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 the
notion of degenerate spin affine Hecke algebras and affine Hecke-Clifford
algebras. For these algebras, we establish the PBW properties, for-
mulate the intertwiners, and describe the centers. We further develop
connections of these algebras with the usual degenerate (i.e. graded)
affine Hecke algebras of Lusztig by introducing a notion of degenerate
covering affine Hecke algebras.
1. Introduction
1.1. The Hecke algebras associated to finite and affine Weyl groups are
ubiquitous in diverse areas, including representation theories over finite
fields, infinite fields of prime characteristic, p-adic fields, and Kazhdan-
Lusztig theory for category O. Lusztig [Lu1, Lu2] introduced the graded
Hecke algebras, also known as the degenerate affine Hecke algebras, associ-
ated to a finite Weyl groupW , and provided a geometric realization in terms
of equivariant homology. The degenerate affine Hecke algebra of type A has
also been defined earlier by Drinfeld [Dr] in connections with Yangians, and
it has recently played an important role in modular representations of the
symmetric group (cf. Kleshchev [Kle]).
In [W1], the second author introduced the degenerate spin affine Hecke
algebra of type A, and related it to the degenerate affine Hecke-Clifford
algebra introduced by Nazarov in his study of the representations of the spin
symmetric group [Naz]. A quantum version of the spin affine Hecke algebra
of type A has been subsequently constructed in [W2], and was shown to be
related 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 the
degenerate affine Hecke-Clifford algebras and degenerate spin affine Hecke
algebras for all classical finite Weyl groups, which goes beyond the type A
case, and then establish some basic properties of these algebras. The notion
of spin Hecke algebras is arguably more fundamental while the notion of
the Hecke-Clifford algebras is crucial for finding the right formulation of the
spin Hecke algebras. We also construct the degenerate covering affine Hecke
algebras which connect to both the degenerate spin affine Hecke algebras
and the degenerate affine Hecke algebras of Lusztig.
http://arxiv.org/abs/0704.0201v3
2 TA KHONGSAP AND WEIQIANG WANG
1.3. Let us describe our constructions in some detail. The Schur multiplier
for each finite Weyl group W has been computed by Ihara and Yokonuma
[IY] (see [Kar]). We start with a distinguished double cover W̃ for any finite
Weyl group W :
1 −→ Z2 −→ W̃ −→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− := CW̃/〈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 on
the 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 W̃ also appeared in Morris [Mo].
We formulate the notion of degenerate affine Hecke-Clifford algebras HcW
and spin affine Hecke algebras H−
, with unequal parameters in type B
case, associated to Weyl groups W of type D and B. The algebra HcW (and
respectively H−
) contain CW ⋊CW (and respectively CW
−) as subalgebras.
We establish the PBW basis properties for these algebras:
∼= 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 both
HcW and H
. The two Hecke algebras HcW and H
are related by a Morita
super-equivalence, i,e. a (super)algebra isomorphism
Φ : HcW
≃−→ CW ⊗ H−W
which extends the isomorphism Φfin. Such an isomorphism holds also for
W of type A [W1].
We generalize the construction in [Naz] of the intertwiners in the affine
Hecke-Clifford algebras HcW of type A to all classical Weyl groups W . We
also generalize the construction of the intertwiners in [W1] for H−
of type
A to all classical Weyl groups W . We further establish the basic properties
of these intertwiners in both HcW and H
. These intertwiners are expected
to play a fundamental role in the future development of the representation
theory of these algebras, as it is indicated by the work of Lusztig, Cherednik
and 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 W̃ 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
and its quotient by
the ideal 〈z−1〉 is identified with Lusztig’s degenerate affine Hecke algebras
associated toW . In this sense, our covering affine Hecke algebra is a natural
affine generalization of the central extension (1.1). A quantum version of
the covering affine Hecke algebra of type A was constructed in [W2].
The results in this paper remain valid over any algebraically closed field
of characteristic p 6= 2 (and in addition p 6= 3 for type G2). In fact, most
of the constructions can be made valid over the ring Z[1
] (occasionally we
need to adjoint
1.4. This paper and [W1] raise many questions, including a geometric re-
alization of the algebras HcW or H
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 HcW in the type A case in-
cluding the modular representations of the spin symmetric group have been
developed by Brundan and Kleshchev [BK] (also cf. [Kle]).
In a sequel [KW] to this paper, we will extend the constructions in this
paper to the setup of rational double affine Hecke algebras (see Etingof-
Ginzburg [EG]), generalizing and improving a main construction initiated
in [W1] for the spin symmetric group. We also hope to quantize these
degenerate spin Hecke algebras, reversing the history of developments from
quantum 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 isomorphism
theorem 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 the
corresponding degenerate spin affine Hecke algebras. We then extend the
isomorphism Φfin to an isomorphism relating these affine Hecke algebras,
establish the PBW properties, and describe the centers of HcW and H
Section 5, we formulate the notion of degenerate covering affine Hecke al-
gebras, and establish the connections to the degenerate spin affine Hecke
algebras and usual affine Hecke algebras.
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 with
the 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}, and
they are specified by the following Coxeter-Dynkin diagrams whose vertices
correspond 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 an
unmarked edge, and mij = 2 if i, j are not connected by an edge.
4 TA KHONGSAP AND WEIQIANG WANG
An ◦ ◦ . . . ◦ ◦
1 2 n− 1 n
Bn(n ≥ 2) ◦ ◦ . . . ◦ ◦
1 2 n− 1 n
Dn(n ≥ 4) ◦ ◦ · · · ◦ ◦
1 2 n− 3
En=6,7,8 ◦ ◦ ◦ . . . ◦ ◦
1 3 4 n− 1 n
F4 ◦ ◦ ◦ ◦
1 2 3 4
G2 ◦ ◦
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]). The
explicit generators and relations for the corresponding covering groups of W
can be found in Karpilovsky [Kar, Table 7.1].
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 CLASSICAL SPIN AFFINE HECKE ALGEBRAS 5
The quotient algebra CW− := CW̃/〈z+1〉 of CW̃ by the ideal generated
by z+1 will be called the spin Weyl group algebra associated to W . Denote
by ti ∈ CW− the image of t̃i. The spin Weyl group algebra CW− has the
following 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 CW̃
is isomorphic to CW .
Example 2.1. Let W be the Weyl group of type An, Bn, or Dn, which will
be assumed in later sections. Then the spin Weyl group algebra CW− is
generated by t1, . . . , tn with the labeling as in the Coxeter-Dynkin diagrams
and the explicit relations summarized in the following table.
Type of W Defining Relations for CW−
i = 1, titi+1ti = ti+1titi+1,
(titj)
2 = −1 if |i− j| > 1
t1, . . . , tn−1 satisfy the relations for CW
n = 1, (titn)
2 = −1 if i 6= n− 1, n,
(tn−1tn)
4 = −1
t1, . . . , tn−1 satisfy the relations for CW
n = 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 representation
of the Weyl groupW (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 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 sub-
space of CN and 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).
Denote by CW the Clifford algebra associated to (h, (−,−)), which is re-
garded as a subalgebra of the Clifford algebra CN associated to (C
N , (−,−)).
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 normalized
with β2i = 1. In particular, CN is generated by c1, . . . , cN subject to the
relations
c2i = 1, cicj = −cjci if i 6= j. (2.3)
6 TA KHONGSAP AND WEIQIANG WANG
The explicit generators for CW are listed in the following table. Note that
CW is naturally a superalgebra with each βi being odd.
Type of W N Generators for CW
An−1 n βi =
(ci − ci+1), 1 ≤ i ≤ n− 1
Bn n βi =
(ci − ci+1), 1 ≤ i ≤ n− 1, βn = cn
Dn n βi =
(ci − ci+1), 1 ≤ i ≤ n− 1, βn = 1√
(cn−1 + cn)
E8 8 β1 =
(c1 + c8 − c2 − c3 − c4 − c5 − c6 − c7)
(c1 + c2), βi =
(ci−1 + ci−2), 3 ≤ i ≤ 8
E7 8 the subset of βi in E8, 1 ≤ i ≤ 7
E6 8 the subset of βi in E8, 1 ≤ i ≤ 6
F4 4 β1 =
(c1 − c2), β2 = 1√2(c2 − c3)
β3 = c3, β4 =
(c4 − c1 − c2 − c3)
G2 3 β1 =
(c1 − c2), β2 = 1√6(−2c1 + c2 + c3)
The action of W on h and h∗ preserves the bilinear form (−,−) and thus
W acts as automorphisms of the algebra CW . This gives rise to a semi-direct
product CW ⋊CW . Moreover, the algebra CW ⋊CW naturally inherits the
superalgebra structure by letting elements inW be even and each βi be odd.
2.4. The basic spin supermodule. The following theorem is due to Mor-
ris [Mo] in full generality, and it goes back to I. Schur [Sch] (cf. [Joz]) in the
type A, namely the symmetric group case. It can be checked case by case
using the explicit formulas of βi in the Table of Section 2.3.
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.
Remark 2.3. In [Mo], W is viewed as a subgroup of the orthogonal Lie group
which preserves (h, (−,−)). The preimage of W in the spin group which
covers the orthogonal group provides the double cover W̃ of W , where the
Atiyah-Bott-Shapiro construction of the spin group in terms of the Clifford
algebra CW was used to describe this double cover of W .
