HECKE-CLIFFORD ALGEBRAS AND SPIN HECKE ALGEBRAS I: THE ... · algebras for all classical ﬁnite Weyl groups, which goes beyond the type A case, and then establish some basic properties

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HECKE-CLIFFORD ALGEBRAS AND SPIN HECKE

ALGEBRAS I: THE CLASSICAL AFFINE TYPE

TA KHONGSAP AND WEIQIANG WANG

Abstract. Associated to the classical Weyl groups, we introduce thenotion of degenerate spin affine Hecke algebras and affine Hecke-Cliffordalgebras. For these algebras, we establish the PBW properties, for-mulate the intertwiners, and describe the centers. We further developconnections of these algebras with the usual degenerate (i.e. graded)affine Hecke algebras of Lusztig by introducing a notion of degeneratecovering affine Hecke algebras.

1. Introduction

1.1. The Hecke algebras associated to finite and affine Weyl groups areubiquitous in diverse areas, including representation theories over finitefields, infinite fields of prime characteristic, p-adic fields, and Kazhdan-Lusztig theory for category O. Lusztig [Lu1, Lu2] introduced the gradedHecke algebras, also known as the degenerate affine Hecke algebras, associ-ated to a finite Weyl groupW , and provided a geometric realization in termsof equivariant homology. The degenerate affine Hecke algebra of type A hasalso been defined earlier by Drinfeld [Dr] in connections with Yangians, andit has recently played an important role in modular representations of thesymmetric group (cf. Kleshchev [Kle]).

In [W1], the second author introduced the degenerate spin affine Heckealgebra of type A, and related it to the degenerate affine Hecke-Cliffordalgebra introduced by Nazarov in his study of the representations of the spinsymmetric group [Naz]. A quantum version of the spin affine Hecke algebraof type A has been subsequently constructed in [W2], and was shown to berelated to the q-analogue of the affine Hecke-Clifford algebra (of type A)defined by Jones and Nazarov [JN].

1.2. The goal of this paper is to provide canonical constructions of thedegenerate affine Hecke-Clifford algebras and degenerate spin affine Heckealgebras for all classical finite Weyl groups, which goes beyond the type Acase, and then establish some basic properties of these algebras. The notionof spin Hecke algebras is arguably more fundamental while the notion ofthe Hecke-Clifford algebras is crucial for finding the right formulation of thespin Hecke algebras. We also construct the degenerate covering affine Heckealgebras which connect to both the degenerate spin affine Hecke algebrasand the degenerate affine Hecke algebras of Lusztig.

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http://arxiv.org/abs/0704.0201v2

2 TA KHONGSAP AND WEIQIANG WANG

1.3. Let us describe our constructions in some detail. The Schur multiplierfor each finite Weyl group W has been computed by Ihara and Yokonuma

[IY] (see [Kar]). We start with a distinguished double cover W for any finiteWeyl group W :

1 −→ Z2 −→ W −→W −→ 1. (1.1)

Denote Z2 = 1, z. Assume that W is generated by s1, . . . , sn subject to

the relations (sisj)mij = 1. The quotient CW− := CW/〈z + 1〉 is then

generated by t1, . . . , tn subject to the relations (titj)mij = 1 for mij odd,

and (titj)mij = −1 for mij even. In the symmetric group case, this double

cover goes back to I. Schur [Sch]. Note that W acts as automorphisms onthe Clifford algebra CW associated to the reflection representation h of W .We establish a (super)algebra isomorphism

Φfin : CW ⋊CW≃−→ CW ⊗ CW−,

extending an isomorphism in the symmetric group case (due to Sergeev[Ser] and Yamaguchi [Yam] independently) to all Weyl groups. That is,the superalgebras CW ⋊ CW and CW− are Morita super-equivalent in the

terminology of [W2]. The double cover W also appeared in Morris [Mo].We formulate the notion of degenerate affine Hecke-Clifford algebras Hc

W

and spin affine Hecke algebras H−W , with unequal parameters in type B

case, associated to Weyl groups W of type D and B. The algebra HcW (and

respectively H−W ) contain CW ⋊CW (and respectively CW−) as subalgebras.

We establish the PBW basis properties for these algebras:

HcW

∼= C[h∗]⊗ CW ⊗CW, H−W

∼= C[h∗]⊗ CW−

where C[h∗] denotes the polynomial algebra and C[h∗] denotes a noncommu-tative skew-polynomial algebra. We describe explicitly the centers for bothHcW and H−

W . The two Hecke algebras HcW and H−

W are related by a Moritasuper-equivalence, i,e. a (super)algebra isomorphism

Φ : HcW

≃−→ CW ⊗ H−W

which extends the isomorphism Φfin. Such an isomorphism holds also forW of type A [W1].

We generalize the construction in [Naz] of the intertwiners in the affineHecke-Clifford algebras Hc

W of type A to all classical Weyl groups W . We

also generalize the construction of the intertwiners in [W1] for H−W of type

A to all classical Weyl groups W . We further establish the basic propertiesof these intertwiners in both Hc

W and H−W . These intertwiners are expected

to play a fundamental role in the future development of the representationtheory of these algebras, as it is indicated by the work of Lusztig, Cherednikand others in the setup of the usual affine Hecke algebras.

We further introduce a notion of degenerate covering affine Hecke algebras

H∼W associated to the double cover W of the Weyl group W of classical

type. The algebra H∼W contains a central element z of order 2 such that the

THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 3

quotient of H∼W by the ideal 〈z+1〉 is identified with H−

W and its quotient bythe ideal 〈z−1〉 is identified with Lusztig’s degenerate affine Hecke algebrasassociated toW . In this sense, our covering affine Hecke algebra is a naturalaffine generalization of the central extension (1.1). A quantum version ofthe covering affine Hecke algebra of type A was constructed in [W2].

The results in this paper remain valid over any algebraically closed fieldof characteristic p 6= 2 (and in addition p 6= 3 for type G2). In fact, mostof the constructions can be made valid over the ring Z[12 ] (occasionally we

need to adjoint√2).

1.4. This paper and [W1] raise many questions, including a geometric re-alization of the algebras Hc

W or H−W in the sense of Lusztig [Lu1, Lu2], the

classification of the simple modules (cf. [Lu3]), the development of the rep-resentation theory, an extension to the exceptional Weyl groups, and so on.We remark that the modular representations of Hc

W in the type A case in-cluding the modular representations of the spin symmetric group has beendeveloped by Brundan and Kleshchev [BK] (also cf. [Kle]).

In a sequel [KW] to this paper, we will extend the constructions in thispaper to the setup of rational double affine Hecke algebras (see Etingof-Ginzburg [EG]), generalizing and improving a main construction initiatedin [W1] for the spin symmetric group. We also hope to quantize thesedegenerate spin Hecke algebras, reversing the history of developments fromquantum to degeneration for the usual Hecke algebras.

