
Theorem 2.7. Fix a 2-cocycle α ∈ H2(W, C∗). We have an isomorphism of superalgebras:
Φ̇α
− : Ch∗ o− CW−α '−→ Ch∗ ⊗ CWα
which extends the identity map on Ch∗ and sends t−i to βiti for each i. The inverse map Ψ̇α
− is
the extension of the identity map on Ch∗ and sends ti to βit
−
i for each i.
Proof. By Lemma 2.5, we have βjβi = −βiβ
si
j . Recall that t−i is odd and ti is even. So we
have βjΦ̇α
−(t−i ) = −Φ̇α
−(t−i ) βsi
j for all admissible i, j. Moreover,
(Φ̇α
−(t−i )Φ̇α
−(t−j ))mij = (βitiβjtj)mij = (βiβj)mij (titj)mij = (−1)mij+1(titj)mij
=
{
1 if mij = 1, 3,
−αij if mij = 2, 4, 6.
Clearly, Φ̇α
− preserves the Z2-grading. Hence, it follows that Φ̇α
− is a surjective superalgebra
homomorphism, and thus an isomorphism by dimension counting. It is clear that Ψ̇α
− is the
inverse of Φ̇α
−. �
Denote by Ch∗ o+ CW−α the algebra generated by subalgebras Ch∗ and CW−α with the
following additional multiplication:
t+i βj = βsi
j t+i ∀ i, j,
where we have denoted by t+i the generators of the subalgebra CW−α of Ch∗ o+ CW−α, in
order to distinguish from the generators ti of CWα. We impose superalgebra structures on the
algebras Ch∗ o+ CW−α and on Ch∗ ⊗ CWα by letting t+i be even, ti be odd, and βi be odd for
all i.
Theorem 2.8. Fix a 2-cocycle α ∈ H2(W, C∗). We have an isomorphism of superalgebras:
Φ̇α
+ : Ch∗ o+ CW−α '−→ Ch∗ ⊗ CWα
which extends the identity map on Ch∗ and sends t+i 7→ −
√
−1βiti. The inverse map Ψ̇α
+ is the
extension of the identity map on Ch∗ and sends each ti to
√
−1βit
+
i .
Proof. By Lemma 2.5, we have βjβi = −βiβ
si
j . Recall that t+i is even while ti is odd. Then
βjΦ̇α
+(t+i ) = Φ̇α
+(t+i ) βsi
j for all admissible i, j. Moreover,
(Φ̇α
+(t+i )Φ̇α
+(t+j ))mij = (−βitiβjtj)mij = (βiβjtitj)mij = (βiβj)mij (titj)mij
=
{
1 if mij = 1, 3,
−αij if mij = 2, 4, 6.
It follows that Φ̇α
+ is an isomorphism of superalgebra with inverse Ψ̇α
+. �
Denote by Ch∗⊕h o+ CWα the algebra generated by subalgebras Ch∗⊕h and CWα with the
following additional multiplication:
tiβj = βsi
j ti, tiνj = νsi
j ti, ∀ i, j.
We impose superalgebra structures on the algebras Ch∗⊕h o+ CWα and on Ch∗⊕h ⊗ CWα by
letting each ti be even, and letting each βi, νi be odd.
8 T. Khongsap and W. Wang
Corollary 2.9. For a 2-cocycle α ∈ H2(W, C∗), we have an isomorphism of superalgebras:
Ch∗⊕h o+ CWα ∼= Ch∗⊕h ⊗ CWα
which extends the identity map on Ch∗⊕h and sends each ti to
√
−1βiνiti.
Remark 2.10. When α = ±1 and CWα becomes the usual group algebra CW or the spin
group algebra CW−, we recover the main results of Section 2.3.
3 The DaHa with two polynomial-Clifford subalgebras
In the remainder of the paper, W is always assumed to be one of the classical Weyl groups of
type An−1, Bn, or Dn, and we shall often write C2n for Ch∗⊕h.
3.1 The definition of Hcc
W
Let W be one of the classical Weyl groups. The goal of this section is to introduce a ratio-
nal double affine Hecke algebra (DaHa) Hcc
W which is generated by CW and two isomorphic
“polynomial-Clifford” subalgebras. Note that this construction is different from the double
affine Hecke–Clifford algebra introduced in [16, 10] which is generated by CW , a polynomial
subalgebra, and a “polynomial-Clifford” subalgebra.
Identify C[h∗] ∼= C[x1, . . . , xn] and C[h] ∼= C[y1, . . . , yn], where xi, yi (1 ≤ i ≤ n) correspond
to the standard orthonormal basis {ei} for h∗ and its dual basis {e∗i } for h, respectively. For x, y
in an algebra A, we denote as usual that
[x, y] = xy − yx ∈ A.
3.1.1 The algebra Hcc
W of type An−1
Definition 3.1. Let t, u ∈ C and W = Sn. The algebra Hcc
W of type An−1 is generated by xi,
yi (1 ≤ i ≤ n), C2n and W , subject to the following additional relations:
xixj = xjxi, yiyj = yjyi (∀ i, j),
σci = cσ
i σ, σei = eσ
i σ,
σxi = xσ
i σ, σyi = yσ
i σ (∀σ ∈ W, ∀ i), (3.1)
eixj = xjei, cixj = (−1)δijxjci (∀ i, j),
ciyj = yjcj , eiyj = (−1)δijyjei (∀ i, j),
[yi, xj ] = u(1 + cicj)(1 + ejei)sij (i 6= j), (3.2a)
[yi, xi] = tciei − u
∑
k 6=i
(1 + ckci)(1 + ekei)ski. (3.2b)
Alternatively, we may view t, u as formal variables and Hcc
W as a C[t, u]-algebra. Similar
remarks apply to other algebras defined in this paper.
3.1.2 The algebra Hcc
W of type Dn
Let W = WDn . Regarding elements in W as even signed permutations of 1, 2, . . . , n as usual,
we identify the generators si ∈ W , 1 ≤ i ≤ n− 1, with transposition (i, i + 1), and sn ∈ W with
the transposition of (n− 1, n) coupled with the sign changes at n− 1, n. For 1 ≤ i 6= j ≤ n, we
denote by sij ≡ (i, j) ∈ W the transposition of i and j, and sij ≡ (i, j) ∈ W the transposition
of i and j coupled with the sign changes at i, j. By convention, we have
sn−1,n ≡ (n− 1, n) = sn, sij ≡ (i, j) = sjnsi,n−1snsi,n−1sjn.
Hecke–Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type 9
Definition 3.2. Let t, u ∈ C and W = WDn . The algebra Hcc
W of type Dn is generated by xi,
yi (1 ≤ i ≤ n), C2n and W , subject to the relations (3.1) with the current W , and (3.3a)–(3.3b)
with i 6= j below:
[yi, xj ] = u(1 + cicj)(1 + ejei)sij − u(1− cicj)(1− ejei)sij , (3.3a)
[yi, xi] = tciei − u
∑
k 6=i
(1 + ckci)(1 + ekei)ski + u
∑
k 6=i
(1− ckci)(1− ekei)ski. (3.3b)
3.1.3 The algebra Hcc
W of type Bn
Let W = WBn . We identify W as usual with the signed permutations on 1, . . . , n. Regarding
WDn as a subgroup of W , we have sij , sij ∈ W for 1 ≤ i 6= j ≤ n. Further denote τi ≡ (i) ∈ W
the sign change at i for 1 ≤ i ≤ n. By definition, we have
τn ≡ (n) = sn, τi ≡ (i) = sinsnsin.
Definition 3.3. Let t, u, v ∈ C, and W = WBn . The algebra Hcc
W of type Bn is generated by xi,
yi (1 ≤ i ≤ n), C2n and W , subject to the relations (3.1) with the current W , and (3.4a)–(3.4b)
with i 6= j below:
[yi, xj ] = u(1 + cicj)(1 + ejei)sij − u(1− cicj)(1− ejei)sij , (3.4a)
[yi, xi] = tciei − u
∑
k 6=i
(1 + ckci)(1 + ekei)ski + u
∑
k 6=i
(1− ckci)(1− ekei)ski − vτi. (3.4b)
3.2 The PBW basis for Hcc
W
We shall denote xα =xa1
1 · · ·xan
n for α=(a1, . . . , an)∈Zn
+, cε =cε1
1 · · · cεn
n for ε = (ε1, . . . , εn)∈Zn
2 .