The superalgebra CW has a unique (up to isomorphism) simple super-
module (i.e. Z2-graded module). By pulling it back via the homomorphism
Ω : CW− → CW , we obtain a distinguished CW−-supermodule, called the
basic spin supermodule. This is a natural generalization of the classical
construction 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 with
multiplication defined by
(a⊗ b)(a′ ⊗ b′) = (−1)|b||a′|(aa′ ⊗ bb′) (a, a′ ∈ A, b, b′ ∈ B) (2.4)
THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 7
where |b| denotes the Z2-degree of b, etc. Also, we shall use short-hand
notation ab for (a⊗ b) ∈ A ⊗ B, a = a⊗ 1, and b = 1⊗ b.
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 sends
ti 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) = 2β
i = 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 for
all 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 counting
argument.
Clearly, Ψ and Φ are inverses of each other. �
Remark 2.7. The type A case of Theorem 2.4 was due to Sergeev and Ya-
maguchi independently [Ser, Yam], and it played a fundamental role in
clarifying the earlier observation in the literature (cf. [Joz, St]) that the
representation 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 the
classical Weyl groups of type A,B, or D.
8 TA KHONGSAP AND WEIQIANG WANG
3. Degenerate affine Hecke-Clifford algebras
In this section, we introduce the degenerate affine Hecke-Clifford algebras
of type D and B, and establish some basic properties. The degenerate affine
Hecke-Clifford algebra associated to the symmetric group Sn was introduced
earlier 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 group
of type An−1. The degenerate affine Hecke-Clifford algebra of type An−1,
denoted by HcW or H
, 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 the
algebra HcW as a C(u)-algebra. Similar remarks apply to various algebras
introduced in this paper. Our convention c2i = 1 differs from Nazarov’s
which sets c2i = −1.
The symmetric group Sn acts as the automorphisms on the symmetric
algebra C[h∗] ∼= C[x1, . . . , xn] by permutation. We shall denote this action
by 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 HcW :
sif = f
sisi + u
f − f si
xi+1 − xi
cici+1f − f sicici+1
xi+1 + xi
It is understood here and in similar expressions below that A
In this sense, both numerators on the right-hand side of the above formula
are (left-)divisible by the corresponding denominators.
Proof. By the definition of HcW , we have that six
j = x
j 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.
THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 9
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
xki+1si + u
(xki − xki+1)
xi+1 − xi
(cici+1x
i − xki+1cici+1)
xi+1 + xi
= xki+1 (xi+1si − u(1− ci+1ci))
(xki − xki+1)
xi+1 − xi
xi + u
(cici+1x
i − xki+1cici+1)
xi+1 + xi
= xk+1i+1 si + u
(xk+1
− xk+1
i+1 )
xi+1 − xi
(cici+1x
− 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 x
i+1. The case l = 0 is established above. Assume the formula
is true for f = xki x
i+1. Then using sixi+1 = xisi+u(1+ci+1ci), we compute
i+1 =
i+1si + u
(xki x
i+1 − xlixki+1)
xi+1 − xi
(cici+1x
i+1 − xlixki+1cici+1)
xi+1 + xi
· xi+1
= xlix
i+1(xisi + u(1 + ci+1ci))
(xki x
i+1 − xlix
i+1 )
xi+1 − xi
(cici+1x
i+1 + x
i+1 cici+1)
xi+1 + xi
= xl+1i x
i+1si + u
(xki x
i+1 − x
xi+1 − xi
(cici+1x
i+1 − x
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+, cǫ = c
1 · · · cǫnn for ǫ =
(ǫ1, . . . , ǫn) ∈ Zn2 .
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
1 from the trivial W -module 1. This induced
module is of independent interest. This approach will then be used for type
D 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
≃−→ HcW .
10 TA KHONGSAP AND WEIQIANG WANG
Equivalently, {xαcǫw|α ∈ Zn+, ǫ ∈ Zn2 , w ∈ W} forms a linear basis for HcW
(called a PBW basis).
Proof. Note that IND := C[x1, . . . , xn] ⊗ Cn admits an algebra structure
by (2.3), (3.1) and (3.2). By the explicit defining relations of HcW , we can
verify that the algebra HcW acts on IND by letting xi and ci act by left
multiplication, and si ∈ Sn act by
si.(fc
ǫ) = f sicsiǫ +
f − f si
xi+1 − xi
cici+1f − f sicici+1
xi+1 + xi
For α = (a1, . . . , an), we denote |α| = a1 + · · · + an. Define a Lexico-
graphic ordering < on the monomials xα, α ∈ Zn+, (or respectively on Zn+),
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 x
αcǫw.
It remains to show that these elements are linearly independent.
Suppose that S :=
xαcǫw = 0 for a finite sum over α, ǫ, w and that
some coefficient a
6= 0; we fix one such ǫ. Now consider the action S
on an element of the form x
2 · · · xNnn for N1 ≫ N2 ≫ · · · ≫ Nn ≫ 0.
Let w̃ be such that (x
2 · · · xNnn )w̃ is maximal among all possible w with
aαǫw 6= 0 for some α. Let α̃ be the largest element among all α with
6= 0. Then among all monomials in S(xN11 x
2 · · · xNnn ), the monomial
xα̃(x
2 · · · xNnn )w̃cǫ appears as a maximal term with coefficient ±aα̃ǫw̃.
It follows from S = 0 that aα̃ǫw̃ = 0. This is a contradiction, 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, s
n = 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 H
W or H
, is the algebra generated
by xi, ci, si, 1 ≤ i ≤ n, subject to the relations (3.1–3.5), (3.6–3.10), and the
THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 11
following 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 generators
si, 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 σ
2 = 1. �
The natural action of Sn on C[h
∗] = C[x1, . . . , xn] is extended to an action
of WDn by letting
xsnn = −xn−1, x
n−1 = −xn, x
= 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 HcW :
(1) sif = f
sisi + u
f − f si
xi+1 − xi
cici+1f − f sicici+1
xi+1 + xi
(2) snf = f
snsn − u
f − f sn
xn + xn−1
cn−1cnf − f sncn−1cn
xn − 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. �
12 TA KHONGSAP AND WEIQIANG WANG
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 relation
for 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 different
conjugacy 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 affine
Hecke-Clifford algebra of type Bn, denoted by H
W or H
, is the algebra
generated 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 = −cnsn
snci = cisn (i 6= n)
snxn + xnsn = −
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 will
write HcW (u, v) for H
W . For any a ∈ C\{0}, we have an isomorphism of
superalgebras ψ : HcW (au, av) → HcW (u, v) given by dilations xi 7→ axi for
1 ≤ i ≤ n, while fixing each si, ci.
The action of Sn on C[h
∗] = C[x1, . . . , xn] can be extended to an action
of WBn by letting
xsnn = −xn, x
i = 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 HcW :
(1) sif = f
sisi + u
f − f si
xi+1 − xi
cici+1f − f sicici+1
xi+1 + xi
(2) snf = f
snsn −
f − f sn
Proof. The proof is similar to type A and D, and will be omitted. �
3.4. PBW basis for HcW . Note that H
W contains C[h
∗],Cn,CW as subal-
gebras. We have the following PBW basis theorem for HcW .
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 ∈ Zn2 , w ∈ W} form a linear
basis for HcW (called a PBW basis).
THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 13
Proof. For W = WDn , we can verify by a direct lengthy computation that
the HcAn−1-action on IND = C[x1, . . . , xn]⊗Cn (see the proof of Theorem 3.4)
naturally extends to an action of HcDn , where (compare Proposition 3.8) sn
acts by
sn.(fc
ǫ) = f sncsnǫ −
f − f sn
xn + xn−1
− ucn−1cnf − f
sncn−1cn
xn − xn−1
Similarly, for W = WBn , the H
-action on IND extends to an action of
HcBn , where (compare Proposition 3.10) sn acts by
sn.(fc
ǫ) = f sncsnǫ −
f − f sn
It is easy to show that, for either W , the elements xαcǫw span HcW . It
remains to show that they are linearly independent. We shall treat theWBn
case in detail and skip the analogous WDn case.
To that end, we shall refer to the argument in the proof of Theorem 3.4
with suitable modification as follows. The w̃ = ((η1, . . . , ηn), σ) ∈ WBn =
{±1}n ⋊ Sn may now not be unique, but the σ and the α̃ are uniquely
determined. Then, by the same argument on the vanishing of a maximal
term, we obtain that
w̃ aα̃ǫw̃x
2 · · · xNnn )w̃ = 0, and hence,
(η1,...,ηn)
aα̃ǫw̃(−1)
i=1 ηiNi = 0.
By choosing N1, . . . , Nn with different parities (2
n choices) and solving the
2n linear equations, we see that all aα̃ǫw̃ = 0. This can also be seen more
explicitly by induction on n. By choosing Nn to be even and odd, we deduce
that for a fixed ηn,
(η1,...,ηn−1)∈{±1}n−1 aα̃ǫw̃(−1)
i=1 ηiNi = 0, which is the
equation for (n− 1) xi’s and the induction applies. �
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(H
of HcW is isomorphic to C[x
1, . . . , x
Proof. We first show that every W -invariant polynomial f in x21, . . . , x
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 3.10
for type Bn, sif = fsi for each i. Since H
W is generated by ci, xi and si for
all i, f is central in HcW and C[x
1, . . . , x
W ⊆ Z(HcW ).