1.5. The paper is organized as follows. In Section 2, we describe the distin-guished covering groups of the Weyl groups, and establish the isomorphismtheorem in the finite-dimensional case. We introduce in Section 3 the degen-erate affine Hecke-Clifford algebras of type D and B, and in Section 4 thecorresponding degenerate spin affine Hecke algebras. We then extend theisomorphism Φfin to an isomorphism relating these affine Hecke algebras,establish the PBW properties, and describe the centers of Hc

W and H−W . In

Section 5, we formulate the notion of degenerate covering affine Hecke al-gebras, and establish the connections to the degenerate spin affine Heckealgebras and usual affine Hecke algebras.

Notations: Z+ denotes 0, 1, 2, · · · , and Z2 denotes Z/2Z.Acknowledgements. W.W. is partially supported by an NSF grant.

2. Spin Weyl groups and Clifford algebras

2.1. The Weyl groups. Let W be an (irreducible) finite Weyl group withthe following presentation:

〈s1, . . . , sn|(sisj)mij = 1, mii = 1, mij = mji ∈ Z≥2, for i 6= j〉 (2.1)

For a Weyl group W , the integers mij take values in 1, 2, 3, 4, 6, andthey are specified by the following Coxeter-Dynkin diagrams whose verticescorrespond to the generators of W . By convention, we only mark the edge


connecting i, j with mij ≥ 4. We have mij = 3 for i 6= j connected by anunmarked edge, and mij = 2 if i, j are not connected by an edge.

An . . . 1 2 n− 1 n

Bn(n ≥ 2) . . . 1 2 n− 1 n

4

Dn(n ≥ 4) · · ·

1 2 n− 3n− 2

n− 1

n

En=6,7,8 . . .

1 3 4 n− 1 n

2

F4 1 2 3 4

4

G2 1 2

6

2.2. A distinguished double covering of Weyl groups. The Schur mul-tipliers for finite Weyl groups W (and actually for all finite Coxeter groups)have been computed by Ihara and Yokonuma [IY] (also cf. [Kar]). Theexplicit generators and relations for the corresponding covering groups of Wcan be found in Karpilovsky [Kar, Table 7.1].

We shall be concerned about a distinguished double covering W of W :

1 −→ Z2 −→ W −→W −→ 1.

We denote by Z2 = 1, z, and by ti a fixed preimage of the generators siof W for each i. The group W is generated by z, t1, . . . , tn with relations


(besides the obvious relation that z is central of order 2) listed in the follow-ing table, which corresponds to setting the αi for all i in Karpilovsky [Kar,Table 7.1] to be z.

W Generators/Relations for Wt2i = 1, 1 ≤ i ≤ n,

An titi+1ti = ti+1titi+1, 1 ≤ i ≤ n− 1titj = ztj ti if mij = 2t2i = 1, 1 ≤ i ≤ n, titi+1ti = ti+1titi+1, 1 ≤ i ≤ n− 2

Bn titj = ztj ti, 1 ≤ i < j ≤ n− 1,mij = 2(n ≥ 2) titn = ztnti, 1 ≤ i ≤ n− 2

(tn−1tn)2 = z(tn tn−1)

2

t2i = 1, 1 ≤ i ≤ n, titj ti = tj titj if mij = 3Dn titj = ztj ti, 1 ≤ i < j ≤ n,mij = 2, i 6= n− 1

(n ≥ 4) tn−1tn = ztntn−1

E6,7,8 t2i = 1, 1 ≤ i ≤ n, titj ti = tj titj if mij = 3titj = ztj ti if mij = 2t2i = 1, 1 ≤ i ≤ 4, titi+1ti = ti+1titi+1 (i = 1, 3)

F4 titj = ztj ti, 1 ≤ i < j ≤ 4,mij = 2(t2t3)

2 = z(t3t2)2

G2 t21 = t22 = 1, (t1t2)3 = z(t2t1)

3

The quotient algebra CW− := CW/〈z+1〉 of CW by the ideal generatedby z+1 will be called the spin Weyl group algebra associated to W . Denoteby ti ∈ CW− the image of ti. The spin Weyl group algebra CW− has thefollowing uniform presentation: CW− is 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.(2.2)

Note that dimCW− = |W |. The algebra CW− has a natural superalgebra(i.e. Z2-graded) structure by letting each ti be odd.

By definition, the quotient by the ideal 〈z − 1〉 of the group algebra CWis isomorphic to CW .

Example 2.1. Let W be the Weyl group of type An, Bn, or Dn, which willbe assumed in later sections. Then the spin Weyl group algebra CW− isgenerated by t1, . . . , tn with the labeling as in the Coxeter-Dynkin diagramsand the explicit relations summarized in the following table.


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

2.3. The Clifford algebra CW . Denote by h the reflection representationof the Weyl groupW (i.e. a Cartan subalgebra of the corresponding complexLie algebra g). In the case of type An−1, we will always choose to work withthe Cartan subalgebra h 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 (−,−). One standard way is to identify h∗

with a suitable subspace of the Euclidean space CN and then describe the

simple roots αi for g using a standard orthonormal basis ei of CN .Denote by CW the Clifford algebra associated to (h, (−,−)), which is re-

garded 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 CW corresponding to the simple root αi normalizedwith β2i = 1. In particular, CN is generated by c1, . . . , cN subject to therelations

c2i = 1, cicj = −cjci if i 6= j. (2.3)

The explicit generators for CW are listed in the following table. Note thatCW is naturally a superalgebra with each βi being odd.

Type of W N Generators for CW

An−1 n βi =1√2(ci − ci+1), 1 ≤ i ≤ n− 1

Bn n βi =1√2(ci − ci+1), 1 ≤ i ≤ n− 1, βn = cn

Dn n βi =1√2(ci − ci+1), 1 ≤ i ≤ n− 1, βn = 1√

2(cn−1 + cn)

E8 8 β1 =1

2√2(c1 + c8 − c2 − c3 − c4 − c5 − c6 − c7)

β2 =1√2(c1 + c2), βi =

1√2(ci−1 + ci−2), 3 ≤ i ≤ 8

E7 8 the subset of βi in E8, 1 ≤ i ≤ 7E6 8 the subset of βi in E8, 1 ≤ i ≤ 6

F4 4 β1 =1√2(c1 − c2), β2 = 1√

2(c2 − c3)

β3 = c3, β4 = 12(c4 − c1 − c2 − c3)

G2 3 β1 =1√2(c1 − c2), β2 = 1√

6(−2c1 + c2 + c3)


The action of W on h and h∗ preserves the bilinear form (−,−) and thusW acts as automorphisms of the algebra CW . This gives rise to a semi-directproduct CW ⋊CW . Moreover, the algebra CW ⋊CW naturally inherits thesuperalgebra structure by letting elements inW be even and each βi be odd.

2.4. The basic spin supermodule.

Theorem 2.2. Let W be a finite Weyl group. Then, there exists a surjective

superalgebra homomorphism CW− Ω−→ CW which sends ti to βi for each i.

Proof. It suffices to show that each βi satisfies the same relations as for theti’s, i.e. (βiβj)

mij = 1 for mij odd and (βiβj)mij = −1 for mij even. This

can be checked case by case using the explicit formulas of βi in the Table ofSection 2.3.