Similarly, we define yα and eε. Note that the algebra Hcc
W contains C[h∗], Ch∗ , C[h], Ch, and CW
as subalgebras.
Theorem 3.4. Let W be WAn−1, WDn or WBn. The multiplication of the subalgebras C[h∗],
C[h], Ch∗, Ch, and CW induces a vector space isomorphism
C[h∗]⊗ Ch∗ ⊗ CW ⊗ C[h]⊗ Ch
'−→ Hcc
W .
Equivalently, the elements {xαcεweε′yγ |α, γ ∈ Zn
+, ε, ε′ ∈ Zn
2 , w ∈ W} form a linear basis for Hcc
W
(the PBW basis).
Proof. Recall that W acts diagonally on V = h∗ ⊕ h. The strategy of proving the theorem
follows the suggestion of [16] to modify [5, Proof of Theorem 1.3] as follows.
Clearly K := C2n o CW is a semisimple algebra. Observe that E := V ⊗C K is a natural
K-bimodule (even though V is not) with the right K-module structure on E given by right
multiplication and the left K-module structure on E by letting
w · (v ⊗ a) = vw ⊗ wa,
ci · (xj ⊗ a) = (−1)δijxj ⊗ (cia),
ci · (yj ⊗ a) = yj ⊗ (cia),
ei · (xj ⊗ a) = xj ⊗ (eia),
ei · (yj ⊗ a) = (−1)δijyj ⊗ (eia),
where v ∈ V , w ∈ W , a ∈ K.
10 T. Khongsap and W. Wang
The rest of the proof can proceed in the same way as in [5, Proof of Theorem 1.3], and it
boils down to the verifications of the conjugation invariance (by ci, ei and W ) of the defining
relations (3.2a)–(3.2b), (3.3a)–(3.3b), or (3.4a)–(3.4b) for type A, D or B respectively, and the
verification of the Jacobi identities among the generators xi and yi for 1 ≤ i ≤ n.
Such verifications are left to Lemmas 3.6, 3.7 and 3.8 below. �
Remark 3.5. The algebra Hcc
W has two different triangular decompositions:
Hcc
W
∼= C[h∗]⊗ (C2n o CW )⊗ C[h],
Hcc
W
∼= (C[h∗]⊗ Ch∗)⊗ CW ⊗ (Ch ⊗ C[h]).
The detailed proofs of Lemmas 3.6, 3.7 and 3.8 below (also compare [10]) are postponed to
the Appendix.
Lemma 3.6. Let W = WAn−1 ,WDn or WBn. Then the relations (3.2a)–(3.2b), (3.3a)–(3.3b),
or (3.4a)–(3.4b) are invariant under the conjugation by ci and ei respectively, 1 ≤ i ≤ n.
Lemma 3.7. The relations (3.2a)–(3.2b), (3.3a)–(3.3b), or (3.4a)–(3.4b) are invariant under
the conjugation by elements in WAn−1, WDn or WBn respectively.
Lemma 3.8. Let W = WAn−1, WDn or WBn. Then the Jacobi identity holds for any triple
among xi, yi in Hcc
W for 1 ≤ i ≤ n.
Remark 3.9. For W = WAn−1 , WDn or WBn , the algebra Hcc
W has a natural superalgebra
structure by letting xi, yi, sj be even and ck, ek be odd for all admissible i, j, k. Moreover, the
map $ : Hcc
W −→ Hcc
W which sends
xi 7→ yi, yi 7→ −xi, ci 7→ ei, ei 7→ −ci, sj 7→ sj ∀ i, j
is an automorphism of Hcc
W .
3.3 The Dunkl representations
Recall K = C2n o CW . Denote by Hy the subalgebra of Hcc
W generated by K and y1, . . . , yn.
A K-module M can be extended to Hy-module by demanding the action of each yi to be trivial.
We define
My := IndHcc
W
Hy
M.
Under the identification of vector spaces
My = C[x1, . . . , xn]⊗M,
the action of Hcc
W on My is transferred to C[x1, . . . , xn]⊗M as follows. K acts on C[x1, . . . , xn]⊗M
by the following formulas:
w · (xj ⊗m) = xw
j ⊗ wm,
ci · (xj ⊗m) = (−1)δijxj ⊗ cim,
ei · (xj ⊗m) = xj ⊗ eim,
where ci, ei ∈ C2n, w ∈ W . Moreover, xi acts by left multiplication in the first tensor factor,
and the action of yi will be given by the so-called Dunkl operators which we compute below
(compare [3, 4]).
A simple choice for a K-module is C2n, whose K-module structure is defined by letting C2n
act by left multiplication and W act diagonally.
Hecke–Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type 11
3.3.1 The Dunkl Operators for type An−1 case
We first prepare a few lemmas. It is understood in this paper that the ratios of two (possibly
noncommutative) operators g and h always means that h
g = 1
g · h.
Lemma 3.10. Let W = WAn−1. Then the following holds in Hcc
W for l ∈ Z+ and i 6= j:
[yi, x
l
j ] = u
(
xl
j − xl
i
xj − xi
+
xl
j − (−1)lxl
i
xj + xi
cicj
)
(1− eiej)sij ,
[yi, x
l
i] = tciei
xl
i − (−xi)l
2xi
− u
∑
k 6=i
(
xl
i − xl
k
xi − xk
+
xl
i − (−xk)l
xi + xk
ckci
)
(1 + ekei)ski.
Proof. This lemma is a type A counterpart of Lemma 3.13 for type B below. A proof can be
simply obtained by modifying the proof of Lemma 3.13 with the removal of those terms involving
sij , ski, τi therein. �
Lemma 3.11. Let W = WAn−1, and f ∈ C[x1, . . . , xn]. Then the following identity holds in
Hcc
W :
[yi, f ] = tciei
f − f τi
2xi
− u
∑
k 6=i
(
f − fski
xi − xk
+
fckci − ckcif
ski
xi + xk
)
(1 + ekei)ski.
Proof. It suffices to check the formula for every monomial f of the form xl1
1 · · ·xln
n , which
follows by Lemma 3.10 and an induction on a based on the identity
[yi, x
l1
1 · · ·x
la
a x
la+1
a+1 ] = [yi, x
l1
1 · · ·x
la
a ]xla+1
a+1 + xl1
1 · · ·x
la
a [yi, x
la+1
a+1 ]. �
Now we are ready to compute the Dunkl operator for yi.
Theorem 3.12. Let W = WAn−1 and M be a (C2n o CW )-module. The action of yi on the
module C[x1, . . . , xn]⊗M is realized as the following Dunkl operators: for any f ∈ C[x1, . . . , xn]
and m ∈ M , we have
yi ◦ (f ⊗m) = tciei
f − f τi
2xi
⊗m− u
∑
k 6=i
(
f − fski
xi − xk
+
fckci − ckcif
ski
xi + xk
)
⊗ (1 + ekei)skim.
Proof. We calculate that
yi ◦ (f ⊗m) = [yi, f ]⊗m + f ⊗ yim = [yi, f ]⊗m.
Now the result follows from Lemma 3.11. �
3.3.2 The Dunkl Operators for type Bn case
The proofs of Lemmas 3.13 and 3.14 are postponed to the Appendix.
Lemma 3.13. Let W = WBn. Then the following holds in Hcc
W for l ∈ Z+ and i 6= j:
[yi, x
l
j ] = u
(
xl
j − xl
i
xj − xi
+
xl
j − (−1)lxl
i
xj + xi
cicj
)
(1− eiej)sij
− u
(
xl
j − (−xi)l
xj + xi
−
xl
j − xl
i
xj − xi
cicj
)
(1 + eiej)sij ,
12 T. Khongsap and W. Wang
[yi, x
l
i] = tciei
xl
i − (−xi)l
2xi
− u
∑
k 6=i
(
xl
i − xl
k
xi − xk
+
xl
i − (−xk)l
xi + xk
ckci
)
(1 + ekei)ski
− u
∑
k 6=i
(
xl
i − (−xk)l
xi + xk
−
xl
i − xl
k
xi − xk
ckci
)
(1− ekei)ski − v
xl
i − (−xi)l
2xi
τi.
Lemma 3.14. Let W = WBn, and f ∈ C[x1, . . . , xn]. Then the following holds in Hcc
W :
[yi, f ] = tciei
f − f τi
2xi
− u
∑
k 6=i
(
f − fski
xi − xk
+
f − fski
xi + xk
ckci
)
(1 + ekei)ski
− v
f − f τi
2xi
τi − u
∑
k 6=i
(
f − fski
xi + xk
− f − fski
xi − xk
ckci
)
(1− ekei)ski.