On the other hand, take an even central element C =
α,ǫ,w
xαcǫw in
HcW . 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 a
α,ǫ,w0
6= 0.
Then x
i 6= xi for some i. By Proposition 3.8 for typeDn or Proposition 3.10
for type Bn, x
iC − Cx2i is equal to
α,ǫ aα,ǫ,w0x
α(x2i − (x
2)cǫw0 plus a
14 TA KHONGSAP AND WEIQIANG WANG
linear combination of monomials not involving w0, hence nonzero. This
contradicts with the fact that C is central. So we can write C =
xαcǫ.
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 type
Dn or Proposition 3.10 for type Bn that C ∈ C[x21, . . . , x2n]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 = (x
i+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(x
i − 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
same formula (3.14) for 1 ≤ i ≤ n− 1 and in addition by letting
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 intertwiners
φi (with 1 ≤ i ≤ n− 1 for type An−1 and 1 ≤ i ≤ n for the other two types)
satisfy the following properties:
(1) φ2i = 2u
2(x2i+1 + x
i )− (x2i+1 − x2i )2 (1 ≤ i ≤ n− 1,∀W );
(2) φ2n = 2u
2(x2n + x
n−1)− (x2n − x2n−1)2, for type Dn;
(3) φ2n = 4x
n − 2v2x2n, for type Bn;
(4) φif = f
siφi (∀f ∈ C[x1, . . . , xn],∀i,∀W );
(5) φicj = c
j φi (1 ≤ j ≤ n,∀i,∀W );
(6) φiφjφi · · ·︸ ︷︷ ︸
= φjφiφj · · ·︸ ︷︷ ︸
Proof. Part (1) follows by a straightforward computation and can also be
found in [Naz] (with u = 1). Part (2) follows from (1) by applying the
involution σ defined in Proposition 3.7. Part (3) and (5) follow by a direct
verification.
Part (4) for WAn−1 follows from clearing the denominators in the formula
in 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 in
Proposition 3.8 in type D and Proposition 3.10 in type B, respectively.
THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 15
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 we
can write
φiφjφi · · · = fw +
pu,wu
for some f ∈ C[x1, . . . , xn], and pu,w ∈ IND. We may rewrite
φiφjφi · · · = fw +
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 x
1≤i<j≤n(x
i − x2j) ∈ IND being invertible. Note that such
a localization is a free module over the corresponding localized ring of
C[x1, . . . , xn]. We can then write
φjφiφj · · · = fw +
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
ru,wφu
for some ru,w. Observe by (4) that ∆p = p
w∆ for any p ∈ C[x1, . . . , xn].
Then we have
pwru,wφu = p
w∆ = ∆p =
ru,wφup =
ru,wp
In other words, (pw − pu)ru,w = 0 for all p ∈ C[x1, . . . , xn] for each given
u < w. This implies that ru,w = 0 for each u, and ∆ = 0. This completes
the proof of (6) modulo Lemma 3.14 below. �
Lemma 3.14. The following identity holds:
i · · ·︸ ︷︷ ︸
= φ0jφ
j · · ·︸ ︷︷ ︸
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.)
16 TA KHONGSAP AND WEIQIANG WANG
Proof. Let W =WBn . For 1 ≤ i ≤ n− 1, mi,i+1 = 3. So we have
i = (x
i+1 − x2i )si(x2i+2 − x2i+1)si+1(x2i+1 − x2i )si
= (x2i+1 − x2i )(x2i+2 − x2i )(x2i+2 − x2i+1)sisi+1si
= (x2i+2 − x2i+1)(x2i+2 − x2i )(x2i+1 − x2i )si+1sisi+1
= (x2i+2 − x2i+1)si+1(x2i+1 − x2i )si(x2i+2 − x2i+1)si+1
= φ0i+1φ
Note that mij = 2 for j 6= i, i+ 1; clearly, in this case, φ0iφ0j = φ0jφ0i .
Noting that mn−1,n = 4, we have
φ0n−1φ
n = 4(x
n − x2n−1)sn−1x2nsn(x2n − x2n−1)sn−1x2nsn
= 4(x2n − x2n−1)x2n−1(x2n−1 − x2n)x2nsn−1snsn−1sn
= 4x2n(x
n − x2n−1)x2n−1(x2n−1 − x2n)snsn−1snsn−1
= 4x2nsn(x
n − x2n−1)sn−1x2nsn(x2n − x2n−1)sn−1
= φ0nφ
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 ∈ HcW given by φw = φi1 · · ·φim where w = si1 · · · sim is any
reduced expression for w. These elements φw should play an important role
for the representation theory of the algebras HcW . It will be very interest-
ing to classify the simple modules of HcW and to find a possible geometric
realization. This was carried out by Lusztig [Lu1, Lu2, Lu3] for the usual
degenerate affine Hecke algebra case.
4. Degenerate spin affine Hecke algebras
In this section we will introduce the degenerate spin affine Hecke algebra
when W is the Weyl group of types Dn or Bn, and then establish the
connections with the corresponding degenerate affine Hecke-Clifford algebras
HcW . See [W1] for the type A case.
4.1. The skew-polynomial algebra. We shall denote by C[b1, . . . , bn] the
C-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 to
this as the skew-polynomial algebra in n variables. This algebra has a linear
basis given by bα := bk11 · · · bknn for α = (k1, . . . , kn) ∈ Zn+, and it contains a
polynomial subalgebra C[b21, . . . , b
THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 17
4.2. The algebra H−
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 affine
Hecke algebra of type Dn, denoted by H
or H−
, 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−
is naturally a superalgebra by letting each ti and bi be
odd generators. It contains the type An−1 degenerate spin affine Hecke
algebra H−
(generated by b1, . . . , bn, t1, . . . , tn−1) as a subalgebra.
Proposition 4.2. The algebra H−
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−
admits an involution σ which swaps tn−1 and tn while
fixing all the remaining generators ti, bi.
Proof. Note that we use the same symbols τ1, τ2, σ to denote the (anti-)
involutions for H−
and HcDn in Proposition 3.7, as those on H
are 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, or
follows by a direct computation as in the proof of Proposition 3.7. �
4.3. The algebra H−
of type Bn.
Definition 4.3. Let u, v ∈ C, and W = WBn . The degenerate spin affine
Hecke algebra of type Bn, denoted by H
or H−
, 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−
(u, v) or H−
(u, v) for H−
or H−
to indicate
the dependence on the parameters u, v.
18 TA KHONGSAP AND WEIQIANG WANG
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−→HcW extends Ψ : Cn⊗CW− −→ Cn⋊CW
(in Theorem 2.4) and sends bi 7−→
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. For
1 ≤ 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, we
have Φ(xi+1si − sixi) = u(1− ci+1ci). Moreover, we have
Φ(snxn + xnsn) =
cntncnbn +
cnbncntn
2cntncnbn +
2cnbncntn
2(tnbn + bncn) = −
Φ(snxj) =
cntncjbj =
2cntncjbj
2cjcntnbj =
2cjbjcntn = Φ(xjsn), for j 6= n.
Thus Φ is a homomorphism of (super)algebras. Similarly, we check that
Ψ is a superalgebra homomorphism. Observe that Φ and Ψ are inverses on
generators and hence they are indeed (inverse) isomorphisms. �
THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 19
4.5. PBW basis for H−
. Note that H−
contains the skew-polynomial
algebra C[b1, . . . , bn] and the spin Weyl group algebra CW
− as subalgebras.
We have the following PBW basis theorem for H−
Theorem 4.5. Let W = WDn or W = WBn. The multiplication of the
subalgebras CW− and C[b1, . . . , bn] induces a vector space isomorphism
C[b1, . . . , bn]⊗ CW−
≃−→ H−
Theorem 4.5 also holds for WAn−1 (see [W1]).
Proof. It follows from the definition that H−
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−→HcW . Observe that the im-
age ψ(bασ) are linearly independent in HcW by the PBW basis Theorem 3.11
for HcW . Hence the elements b
ασ are linearly independent in H−
4.6. The even center for H−
Proposition 4.6. Let W =WDn or W = WBn . The even center of H
isomorphic to C[b21, . . . , b
Proof. By the isomorphism Φ : HcW → Cn⊗H
(see Theorems 4.4) and the
description of the center Z(HcW ) (see Proposition 3.12), we have
Z(Cn ⊗ H−W ) = Φ(Z(H
W )) = Φ(C[x
1, . . . , x
W ) = C[b21, . . . , b
Thus, C[b21, . . . , b
W ⊆ Z(H−
Now let C ∈ Z(H−
). Since C is even, C commutes with Cn and thus com-
mutes with the algebra Cn ⊗ H−W . Then Ψ(C) ∈ Z(H
W ) = C[x
1, . . . , x
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 the
simple modules of the spin affine Hecke algebra H−
is equivalent to the
classification problem for the affine Hecke-Clifford algebra HcW . It remains
to be seen whether it is more convenient to find the geometric realization of
instead of HcW .