Remark 2.3. In the type A, namely the symmetric group case, Theorem 2.2goes back to I. Schur [Sch] (cf. [Joz]). Theorem 2.2 has appeared in asomewhat different form in Morris [Mo]. In [Mo], W is viewed as a subgroupof the orthogonal Lie group which preserves (h, (−,−)). The preimage ofW in the spin group which covers the orthogonal group provides the double

cover W ofW , where the Atiyah-Bott-Shapiro construction of the spin groupin terms of the Clifford algebra CW was used to describe this double coverof W .

The superalgebra CW has a unique (up to isomorphism) simple super-module (i.e. Z2-graded module). By pulling it back via the homomorphismΩ : CW− → CW , we obtain a distinguished CW−-supermodule, called thebasic spin supermodule. This is a natural generalization of the classicalconstruction for CS−

n due to Schur [Sch] (see [Joz]).

2.5. A superalgebra isomorphism. Given two superalgebras A and B,we view the tensor product of superalgebras A ⊗ B as a superalgebra withmultiplication defined by

(a⊗ b)(a′ ⊗ b′) = (−1)|b||a′|(aa′ ⊗ bb′) (a, a′ ∈ A, b, b′ ∈ B) (2.4)

where |b| denotes the Z2-degree of b, etc. Also, we shall use short-handnotation ab for (a⊗ b) ∈ A ⊗ B, a = a⊗ 1, and b = b⊗ 1.

We have the following Morita super-equivalence in the sense of [W2] be-tween the superalgebras CW ⋊CW and CW−.

Theorem 2.4. We have an isomorphism of superalgebras:

Φ : CW ⋊CW≃−→ CW ⊗ CW−

which extends the identity map on CW and sends si 7→ −√−1βiti. The

inverse map Ψ is the extension of the identity map on CW which sendsti 7→

√−1βisi.

We first prepare some lemmas.

Lemma 2.5. We have (Φ(si)Φ(sj))mij = 1.


Proof. Theorem 2.2 says that (titj)mij = (βiβj)

mij = ±1. Thanks to the

identities βjti = −tiβj and Φ(si) = −√−1βiti, we have

(Φ(si)Φ(sj))mij = (−βitiβjtj)mij

= (βiβjtitj)mij = (βiβj)

mij (titj)mij = 1.

Lemma 2.6. We have βjΦ(si) = Φ(si) si(βj) for all i, j.

Proof. Note that (βi, βi) = 2β2i = 2, and hence

βjβi = −βiβj + (βj , βi) = −βiβj +2(βj , βi)

(βi, βi)β2i = −βisi(βj).

Thus, we have

βjΦ(si) = −√−1βjβiti

= −√−1tiβjβi =

√−1tiβisi(βj) = Φ(si) si(βj).

Proof of Theorem 2.4. The algebra CW ⋊CW is generated by βi and si forall i. Lemmas 2.5 and 2.6 imply that Φ is a (super) algebra homomorphism.Clearly Φ is surjective, and thus an isomorphism by a dimension countingargument.

Clearly, Ψ and Φ are inverses of each other.

Remark 2.7. We were led to consider the distinguished double cover W insearch of an isomorphism as in Theorem 2.4 and found Theorem 2.2 beforelearning about [Mo]. The type A case of Theorem 2.4 was due to Sergeevand Yamaguchi independently [Ser, Yam], and it played a fundamental rolein clarifying the earlier observation in the literature (cf. [Joz, St]) that therepresentation theories of CS−

n and Cn ⋊CSn are essentially the same.

In the remainder of the paper, W is always assumed to be one of theclassical Weyl groups of type A,B, or D.

3. Degenerate affine Hecke-Clifford algebras

In this section, we introduce the degenerate affine Hecke-Clifford algebrasof type D and B, and establish some basic properties. The degenerate affineHecke-Clifford algebra associated to the symmetric group Sn was introducedearlier by Nazarov under the terminology of the affine Sergeev algebra [Naz].

3.1. The algebra HcW of type An−1.

Definition 3.1. [Naz] Let u ∈ C, and W =WAn−1 = Sn be the Weyl groupof type An−1. The degenerate affine Hecke-Clifford algebra of type An−1,


denoted by HcW or Hc

An−1, is the algebra generated by x1, . . . , xn, c1, . . . , cn,

and Sn subject to the relation (2.3) and the following relations:

xixj = xjxi (∀i, j) (3.1)

xici = −cixi, xicj = cjxi (i 6= j) (3.2)

σci = cσiσ (1 ≤ i ≤ n, σ ∈ Sn) (3.3)

xi+1si − sixi = u(1− ci+1ci) (3.4)

xjsi = sixj (j 6= i, i+ 1) (3.5)

Remark 3.2. Alternatively, we may view u as a formal parameter and thealgebra Hc

W as a C(u)-algebra. Similar remarks apply to various algebrasintroduced in this paper. Our convention c2i = 1 differs from Nazarov’swhich sets c2i = −1.

The symmetric group Sn acts as the automorphisms on the symmetricalgebra C[h∗] ∼= C[x1, . . . , xn] by permutation. We shall denote this actionby f 7→ fσ for σ ∈ Sn, f ∈ C[x1, . . . , xn].

Proposition 3.3. Let W = WAn−1. Given f ∈ C[x1, . . . , xn] and 1 ≤ i ≤n− 1, the following identity holds in Hc

W :

sif = f sisi + uf − f si

xi+1 − xi+ u

cici+1f − f sicici+1

xi+1 + xi.

It is understood here and in similar expressions below that Ag(x) = 1

g(x) · A.In this sense, both numerators on the right-hand side of the above formulaare (left-)divisible by the corresponding denominators.

Proof. By the definition of HcW , we have that six

kj = xkj si for any k if

j 6= i, i + 1. So it suffices to check the identity for f = xki xli+1. We will

proceed by induction.First, consider f = xki , i.e. l = 0. For k = 1, this follows from (3.4). Now

assume that the statement is true for k. Then

sixk+1i =

(xki+1si + u

(xki − xki+1)

xi+1 − xi+ u

(cici+1xki − xki+1cici+1)

xi+1 + xi

)xi

= xki+1 (xi+1si − u(1− ci+1ci))

+u(xki − xki+1)

xi+1 − xixi + u

(cici+1xki − xki+1cici+1)

xi+1 + xixi

= xk+1i+1 si + u

(xk+1i − xk+1

i+1 )

xi+1 − xi+ u

(cici+1xk+1i − xk+1

i+1 cici+1)

xi+1 + xi,

where the last equality is obtained by using (3.2) and (3.4) repeatedly.An induction on l will complete the proof of the proposition for the mono-

mial f = xki xli+1. The case l = 0 is established above. Assume the formula is


true for f = xki xli+1. Then using sixi+1 = (xisi+u(1+ ci+1ci)), we compute

that

sixki x

l+1i+1 =

(xlix

ki+1si + u

(xki xli+1 − xlix

ki+1)

xi+1 − xi

+u(cici+1x

ki x

li+1 − xlix

ki+1cici+1)

xi+1 + xi

)· xi+1

= xlixki+1(xisi + u(1 + ci+1ci))

+u(xki x

l+1i+1 − xlix

k+1i+1 )

xi+1 − xi+ u

(cici+1xki x

l+1i+1 + xlix

k+1i+1 cici+1)

xi+1 + xi

= xl+1i xki+1si + u

(xki xl+1i+1 − xl+1

i xki+1)

xi+1 − xi

+u(cici+1x

ki x

l+1i+1 − xl+1

i xki+1cici+1)

xi+1 + xi.