Now we are ready to compute the Dunkl operator for yi.
Theorem 3.15. Let W = WBn and M be a (C2n oCW )-module. The action of yi on the module
C[x1, . . . , xn] ⊗ M is realized as the following Dunkl operators: for any f ∈ C[x1, . . . , xn] and
m ∈ M , we have
yi ◦ (f ⊗m) = tciei
f − f τi
2xi
⊗m− u
∑
k 6=i
(
f − fski
xi − xk
+
f − fski
xi + xk
ckci
)
⊗ (1 + ekei)skim
− u
∑
k 6=i
(
f − fski
xi + xk
− f − fski
xi − xk
ckci
)
⊗ (1− ekei)skim− v
f − f τi
2xi
⊗ τim.
Proof. We observe that
yi ◦ (f ⊗m) = [yi, f ]⊗m + f ⊗ yim = [yi, f ]⊗m.
Now the result follows from Lemma 3.14. �
3.3.3 The Dunkl Operators for type Dn case
Due to the similarity of the bracket relations [−,−] in Dn and Bn cases (e.g. compare (3.3b)
with (3.4b)), the formula below for type Dn is obtained from its type Bn counterpart in the
previous subsection by dropping the terms involving the parameter v.
Theorem 3.16. Let W = WDn, and let M be a (C2n o CW )-module. The action of yi on
C[x1, . . . , xn] ⊗ M is realized as the following Dunkl operators: for any f ∈ C[x1, . . . , xn] and
m ∈ M , we have
yi ◦ (f ⊗m) = tciei
f − f τi
2xi
⊗m− u
∑
k 6=i
(
f − fski
xi − xk
+
f − fski
xi + xk
ckci
)
⊗ (1 + ekei)skim
− u
∑
k 6=i
(
f − fski
xi + xk
− f − fski
xi − xk
ckci
)
⊗ (1− ekei)skim.
3.4 The even center for Hcc
W
Recall that the even center Z(A) of a superalgebra A consists of the even central elements of A.
It turns out the algebra Hcc
W has a large center.
Hecke–Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type 13
Proposition 3.17. Let W be either WAn−1, WDn or WBn. The even center Z(Hcc
W ) contains
C[x2
1, . . . , x
2
n]W and C[y2
1, . . . , y
2
n]W as subalgebras. In particular, Hcc
W is module-finite over its
even center.
Proof. Let f ∈ C[x2
1, . . . , x
2
n]W . Then f−f τi = 0 for each i. Moreover, by the definition of Hcc
W ,
f commutes with C2n, W , and xi for all 1 ≤ i ≤ n. Since f = fw for all w ∈ W , it follows
from Lemmas 3.11 and 3.14 that [yi, f ] = 0 for each i. Hence f commutes with C[y1, . . . , yn].
Therefore f is in the even center Z(Hcc
W ). It follows from the automorphism $ of Hcc
W defined
in Remark 3.9 that C[y2
1, . . . , y
2
n]W must also be in the even center Z(Hcc
W ). �
4 The spin double affine Hecke–Clifford algebras
Recall that W is one of the classical Weyl groups of type An−1, Bn, or Dn. The goal of this
section is to introduce and study the spin double affine Hecke–Clifford algebra (sDaHCa) H−c
W ,
which is, roughly speaking, obtained by decoupling the Clifford algebra Ch from the DaHa Hcc
W
in Section 3. The spin Weyl group algebra CW− appears naturally in the process. We remark
that the algebra H−c
W is different from either the spin double affine Hecke algebra or the double
affine Hecke–Clifford algebra introduced in [16, 10].
4.1 The definition of sDaHCa H−c
W
Following [10], we introduce the notation
ti↑j =
{
titi+1 · · · tj , if i ≤ j,
1, otherwise,
ti↓j =
{
titi−1 · · · tj , if i ≥ j,
1, otherwise.
Define the following odd elements in CW− of order 2, which are an analogue of reflections in W ,
for 1 ≤ i < j ≤ n:
tij ≡ [i, j] = (−1)j−i−1tj−1 · · · ti+1titi+1 · · · tj−1,
tji ≡ [j, i] = −[i, j],
tij ≡ [i, j] =
{
(−1)j−i−1tj↑n−1ti↑n−2tntn−2↓itn−1↓j , for type Dn,
(−1)j−itj↑n−1ti↑n−2tntn−1tntn−2↓itn−1↓j , for type Bn,
tji ≡ [j, i] = [i, j],
ti ≡ [i] = (−1)n−iti · · · tn−1tntn−1 · · · ti (1 ≤ i ≤ n).
The notations [i, j], [i, j] here are consistent with the inclusions of algebras CW−
An−1
≤ CW−
Dn
≤
CW−
Bn
.
As in [16] (also cf. [9, 10]), a skew-polynomial algebra is the C-algebra generated by b1, . . . , bn
subject to the relations bibj + bjbi = 0, (i 6= j). This algebra, denoted by C[b1, . . . , bn], is
naturally a superalgebra by letting each bi be odd, and it has a linear basis given by bα :=
bk1
1 · · · bkn
n for α = (k1, . . . , kn) ∈ Zn
+.
Consider the group homomorphism ρ : WBn → Sn defined by ρ(si) = si and ρ(sn) = 1 for
1 ≤ i ≤ n− 1. By restriction if needed, we have a group homomorphism
ρ : W −→ Sn, σ 7→ ρ(σ) = σ∗
for W = WAn−1 ,WBn or WDn . Observe that τ∗i = 1 and s∗ij = sij for all 1 ≤ i 6= j ≤ n.
14 T. Khongsap and W. Wang
Definition 4.1. Let t, u, v ∈ C, and W = WAn−1 ,WDn , or WBn . The sDaHCa H−c
W is the
algebra generated by xi, ηi (1 ≤ i ≤ n) and Ch∗ o CW−, subject to the relations
ηiηj = −ηjηi, xixj = xjxi (i 6= j),
ciηj = −ηjci, cixj = (−1)δijxjci (∀ i, j),
tixj = xsi
j ti, tiηj = −η
s∗i
j ti (ti ∈ CW−)
and the following additional relations:
Type A:
[ηi, xj ] = u(1 + cicj)[i, j] (i 6= j),
[ηi, xi] = tci + u
∑
k 6=i
(1 + ckci)[k, i],
Type D:
[ηi, xj ] = u((1 + cicj)[i, j]− (1− cicj)[i, j]) (i 6= j),
[ηi, xi] = tci + u
∑
k 6=i
(
(1 + ckci)[k, i]− (1− ckci)[k, i]
)
,
Type B:
[ηi, xj ] = u((1 + cicj)[i, j]− (1− cicj)[i, j]) (i 6= j),
[ηi, xi] = tci + u
∑
k 6=i
(
(1 + ckci)[k, i]− (1− ckci)[k, i]
)
+ v[i].
4.2 Isomorphism of superalgebras
For W = WAn−1 ,WBn , or WDn , we recall an algebra isomorphism (see [10, Lemma 5.4])
Φ : Ch o CW → Ch ⊗ CW−
which sends
(ek − ei)sik 7−→ −
√
−2 [k, i],
(ek + ei)sik 7−→ −
√
−2 [k, i], (4.1)
eiτi 7−→ −
√
−1 [i]
for i 6= k, whenever it is applicable. The inverse of Φ is denoted by Ψ.
Note that the algebra H−c
W has a natural superalgebra structure by letting each ηi, ci, tj be
odd and xi be even for all admissible i, j.
Theorem 4.2. Let W be WAn−1, WDn or WBn. Then,
1) there exists an isomorphism of superalgebras
Φ : Hcc
W (t, u, v) −→ Ch ⊗H−c
W (−t,−
√
−2u,
√
−1v)
which extends Φ : Ch o CW → Ch ⊗ CW− and sends
yi 7→ eiηi, xi 7→ xi, ci 7→ ci, ∀ i;
2) the inverse
Ψ : Ch ⊗H−c
W (−t,−
√
−2u,
√
−1v) −→ Hcc
W (t, u, v)
extends Ψ : Ch ⊗ CW− → Ch o CW and sends
ηi 7→ eiyi, xi 7→ xi, ci 7→ ci, ∀ i.