4.7. The intertwiners in H−
. The intertwiners Ii ∈ H−W (1 ≤ i ≤ n− 1)
for W =WAn−1 were introduced in [W1] (with u = 1):
Ii = (b
i+1 − b2i )ti − u(bi+1 − bi). (4.1)
The commutation relations in Definition 4.1 gives us another equivalent
expression for Ii:
Ii = ti(b
i − b2i+1) + u(bi+1 − bi).
We define the intertwiners Ii ∈ H−W for W = WDn (1 ≤ i ≤ n) by the
same formula (4.1) for 1 ≤ i ≤ n− 1 and in addition by letting
In ≡ IDn = (b2n − b2n−1)tn − u(bn − bn−1). (4.2)
20 TA KHONGSAP AND WEIQIANG WANG
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−
, for W = WAn−1 ,
WBn, or WDn:
(1) Iibi = −bi+1Ii,Iibi+1 = −biIi, and Iibj = −bjIi (j 6= i, i + 1), for
1 ≤ 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 = (b
i+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 = (b
n − 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
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
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.
THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 21
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)(b2i+1 − b2i )ti − u
−2(ci+1bi+1 − cibi)
−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)(b2n − b2n−1)tn + u
−2(cnbn − cn−1bn−1)
−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 less
complicated. �
Proposition 4.9. The following identities hold in H−
, for W = WAn−1 ,
WBn, or WDn:
(1) I2i = u
2(b2i+1 + b
i )− (b2i+1 − b2i )2, for 1 ≤ i ≤ n− 1 and every type of
(2) I2n = u
2(b2n + b
n−1)− (b2n − b2n−1)2, for type Dn.
(3) I2n = 4b
n − v2b2n, for type Bn.
Proof. It follows from the counterparts in Theorem 3.13 via the explicit
correspondences under the isomorphism Φ (see Theorem 4.8). It can of
course also be proved by a direct computation. �
Proposition 4.10. For W =WAn−1, WBn , or WDn , we have
IiIjIi · · ·︸ ︷︷ ︸
= (−1)mij+1 IjIiIj · · ·︸ ︷︷ ︸
Proof. By Theorem 2.2, we have
βiβjβi · · ·︸ ︷︷ ︸
= (−1)mij+1 βjβiβj · · ·︸ ︷︷ ︸
Now the statement follows from the above equation and Theorem 3.13 (6)
via the correspondence of the intertwiners under the isomorphism Φ (see
Theorem 4.8). �
Remark 4.11. Proposition 4.7, Theorem 4.8, and Proposition 4.10 for H−
can be found in [W1].
22 TA KHONGSAP AND WEIQIANG WANG
5. Degenerate covering affine Hecke algebras
In this section, the degenerate covering affine Hecke algebras associated
to the double covers W̃ of classical Weyl groups W are introduced. It has
as its natural quotients the usual degenerate affine Hecke algebras HW [Dr,
Lu1, Lu2] and the spin degenerate affine Hecke algebras H−
introduced by
the authors.
Recall the distinguished double cover W̃ 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 covering
affine Hecke algebra of type An−1, denoted by H
W or H
, is the algebra
generated by x̃1, . . . , x̃n and z, t̃1, . . . , t̃n−1, subject to the relations for W̃
and the additional relations:
zx̃i = x̃iz, z is central of order 2 (5.1)
x̃ix̃j = zx̃jx̃i (i 6= j) (5.2)
t̃ix̃j = zx̃j t̃i (j 6= i, i + 1) (5.3)
t̃ix̃i+1 = zx̃it̃i + u. (5.4)
Clearly H∼W contains CW̃ as a subalgebra.
5.2. The algebra H∼W of type Dn.
Definition 5.2. Let W = WDn , and let u ∈ C. The degenerate covering
affine Hecke algebra of type Dn, denoted by H
W or H
, is the algebra
generated by x̃1, . . . , x̃n and z, t̃1, . . . , t̃n, subject to the relations (5.1–5.4)
and the following additional relations:
t̃nx̃i = zx̃it̃n (i 6= n− 1, n)
t̃nx̃n = −x̃n−1t̃n + u.
5.3. The algebra H∼W of type Bn.
Definition 5.3. Let W = WBn , and let u, v ∈ C. The degenerate covering
affine Hecke algebra of type Bn, denoted by H
W or H
, is the algebra
generated by x̃1, . . . , x̃n and z, t̃1, . . . , t̃n, subject to the relations (5.1–5.4)
and the following additional relations:
t̃nx̃i = zx̃it̃n (i 6= n)
t̃nx̃n = −x̃nt̃n + v.
5.4. PBW basis for H∼W .
Proposition 5.4. Let W = WAn−1 ,WDn , or WBn . Then the quotient of
the covering affine Hecke algebra H∼W by the ideal 〈z − 1〉 (respectively, by
the ideal 〈z+1〉) is isomorphic to the usual degenerate affine Hecke algebras
HW (respectively, the spin degenerate affine Hecke algebras H
THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 23
Proof. Follows by the definitions in terms of generators and relations of all
the algebras involved. �
Theorem 5.5. Let W = WAn−1 ,WDn , or WBn. Then the elements x̃
where α ∈ Zn+ and w̃ ∈ W̃ , form a basis for H∼W (called a PBW basis).
Proof. By the defining relations, it is easy to see that the elements x̃αw̃
form a spanning set for H∼W . So it remains to show that they are linearly
independent.
For each element t ∈ W , denote the two preimages in W̃ of t by {t̃, zt̃}.
Now suppose that
aα,t̃x̃
αt̃+ bα,t̃zx̃
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
. Consider the
projections:
Υ+ : H
W −→ H∼W /I+, Υ− : H∼W −→ H∼W /I−.
By abuse of notation, denote the image of x̃α in HW by x
α. Observe that
0 = Υ+
(aα,t̃x̃
αt̃+ bα,t̃x̃
αzt̃)
(aα,t̃ + bα,t̃)x
αt ∈ 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,t̃ = −bα,t̃ for all α and t.
Similarly, denoting the image in CW− of t̃ by t̄, we have
0 = Υ−
(aα,t̃x̃
αt̃+ bα,t̃x̃
αzt̃)
(aα,t̃ − bα,t̃)x
αt̄ ∈ H−
Since {xαt̄} is a basis for the spin degenerate affine Hecke algebra H−
have aα,t̃ = bα,t̃ for all α and t. Hence, aα,t̃ = bα,t̃ = 0, and the linear
independence is proved. �
References
[BK] J. Brundan, A Kleshchev, Hecke-Clifford superalgebras, crystals of type A
2l and
modular branching rules for Ŝn, Represent. Theory 5 (2001), 317–403.
[Dr] V. Drinfeld, Degenerate affine Hecke algebras and Yangians, Funct. Anal. Appl.
20 (1986), 58–60.
[EG] P. Etingof and V. Ginzburg, Symplectic reflection algebras, Calogero-Moser space,
and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243–348.
[IY] S. Ihara and T. Yokonuma, On the second cohomology groups (Schur multipliers)
of finite reflection groups, J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 11 (1965),
155–171.
[JN] A. Jones and M. Nazarov, Affine Sergeev algebra and q-analogues of the Young
symmetrizers for projective representations of the symmetric group, Proc. London
Math. Soc. 78 (1999), 481–512.
[Joz] T. Józefiak, A class of projective representations of hyperoctahedral groups and
Schur Q-functions, Topics in Algebra, Banach Center Publ., 26, Part 2, PWN-
Polish Scientific Publishers, Warsaw (1990), 317–326.
[Kar] G. Karpilovsky, The Schur multiplier, London Math. Soc. Monagraphs, New Series
2, Oxford University Press, 1987.
24 TA KHONGSAP AND WEIQIANG WANG
[Kle] A. Kleshchev, Linear and projective representations of symmetric groups, Cam-
bridge Tracts in Mathematics 163, Cambridge University Press, 2005.
[KW] T. Khongsap and W. Wang, Hecke-Clifford algebras and spin Hecke algebras II:
the rational double affine type, Preprint 2007.
[Lu1] G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2
(1989), 599–635.
[Lu2] ———, Cuspidal local systems and graded Hecke algebras I, Publ. IHES 67 (1988),
145–202.
[Lu3] ———, Cuspidal local systems and graded Hecke algebras III, Represent. Theory
6 (2002), 202–242.
[Mo] A. Morris, Projective representations of reflection groups, Proc. London Math. Soc
32 (1976), 403–420.
[Naz] M. Nazarov, Young’s symmetrizers for projective representations of the symmetric
group, Adv. in Math. 127 (1997), 190–257.
[Sch] I. Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppe
durch gebrochene lineare Substitutionen, J. reine angew. Math. 139 (1911), 155–
[Ser] A. Sergeev, The Howe duality and the projective representations of symmetric
groups, Represent. Theory 3 (1999), 416–434.
[St] J. Stembridge, The projective representations of the hyperoctahedral group, J. Al-
gebra 145 (1992), 396–453.
[W1] W. Wang, Double affine Hecke algebras for the spin symmetric group, preprint
2006, math.RT/0608074.
[W2] ———, Spin Hecke algebras of finite and affine types, Adv. in Math. 212 (2007),
723–748.
[Yam] M. Yamaguchi, A duality of the twisted group algebra of the symmetric group and
a Lie superalgebra, J. Algebra 222 (1999), 301–327.
