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Hecke-Clifford algebras and spin Hecke algebras I the classical affine type


HECKE-CLIFFORD ALGEBRAS AND SPIN HECKE ALGEBRAS I: THE CLASSICAL AFFINE TYPE

arXiv:0704.0201v3 [math.RT] 17 Oct 2007

TA KHONGSAP AND WEIQIANG WANG Abstract. Associated to the classical Weyl groups, we introduce the notion of degenerate spin a?ne Hecke algebras and a?ne Hecke-Cli?ord 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) a?ne Hecke algebras of Lusztig by introducing a notion of degenerate covering a?ne Hecke algebras.

1. Introduction 1.1. The Hecke algebras associated to ?nite and a?ne Weyl groups are ubiquitous in diverse areas, including representation theories over ?nite ?elds, in?nite ?elds of prime characteristic, p-adic ?elds, and KazhdanLusztig theory for category O. Lusztig [Lu1, Lu2] introduced the graded Hecke algebras, also known as the degenerate a?ne Hecke algebras, associated to a ?nite Weyl group W , and provided a geometric realization in terms of equivariant homology. The degenerate a?ne Hecke algebra of type A has also been de?ned 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 a?ne Hecke algebra of type A, and related it to the degenerate a?ne Hecke-Cli?ord algebra introduced by Nazarov in his study of the representations of the spin symmetric group [Naz]. A quantum version of the spin a?ne Hecke algebra of type A has been subsequently constructed in [W2], and was shown to be related to the q-analogue of the a?ne Hecke-Cli?ord algebra (of type A) de?ned by Jones and Nazarov [JN]. 1.2. The goal of this paper is to provide canonical constructions of the degenerate a?ne Hecke-Cli?ord algebras and degenerate spin a?ne Hecke algebras for all classical ?nite 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-Cli?ord algebras is crucial for ?nding the right formulation of the spin Hecke algebras. We also construct the degenerate covering a?ne Hecke algebras which connect to both the degenerate spin a?ne Hecke algebras and the degenerate a?ne Hecke algebras of Lusztig.
1

2

TA KHONGSAP AND WEIQIANG WANG

1.3. Let us describe our constructions in some detail. The Schur multiplier for each ?nite Weyl group W has been computed by Ihara and Yokonuma [IY] (see [Kar]). We start with a distinguished double cover W for any ?nite Weyl group W : Denote Z2 = {1, z}. Assume that W is generated by s1 , . . . , sn subject to the relations (si sj )mij = 1. The quotient CW ? := CW / z + 1 is then generated by t1 , . . . , tn subject to the relations (ti tj )mij = 1 for mij odd, and (ti tj )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 Cli?ord algebra CW associated to the re?ection representation h of W . We establish a (super)algebra isomorphism extending an isomorphism in the symmetric group case (due to Sergeev [Ser] and Yamaguchi [Yam] independently) to all Weyl groups. That is, the superalgebras CW ? CW and CW ? are Morita super-equivalent in the terminology of [W2]. The double cover W also appeared in Morris [Mo]. We formulate the notion of degenerate a?ne Hecke-Cli?ord algebras Hc W and spin a?ne Hecke algebras H? , with unequal parameters in type B W case, associated to Weyl groups W of type D and B. The algebra Hc (and W respectively H? ) contain CW ?CW (and respectively CW ? ) as subalgebras. W We establish the PBW basis properties for these algebras: Hc ? C[h? ] ? CW ? CW, H? ? C[h? ] ? CW ? = =
W W

1 ?→ Z2 ?→ W ?→ W ?→ 1.

(1.1)

Φf in : CW ? CW ?→ CW ? CW ? ,

?

where denotes the polynomial algebra and C[h? ] denotes a noncommutative skew-polynomial algebra. We describe explicitly the centers for both Hc and H? . The two Hecke algebras Hc and H? are related by a Morita W W W W super-equivalence, i,e. a (super)algebra isomorphism Φ : Hc ?→ CW ? H? W W
?