Hecke–Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type 15
Proof. We need to check that Φ preserves the relations (3.1), (3.2a)–(3.2b), (3.3a)–(3.3b), and
(3.4a)–(3.4b) for W = WAn−1 ,WDn , and WBn respectively.
First, we shall verify that Φ preserves (3.4a)–(3.4b) with W = WBn . Indeed, by (4.1) or [10,
Lemma 5.4], we have
Φ(l.h.s. of (3.4a)) = ei[ηi, xj ] = −
√
−2uei
(
(1− cjci)[i, j]− (1 + cjci)[i, j]
)
= Φ
(
u
(
(1 + cicj)(1 + ejei)sji − (1− cicj)(1− ejei)sij
))
= Φ(r.h.s. of (3.4a)).
Also, we have
Φ(l.h.s. of (3.4b)) = ei[ηi, xi]
= −t · eici −
√
−2uei
∑
k 6=i
(
(1 + ckci)[k, i]− (1− ckci)[k, i]
)
+
√
−1vei[i]
= Φ
(
tciei − u
∑
k 6=i
(
(1 + ckci)(1 + ekei)ski + (1− ckci)(1− ekei)ski
)
− vτi
)
= Φ(r.h.s. of (3.4b)).
It is easy to check that Φ preserves (3.1), and we will restrict ourselves to verify just a few
relations among (3.1). For j 6= i, i + 1, we have
Φ(siyj) = −
√
−1νitiejηj = −
√
−1ejηjνiti = Φ(yjsi).
Moreover,
Φ(snyn) = −
√
−1νntnenηn =
√
−1tnηn = −
√
−1ηntn = Φ(−ynsn).
This proves that Φ is an algebra homomorphism for type Bn.
By dropping the terms involving v in the above equations, we verify that the relations (3.3a)–
(3.3b) with W = WDn are preserved by Φ. By further dropping the terms involving [ij], sij etc.,
we also verify (3.2a)–(3.2b) with W = WAn−1 . So, the homomorphism Φ is well defined in all
cases.
Similarly, one shows that Ψ is a well-defined algebra homomorphism. Since Φ and Ψ are
inverses on generators, they are (inverse) algebra isomorphisms. �
The isomorphism in Theorem 4.2 exactly means that the superalgebras Hcc
W and H−c
W are
Morita super-equivalent in the sense of [17].
Corollary 4.3. Let W be one of the Weyl groups WAn−1, WDn or WBn. The even center
Z(H−c
W ) of H−c
W contains C[η2
1, . . . , η
2
n]W and C[x2
1, . . . , x
2
n]W . In particular, H−c
W is module-finite
over its even center.
Proof. By the isomorphism Φ in Theorem 4.2 and the Proposition 3.17, we have that Z(Ch ⊗
H−c
W ) contains the subalgebras C[η2
1, . . . , η
2
n]W and C[x2
1, . . . , x
2
n]W , and so does Z(H−c
W ). �
4.3 The PBW property for H−c
W
We have the following PBW type property for the algebra H−c
W .
Theorem 4.4. Let W be one of the Weyl groups WAn−1, WDn or WBn. The multiplication of
the subalgebras induces an isomorphism of vector spaces
C[η1, . . . , ηn]⊗ Ch∗ ⊗ CW− ⊗ C[h∗] −→ H−c
W .
Equivalently, the set {ηαcεσxγ} forms a basis for H−c
W , where σ runs over a basis for CW−,
ε ∈ Zn
2 , and α, γ ∈ Zn
+.
16 T. Khongsap and W. Wang
Proof. It follows from the defining relations that H−c
W is spanned by the elements ηαcεσxγ where
σ runs over a basis for CW−, α, γ ∈ Zn
+, and ε ∈ Zn
2 . By the isomorphism Ψ : Ch⊗H−c
W −→ Hcc
W
in Theorem 4.2, we see that the image Ψ(ηαcεσxγ) are linearly independent in Hcc
W by the PBW
property for Hcc
W (see Theorem 3.4). So the elements ηαcεσxγ are linearly independent in H−c
W .
Therefore, the set {ηαcεσxγ} forms a basis for H−c
W . �
4.4 The Dunkl operators for H−c
W
Denote by hη the subalgebra of H−c
W generated by ηi (1 ≤ i ≤ n) and Ch∗ o− CW−. A (Ch∗ o−
CW−)-module V can be extended to a hη-modules by letting the actions of ηi on V to be trivial
for each i. We define
Vη := IndH−c
W
hη
V ∼= C[x1, . . . , xn]⊗ V.
On C[x1, . . . , xn] ⊗ V , the element ti ∈ CW− acts as si ⊗ ti, ci ∈ Ch∗ acts by ci · (xj ⊗ v) =
(−1)δijxj⊗civ, and xi acts by left multiplication, and ηi acts as anti-commuting Dunkl operators,
which we will describe in this section.
Under the superalgebra isomorphism Φ : Hcc
W → Cn ⊗ H−c
W in Theorem 4.2, we obtain anti-
commuting Dunkl operators ηi by fairly straightforward computation. They are counterparts of
those in Section 3, and we omit the proofs.
4.4.1 Dunkl operator for type An−1
The following is a counterpart of Theorem 3.12.
Proposition 4.5. Let W = WAn−1 and V be a Cn o CW−-module. The action of ηi on the
H−c
W -module C[x1, . . . , xn] ⊗ V is realized as a Dunkl operator as follows. For any polynomial
f ∈ C[x1, . . . , xn] and m ∈ V , we have
ηi ◦ (f ⊗m) = tci
f − f τi
2xi
⊗m + u
∑
k 6=i
(
f − fski
xi − xk
+
fckci − ckckf
ski
xi + xk
)
⊗ [k, i]m.
4.4.2 Dunkl operator for type Bn
The following is a counterpart of Theorem 3.15
Proposition 4.6. Let W = WBn and V be a (Cn o CW−)-module. The action of ηi on the
H−c
W -module C[x1, . . . , xn] ⊗ V is realized as a Dunkl operator as follows. For any polynomial
f ∈ C[x1, . . . , xn] and m ∈ V , we have
ηi ◦ (f ⊗m) = tci
f − f τi
2xi
⊗m + u
∑
k 6=i
(
f − fski
xi − xk
+
f − fski
xi + xk
ckci
)
⊗ [k, i]m
− u
∑
k 6=i
(
f − fski
xi + xk
− f − fski
xi − xk
ckci
)
⊗ [k, i]m + v
f − f τi
2xi
⊗ [i]m.
4.4.3 Dunkl operator for type Dn
Proposition 4.7. Let W = WDn and V be a Cn o CW−-module. The action of ηi on the
H−c
W -module C[x1, . . . , xn] ⊗ V is realized as a Dunkl operator as follows. For any polynomial
f ∈ C[x1, . . . , xn] and m ∈ V , we have
ηi ◦ (f ⊗m) = tci
f − f τi
2xi
⊗m + u
∑
k 6=i
(
f − fski
xi − xk
+
f − fski
xi + xk
ckci
)
⊗ [k, i]m
Hecke–Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type 17
− u
∑
k 6=i
(
f − fski
xi + xk
− f − fski
xi − xk
ckci
)
⊗ [k, i]m.
Remark 4.8. The general formula for the Dunkl operators ηi for H−c
W resembles the Dunkl
operator yi for Ḧc
W which appeared in [10, Theorems 4.4, 4.10, 4.14]. However, ηiηj = −ηjηi,
while yiyj = yjyi for i 6= j.
5 The odd double affine Hecke algebras
In this section, we shall introduce an odd double affine Hecke algebra HW which is generated
by CW and two isomorphic skew-polynomial subalgebras. Recall that W is assumed to be one
of the classical Weyl groups of type An−1, Bn, or Dn.
Recall also the group homomorphism ρ : W −→ Sn defined in Section 4 which sends σ 7→ σ∗
for all σ ∈ W . We shall need two (isomorphic) skew-polynomial algebras C{h∗} = C[ξ1, . . . , ξn]
and C{h} = C[η1, . . . , ηn], which are naturally acted upon by the symmetric group Sn or the
group WBn by permuting the indices possibly coupled with sign changes. We shall denote the
action of σ ∈ WBn by f 7→ fσ.
5.1 The definition of HW
As usual we denote [ξ, η]+ = ξη + ηξ.