This completes the proof of the proposition.

The algebra HcW contains C[h∗],Cn, and CW as subalgebras. We shall

denote xα = xa11 · · · xann for α = (a1, . . . , an) ∈ Zn+, c

ǫ = cǫ11 · · · cǫnn for ǫ =(ǫ1, . . . , ǫn) ∈ Z

n2 .

Below we give a new proof of the PBW basis theorem for HcW (which

has been established by different methods in [Naz, Kle]), using in effect the

induced HcW -module Ind

HcW

W 1 from the trivial W -module 1. This inducedmodule is of independent interest. This approach will then be used for typeD and B.

Theorem 3.4. LetW =WAn−1 . The multiplication of subalgebras C[h∗],Cn,and CW induces a vector space isomorphism

C[h∗]⊗ Cn ⊗ CW≃−→ Hc

W .

Equivalently, xαcǫw|α ∈ Zn+, ǫ ∈ Z

n2 , w ∈ W forms a linear basis for Hc

W

(called a PBW basis).

Proof. Note that IND := C[x1, . . . , xn] ⊗ Cn admits an algebra structureby (2.3), (3.1) and (3.2). By the explicit defining relations of Hc

W , we canverify that the algebra Hc

W acts on IND by letting xi and ci act by leftmultiplication, and si ∈ Sn act by

si.(fcǫ) = f sicsiǫ +

(uf − f si

xi+1 − xi+ u


xi+1 + xi

)cǫ.

For α = (a1, . . . , an), we denote |α| = a1 + · · · + an. Define a dictionarytotal ordering < on the monomials xα, α ∈ Z

n+, (or respectively on Z

n+), by

declaring xα < xα′

, (or respectively α < α′), if |α| < |α′|, or if |α| = |α′|then there exists an 1 ≤ i ≤ n such that ai < a′i and aj = a′j for each j < i.


Note that the algebra HcW is spanned by the elements of the form xαcǫw.

It remains to show that these elements are linearly independent.Suppose S :=

∑aαǫwx

αcǫw = 0 for a finite sum over α, ǫ, w. Now consider

the action S on an element of the form xN1 xN2

2 · · · xNn

n for N ≫ 0. Let w be

such that (xN1 xN2

2 · · · xNn

n )w is minimal among all possible w with aαǫw 6= 0for some α, ǫ. Let α be the smallest element among all α with a

αǫw6= 0

for some ǫ. Then among all monomials in S(xN1 xN2

2 · · · xNn

n ), the monomial

xα(xN1 xN2

2 · · · xNn

n )wcǫ appears as a minimal term with coefficient ±aαǫw. Itfollows from S = 0 that aαǫw = 0. This is only possible when all aαǫw = 0,and hence the elements xαcǫw are linearly independent.

Remark 3.5. By the PBW Theorem 3.4, the HcW -module IND introduced

in the above proof can be identified with the HcW -module induced from the

trivial CW -module. The same remark applies below to type D and B.

3.2. The algebra HcW of type Dn. Let W = WDn be the Weyl group of

type Dn. It is generated by s1, . . . , sn, subject to the following relations:

s2i = 1 (i ≤ n− 1) (3.6)

sisi+1si = si+1sisi+1 (i ≤ n− 2) (3.7)

sisj = sjsi (|i− j| > 1, i, j 6= n) (3.8)

sisn = snsi (i 6= n− 2) (3.9)

sn−2snsn−2 = snsn−2sn, s2n = 1. (3.10)

In particular, Sn is generated by s1, . . . , sn−1 subject to the relations (3.6–3.8) above.

Definition 3.6. Let u ∈ C, and letW =WDn . The degenerate affine Hecke-Clifford algebra of type Dn, denoted by Hc

W or HcDn

, is the algebra generatedby xi, ci, si, 1 ≤ i ≤ n, subject to the relations (3.1–3.5), (3.6–3.10), and thefollowing additional relations:

sncn = −cn−1sn

snci = cisn (i 6= n− 1, n)

snxn + xn−1sn = −u(1 + cn−1cn) (3.11)

snxi = xisn (i 6= n− 1, n).

Proposition 3.7. The algebra HcDn

admits anti-involutions τ1, τ2 defined by

τ1 : si 7→ si, cj 7→ cj , xj 7→ xj , (1 ≤ i ≤ n);

τ2 : si 7→ si, cj 7→ −cj, xj 7→ xj , (1 ≤ i ≤ n).

Also, the algebra HcDn

admits an involution σ which fixes all generatorssi, xi, ci except the following 4 generators:

σ : sn 7→ sn−1, sn−1 7→ sn, xn 7→ −xn, cn 7→ −cn.


Proof. We leave the easy verifications on τ1, τ2 to the reader.It remains to check that σ preserves the defining relations. Almost all the

relations are obvious except (3.4) and (3.11). We see that σ preserves (3.4)as follows: for i ≤ n− 2,

σ(xi+1si − sixi) = xi+1si − sixi

= u(1− ci+1ci) = σ(u(1 − ci+1ci));

σ(xnsn−1 − sn−1xn−1) = −xnsn − snxn−1

= u(1 + cncn−1) = σ(u(1− cncn−1)).

Also, σ preserves (3.11) since

σ(snxn + xn−1sn) = −sn−1xn + xn−1sn−1

= −u(1− cn−1cn) = σ(−u(1 + cn−1cn)).

Hence, σ is an automorphism of HcDn

. Clearly σ2 = 1.

The natural action of Sn on C[h∗] = C[x1, . . . , xn] is extended to an actionof WDn by letting

xsnn = −xn−1, xsnn−1 = −xn, xsni = xi (i 6= n− 1, n).

Proposition 3.8. Let W = WDn , 1 ≤ i ≤ n − 1, and f ∈ C[x1, . . . , xn].Then the following identities hold in Hc

W :

(1) sif = f sisi + uf − f si

xi+1 − xi+ u


xi+1 + xi,

(2) snf = f snsn − uf − f sn

xn + xn−1+ u

cn−1cnf − f sncn−1cnxn − xn−1

.

Proof. Formula (1) has been established by induction as in type An−1. For-mula (2) can be verified by a similar induction.

3.3. The algebra HcW of type Bn. Let W = WBn be the Weyl group of

type Bn, which is generated by s1, . . . , sn, subject to the defining relationfor Sn on s1, . . . , sn−1 and the following additional relations:

sisn = snsi (1 ≤ i ≤ n− 2) (3.12)

(sn−1sn)4 = 1, s2n = 1. (3.13)

We note that the simple reflections s1, . . . , sn belongs to two differentconjugacy classes in WBn , with s1, . . . , sn−1 in one and sn in the other.

Definition 3.9. Let u, v ∈ C, and let W = WBn . The degenerate affineHecke-Clifford algebra of type Bn, denoted by Hc

W or HcBn

, is the algebragenerated by xi, ci, si, 1 ≤ i ≤ n, subject to the relations (3.1–3.5), (3.6–3.8),


(3.12–3.13), and the following additional relations:

sncn = −cnsnsnci = cisn (i 6= n)

snxn + xnsn = −√2 v

snxi = xisn (i 6= n).