Department of Math., University of Virginia, Charlottesville, VA 22904
E-mail address: tk7p@virginia.edu (Khongsap); ww9c@virginia.edu (Wang)
http://arxiv.org/abs/math/0608074
	1. Introduction
	1.1. 
	1.2. 
	1.3. 
	1.4. 
	1.5. 
	2. Spin Weyl groups and Clifford algebras
	2.1. The Weyl groups
	2.2. A distinguished double covering of Weyl groups
	2.3. The Clifford algebra CW
	2.4. The basic spin supermodule
	2.5. A superalgebra isomorphism
	3. Degenerate affine Hecke-Clifford algebras
	3.1. The algebra HcW of type An-1
	3.2. The algebra HcW of type Dn
	3.3. The algebra HcW of type Bn
	3.4. PBW basis for HcW
	3.5. The even center for HcW
	3.6. The intertwiners in HcW
	4. Degenerate spin affine Hecke algebras
	4.1. The skew-polynomial algebra
	4.2. The algebra H-W of type Dn
	4.3. The algebra H-W of type Bn
	4.4. A superalgebra isomorphism
	4.5. PBW basis for H-W
	4.6. The even center for H-W
	4.7. The intertwiners in H-W
	5. Degenerate covering affine Hecke algebras
	5.1. The algebra HW of type An-1
	5.2. The algebra HW of type Dn
	5.3. The algebra HW of type Bn
	5.4. PBW basis for HW
	References

  Associated to the classical Weyl groups, we introduce the notion of
degenerate spin affine Hecke algebras and affine Hecke-Clifford algebras. For
these algebras, we establish the PBW properties, formulate the intertwiners,
and describe the centers. We further develop connections of these algebras with
the usual degenerate (i.e. graded) affine Hecke algebras of Lusztig by
introducing a notion of degenerate covering affine Hecke algebras.

Introduction
1.1. The Hecke algebras associated to finite and affine Weyl groups are
ubiquitous in diverse areas, including representation theories over finite
fields, infinite fields of prime characteristic, p-adic fields, and Kazhdan-
Lusztig theory for category O. Lusztig [Lu1, Lu2] introduced the graded
Hecke algebras, also known as the degenerate affine Hecke algebras, associ-
ated to a finite Weyl groupW , and provided a geometric realization in terms
of equivariant homology. The degenerate affine Hecke algebra of type A has
also been defined earlier by Drinfeld [Dr] in connections with Yangians, and
it has recently played an important role in modular representations of the
symmetric group (cf. Kleshchev [Kle]).
In [W1], the second author introduced the degenerate spin affine Hecke
algebra of type A, and related it to the degenerate affine Hecke-Clifford
algebra introduced by Nazarov in his study of the representations of the spin
symmetric group [Naz]. A quantum version of the spin affine Hecke algebra
of type A has been subsequently constructed in [W2], and was shown to be
related 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 the
degenerate affine Hecke-Clifford algebras and degenerate spin affine Hecke
algebras for all classical finite Weyl groups, which goes beyond the type A
case, and then establish some basic properties of these algebras. The notion
of spin Hecke algebras is arguably more fundamental while the notion of
the Hecke-Clifford algebras is crucial for finding the right formulation of the
spin Hecke algebras. We also construct the degenerate covering affine Hecke
algebras which connect to both the degenerate spin affine Hecke algebras
and the degenerate affine Hecke algebras of Lusztig.
http://arxiv.org/abs/0704.0201v3
2 TA KHONGSAP AND WEIQIANG WANG
1.3. Let us describe our constructions in some detail. The Schur multiplier
for each finite Weyl group W has been computed by Ihara and Yokonuma
[IY] (see [Kar]). We start with a distinguished double cover W̃ for any finite
Weyl group W :
1 −→ Z2 −→ W̃ −→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− := CW̃/〈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 on
the 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 W̃ also appeared in Morris [Mo].
We formulate the notion of degenerate affine Hecke-Clifford algebras HcW
and spin affine Hecke algebras H−
, with unequal parameters in type B
case, associated to Weyl groups W of type D and B. The algebra HcW (and
respectively H−
) contain CW ⋊CW (and respectively CW
−) as subalgebras.
We establish the PBW basis properties for these algebras:
∼= 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 both
HcW and H
. The two Hecke algebras HcW and H
are related by a Morita
super-equivalence, i,e. a (super)algebra isomorphism
Φ : HcW
≃−→ CW ⊗ H−W
which extends the isomorphism Φfin. Such an isomorphism holds also for
W of type A [W1].
We generalize the construction in [Naz] of the intertwiners in the affine
Hecke-Clifford algebras HcW of type A to all classical Weyl groups W . We
also generalize the construction of the intertwiners in [W1] for H−
of type
A to all classical Weyl groups W . We further establish the basic properties
of these intertwiners in both HcW and H
. These intertwiners are expected
to play a fundamental role in the future development of the representation
theory of these algebras, as it is indicated by the work of Lusztig, Cherednik
and 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 W̃ 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
and its quotient by
the ideal 〈z−1〉 is identified with Lusztig’s degenerate affine Hecke algebras
associated toW . In this sense, our covering affine Hecke algebra is a natural
affine generalization of the central extension (1.1). A quantum version of
the covering affine Hecke algebra of type A was constructed in [W2].
The results in this paper remain valid over any algebraically closed field
of characteristic p 6= 2 (and in addition p 6= 3 for type G2). In fact, most
of the constructions can be made valid over the ring Z[1
] (occasionally we
need to adjoint
1.4. This paper and [W1] raise many questions, including a geometric re-
alization of the algebras HcW or H
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 HcW in the type A case in-
cluding the modular representations of the spin symmetric group have been
developed by Brundan and Kleshchev [BK] (also cf. [Kle]).
In a sequel [KW] to this paper, we will extend the constructions in this
paper to the setup of rational double affine Hecke algebras (see Etingof-
Ginzburg [EG]), generalizing and improving a main construction initiated
in [W1] for the spin symmetric group. We also hope to quantize these
degenerate spin Hecke algebras, reversing the history of developments from
quantum 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 isomorphism
theorem 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 the
corresponding degenerate spin affine Hecke algebras. We then extend the
isomorphism Φfin to an isomorphism relating these affine Hecke algebras,
establish the PBW properties, and describe the centers of HcW and H
Section 5, we formulate the notion of degenerate covering affine Hecke al-
gebras, and establish the connections to the degenerate spin affine Hecke
algebras and usual affine Hecke algebras.
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 with
the 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}, and
they are specified by the following Coxeter-Dynkin diagrams whose vertices
correspond 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 an
unmarked edge, and mij = 2 if i, j are not connected by an edge.
4 TA KHONGSAP AND WEIQIANG WANG
An ◦ ◦ . . . ◦ ◦
1 2 n− 1 n
Bn(n ≥ 2) ◦ ◦ . . . ◦ ◦
1 2 n− 1 n
Dn(n ≥ 4) ◦ ◦ · · · ◦ ◦
1 2 n− 3
En=6,7,8 ◦ ◦ ◦ . . . ◦ ◦
1 3 4 n− 1 n
F4 ◦ ◦ ◦ ◦
1 2 3 4
G2 ◦ ◦
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]). The
explicit generators and relations for the corresponding covering groups of W
can be found in Karpilovsky [Kar, Table 7.1].
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 CLASSICAL SPIN AFFINE HECKE ALGEBRAS 5
The quotient algebra CW− := CW̃/〈z+1〉 of CW̃ by the ideal generated
by z+1 will be called the spin Weyl group algebra associated to W . Denote
by ti ∈ CW− the image of t̃i. The spin Weyl group algebra CW− has the
following 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 CW̃
is isomorphic to CW .
Example 2.1. Let W be the Weyl group of type An, Bn, or Dn, which will
be assumed in later sections. Then the spin Weyl group algebra CW− is
generated by t1, . . . , tn with the labeling as in the Coxeter-Dynkin diagrams
and the explicit relations summarized in the following table.
Type of W Defining Relations for CW−
i = 1, titi+1ti = ti+1titi+1,
(titj)
2 = −1 if |i− j| > 1
t1, . . . , tn−1 satisfy the relations for CW
n = 1, (titn)
2 = −1 if i 6= n− 1, n,
(tn−1tn)
4 = −1
t1, . . . , tn−1 satisfy the relations for CW
n = 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 representation
of the Weyl groupW (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 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 sub-
space of CN and 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).
Denote by CW the Clifford algebra associated to (h, (−,−)), which is re-
garded as a subalgebra of the Clifford algebra CN associated to (C
N , (−,−)).
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 normalized
with β2i = 1. In particular, CN is generated by c1, . . . , cN subject to the
relations
c2i = 1, cicj = −cjci if i 6= j. (2.3)
6 TA KHONGSAP AND WEIQIANG WANG
The explicit generators for CW are listed in the following table. Note that
CW is naturally a superalgebra with each βi being odd.