Definition 5.1. Let t, u, v ∈ C and W be WAn−1 ,WDn , or WBn . The odd DaHa HW is the
algebra generated by ξi, ηi (1 ≤ i ≤ n) and CW , subject to the relations
ηiηj = −ηjηi, ξiξj = −ξjξi (i 6= j),
σξj = ξσ∗
j σ, σηj = ησ∗
j σ (σ ∈ W )
and the following additional relations:
Type A:
[ηi, ξj ]+ = usij (i 6= j),
[ηi, ξi]+ = t · 1 + u
∑
k 6=i
ski,
Type D:
[ηi, ξj ]+ = u (sij + sij) (i 6= j),
[ηi, ξi]+ = t · 1 + u
∑
k 6=i
(ski + sij) ,
Type B:
[ηi, ξj ]+ = u (sij + sij) (i 6= j),
[ηi, ξi]+ = t · 1 + u
∑
k 6=i
(ski + sij) + vτi.
The algebra HW has a natural superalgebra structure by letting sj be even and ηi, ξi be odd
for all i, j.
Remark 5.2. The defining relations for the algebra HW differ from those for the usual rational
DaHa (also known as rational Cherednik algebra) ḦW [5] by signs. One can introduce a so-called
“covering algebra” H̃ (as done in [17, 10] in similar setups) which contains a central element z of
order 2, so that the algebras ḦW and HW are simply the quotients of H̃ by the ideal generated
by z − 1 and z + 1 respectively.
The definition of HW is motivated by the Morita (super)equivalence with Hcc
W and H−c
W . The
defining relations above suggest a further extension of odd DaHa associated to the infinite series
complex reflection groups.
18 T. Khongsap and W. Wang
5.2 Isomorphism of superalgebras
Lemma 5.3. Let W be one of the Weyl groups WAn−1, WDn or WBn. The isomorphism Φ̇ :
Cn o− CW− → Cn ⊗ CW (see Theorem 2.3) sends
(ck − ci)[k, i] 7−→
√
2 ski, (ck + ci)[k, i] 7−→
√
2 sik, ci[i] 7−→ τi.
Proof. The lemma can be proved by induction very similar to [10, Lemma 5.4], and we skip
the detail. �
Theorem 5.4. Let W be one of the Weyl groups WAn−1, WDn or WBn. Then,
1) there exists an isomorphism of superalgebras
Φ̇ : H−c
W (t, u, v) −→ Cn ⊗HW (−t,
√
2u,−v)
which extends Φ̇ : Cn o− CW− → Cn ⊗ CW and sends
ηi 7→ ηi, xi 7→ ciξi, ∀ i;
2) the inverse
Ψ̇ : Cn ⊗HW (−t,
√
2u,−v) −→ H−c
W (t, u, v)
extends Ψ̇ : Cn ⊗ CW → Cn o− CW− and sends
ηi 7→ ηi, ξi 7→ cixi, ∀ i.
Proof. We first need to check that Φ̇ preserves the defining relations of H−c
W (t, u, v) and so Φ
is a well-defined homomorphism. Using Lemma 5.3, we shall check a few cases in type Bn case,
and leave the rest for the reader to verify. For i 6= j, we have
Φ̇([ηi, xj ]) = −cj [ηi, ξj ]+ = −
√
2ucj(sij + sij)
=
u√
2
(
(1 + cicj)(ci − cj)sij − (1− cicj)(ci + cj)sij
)
= Φ̇
(
u
(
(1 + cicj)[i, j]− (1− cicj)[i, j]
))
,
Φ̇([ηi, xi]) = −ci[ηi, ξi]+ = −ci
(
− t +
√
2u
∑
k 6=i
(
sik + sik
)
− vτi
)
= tci −
√
2uci
∑
k 6=i
(
sik + sik
)
+ vciτi
= Φ̇
(
tci + u
∑
k 6=i
(
(1 + ckci)[k, i] + (1− ckci)[k, i]
)
+ vτi
)
.
Also, if j 6= n, we have
Φ̇(tnxj) = cnsncjξj = cjξjcnsn = Φ̇(xjtn),
Φ̇(tnxn) = cnsncnξn = snξn = Φ̇(−xntn),
Φ̇(tnηj) = cnsnηj = −ηjcnsn = Φ̇(−ηjtn),
Φ̇(tnηn) = cnsnηn = −ηncnsn = Φ̇(−ηntn).
Similarly, one shows that Ψ̇ is a well-defined algebra homomorphism. Since Φ̇ and Ψ̇ are
inverses on generators, they are (inverse) algebra isomorphisms. �
Hecke–Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type 19
The next corollary can be proved similarly to Corollary 4.3.
Corollary 5.5. Let W be one of the Weyl groups WAn−1, WDn or WBn. The even center for
HW contains C[η2
1, . . . , η
2
n]W and C[ξ2
1 , . . . , ξ
2
n]W . In particular, HW is module-finite over its
even center.
Example 5.6. Usually there are other central elements beyond those given in the above corol-
lary. For example, ξ2
1η
2
2 + ξ2
2η
2
1 − us1(ξ1 − ξ2)(η1 − η2) lies in Z(HWA1
).
5.3 The PBW property for HW
We have the following PBW type property for the algebra HW which can be proved similarly
to Theorem 4.4, using now the isomorphism Φ̇.
Theorem 5.7. Let W be one of the Weyl groups WAn−1, WDn or WBn. The multiplication of
the subalgebras induces an isomorphism of vector spaces
C[ξ1, . . . , ξn]⊗ CW ⊗ C[η1, . . . , ηn] −→ HW .
Equivalently, the set {ξασηγ} forms a basis for HW , where σ ∈ W , and α, γ ∈ Zn
+.
5.4 The Dunkl operators for HW
Denote by hη the subalgebra of HW generated by ηi (1 ≤ i ≤ n) and CW . Let V be the trivial
CW -module, and extend V a hη-module by letting the actions of each ηi on V be trivial. Define
Vη := IndHW
hη
V ∼= C[ξ1, . . . , ξn].
On C[ξ1, . . . , ξn], σ ∈ W acts as ρ(σ) = σ∗, ξi acts by left multiplication, and ηi acts as anti-
commuting Dunkl operators which we establish below. (It is easy to replace the trivial module
above by any CW -module.)
5.4.1 Dunkl operator for type A case
For each i, we introduce a super derivation ∂ξi
on C[ξ1, . . . , ξn] defined inductively by ∂ξi
(ξj) = δij
and
∂ξi
(ξa1 · · · ξal
) =
∑
k
(−1)k−1ξa1 · · · ξak−1
∂ξi
(ξak
)ξak+1
· · · ξal
.
The formulas below for type An−1 case can be obtained from Lemmas 5.11, 5.12, and Theo-
rem 5.13 with the removal of those terms involving sij , ski, and the parameter v therein.
Lemma 5.8. Let W = WAn−1. Then the following holds in HW for l ∈ Z+ and i 6= j:
[ηi, ξ
l
j ]+ =
u
ξ2
i − ξ2
j
(
ξl+1
i − ξjξ
l
i − ξiξ
l
j + (−1)lξl+1
j
)
sij ,
[ηi, ξ
l
i]+ = t
ξl
i − (−ξi)l
2ξi
+ u
∑
k 6=i
1
ξ2
i − ξ2
k
(
ξiξ
l
k − ξl+1
k − (−1)lξl+1
i + ξkξ
l
i
)
sik.
Lemma 5.9. Let W = WAn−1, and f ∈ C[ξ1, . . . , ξn]. Then the following identity holds in HW :
[ηi, f ]+ = t
f − f τi
2ξi
+ u
∑
k 6=i
1
ξ2
i − ξ2
k
((
ξi − ξk
)
fsik −
(
ξif
τi − ξkf
τk
))
ski.
Theorem 5.10. Let W = WAn−1. The action of ηi on C[ξ1, . . . , ξn] is realized as Dunkl operators
as follows:
ηi = t∂ξi
+ u
∑
k 6=i
1
ξ2
i − ξ2
k
((
ξi − ξk
)
sik −
(
ξiτi − ξkτk
))
.
20 T. Khongsap and W. Wang
5.4.2 Dunkl operator for type Bn case
The proofs of Lemma 5.11 and 5.12 are given in the Appendix.