The factor√2 above is inserted for the convenience later in relation to

the spin affine Hecke algebras. When it is necessary to indicate u, v, we willwrite Hc

W (u, v) for HcW . For any a ∈ C\0, we have an isomorphism of

superalgebras ψ : HcW (au, av) → Hc

W (u, v) given by dilations xi 7→ axi for1 ≤ i ≤ n, while fixing each si, ci.

The action of Sn on C[h∗] = C[x1, . . . , xn] can be extended to an actionof WBn by letting

xsnn = −xn, xsni = xi, (i 6= n).

Proposition 3.10. Let W = WBn. Given f ∈ C[x1, . . . , xn] and 1 ≤ i ≤n− 1, the following identities hold in Hc

W :

(1) sif = f sisi + uf − f si

xi+1 − xi+ u


xi+1 + xi,

(2) snf = f snsn −√2vf − f sn

2xn.

Proof. The proof is similar to type A and D, and will be omitted.

3.4. PBW basis for HcW . Note that Hc

W contains C[h∗],Cn,CW as subal-gebras. We have the following PBW basis theorem for Hc

W .

Theorem 3.11. Let W =WDn or W = WBn. The multiplication of subal-gebras C[h∗],Cn, and CW induces a vector space isomorphism

C[h∗]⊗ Cn ⊗ CW −→ HcW .

Equivalently, the elements xαcǫw|α ∈ Zn+, ǫ ∈ Z

n2 , w ∈ W form a linear

basis for HcW (called a PBW basis).

Proof. Let us first assume W = WDn . We can verify by a direct lengthycomputation that the Hc

An−1-action on IND = C[x1, . . . , xn] ⊗ Cn (see the

proof of Theorem 3.4) naturally extends to an action of HcDn

, where (compareProposition 3.8) sn acts by

sn.(fcǫ) = f sncsnǫ −

(uf − f sn

xn + xn−1− u

cn−1cnf − f sncn−1cnxn − xn−1

)cǫ.

Clearly, the elements xαcǫw, where α ∈ Zn+, ǫ ∈ Z

n2 , w ∈ W, span Hc

W . It isknown thatW acts on C[x1, . . . , xn] as linearly independent operators. Nowa similar argument as in the proof of Theorem 3.4 applies here to establishthe linear independence of xαcǫw.


The proof for W = WBn is entirely analogous. We will omit the detailsexcept mentioning that the Hc

An−1-action on IND extends to an action of

HcBn

, where (compare Proposition 3.10) sn acts by

sn.(fcǫ) = f sncsnǫ −

√2vf − f sn

2xncǫ.

3.5. The even center for HcW . The even center of a superalgebra A, de-

noted by Z(A), is the subalgebra of even central elements of A.

Proposition 3.12. Let W = WDn or W = WBn . The even center Z(HcW )

of HcW is isomorphic to C[x21, . . . , x

2n]

W .

Proof. We first show that every W -invariant polynomial f in x21, . . . , x2n is

central in HcW . Indeed, f commutes with each ci by (3.2) and clearly f

commutes with each xi. By Proposition 3.8 for type Dn or Proposition 3.10for type Bn, sif = fsi for each i. Since Hc

W is generated by ci, xi and si for

all i, f is central in HcW and C[x21, . . . , x

2n]

W ⊆ Z(HcW ).

On the other hand, take an even central element C =∑aα,ǫ,wx

αcǫw inHcW . We claim that w = 1 whenever aα,ǫ,w 6= 0. Otherwise, let 1 6= w0 ∈ W

be maximal with respect to the Bruhat ordering in W such that aα,ǫ,w06= 0.

Then xw0i 6= xi for some i. By Proposition 3.8 for typeDn or Proposition 3.10

for type Bn, x2iC − Cx2i is equal to aα,ǫ,w0

xα(x2i − (xw0i )2)cǫw0 plus a linear

combination of monomials not involving w0, hence nonzero. This contradictsto the fact that C is central. So we can write C =

∑aα,ǫx

αcǫ.Since xiC = Cxi for each i, then (3.2) forces C to be in C[x1, . . . , xn].

Now by (3.2) and ciC = Cci for each i we have that C ∈ C[x21, . . . , x2n].

Since siC = Csi for each i, we then deduce from Proposition 3.8 for typeDn or Proposition 3.10 for type Bn that C ∈ C[x21, . . . , x

2n]

W .This completes the proof of the proposition.

3.6. The intertwiners in HcW . In this subsection, we will define the inter-

twiners in the degenerate affine Hecke-Clifford algebras HcW .

The following intertwiners φi ∈ HcW (with u = 1) for W = WAn−1 were

introduced by Nazarov [Naz] (also cf. [Kle]), where 1 ≤ i ≤ n− 1:

φi = (x2i+1 − x2i )si − u(xi+1 + xi)− u(xi+1 − xi)cici+1. (3.14)

A direct computation using (3.4) provides another equivalent formula for φi:

φi = si(x2i − x2i+1) + u(xi+1 + xi) + u(xi+1 − xi)cici+1.

We define the intertwiners φi ∈ HcW for W = WDn (1 ≤ i ≤ n) by the

same formula (3.14) for 1 ≤ i ≤ n− 1 and in addition by letting

φn ≡ φDn = (x2n − x2n−1)sn + u(xn − xn−1)− u(xn + xn−1)cn−1cn. (3.15)


We define the intertwiners φi ∈ HcW for W = WBn (1 ≤ i ≤ n) by the


φn ≡ φBn = 2x2nsn +√2vxn. (3.16)

The following generalizes the type An−1 results of Nazarov [Naz].

Theorem 3.13. Let W be either WAn−1 , WDn, or WBn. The intertwinersφi (with 1 ≤ i ≤ n− 1 for type An−1 and 1 ≤ i ≤ n for the other two types)satisfy the following properties:

(1) φ2i = 2u2(x2i+1 + x2i )− (x2i+1 − x2i )2 (1 ≤ i ≤ n− 1,∀W );

(2) φ2n = 2u2(x2n + x2n−1)− (x2n − x2n−1)2, for type Dn;

(3) φ2n = 4x4n − 2v2x2n, for type Bn;(4) φif = f siφi (∀f ∈ C[x1, . . . , xn],∀i,∀W );(5) φicj = csij φi (1 ≤ j ≤ n,∀i,∀W );

(6) φiφjφi · · ·︸︷︷︸mij

= φjφiφj · · ·︸︷︷︸mij

.

Proof. Part (1) follows by a straightforward computation and can also befound in [Naz] (with u = 1). Part (2) follows from (1) by applying theinvolution σ defined in Proposition 3.7. Part (3) and (5) follow by a directverification.

Part (4) for WAn−1 follows from clearing the denominators in the formulain Proposition 3.3 and then rewriting in terms of φi as defined in (3.14).Similarly, (4) for WDn and WBn follows by rewriting the formulas given inProposition 3.8 in type D and Proposition 3.10 in type B, respectively.