Type of W N Generators for CW
An−1 n βi =
(ci − ci+1), 1 ≤ i ≤ n− 1
Bn n βi =
(ci − ci+1), 1 ≤ i ≤ n− 1, βn = cn
Dn n βi =
(ci − ci+1), 1 ≤ i ≤ n− 1, βn = 1√
(cn−1 + cn)
E8 8 β1 =
(c1 + c8 − c2 − c3 − c4 − c5 − c6 − c7)
(c1 + c2), βi =
(ci−1 + ci−2), 3 ≤ i ≤ 8
E7 8 the subset of βi in E8, 1 ≤ i ≤ 7
E6 8 the subset of βi in E8, 1 ≤ i ≤ 6
F4 4 β1 =
(c1 − c2), β2 = 1√2(c2 − c3)
β3 = c3, β4 =
(c4 − c1 − c2 − c3)
G2 3 β1 =
(c1 − c2), β2 = 1√6(−2c1 + c2 + c3)
The action of W on h and h∗ preserves the bilinear form (−,−) and thus
W acts as automorphisms of the algebra CW . This gives rise to a semi-direct
product CW ⋊CW . Moreover, the algebra CW ⋊CW naturally inherits the
superalgebra structure by letting elements inW be even and each βi be odd.
2.4. The basic spin supermodule. The following theorem is due to Mor-
ris [Mo] in full generality, and it goes back to I. Schur [Sch] (cf. [Joz]) in the
type A, namely the symmetric group case. It can be checked case by case
using the explicit formulas of βi in the Table of Section 2.3.
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.
Remark 2.3. In [Mo], W is viewed as a subgroup of the orthogonal Lie group
which preserves (h, (−,−)). The preimage of W in the spin group which
covers the orthogonal group provides the double cover W̃ of W , where the
Atiyah-Bott-Shapiro construction of the spin group in terms of the Clifford
algebra CW was used to describe this double cover of W .
The superalgebra CW has a unique (up to isomorphism) simple super-
module (i.e. Z2-graded module). By pulling it back via the homomorphism
Ω : CW− → CW , we obtain a distinguished CW−-supermodule, called the
basic spin supermodule. This is a natural generalization of the classical
construction 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 with
multiplication defined by
(a⊗ b)(a′ ⊗ b′) = (−1)|b||a′|(aa′ ⊗ bb′) (a, a′ ∈ A, b, b′ ∈ B) (2.4)
THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 7
where |b| denotes the Z2-degree of b, etc. Also, we shall use short-hand
notation ab for (a⊗ b) ∈ A ⊗ B, a = a⊗ 1, and b = 1⊗ b.
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 sends
ti 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) = 2β
i = 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 for
all 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 counting
argument.
Clearly, Ψ and Φ are inverses of each other. �
Remark 2.7. The type A case of Theorem 2.4 was due to Sergeev and Ya-
maguchi independently [Ser, Yam], and it played a fundamental role in
clarifying the earlier observation in the literature (cf. [Joz, St]) that the
representation 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 the
classical Weyl groups of type A,B, or D.
8 TA KHONGSAP AND WEIQIANG WANG
3. Degenerate affine Hecke-Clifford algebras
In this section, we introduce the degenerate affine Hecke-Clifford algebras
of type D and B, and establish some basic properties. The degenerate affine
Hecke-Clifford algebra associated to the symmetric group Sn was introduced
earlier 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 group
of type An−1. The degenerate affine Hecke-Clifford algebra of type An−1,
denoted by HcW or H
, 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 the
algebra HcW as a C(u)-algebra. Similar remarks apply to various algebras
introduced in this paper. Our convention c2i = 1 differs from Nazarov’s
which sets c2i = −1.
The symmetric group Sn acts as the automorphisms on the symmetric
algebra C[h∗] ∼= C[x1, . . . , xn] by permutation. We shall denote this action
by 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 HcW :
sif = f
sisi + u
f − f si
xi+1 − xi
cici+1f − f sicici+1
xi+1 + xi
It is understood here and in similar expressions below that A
In this sense, both numerators on the right-hand side of the above formula
are (left-)divisible by the corresponding denominators.
Proof. By the definition of HcW , we have that six
j = x
j 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.
THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 9
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
xki+1si + u
(xki − xki+1)
xi+1 − xi
(cici+1x
i − xki+1cici+1)
xi+1 + xi
= xki+1 (xi+1si − u(1− ci+1ci))
(xki − xki+1)
xi+1 − xi
xi + u
(cici+1x
i − xki+1cici+1)
xi+1 + xi
= xk+1i+1 si + u
(xk+1
− xk+1
i+1 )
xi+1 − xi
(cici+1x
− 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 x
i+1. The case l = 0 is established above. Assume the formula
is true for f = xki x
i+1. Then using sixi+1 = xisi+u(1+ci+1ci), we compute
i+1 =
i+1si + u
(xki x
i+1 − xlixki+1)
xi+1 − xi
(cici+1x
i+1 − xlixki+1cici+1)
xi+1 + xi
· xi+1
= xlix
i+1(xisi + u(1 + ci+1ci))
(xki x
i+1 − xlix
i+1 )
xi+1 − xi
(cici+1x
i+1 + x
i+1 cici+1)
xi+1 + xi
= xl+1i x
i+1si + u
(xki x
i+1 − x
xi+1 − xi
(cici+1x
i+1 − x
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+, cǫ = c
1 · · · cǫnn for ǫ =
(ǫ1, . . . , ǫn) ∈ Zn2 .
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
1 from the trivial W -module 1. This induced
module is of independent interest. This approach will then be used for type
D 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
≃−→ HcW .
10 TA KHONGSAP AND WEIQIANG WANG
Equivalently, {xαcǫw|α ∈ Zn+, ǫ ∈ Zn2 , w ∈ W} forms a linear basis for HcW
(called a PBW basis).
Proof. Note that IND := C[x1, . . . , xn] ⊗ Cn admits an algebra structure
by (2.3), (3.1) and (3.2). By the explicit defining relations of HcW , we can
verify that the algebra HcW acts on IND by letting xi and ci act by left
multiplication, and si ∈ Sn act by
si.(fc
ǫ) = f sicsiǫ +
f − f si
xi+1 − xi
cici+1f − f sicici+1
xi+1 + xi
For α = (a1, . . . , an), we denote |α| = a1 + · · · + an. Define a Lexico-
graphic ordering < on the monomials xα, α ∈ Zn+, (or respectively on Zn+),
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 x
αcǫw.
It remains to show that these elements are linearly independent.
Suppose that S :=
xαcǫw = 0 for a finite sum over α, ǫ, w and that
some coefficient a
6= 0; we fix one such ǫ. Now consider the action S
on an element of the form x
2 · · · xNnn for N1 ≫ N2 ≫ · · · ≫ Nn ≫ 0.
Let w̃ be such that (x
2 · · · xNnn )w̃ is maximal among all possible w with
aαǫw 6= 0 for some α. Let α̃ be the largest element among all α with
6= 0. Then among all monomials in S(xN11 x
2 · · · xNnn ), the monomial
xα̃(x
2 · · · xNnn )w̃cǫ appears as a maximal term with coefficient ±aα̃ǫw̃.
It follows from S = 0 that aα̃ǫw̃ = 0. This is a contradiction, 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, s
n = 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 H
W or H
, is the algebra generated
by xi, ci, si, 1 ≤ i ≤ n, subject to the relations (3.1–3.5), (3.6–3.10), and the
THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 11
following 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 generators
si, 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 σ
2 = 1. �
The natural action of Sn on C[h
∗] = C[x1, . . . , xn] is extended to an action
of WDn by letting
xsnn = −xn−1, x
n−1 = −xn, x
= 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 HcW :
(1) sif = f
sisi + u
f − f si
xi+1 − xi
cici+1f − f sicici+1
xi+1 + xi
(2) snf = f
snsn − u
f − f sn
xn + xn−1
cn−1cnf − f sncn−1cn
xn − 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. �
12 TA KHONGSAP AND WEIQIANG WANG
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 relation
for 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 different
conjugacy 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 affine
Hecke-Clifford algebra of type Bn, denoted by H
W or H
, is the algebra
generated 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 = −cnsn
snci = cisn (i 6= n)
snxn + xnsn = −
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 will
write HcW (u, v) for H
W . For any a ∈ C\{0}, we have an isomorphism of
superalgebras ψ : HcW (au, av) → HcW (u, v) given by dilations xi 7→ axi for
1 ≤ i ≤ n, while fixing each si, ci.
The action of Sn on C[h
∗] = C[x1, . . . , xn] can be extended to an action
of WBn by letting
xsnn = −xn, x
i = 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 HcW :
(1) sif = f
sisi + u
f − f si
xi+1 − xi
cici+1f − f sicici+1
xi+1 + xi
(2) snf = f
snsn −
f − f sn
Proof. The proof is similar to type A and D, and will be omitted. �
3.4. PBW basis for HcW . Note that H
W contains C[h
∗],Cn,CW as subal-
gebras. We have the following PBW basis theorem for HcW .
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 ∈ Zn2 , w ∈ W} form a linear
basis for HcW (called a PBW basis).
THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 13
Proof. For W = WDn , we can verify by a direct lengthy computation that
the HcAn−1-action on IND = C[x1, . . . , xn]⊗Cn (see the proof of Theorem 3.4)
naturally extends to an action of HcDn , where (compare Proposition 3.8) sn
acts by
sn.(fc
ǫ) = f sncsnǫ −
f − f sn
xn + xn−1
− ucn−1cnf − f
sncn−1cn
xn − xn−1
Similarly, for W = WBn , the H
-action on IND extends to an action of
HcBn , where (compare Proposition 3.10) sn acts by
sn.(fc
ǫ) = f sncsnǫ −
f − f sn
It is easy to show that, for either W , the elements xαcǫw span HcW . It
remains to show that they are linearly independent. We shall treat theWBn
case in detail and skip the analogous WDn case.