Lemma 5.11. Let W = WBn. Then the following holds in HW for l ∈ Z+ and i 6= j:
[ηi, ξ
l
j ]+ = u
(
ξl−1
i − ξjξ
l−2
i + · · ·+ (−1)l−1ξl−1
j
)(
sij + sij
)
= u
( 1
ξ2
i − ξ2
j
(
ξl+1
i − ξjξ
l
i − ξiξ
l
j + (−1)lξl+1
j
))(
sij + sij
)
,
[ηi, ξ
l
i]+ = t
ξl
i − (−ξi)l
2ξi
+ v
ξl
i − (−ξi)l
2ξi
τi
+ u
∑
k 6=i
(
ξl−1
k − ξiξ
l−2
k + · · ·+ (−1)l−1ξl−1
i
)(
sik + sik
)
= t
ξl
i − (−ξi)l)
2ξi
+ v
ξl
i − (−ξi)l
2ξi
τi
+ u
∑
k 6=i
1
ξ2
i − ξ2
k
(
ξiξ
l
k − ξl+1
k − (−1)lξl+1
i + ξkξ
l
i
)(
sik + sik
)
.
Lemma 5.12. Let W = WBn, and f ∈ C[ξ1, . . . , ξn]. Then the following identity holds in HW :
[ηi, f ]+ = t
f − f τi
2ξi
+ v
f − f τi
2ξi
τi
+ u
∑
k 6=i
1
ξ2
i − ξ2
k
((
ξi − ξk
)
fsik −
(
ξif
τi − ξkf
τk
))(
sik + sik
)
.
Theorem 5.13. Let W = WBn. The action of ηi on C[ξ1, . . . , ξn] is realized as operators as
follows:
ηi = t∂ξi
+ v
1− τi
2ξi
+ u
∑
k 6=i
2
ξ2
i − ξ2
k
((
ξi − ξk
)
sik −
(
ξiτi − ξkτk
))
.
Proof. It suffices to check the formula for every monomial f . Consider f = ξa1
1 · · · ξan
n where
ai ∈ Z+, and observe that
∂ξi
(f) =
f − f τi
2ξi
, ηi · f = [ηi, f ]+ + (−1)a1+···+anf · ηi = [ηi, f ]+.
The theorem now follows by Lemma 5.12. �
5.4.3 Dunkl operator for type Dn case
The formula below for the Dunkl operator type Dn case is obtained from their type Bn coun-
terparts (see Theorem 5.13) by dropping the terms involving the parameter v.
Theorem 5.14. Let W = WDn. The action of ηi on C[ξ1, . . . , ξn] is realized as Dunkl operators
as follows:
ηi = t∂ξi
+ u
∑
k 6=i
2
ξ2
i − ξ2
k
((
ξi − ξk
)
sik −
(
ξiτi − ξkτk
))
.
Remark 5.15. Let W = WAn−1 ,WBn , or WDn . The Dunkl operators ηi anti-commute, i.e.
ηiηj = −ηjηi (i 6= j). It is not easy to check this directly.
Hecke–Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type 21
5.5 An affine Hecke subalgebra
In this subsection, we will show that the odd DaHa of type A contains as a subalgebra the
degenerate affine Hecke algebra of type A introduced by Drinfeld and Lusztig [2, 11]. Let
zi = −ξiηi + u
∑
k<i
ski.
Lemma 5.16. We have [zi, zj ] = 0, ∀ i, j.
Proof. Let us assume i < j. Then,
[zi, zj ] =
[
− ξiηi,−ξjηj + u
∑
k<j
skj
]
= (ξi[ηi, ξj ]+ηj − ξj [ηj , ξi]+ηi)− u[ξiηi, sij ]
= u(ξisijηj − ξjsijηi)− u(ξiηi − ξjηj)sij = 0. �
Lemma 5.17. The following identities hold:
sizi = zi+1si − u, sizj = zjsi (j 6= i, i + 1).
Proof. Recall that Li :=
∑
k<i
ski is the Jucys–Murphy element, and it is known that siLi =
Li+1si − 1 and siLj = Ljsi for j 6= i, i + 1. The lemma follows from these relations. �
Proposition 5.18. The zi (1 ≤ i ≤ n) and Sn generate the degenerate affine Hecke algebra.
Proof. The proposition follows from Theorem 5.7 and Lemma 5.17. �
A Appendix: proofs of several lemmas
A.1 Proofs of Lemmas in Section 3
A.1.1 Proof of Lemma 3.6
We will show that the relations (3.4a) and (3.4b) are invariant under the conjugation by ele-
ments cl and el, 1 ≤ l ≤ n. We will only verify for the cl and leave the similar verification
for the el to the reader. Also, the verifications for the invariants in type A and D under the
conjugation by cl and el are similar and will be omitted.
Consider the relation (3.4a) first. Clearly, (3.4a) is invariant under the conjugation by cl,
and el if l 6= i, j. Moreover, we calculate that
ci(r.h.s. of (3.4a))ci = u
(
(1 + cicj)(1 + ejei)sij − (1− cicj)(1− ejei)sij
)
= [yi, xj ] = ci(l.h.s. of (3.4a))ci,
cj(r.h.s. of (3.4a))cj = u
(
(cjci − 1)(1 + ejei)sji − (−cjci − 1)(1− ejei)sij
)
= −[yi, xj ] = cj(l.h.s. of (3.4a))cj .
Thus, (3.4a) is conjugation-invariant by all cl.
Next, we will show that the relation (3.4b) is invariant under the conjugation by each cl.
Indeed, we have
ci(r.h.s. of (3.4b))ci
= teici − vciτici − u
∑
k 6=i
ci((1 + ckci)(1 + ekei)ski + (1− ckci)(1− ekei)ski)ci
22 T. Khongsap and W. Wang
= −tciei + vτi − u
∑
k 6=i
((cick − 1)(1 + ekei)ski + (−cick − 1)(1− ekei)ski)
= −tciei + vτi + u
∑
k 6=i
((1 + ckci)(1 + ekei)ski + (1− ckci)(1− ekei)ski)
= −[yi, xi] = ci(l.h.s. of (3.4b))ci.
For j 6= i, we have
cj(r.h.s. of (3.4b))cj
= tciei − vτi − ucj((1 + cjci)(1 + ejei)sji + (1− cjci)(1− ejei)sji)cj
− u
∑
k 6=i,j
cj((1 + ckci)(1 + ekei)ski + (1− ckci)(1− ekei)ski)cj
= tciei − vτi − u((cjci + 1)(1 + ejei)sji + (−cjci + 1)(1− ejei)sji)cj
− u
∑
k 6=i,j
((1 + ckci)(1 + ekei)ski + (1− ckci)(1− ekei)ski)
= cj(l.h.s. of (3.4b))cj .
Therefore, the lemma is proved.
A.1.2 Proof of Lemma 3.7
We will show below that the relations (3.4a)–(3.4b) are invariant under the conjugation by
elements in WBn . The proof can be readily modified to yield the Weyl group invariance of the
relations (3.2a)–(3.2b) and (3.3a)–(3.3b) in type A and D cases respectively, and we leave the
details to the reader.
(i) We check the invariance of (3.4a) under WBn .
Consider first the conjugation invariance by the transposition slk. If {l, k}∩ {i, j} = ∅, then
we have
slk(r.h.s. of (3.4a))slk = u
(
(1 + cicj)(1 + ejei)sij − (1− cicj)(1− ejei)sij
)
= [yi, xj ] = slk(l.h.s. of (3.4a))slk.
If {l, k} ∩ {i, j} = {j}, then we may assume l = j and we have
sjk(r.h.s. of (3.4a))sjk = u
(
(1 + cick)(1 + ekei)sik − (1− cick)(1− ekei)sik
)
= [yi, xk] = sjk(l.h.s. of (3.4a))sjk.
We leave an entirely analogous computation when {l, k} ∩ {i, j} = {i} to the reader.
Now, if {l, k} = {i, j}, then
sij(r.h.s. of (3.4a))sij = u
(
(1 + cjci)(1 + eiej)sij − (1− cjci)(1− eiej)sij
)
= [yj , xi] = sij(l.h.s. of (3.4a))sij .
So (3.4a) is invariant under the conjugation by each transposition slk.
It remains to show that (3.4a) is invariant under the conjugation by the simple reflection
sn = τn. Observe that (3.4a) is clearly invariant under the conjugation by sn for n 6= i, j.
Moreover, if j = n then we have
sn(r.h.s. of (3.4a))sn = u
(
(1− cicj)(1− ejei)sij − (1 + cicj)(1 + ejei)sij
)
= −[yi, xj ] = sn(l.h.s. of (3.4a))sn.