It remains to prove (6) which is less trivial. Recall that

mij︷︸︸︷sisjsi · · · =

mij︷︸︸︷sjsisj · · ·,

(denoting this element by w). Let IND be the subalgebra of HcW generated

by C[x1, . . . , xn] and Cn. Denote by ≤ the Bruhat ordering on W . Then wecan write

φiφjφi · · · = fw +∑

u<w

pu,wu

for some f ∈ C[x1, . . . , xn], and pu,w ∈ IND. We may rewrite

φiφjφi · · · = fw +∑

u<w

r′u,wφu

where φu := φaφb · · · for any subword u = sasb · · · of w = sisjsi · · · ,and r′u,w is in some suitable localization of IND with the central element∏

1≤k≤n x2k

∏1≤i<j≤n(x

2i − x2j) ∈ IND being invertible. We can then write

φjφiφj · · · = fw +∑

u<w

r′′u,wφu


with the same coefficient of w as for φiφjφi · · ·, according to Lemma 3.14.The difference ∆ := (φiφjφi · · · − φjφiφj · · ·) is of the form

∆ =∑

u<w

ru,wφu

for some ru,w. Observe by (4) that ∆p = pw∆ for any p ∈ C[x1, . . . , xn].Then we have∑

u<w

pwru,wφu = pw∆ = ∆p =∑

u<w

ru,wφup =∑

u<w

ru,wpuφu.

In other words, (pw − pu)ru,w = 0 for all p ∈ C[x1, . . . , xn] for each givenu < w. This implies that ru,w = 0 for each u, and thus ∆ = 0. Thiscompletes the proof of (6) modulo Lemma 3.14 below.

Lemma 3.14. The following identity holds:

φ0iφ0jφ

0i · · ·︸︷︷︸

mij

= φ0jφ0iφ

0j · · ·︸︷︷︸

mij

where φ0i denotes the specialization φi|u=0 of φ at u = 0 (or rather φBn |v=0

when i = n in the type Bn case.)

Proof. Let W =WBn . For 1 ≤ i ≤ n− 1, mi,i+1 = 3. So we have

φ0iφ0i+1φ

0i = (x2i+1 − x2i )si(x

2i+2 − x2i+1)si+1(x

2i+1 − x2i )si

= (x2i+1 − x2i )(x2i+2 − x2i )(x

2i+2 − x2i+1)sisi+1si

= (x2i+2 − x2i+1)(x2i+2 − x2i )(x

2i+1 − x2i )si+1sisi+1

= (x2i+2 − x2i+1)si+1(x2i+1 − x2i )si(x

2i+2 − x2i+1)si+1

= φ0i+1φ0iφ

0i+1.

Note that mij = 2 for j 6= i, i+ 1; clearly, in this case, φ0iφ0j = φ0jφ

0i .

Noting that mn−1,n = 4, we have

φ0n−1φ0nφ

0n−1φ

0n = 4(x2n − x2n−1)sn−1x

2nsn(x

2n − x2n−1)sn−1x

2nsn

= 4(x2n − x2n−1)x2n−1(x

2n−1 − x2n)x

2nsn−1snsn−1sn

= 4x2n(x2n − x2n−1)x

2n−1(x

2n−1 − x2n)snsn−1snsn−1

= 4x2nsn(x2n − x2n−1)sn−1x

2nsn(x

2n − x2n−1)sn−1

= φ0nφ0n−1φ

0nφ

0n−1.

This completes the proof for type Bn.The similar proofs for types An−1 and Dn are skipped.

Theorem 3.13 implies that for every w ∈ W we have a well-defined el-ement φw ∈ Hc

W given by φw = φi1 · · ·φim where w = si1 · · · sim is anyreduced expression for w. These elements φw should play an important rolefor the representation theory of the algebras Hc

W . It will be very interest-ing to classify the simple modules of Hc

W and to find a possible geometric


realization. This was carried out by Lusztig [Lu1, Lu2, Lu3] for the usualdegenerate affine Hecke algebra case.

4. Degenerate spin affine Hecke algebras

In this section we will introduce the degenerate spin affine Hecke algebraH−W when W is the Weyl group of types Dn or Bn, and then establish the

connections with the corresponding degenerate affine Hecke-Clifford algebrasHcW . See [W1] for the type A case.

4.1. The skew-polynomial algebra. We shall denote by C[b1, . . . , bn] theC-algebra generated by b1, . . . , bn subject to the relations

bibj + bjbi = 0 (i 6= j).

This is naturally a superalgebra by letting each bi be odd. We will refer tothis as the skew-polynomial algebra in n variables. This algebra has a linearbasis given by bα := bk11 · · · bknn for α = (k1, . . . , kn) ∈ Z

n+, and it contains a

polynomial subalgebra C[b21, . . . , b2n].

4.2. The algebra H−W of type Dn. Recall that the spin Weyl group CW−

associated to a Weyl group W is generated by t1, . . . , tn subject to the rela-tions as specified in Example 2.1.

Definition 4.1. Let u ∈ C and let W = WDn . The degenerate spin affineHecke algebra of type Dn, denoted by H−

W or H−Dn

, is the algebra generated

by C[b1, . . . , bn] and CW− subject to the following relations:

tibi + bi+1ti = u (1 ≤ i ≤ n− 1)

tibj = −bjti (j 6= i, i+ 1, 1 ≤ i ≤ n− 1)

tnbn + bn−1tn = u

tnbi = −bitn (i 6= n− 1, n).

The algebra H−W is naturally a superalgebra by letting each ti and bi be

odd generators. It contains the type An−1 degenerate spin affine Heckealgebra H−

An−1(generated by b1, . . . , bn, t1, . . . , tn−1) as a subalgebra.

Proposition 4.2. The algebra H−Dn

admits anti-involutions τ1, τ2 defined by

τ1 : ti 7→ −ti, bi 7→ −bi (1 ≤ i ≤ n);

τ2 : ti 7→ ti, bi 7→ bi (1 ≤ i ≤ n).

Also, the algebra H−Dn

admits an involution σ which swaps tn−1 and tn whilefixing all the remaining generators ti, bi.

Proof. Note that we use the same symbols τ1, τ2, σ to denote the (anti-)involutions for H−

Dnand Hc

Dnin Proposition 3.7, as those on H−

Dnare the

restrictions from those on HcDn

via the isomorphism in Theorem 4.4 below.The proposition is thus established via the isomorphism in Theorem 4.4, orfollows by a direct computation as in the proof of Proposition 3.7.


4.3. The algebra H−W of type Bn.

Definition 4.3. Let u, v ∈ C, and W = WBn . The degenerate spin affineHecke algebra of type Bn, denoted by H−

W or H−Bn

, is the algebra generated

by C[b1, . . . , bn] and CW− subject to the following relations:

tibi + bi+1ti = u (1 ≤ i ≤ n− 1)

tibj = −bjti (j 6= i, i+ 1, 1 ≤ i ≤ n− 1)

tnbn + bntn = v

tnbi = −bitn (i 6= n).

Sometimes, we will write H−W (u, v) or H−

Bn(u, v) for H−

W or H−Bn

to indicatethe dependence on the parameters u, v.