To that end, we shall refer to the argument in the proof of Theorem 3.4
with suitable modification as follows. The w̃ = ((η1, . . . , ηn), σ) ∈ WBn =
{±1}n ⋊ Sn may now not be unique, but the σ and the α̃ are uniquely
determined. Then, by the same argument on the vanishing of a maximal
term, we obtain that
w̃ aα̃ǫw̃x
2 · · · xNnn )w̃ = 0, and hence,
(η1,...,ηn)
aα̃ǫw̃(−1)
i=1 ηiNi = 0.
By choosing N1, . . . , Nn with different parities (2
n choices) and solving the
2n linear equations, we see that all aα̃ǫw̃ = 0. This can also be seen more
explicitly by induction on n. By choosing Nn to be even and odd, we deduce
that for a fixed ηn,
(η1,...,ηn−1)∈{±1}n−1 aα̃ǫw̃(−1)
i=1 ηiNi = 0, which is the
equation for (n− 1) xi’s and the induction applies. �
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(H
of HcW is isomorphic to C[x
1, . . . , x
Proof. We first show that every W -invariant polynomial f in x21, . . . , x
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 3.10
for type Bn, sif = fsi for each i. Since H
W is generated by ci, xi and si for
all i, f is central in HcW and C[x
1, . . . , x
W ⊆ Z(HcW ).
On the other hand, take an even central element C =
α,ǫ,w
xαcǫw in
HcW . 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 a
α,ǫ,w0
6= 0.
Then x
i 6= xi for some i. By Proposition 3.8 for typeDn or Proposition 3.10
for type Bn, x
iC − Cx2i is equal to
α,ǫ aα,ǫ,w0x
α(x2i − (x
2)cǫw0 plus a
14 TA KHONGSAP AND WEIQIANG WANG
linear combination of monomials not involving w0, hence nonzero. This
contradicts with the fact that C is central. So we can write C =
xαcǫ.
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 type
Dn or Proposition 3.10 for type Bn that C ∈ C[x21, . . . , x2n]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 = (x
i+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(x
i − 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
same formula (3.14) for 1 ≤ i ≤ n− 1 and in addition by letting
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 intertwiners
φi (with 1 ≤ i ≤ n− 1 for type An−1 and 1 ≤ i ≤ n for the other two types)
satisfy the following properties:
(1) φ2i = 2u
2(x2i+1 + x
i )− (x2i+1 − x2i )2 (1 ≤ i ≤ n− 1,∀W );
(2) φ2n = 2u
2(x2n + x
n−1)− (x2n − x2n−1)2, for type Dn;
(3) φ2n = 4x
n − 2v2x2n, for type Bn;
(4) φif = f
siφi (∀f ∈ C[x1, . . . , xn],∀i,∀W );
(5) φicj = c
j φi (1 ≤ j ≤ n,∀i,∀W );
(6) φiφjφi · · ·︸ ︷︷ ︸
= φjφiφj · · ·︸ ︷︷ ︸
Proof. Part (1) follows by a straightforward computation and can also be
found in [Naz] (with u = 1). Part (2) follows from (1) by applying the
involution σ defined in Proposition 3.7. Part (3) and (5) follow by a direct
verification.
Part (4) for WAn−1 follows from clearing the denominators in the formula
in 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 in
Proposition 3.8 in type D and Proposition 3.10 in type B, respectively.
THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 15
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 we
can write
φiφjφi · · · = fw +
pu,wu
for some f ∈ C[x1, . . . , xn], and pu,w ∈ IND. We may rewrite
φiφjφi · · · = fw +
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 x
1≤i<j≤n(x
i − x2j) ∈ IND being invertible. Note that such
a localization is a free module over the corresponding localized ring of
C[x1, . . . , xn]. We can then write
φjφiφj · · · = fw +
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
ru,wφu
for some ru,w. Observe by (4) that ∆p = p
w∆ for any p ∈ C[x1, . . . , xn].
Then we have
pwru,wφu = p
w∆ = ∆p =
ru,wφup =
ru,wp
In other words, (pw − pu)ru,w = 0 for all p ∈ C[x1, . . . , xn] for each given
u < w. This implies that ru,w = 0 for each u, and ∆ = 0. This completes
the proof of (6) modulo Lemma 3.14 below. �
Lemma 3.14. The following identity holds:
i · · ·︸ ︷︷ ︸
= φ0jφ
j · · ·︸ ︷︷ ︸
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.)
16 TA KHONGSAP AND WEIQIANG WANG
Proof. Let W =WBn . For 1 ≤ i ≤ n− 1, mi,i+1 = 3. So we have
i = (x
i+1 − x2i )si(x2i+2 − x2i+1)si+1(x2i+1 − x2i )si
= (x2i+1 − x2i )(x2i+2 − x2i )(x2i+2 − x2i+1)sisi+1si
= (x2i+2 − x2i+1)(x2i+2 − x2i )(x2i+1 − x2i )si+1sisi+1
= (x2i+2 − x2i+1)si+1(x2i+1 − x2i )si(x2i+2 − x2i+1)si+1
= φ0i+1φ
Note that mij = 2 for j 6= i, i+ 1; clearly, in this case, φ0iφ0j = φ0jφ0i .
Noting that mn−1,n = 4, we have
φ0n−1φ
n = 4(x
n − x2n−1)sn−1x2nsn(x2n − x2n−1)sn−1x2nsn
= 4(x2n − x2n−1)x2n−1(x2n−1 − x2n)x2nsn−1snsn−1sn
= 4x2n(x
n − x2n−1)x2n−1(x2n−1 − x2n)snsn−1snsn−1
= 4x2nsn(x
n − x2n−1)sn−1x2nsn(x2n − x2n−1)sn−1
= φ0nφ
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 ∈ HcW given by φw = φi1 · · ·φim where w = si1 · · · sim is any
reduced expression for w. These elements φw should play an important role
for the representation theory of the algebras HcW . It will be very interest-
ing to classify the simple modules of HcW and to find a possible geometric
realization. This was carried out by Lusztig [Lu1, Lu2, Lu3] for the usual
degenerate affine Hecke algebra case.
4. Degenerate spin affine Hecke algebras
In this section we will introduce the degenerate spin affine Hecke algebra
when W is the Weyl group of types Dn or Bn, and then establish the
connections with the corresponding degenerate affine Hecke-Clifford algebras
HcW . See [W1] for the type A case.
4.1. The skew-polynomial algebra. We shall denote by C[b1, . . . , bn] the
C-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 to
this as the skew-polynomial algebra in n variables. This algebra has a linear
basis given by bα := bk11 · · · bknn for α = (k1, . . . , kn) ∈ Zn+, and it contains a
polynomial subalgebra C[b21, . . . , b
THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 17
4.2. The algebra H−
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 affine
Hecke algebra of type Dn, denoted by H
or H−
, 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−
is naturally a superalgebra by letting each ti and bi be
odd generators. It contains the type An−1 degenerate spin affine Hecke
algebra H−
(generated by b1, . . . , bn, t1, . . . , tn−1) as a subalgebra.
Proposition 4.2. The algebra H−
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−
admits an involution σ which swaps tn−1 and tn while
fixing all the remaining generators ti, bi.
Proof. Note that we use the same symbols τ1, τ2, σ to denote the (anti-)
involutions for H−
and HcDn in Proposition 3.7, as those on H
are 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, or
follows by a direct computation as in the proof of Proposition 3.7. �
4.3. The algebra H−
of type Bn.
Definition 4.3. Let u, v ∈ C, and W = WBn . The degenerate spin affine
Hecke algebra of type Bn, denoted by H
or H−
, 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−
(u, v) or H−
(u, v) for H−
or H−
to indicate
the dependence on the parameters u, v.
18 TA KHONGSAP AND WEIQIANG WANG
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−→HcW extends Ψ : Cn⊗CW− −→ Cn⋊CW
(in Theorem 2.4) and sends bi 7−→
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. For
1 ≤ 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, we
have Φ(xi+1si − sixi) = u(1− ci+1ci). Moreover, we have
Φ(snxn + xnsn) =
cntncnbn +
cnbncntn
2cntncnbn +
2cnbncntn
2(tnbn + bncn) = −
Φ(snxj) =
cntncjbj =
2cntncjbj
2cjcntnbj =
2cjbjcntn = Φ(xjsn), for j 6= n.
Thus Φ is a homomorphism of (super)algebras. Similarly, we check that
Ψ is a superalgebra homomorphism. Observe that Φ and Ψ are inverses on
generators and hence they are indeed (inverse) isomorphisms. �
THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 19
4.5. PBW basis for H−
. Note that H−
contains the skew-polynomial
algebra C[b1, . . . , bn] and the spin Weyl group algebra CW
− as subalgebras.
We have the following PBW basis theorem for H−
Theorem 4.5. Let W = WDn or W = WBn. The multiplication of the
subalgebras CW− and C[b1, . . . , bn] induces a vector space isomorphism
C[b1, . . . , bn]⊗ CW−
≃−→ H−
Theorem 4.5 also holds for WAn−1 (see [W1]).