Hecke–Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type 23
If i = n, then we have
sn(r.h.s. of (3.4a))sn = u
(
(1− cicj)(1− ejei)sji − (1 + cicj)(1 + ejei)sij
)
= −[yi, xj ] = sn(l.h.s. of (3.4a))sn.
This completes (i).
(ii) We check the invariance of (3.4b) under WBn .
Consider first the conjugation invariance by sjl. If {j, l} ∩ {i} = ∅, then we have
sjl(r.h.s. of (3.4b))sjl
= tciei − vτi − u
∑
k 6=i,j,l
sjl((1 + ckci)(1 + ekei)ski + (1− ckci)(1− ekei)ski)sjl
− usjl ((1 + cjci)(1 + ejei)sji + (1− cjci)(1− ejei)sji) sjl
− usjl ((1 + clci)(1 + elei)sli + (1− clci)(1− elei)sli) sjl
= [yi, xi] = sjl(l.h.s. of (3.4b))sjl.
If {j, l} ∩ {i} = {i}, we may assume that j = i, and then we have
sil(r.h.s. of (3.4b))sil
= tclel − usil ((1 + clci)(1 + elei)sji + (1− clci)(1− elei)sli) sil
− u
∑
k 6=i,l
sil((1 + ckcl)(1 + ekei)skl + (1− ckcl)(1− ekei)skl)sil − vτl
= [yl, xl] = sil(l.h.s. of (3.4b))sil.
It remains to show that (3.4b) is invariant under the conjugation by the simple reflection
sn ≡ τn ∈ WBn . If i 6= n, we have
sn(r.h.s. of (3.4b))sn
= tciei − vτi − usn ((1 + cnci)(1 + enei)sni + (1− cnci)(1− enei)sni)) sn
− u
∑
k 6=i,n
sn((1 + ckci)(1 + ekei)ski + (1− ckci)(1− ekei)ski)sn
= −vτi − u ((1− cnci)sni + (1 + cnci)sni))− u
∑
k 6=i,n
((1 + ckci)ski + (1− ckci)ski)
= [yi, xi] = sn(l.h.s. of (3.4b))sn.
If i = n, then
sn(r.h.s. of (3.4b))sn
= tcnen − vτn − u
∑
k 6=n
((1− ckcn)(1− eken)skn + (1 + ckcn)(1 + eken)skn)
= [yn, xn] = sn(l.h.s. of (3.4b))sn.
This completes the proof of (ii). Hence the lemma is proved.
A.1.3 Proof of Lemma 3.8
We will establish the Jacobi identity for W = WBn . The proof can be easily modified for the
cases of type A and D, and we leave the details to the reader.
24 T. Khongsap and W. Wang
The Jacobi identity trivially holds among triple xi’s or triple yi’s.
Now, we consider the triple with two y’s and one x. The case with two identical yi is trivial.
So we first consider xi, yj , and yl where i, j, l are all distinct. The Jacobi identity holds in this
case since
[xi, [yj , yl]] + [yl, [xi, yj ]] + [yj , [yl, xi]]
= 0 + [yl,−u
(
(1 + cjci)(1 + eiej)sji − (1− cjci)(1− eiej)sij
)
]
+ [yj , u ((1 + clci)(1− eiel)sli − (1− clci)(1− eiel)sil)] = 0.
Now for i 6= j, we have
[xi, [yi, yj ]] + [yj , [xi, yi]] + [yi, [yj , xi]]
=
[
yj ,−tciei + u
∑
k 6=i
(
(1 + ckci)(1 + ekei)ski + (1− ckci)(1− ekei)ski
)
+ vτi
]
+ [yi, u ((1 + cjci)(1 + eiej)sij − (1− cjci)(1− eiej)sij)]
=
[
yj , u
∑
k 6=i,j
(
(1 + ckci)(1 + ekei)ski + (1− ckci)(1− ekei)ski
)]
+ [yj , u
(
(1 + cjci)(1 + ejei)sji + (1− cjci)(1− ejei)sji
)
]
+ [yi, u ((1 + cjci)(1 + eiej)sij − (1− cjci)(1− eiej)sij)]
= 0 + u
(
yj(1 + cjci)(1 + ejei)sji + yj(1− cjci)(1− ejei)sji
)
− u
(
(1 + cjci)(1 + ejei)sjiyj + (1− cjci)(1− ejei)sjiyj
)
+ u (yi(1 + cjci)(1 + eiej)sji − yi(1− cjci)(1− eiej)sij)
− u
(
(1 + cjci)(1 + eiej)sjiyi − (1− cjci)(1− eiej)sijyi
)
= 0.
Thanks to the automorphism $ of Hcc
W which switches xi and yi, we obtain the Jacobi identity
with one y and two x’s from the above calculation. This completes the proof of Lemma 3.8.
A.1.4 Proof of Lemma 3.13
We will proceed by induction on l. For l = 1, then the equations hold by (3.4a) and (3.4b). Now
assume that the statement is true for l. Then
[yi, x
l+1
j ] = [yi, x
l
j ]xj + xl
j [yi, xj ]
= u
(
xl
j − xl
i
xj − xi
+
xl
j − (−xi)l
xj + xi
cicj
)
(1− eiej)sijxj
− u
(
xl
j − (−xi)l
xj + xi
−
xl
j − xl
i
xj − xi
cicj
)
(1 + eiej)sijxj
+ xl
ju
(
(1 + cicj)(1 + ejei)sij − (1− cicj)(1− ejei)sij
)
= u
(
xl+1
j − xl+1
i
xj − xi
+
xl+1
j − (−xi)l+1
xj + xi
cicj
)
(1− eiej)sij
− u
(
xl+1
j − (−xi)l+1
xj + xi
−
xl+1
j − xl+1
i
xj − xi
cicj
)
(1 + eiej)sij ,
[yi, x
l+1
i ] = [yi, x
l
i]xi + xl
i[yi, xi]
= tciei
xl
i − (xl
i)
τi
2
− v
xl
i − (xl
i)
τi
2
τi
Hecke–Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type 25
− u
∑
k 6=i
(
xl
i − xl
k
xi − xk
+
xl
i − (−xk)l
xi + xk
ckci
)
(1 + ekei)skixi
− u
∑
k 6=i
(
xl
i − (−xk)l
xi + xk
−
xl
i − xl
k
xi − xk
ckci
)
(1− ekei)skixi
− uxl
i
∑
k 6=i
((1 + ckci)(1 + ekei)ski + (1− ckci)(1− ekei)ski) + txl
iciei − vxl
iτi
= tciei
xl+1
i − (xl+1
i )τi
2xi
− v
xl+1
i − (xl+1
i )τi
2xi
τi
− u
∑
k 6=i
(
xl+1
i − xl+1
k
xi − xk
+
xl+1
i − (−xk)l+1
xi + xk
ckci
)
(1 + ekei)ski
− u
∑
k 6=i
(
xl+1
i − (−xk)l+1
xi + xk
−
xl+1
i − xl+1
k
xi − xk
ckci
)
(1− ekei)ski.
This completes the proof.
A.1.5 Proof of Lemma 3.14
It suffices to check the formula for every monomial f . First, we consider the monomial g =∏
j 6=i
x
aj
j . By induction and Lemma 3.13, we can show that the formula holds for the monomial
of the form g =
∏
j 6=i
x
aj
j (the detail of the induction step does not differ much from the following
calculation). Now consider the monomial f = xl
ig.
[yi, f ] = [yi, x
l
i]g + xl
i[yi, g]
= tciei
xl
i − (−xi)l
2xi
g − v
xl
i − (−xi)l
2xi
τig
− u
∑
k 6=i
(
xl
i − xl
k
xi − xk
+
xl
i − (−xk)l
xi + xk
ckci
)
(1 + ekei)skig
− u
∑
k 6=i
(
xl
i − (−xk)l
xi + xk
−
xl
i − xl
k
xi − xk
ckci
)
(1− ekei)skig
+ txl
iciei
g − gτi
2xi
− vxl
i
g − gτi
2xi
τi
− u
∑
k 6=i
xl
i
(
g − gski
xi − xk
+
g − gski
xi + xk
ckci
)
(1 + ekei)ski
− u
∑
k 6=i
xl
i
(
g − gski
xi + xk
− g − gski
xi − xk
ckci
)
(1− ekei)ski
= tciei
f − f τi
2xi
− v
f − f τi
2xi
τi − u
∑
k 6=i
(
f − fski
xi − xk
+
f − fski
xi + xk
ckci
)
(1 + ekei)ski
− u
∑
k 6=i
(
f − fski
xi + xk
− f − fski
xi − xk
ckci
)
(1− ekei)ski.