4.4. A superalgebra isomorphism.

Theorem 4.4. Let W =WDn or W =WBn . Then,

(1) there exists an isomorphism of superalgebras

Φ : HcW−→Cn ⊗ H−

W

which extends the isomorphism Φ : Cn ⋊ CW −→ Cn ⊗ CW− (in

Theorem 2.4) and sends xi 7−→√−2cibi for each i;

(2) the inverse Ψ : Cn⊗H−W−→Hc

W extends Ψ : Cn⊗CW− −→ Cn⋊CW

(in Theorem 2.4) and sends bi 7−→1√−2

cixi for each i.

Theorem 4.4 also holds for WAn−1 (see [W1]).

Proof. We only need to show that Φ preserves the defining relations in HcW

which involve xi’s.Let W = WDn . Here, we will verify two such relations below. The

verification of the remaining relations is simpler and will be skipped. For1 ≤ i ≤ n− 1, we have

Φ(xi+1si − sixi) = ci+1bi+1(ci − ci+1)ti − (ci − ci+1)ticibi

= (1− ci+1ci)bi+1ti + (1− ci+1ci)tibi

= u(1− ci+1ci),

Φ(snxn + xn−1sn) = (cn−1 + cn)tncnbn + cn−1bn−1(cn−1 + cn)tn

= −(1 + cn−1cn)tnbn − (1 + cn−1cn)bn−1tn

= −u(1 + cn−1cn).

Now let W = WBn . For 1 ≤ i ≤ n − 1, as in the proof in type Dn, wehave Φ(xi+1si − sixi) = u(1− ci+1ci). Moreover, we have

Φ(snxn + xnsn) =

√−2√−1

cntncnbn +

√−2√−1

cnbncntn


=√2cntncnbn +

√2cnbncntn

= −√2(tnbn + bncn) = −

√2v,

Φ(snxj) =

√−2√−1

cntncjbj =√2cntncjbj

=√2cjcntnbj =

√2cjbjcntn = Φ(xjsn), for j 6= n.

Thus Φ is a homomorphism of (super)algebras. Similarly, we check thatΨ is a superalgebra homomorphism. Observe that Φ and Ψ are inverses ongenerators and hence they are indeed (inverse) isomorphisms.

4.5. PBW basis for H−W . Note that H−

W contains the skew-polynomialalgebra C[b1, . . . , bn] and the spin Weyl group algebra CW− as subalgebras.We have the following PBW basis theorem for H−

W .

Theorem 4.5. Let W = WDn or W = WBn. The multiplication of thesubalgebras CW− and C[b1, . . . , bn] induces a vector space isomorphism

C[b1, . . . , bn]⊗ CW− ≃−→ H−W .

Theorem 4.5 also holds for WAn−1 (see [W1]).

Proof. It follows from the definition that H−W is spanned by the elements of

the form bασ where σ runs over a basis for CW− and α ∈ Zn+. By Theo-

rem 4.4, we have an isomorphism ψ : Cn⊗H−W−→Hc

W . Observe that the im-age ψ(bασ) are linearly independent in Hc

W by the PBW basis Theorem 3.11for Hc

W . Hence the elements bασ are linearly independent in H−W .

4.6. The even center for H−W .

Proposition 4.6. Let W =WDn or W = WBn . The even center of H−W is

isomorphic to C[b21, . . . , b2n]

W .

Proof. By the isomorphism Φ : HcW → Cn⊗H−

W (see Theorems 4.4) and thedescription of the center Z(Hc

W ) (see Proposition 3.12), we have

Z(Cn ⊗ H−W ) = Φ(Z(Hc

W )) = Φ(C[x21, . . . , x2n]

W ) = C[b21, . . . , b2n]

W .

Thus, C[b21, . . . , b2n]

W ⊆ Z(H−W ).

Now let C ∈ Z(H−W ). Since C is even, C commutes with Cn and thus com-

mutes with the algebra Cn ⊗ H−W . Then Ψ(C) ∈ Z(Hc

W ) = C[x21, . . . , x2n]

W ,

and thus, C = ΦΨ(C) ∈ Φ(C[x21, . . . , x2n]

W ) = C[b21, . . . , b2n]

W .

In light of the isomorphism Theorem 4.4, the problem of classifying thesimple modules of the spin affine Hecke algebra H−

W is equivalent to theclassification problem for the affine Hecke-Clifford algebra Hc

W . It remainsto be seen whether it is more convenient to find the geometric realization ofH−W instead of Hc

W .


4.7. The intertwiners in H−W . The intertwiners Ii ∈ H−

W (1 ≤ i ≤ n− 1)for W =WAn−1 were introduced in [W1] (with u = 1):

Ii = (b2i+1 − b2i )ti − u(bi+1 − bi). (4.1)

The commutation relations in Definition 4.1 gives us another equivalentexpression for Ii:

Ii = ti(b2i − b2i+1) + u(bi+1 − bi).

We define the intertwiners Ii ∈ H−W for W = WDn (1 ≤ i ≤ n) by the


In ≡ IDn = (b2n − b2n−1)tn − u(bn − bn−1). (4.2)

Also, we define the intertwiners Ii ∈ H−W for W = WBn (1 ≤ i ≤ n) by

the same formula (4.1) for 1 ≤ i ≤ n− 1 and in addition by letting

In ≡ IBn = 2b2ntn − vbn. (4.3)

Proposition 4.7. The following identities hold in H−W , for W = WAn−1 ,

WBn, or WDn:

(1) Iibi = −bi+1Ii,Iibi+1 = −biIi, and Iibj = −bjIi (j 6= i, i + 1), for1 ≤ i ≤ n− 1, 1 ≤ j ≤ n, and any W ;In addition,

(2) Inbn−1 = −bnIn,Inbn = −bn−1In, and Inbi = −biIn (i 6= n− 1, n),for type Dn;

(3) Inbn = −bnIn, and Inbi = −biIn (i 6= n), for type Bn.

Proof. (1) We first prove the case when j = i:

Iibi = (b2i+1 − b2i )tibi − u(bi+1 − bi)bi

= (b2i+1 − b2i )(−bi+1ti + u)− u(bi+1bi − b2i )

= −bi+1

((b2i+1 − b2i )ti − u(bi+1 − bi)

)

= −bi+1Ii.

The proof for Iibi+1 = −biIi is similar and thus skipped.For j 6= i, i+ 1, we have tibj = −bjti, and hence Iibj = −bjIi.(2) We prove only the first identity. The proofs of the remaining two

identities are similar and will be skipped.

Inbn−1 = (b2n − b2n−1)tnbn−1 − u(bn − bn−1)bn−1

= (b2n − b2n−1)(−bntn + u)− u(bnbn−1 − b2n−1)

= −bn((b2n − b2n−1)tn − u(bn − bn−1)

)

= −bnIn.The proof of (3) is analogous to (2), and is thus skipped.

Recall the superalgebra isomorphism Φ : HcW−→Cn ⊗H−

W defined in Sec-tion 4 and the elements βi ∈ Cn defined in Section 2.


Theorem 4.8. Let W be either WAn−1 , WDn, or WBn . The isomorphism

Φ : HcW −→ Cn ⊗H−

W sends φi 7→ −2√−1βiIi for each i. More explicitly, Φ

sends

φi 7−→ −√−2(ci − ci+1)⊗ Ii (1 ≤ i ≤ n− 1);

φn 7−→ −√−2(cn−1 + cn)⊗ In for type Dn;

φn 7−→ −2√−1cn ⊗ In for type Bn.