Proof. It follows from the definition that H−
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−→HcW . Observe that the im-
age ψ(bασ) are linearly independent in HcW by the PBW basis Theorem 3.11
for HcW . Hence the elements b
ασ are linearly independent in H−
4.6. The even center for H−
Proposition 4.6. Let W =WDn or W = WBn . The even center of H
isomorphic to C[b21, . . . , b
Proof. By the isomorphism Φ : HcW → Cn⊗H
(see Theorems 4.4) and the
description of the center Z(HcW ) (see Proposition 3.12), we have
Z(Cn ⊗ H−W ) = Φ(Z(H
W )) = Φ(C[x
1, . . . , x
W ) = C[b21, . . . , b
Thus, C[b21, . . . , b
W ⊆ Z(H−
Now let C ∈ Z(H−
). Since C is even, C commutes with Cn and thus com-
mutes with the algebra Cn ⊗ H−W . Then Ψ(C) ∈ Z(H
W ) = C[x
1, . . . , x
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 the
simple modules of the spin affine Hecke algebra H−
is equivalent to the
classification problem for the affine Hecke-Clifford algebra HcW . It remains
to be seen whether it is more convenient to find the geometric realization of
instead of HcW .
4.7. The intertwiners in H−
. The intertwiners Ii ∈ H−W (1 ≤ i ≤ n− 1)
for W =WAn−1 were introduced in [W1] (with u = 1):
Ii = (b
i+1 − b2i )ti − u(bi+1 − bi). (4.1)
The commutation relations in Definition 4.1 gives us another equivalent
expression for Ii:
Ii = ti(b
i − b2i+1) + u(bi+1 − bi).
We define the intertwiners Ii ∈ H−W for W = WDn (1 ≤ i ≤ n) by the
same formula (4.1) for 1 ≤ i ≤ n− 1 and in addition by letting
In ≡ IDn = (b2n − b2n−1)tn − u(bn − bn−1). (4.2)
20 TA KHONGSAP AND WEIQIANG WANG
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−
, for W = WAn−1 ,
WBn, or WDn:
(1) Iibi = −bi+1Ii,Iibi+1 = −biIi, and Iibj = −bjIi (j 6= i, i + 1), for
1 ≤ 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 = (b
i+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 = (b
n − 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
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
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.
THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 21
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)(b2i+1 − b2i )ti − u
−2(ci+1bi+1 − cibi)
−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)(b2n − b2n−1)tn + u
−2(cnbn − cn−1bn−1)
−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 less
complicated. �
Proposition 4.9. The following identities hold in H−
, for W = WAn−1 ,
WBn, or WDn:
(1) I2i = u
2(b2i+1 + b
i )− (b2i+1 − b2i )2, for 1 ≤ i ≤ n− 1 and every type of
(2) I2n = u
2(b2n + b
n−1)− (b2n − b2n−1)2, for type Dn.
(3) I2n = 4b
n − v2b2n, for type Bn.
Proof. It follows from the counterparts in Theorem 3.13 via the explicit
correspondences under the isomorphism Φ (see Theorem 4.8). It can of
course also be proved by a direct computation. �
Proposition 4.10. For W =WAn−1, WBn , or WDn , we have
IiIjIi · · ·︸ ︷︷ ︸
= (−1)mij+1 IjIiIj · · ·︸ ︷︷ ︸
Proof. By Theorem 2.2, we have
βiβjβi · · ·︸ ︷︷ ︸
= (−1)mij+1 βjβiβj · · ·︸ ︷︷ ︸
Now the statement follows from the above equation and Theorem 3.13 (6)
via the correspondence of the intertwiners under the isomorphism Φ (see
Theorem 4.8). �
Remark 4.11. Proposition 4.7, Theorem 4.8, and Proposition 4.10 for H−
can be found in [W1].
22 TA KHONGSAP AND WEIQIANG WANG
5. Degenerate covering affine Hecke algebras
In this section, the degenerate covering affine Hecke algebras associated
to the double covers W̃ of classical Weyl groups W are introduced. It has
as its natural quotients the usual degenerate affine Hecke algebras HW [Dr,
Lu1, Lu2] and the spin degenerate affine Hecke algebras H−
introduced by
the authors.
Recall the distinguished double cover W̃ 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 covering
affine Hecke algebra of type An−1, denoted by H
W or H
, is the algebra
generated by x̃1, . . . , x̃n and z, t̃1, . . . , t̃n−1, subject to the relations for W̃
and the additional relations:
zx̃i = x̃iz, z is central of order 2 (5.1)
x̃ix̃j = zx̃jx̃i (i 6= j) (5.2)
t̃ix̃j = zx̃j t̃i (j 6= i, i + 1) (5.3)
t̃ix̃i+1 = zx̃it̃i + u. (5.4)
Clearly H∼W contains CW̃ as a subalgebra.
5.2. The algebra H∼W of type Dn.
Definition 5.2. Let W = WDn , and let u ∈ C. The degenerate covering
affine Hecke algebra of type Dn, denoted by H
W or H
, is the algebra
generated by x̃1, . . . , x̃n and z, t̃1, . . . , t̃n, subject to the relations (5.1–5.4)
and the following additional relations:
t̃nx̃i = zx̃it̃n (i 6= n− 1, n)
t̃nx̃n = −x̃n−1t̃n + u.
5.3. The algebra H∼W of type Bn.
Definition 5.3. Let W = WBn , and let u, v ∈ C. The degenerate covering
affine Hecke algebra of type Bn, denoted by H
W or H
, is the algebra
generated by x̃1, . . . , x̃n and z, t̃1, . . . , t̃n, subject to the relations (5.1–5.4)
and the following additional relations:
t̃nx̃i = zx̃it̃n (i 6= n)
t̃nx̃n = −x̃nt̃n + v.
5.4. PBW basis for H∼W .
Proposition 5.4. Let W = WAn−1 ,WDn , or WBn . Then the quotient of
the covering affine Hecke algebra H∼W by the ideal 〈z − 1〉 (respectively, by
the ideal 〈z+1〉) is isomorphic to the usual degenerate affine Hecke algebras
HW (respectively, the spin degenerate affine Hecke algebras H
THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 23
Proof. Follows by the definitions in terms of generators and relations of all
the algebras involved. �
Theorem 5.5. Let W = WAn−1 ,WDn , or WBn. Then the elements x̃
where α ∈ Zn+ and w̃ ∈ W̃ , form a basis for H∼W (called a PBW basis).
Proof. By the defining relations, it is easy to see that the elements x̃αw̃
form a spanning set for H∼W . So it remains to show that they are linearly
independent.
For each element t ∈ W , denote the two preimages in W̃ of t by {t̃, zt̃}.
Now suppose that
aα,t̃x̃
αt̃+ bα,t̃zx̃
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
. Consider the
projections:
Υ+ : H
W −→ H∼W /I+, Υ− : H∼W −→ H∼W /I−.
By abuse of notation, denote the image of x̃α in HW by x
α. Observe that
0 = Υ+
(aα,t̃x̃
αt̃+ bα,t̃x̃
αzt̃)
(aα,t̃ + bα,t̃)x
αt ∈ 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,t̃ = −bα,t̃ for all α and t.
Similarly, denoting the image in CW− of t̃ by t̄, we have
0 = Υ−
(aα,t̃x̃
αt̃+ bα,t̃x̃
αzt̃)
(aα,t̃ − bα,t̃)x
αt̄ ∈ H−
Since {xαt̄} is a basis for the spin degenerate affine Hecke algebra H−
have aα,t̃ = bα,t̃ for all α and t. Hence, aα,t̃ = bα,t̃ = 0, and the linear
independence is proved. �
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[W1] W. Wang, Double affine Hecke algebras for the spin symmetric group, preprint
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Department of Math., University of Virginia, Charlottesville, VA 22904
E-mail address: tk7p@virginia.edu (Khongsap); ww9c@virginia.edu (Wang)
http://arxiv.org/abs/math/0608074
	1. Introduction
	1.1. 
	1.2. 
	1.3. 
	1.4. 
	1.5. 
	2. Spin Weyl groups and Clifford algebras
	2.1. The Weyl groups
	2.2. A distinguished double covering of Weyl groups
	2.3. The Clifford algebra CW
	2.4. The basic spin supermodule
	2.5. A superalgebra isomorphism
	3. Degenerate affine Hecke-Clifford algebras
	3.1. The algebra HcW of type An-1
	3.2. The algebra HcW of type Dn
	3.3. The algebra HcW of type Bn
	3.4. PBW basis for HcW
	3.5. The even center for HcW
	3.6. The intertwiners in HcW
	4. Degenerate spin affine Hecke algebras
	4.1. The skew-polynomial algebra
	4.2. The algebra H-W of type Dn
	4.3. The algebra H-W of type Bn
	4.4. A superalgebra isomorphism
	4.5. PBW basis for H-W
	4.6. The even center for H-W
	4.7. The intertwiners in H-W
	5. Degenerate covering affine Hecke algebras
	5.1. The algebra HW of type An-1
	5.2. The algebra HW of type Dn
	5.3. The algebra HW of type Bn
	5.4. PBW basis for HW
	References