So the lemma is proved.
26 T. Khongsap and W. Wang
A.2 Proofs of Lemmas in Section 5
A.2.1 Proof of Lemma 5.11
We will proceed by induction on l. For l = 1, then the equations hold by the definition of HW .
Now assume that the statement is true for l. Then
[ηi, ξ
l+1
j ]+ = [ηi, ξ
l
j ]+ξj + (−1)lξl
j [ηi, ξj ]+
= u
1
ξ2
i − ξ2
j
(
ξl+1
i − ξjξ
l
i − ξiξ
l
j + (−1)lξl+1
j
)(
sij + sij
)
ξj
+ u
(−ξj)l
ξ2
i − ξ2
j
(ξ2
i − ξ2
j )
(
sij + sij
)
= u
1
ξ2
i − ξ2
j
(
ξl+2
i − ξjξ
l+1
i − ξiξ
l+1
j + (−1)l+1ξl+2
j
)(
sij + sij
)
,
[ηi, ξ
l+1
i ]+ = [ηi, ξ
l
i]+ξi + (−1)lξl
i[ηi, ξi]+
= t
ξl
i − (ξl
i)
τi
2ξi
ξi + v
ξl
i − (ξl
i)
τi
2ξi
τiξi
+ u
∑
k 6=i
1
ξ2
i − ξ2
k
(
ξiξ
l
k − ξl+1
k − (−1)lξl+1
i + ξkξ
l
i
)(
sik + sik
)
ξi
+ t(−ξi)l + v(−ξi)lτi + u
∑
k 6=i
(−ξi)l
ξ2
i − ξ2
k
(
ξ2
i − ξ2
k
)(
ski + sij
)
= t
ξl+1
i − (ξl+1
i )τi
2ξi
+ v
ξl+1
i − (ξl+1
i )τi
2ξi
τi
+ u
∑
k 6=i
1
ξ2
i − ξ2
k
(
ξiξ
l+1
k − ξl+2
k − (−1)l+1ξl+2
i + ξkξ
l+1
i
)(
sik + sik
)
.
This completes the proof.
A.2.2 Proof of Lemma 5.12
It suffices to check the formula for every monomial f . First, we consider the monomial g =∏
j 6=i
ξ
aj
j . By induction and Lemma 5.11, we can show that the formula holds for the monomial
of the form g =
∏
j 6=i
ξ
aj
j (the detail of the induction step does not differ much from the following
calculation). Now consider the monomial f = ξl
ig.
[ηi, f ]+ = [ηi, ξ
l
i]+g + (−1)lξl
i[ηi, g]+
= t
ξl
ig − (ξl
ig)τi
2ξi
+ v
ξl
ig − (ξl
ig)τi
2ξi
τi
+ u
∑
k 6=i
1
ξ2
i − ξ2
k
(
ξiξ
l
k − ξl+1
k − (−1)lξl+1
i + ξkξ
l
i
)(
sik + sik
)
g
+ u
∑
k 6=i
(−ξi)l
ξ2
i − ξ2
k
((
ξi − ξk
)
gsik −
(
ξig
τi − ξkg
τk
))(
sik + sik
)
= t
ξl
ig − (ξl
ig)τi
2ξi
+ v
ξl
ig − (ξl
ig)τi
2ξi
τi
+ u
∑
k 6=i
1
ξ2
i − ξ2
k
(
(ξi − ξk)(ξl
ig)sik − (ξi(ξl
i)
τi − ξk(ξl
i)
τk)gsik
)(
sik + sik
)
Hecke–Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type 27
+ u
∑
k 6=i
(−ξi)l
ξ2
i − ξ2
k
((
ξi − ξk
)
gsik −
(
ξig
τi − ξkg
τk
))(
sik + sik
)
= t
ξl
ig − (ξl
ig)τi
2ξi
+ v
ξl
ig − (ξl
ig)τi
2ξi
τi
+ u
∑
k 6=i
1
ξ2
i − ξ2
k
((
ξi − ξk
)
(ξl
ig)sik −
(
ξi(ξl
ig)τi − ξk(ξl
ig)τk
))(
sik + sik
)
.
So the lemma is proved.
Acknowledgements
This research is partially supported by NSF grant DMS-0800280. The main results of this paper
for type A were obtained at MSRI in 2006.
References
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Clarendon Press, Oxford University Press, New York, 1987.
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form. Groups 13 (2008), 389–412, arXiv:0704.0201.
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http://arxiv.org/abs/0806.0867
http://arxiv.org/abs/math.RT/0108185
http://arxiv.org/abs/math.AG/0011114
http://arxiv.org/abs/0808.2951
http://arxiv.org/abs/0704.0201
http://arxiv.org/abs/0710.5877
http://arxiv.org/abs/math.RT/0504600
http://arxiv.org/abs/math.RT/0608074
http://arxiv.org/abs/math.RT/0611950
1 Introduction
2 Schur multipliers of Weyl groups and Clifford algebras
2.1 A distinguished double cover
2.2 Clifford algebra
2.3 A superalgebra isomorphism
2.4 The case of general 2-cocycles
3 The DaHa with two polynomial-Clifford subalgebras
3.1 The definition of H^{cc}_W
3.1.1 The algebra H^{cc}_W of type A_{n-1}
3.1.2 The algebra H^{cc}_W of type D_n
3.1.3 The algebra H^{cc}_W of type B_n
3.2 The PBW basis for H^{cc}_W
3.3 The Dunkl representations
3.3.1 The Dunkl Operators for type A_{n-1} case
3.3.2 The Dunkl Operators for type B_n case
3.3.3 The Dunkl Operators for type D_n case
3.4 The even center for H^{cc}_W
4 The spin double affine Hecke-Clifford algebras
4.1 The definition of sDaHCa H^{-c}_W
4.2 Isomorphism of superalgebras
4.3 The PBW property for H^{-c}_W
4.4 The Dunkl operators for H^{-c}_W
4.4.1 Dunkl operator for type A_{n-1}
4.4.2 Dunkl operator for type B_n
4.4.3 Dunkl operator for type D_n
5 The odd double affine Hecke algebras
5.1 The definition of H_W
5.2 Isomorphism of superalgebras
5.3 The PBW property for H_W
5.4 The Dunkl operators for H_W
5.4.1 Dunkl operator for type A case
5.4.2 Dunkl operator for type B_n case
5.4.3 Dunkl operator for type D_n case
5.5 An affine Hecke subalgebra
A Appendix: proofs of several lemmas
A.1 Proofs of Lemmas in Section 3
A.1.1 Proof of Lemma 3.6
A.1.2 Proof of Lemma 3.7
A.1.3 Proof of Lemma 3.8
A.1.4 Proof of Lemma 3.13
A.1.5 Proof of Lemma 3.14
A.2 Proofs of Lemmas in Section 5
A.2.1 Proof of Lemma 5.11
A.2.2 Proof of Lemma 5.12
References
|
| id | nasplib_isofts_kiev_ua-123456789-149249 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:57:07Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Khongsap, T. Wang, W. 2019-02-19T19:21:36Z 2019-02-19T19:21:36Z 2009 Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type / T. Khongsap, W. Wang // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 17 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 20C08 https://nasplib.isofts.kiev.ua/handle/123456789/149249 We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skew-polynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W and two polynomial-Clifford subalgebras. There is yet a third algebra containing a spin Weyl group algebra which is Morita (super)equivalent to the above two algebras. We establish the PBW properties and construct Verma-type representations via Dunkl operators for these algebras. This paper is a contribution to the Special Issue on Dunkl Operators and Related Topics. This research is partially supported by NSF grant DMS-0800280. The main results of this paper for type A were obtained at MSRI in 2006. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type Article published earlier |
| spellingShingle | Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type Khongsap, T. Wang, W. |
| title | Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type |
| title_full | Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type |
| title_fullStr | Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type |
| title_full_unstemmed | Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type |
| title_short | Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type |
| title_sort | hecke-clifford algebras and spin hecke algebras iv: odd double affine type |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149249 |
| work_keys_str_mv | AT khongsapt heckecliffordalgebrasandspinheckealgebrasivodddoubleaffinetype AT wangw heckecliffordalgebrasandspinheckealgebrasivodddoubleaffinetype |