Proof. Recall that the isomorphism Φ sends si 7→ −√−1βiti, xi 7→

√−2cibi

for each i. So, for 1 ≤ i ≤ n− 1, we have the following

Φ(φi) = Φ((x2i+1 − x2i )si − u(xi+1 + xi)− u(xi+1 − xi)cici+1

)

= −√−2(ci − ci+1)(b

2i+1 − b2i )ti − u

√−2(ci+1bi+1 − cibi)

−u√−2(ci+1bi+1 − cibi)cici+1

= −√−2(ci − ci+1)

((b2i+1 − b2i )ti − u(bi+1 − bi)

)

= −√−2(ci − ci+1)⊗ Ii.

Next for φn ∈ HcDn

, we have

Φ(φn) = Φ((x2n − x2n−1)sn + u(xn − xn−1)− u(xn + xn−1)cn−1cn

)

= −√−2(cn + cn−1)(b

2n − b2n−1)tn + u

√−2(cnbn − cn−1bn−1)

−u√−2(cnbn − cn−1bn−1)cn−1cn

= −√−2(cn + cn−1)

((b2n − b2n−1)tn − u(bn − bn−1)

)

= −√−2(cn−1 + cn)⊗ In.

We skip the computation for φn ∈ HcBn

which is very similar but lesscomplicated.

Proposition 4.9. The following identities hold in H−W , for W = WAn−1 ,

WBn, or WDn:

(1) I2i = u2(b2i+1 + b2i )− (b2i+1 − b2i )2, for 1 ≤ i ≤ n− 1 and every type of

W .(2) I2n = u2(b2n + b2n−1)− (b2n − b2n−1)

2, for type Dn.

(3) I2n = 4b4n − v2b2n, for type Bn.

Proof. It follows from the counterparts in Theorem 3.13 via the explicitcorrespondences under the isomorphism Φ (see Theorem 4.8). It can ofcourse also be proved by a direct computation.

Proposition 4.10. For W =WAn−1, WBn , or WDn , we have

IiIjIi · · ·︸︷︷︸mij

= (−1)mij+1 IjIiIj · · ·︸︷︷︸mij

.


Proof. By Theorem 2.2, we have

βiβjβi · · ·︸︷︷︸mij

= (−1)mij+1 βjβiβj · · ·︸︷︷︸mij

.

Now the statement follows from the above equation and Theorem 3.13 (6)via the correspondence of the intertwiners under the isomorphism Φ (seeTheorem 4.8).

Remark 4.11. Proposition 4.7, Theorem 4.8, and Proposition 4.10 for H−An−1

can be found in [W1].

5. Degenerate covering affine Hecke algebras

In this section, the degenerate covering affine Hecke algebras associated

to the double covers W of classical Weyl groups W are introduced. It hasas its natural quotients the usual degenerate affine Hecke algebras HW [Dr,Lu1, Lu2] and the spin degenerate affine Hecke algebras H−

W introduced bythe authors.

Recall the distinguished double cover W of a Weyl group W from Sec-tion 2.2.

5.1. The algebra H∼W of type An−1.

Definition 5.1. Let W = WAn−1 , and let u ∈ C. The degenerate coveringaffine Hecke algebra of type An−1, denoted by H∼

W or H∼An−1

, is the algebra

generated by x1, . . . , xn and z, t1, . . . , tn−1, subject to the relations for Wand the additional relations:

zxi = xiz, z is central of order 2 (5.1)

xixj = zxjxi (i 6= j) (5.2)

tixj = zxj ti (j 6= i, i + 1) (5.3)

tixi+1 = zxiti + u. (5.4)

Clearly H∼W contains CW as a subalgebra.

5.2. The algebra H∼W of type Dn.

Definition 5.2. Let W = WDn , and let u ∈ C. The degenerate coveringaffine Hecke algebra of type Dn, denoted by H∼

W or H∼Dn

, is the algebra

generated by x1, . . . , xn and z, t1, . . . , tn, subject to the relations (5.1–5.4)and the following additional relations:

tnxi = zxitn (i 6= n− 1, n)

tnxn = −xn−1tn + u.


5.3. The algebra H∼W of type Bn.

Definition 5.3. Let W = WBn , and let u, v ∈ C. The degenerate coveringaffine Hecke algebra of type Bn, denoted by H∼

W or H∼Bn

, is the algebra

generated by x1, . . . , xn and z, t1, . . . , tn, subject to the relations (5.1–5.4)and the following additional relations:

tnxi = zxitn (i 6= n)

tnxn = −xntn + v.

5.4. PBW basis for H∼W .

Proposition 5.4. Let W = WAn−1 ,WDn , or WBn . Then the quotient ofthe covering affine Hecke algebra H∼

W by the ideal 〈z − 1〉 (respectively, bythe ideal 〈z+1〉) is isomorphic to the usual degenerate affine Hecke algebrasHW (respectively, the spin degenerate affine Hecke algebras H−

W ).

Proof. Follows by the definitions in terms of generators and relations of allthe algebras involved.

Theorem 5.5. Let W = WAn−1 ,WDn , or WBn. Then the elements xαw,

where α ∈ Zn+ and w ∈ W , form a basis for H∼

W (called a PBW basis).

Proof. By the defining relations, it is easy to see that the elements xαwform a spanning set for H∼

W . So it remains to show that they are linearlyindependent.

For each element t ∈ W , denote the two preimages in W of t by t, zt.Now suppose that

0 =∑

aα,txαt+ bα,tzx

αt.

Let I+ and I− be the ideals of H∼W generated by z−1 and z+1 respectively.

Then by Proposition 5.4, H∼W /I

+ ∼= HW and H∼W /I

− ∼= H−W . Consider the

projections:

Υ+ : H∼W −→ H∼

W /I+, Υ− : H∼

W −→ H∼W /I

−.

By abuse of notation, denote the image of xα in HW by xα. Observe that

0 = Υ+

(∑(aα,tx

αt+ bα,txαzt)

)=∑

(aα,t + bα,t)xαt ∈ HW .

Since it is known [Lu1] that xαt|α ∈ Zn+ and t ∈ W form a basis for

the usual degenerate affine Hecke algebra HW , aα,t = −bα,t for all α and t.

Similarly, denoting the image in CW− of t by t, we have

0 = Υ−(∑

(aα,txαt+ bα,tx

αzt))=∑

(aα,t − bα,t)xαt ∈ H−

W .

Since xαt is a basis for the spin degenerate affine Hecke algebra H−W , we

have aα,t = bα,t for all α and t. Hence, aα,t = bα,t = 0, and the linearindependence is proved.


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Department of Math., University of Virginia, Charlottesville, VA 22904

E-mail address: [email protected] (Khongsap); [email protected] (Wang)

http://arxiv.org/abs/math/0608074

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HECKE-CLIFFORD ALGEBRAS AND SPIN HECKE ALGEBRAS I: THE ... · algebras for all classical ﬁnite Weyl groups, which goes beyond the type A case, and then establish some basic properties