C[h? ]

which extends the isomorphism Φf in . Such an isomorphism holds also for W of type A [W1]. We generalize the construction in [Naz] of the intertwiners in the a?ne Hecke-Cli?ord algebras Hc of type A to all classical Weyl groups W . We W also generalize the construction of the intertwiners in [W1] for H? of type W A to all classical Weyl groups W . We further establish the basic properties of these intertwiners in both Hc and H? . These intertwiners are expected W W 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 a?ne Hecke algebras. We further introduce a notion of degenerate covering a?ne Hecke algebras ? associated to the double cover W of the Weyl group W of classical HW type. The algebra H? contains a central element z of order 2 such that the W

THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS

3

quotient of H? by the ideal z + 1 is identi?ed with H? and its quotient by W W the ideal z ? 1 is identi?ed with Lusztig’s degenerate a?ne Hecke algebras associated to W . In this sense, our covering a?ne Hecke algebra is a natural a?ne generalization of the central extension (1.1). A quantum version of the covering a?ne Hecke algebra of type A was constructed in [W2]. The results in this paper remain valid over any algebraically closed ?eld of characteristic p = 2 (and in addition p = 3 for type G2 ). In fact, most 1 of the constructions can be made valid over the ring Z[ 2 ] (occasionally we √ need to adjoint 2). 1.4. This paper and [W1] raise many questions, including a geometric realization of the algebras Hc or H? in the sense of Lusztig [Lu1, Lu2], the W W classi?cation of the simple modules (cf. [Lu3]), the development of the representation theory, an extension to the exceptional Weyl groups, and so on. We remark that the modular representations of Hc in the type A case inW 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 a?ne Hecke algebras (see EtingofGinzburg [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 distinguished covering groups of the Weyl groups, and establish the isomorphism theorem in the ?nite-dimensional case. We introduce in Section 3 the degenerate a?ne Hecke-Cli?ord algebras of type D and B, and in Section 4 the corresponding degenerate spin a?ne Hecke algebras. We then extend the isomorphism Φf in to an isomorphism relating these a?ne Hecke algebras, establish the PBW properties, and describe the centers of Hc and H? . In W W Section 5, we formulate the notion of degenerate covering a?ne Hecke algebras, and establish the connections to the degenerate spin a?ne Hecke algebras and usual a?ne 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) ?nite Weyl group with the following presentation: For a Weyl group W , the integers mij take values in {1, 2, 3, 4, 6}, and they are speci?ed 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 = j connected by an unmarked edge, and mij = 2 if i, j are not connected by an edge. s1 , . . . , sn |(si sj )mij = 1, mii = 1, mij = mji ∈ Z≥2 , for i = j (2.1)

4

TA KHONGSAP AND WEIQIANG WANG

An

? 1

? 2

...

? n?1 ? 4 n?1

? n

Bn (n ≥ 2)

? 1

? 2

...

? n
? ?

?n

Dn (n ≥ 4)

? 1

? 2

···

? n?3

? n?2
d d d? n ? 1

?

En=6,7,8

1 ?

3 ?

4 ? ? 2

...

n?1 ?

n ?

F4

? 1

? 2

4

? 3

? 4

G2

? 1

6

? 2

2.2. A distinguished double covering of Weyl groups. The Schur multipliers for ?nite Weyl groups W (and actually for all ?nite 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 W of W : 1 ?→ Z2 ?→ W ?→ W ?→ 1. ? We denote by Z2 = {1, z}, and by ti a ?xed preimage of the generators si of ? ? W for each i. The group W is generated by z, t1 , . . . , tn with relations z 2 = 1, ?? (ti tj )mij = 1, if mij = 1, 3 z, if mij = 2, 4, 6.

THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS

5

The quotient algebra CW ? := CW / z + 1 of CW 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 ti . 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 (ti tj )mij = (?1)mij +1 ≡ 1, if mij = 1, 3 ?1, if mij = 2, 4, 6. (2.2)

Note that dim CW ? = |W |. The algebra CW ? has a natural superalgebra (i.e. Z2 -graded) structure by letting each ti be odd. By de?nition, the quotient by the ideal z ? 1 of the group algebra CW 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 An De?ning Relations for CW ? t2 = 1, ti ti+1 ti = ti+1 ti ti+1 , i (ti tj )2 = ?1 if |i ? j| > 1 ? t1 , . . . , tn?1 satisfy the relations for CWAn?1 , t2 = 1, (ti tn )2 = ?1 if i = n ? 1, n, n (tn?1 tn )4 = ?1 ? t1 , . . . , tn?1 satisfy the relations for CWAn?1 , t2 = 1, (ti tn )2 = ?1 if i = n ? 2, n, n tn?2 tn tn?2 = tn tn?2 tn

Bn

Dn

2.3. The Cli?ord algebra CW . Denote by h the re?ection representation of the Weyl group W (i.e. a Cartan subalgebra of the corresponding complex Lie algebra g). In the case of type An?1 , we will always choose to work with the Cartan subalgebra h of gln instead of sln in this paper. Note that h carries a W -invariant nondegenerate bilinear form (?, ?), which gives rise to an identi?cation h? ? h and also a bilinear form on h? = which will be again denoted by (?, ?). We identify h? with a suitable subspace 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 Cli?ord algebra associated to (h, (?, ?)), which is regarded as a subalgebra of the Cli?ord algebra CN associated√ (CN , (?, ?)). to 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 2 with βi = 1. In particular, CN is generated by c1 , . . . , cN subject to the relations c2 = 1, i ci cj = ?cj ci if i = j. (2.3)

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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 An?1 Bn Dn E8 E7 E6 F4 G2 N n n n 8 8 8 4 3 Generators for CW 1 βi = √2 (ci ? ci+1 ), 1 ≤ i ≤ n ? 1 1 βi = √2 (ci ? ci+1 ), 1 ≤ i ≤ n ? 1, βn = cn 1 1 βi = √2 (ci ? ci+1 ), 1 ≤ i ≤ n ? 1, βn = √2 (cn?1 + cn ) 1 β1 = 2√2 (c1 + c8 ? c2 ? c3 ? c4 ? c5 ? c6 ? c7 ) 1 1 β2 = √2 (c1 + c2 ), βi = √2 (ci?1 + ci?2 ), 3 ≤ i ≤ 8 the subset of βi in E8 , 1 ≤ i ≤ 7 the subset of βi in E8 , 1 ≤ i ≤ 6 1 1 β1 = √2 (c1 ? c2 ), β2 = √2 (c2 ? c3 ) β3 = c3 , β4 = 1 (c4 ? c1 ? c2 ? c3 ) 2 1 1 β1 = √2 (c1 ? c2 ), β2 = √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 in W be even and each βi be odd. 2.4. The basic spin supermodule. The following theorem is due to Morris [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 ?nite Weyl group. Then, there exists a surjective ? superalgebra homomorphism CW ? ?→ 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 W of W , where the Atiyah-Bott-Shapiro construction of the spin group in terms of the Cli?ord algebra CW was used to describe this double cover of W . The superalgebra CW has a unique (up to isomorphism) simple supermodule (i.e. Z2 -graded module). By pulling it back via the homomorphism ? : CW ? → CW , we obtain a distinguished CW ? -supermodule, called the basic spin supermodule. This is a natural generalization of the classical ? construction for CSn 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 de?ned 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] between the superalgebras CW ? CW and CW ? . Theorem 2.4. We have an isomorphism of superalgebras: √ which extends the identity map on CW and sends si → ? ?1βi ti . The inverse map Ψ is the extension of the identity map on CW which sends √ ti → ?1βi si . We ?rst prepare some lemmas. Lemma 2.5. We have (Φ(si )Φ(sj ))mij = 1.
ij Proof. Theorem 2.2 says that (ti tj )m√ = (βi βj )mij = ±1. Thanks to the identities βj ti = ?ti βj and Φ(si ) = ? ?1βi ti , we have

Φ : CW ? CW ?→ CW ? CW ?

?

(Φ(si )Φ(sj ))mij = (?βi ti βj tj )mij

= (βi βj ti tj )mij = (βi βj )mij (ti tj )mij = 1.

Lemma 2.6. We have βj Φ(si ) = Φ(si ) si (βj ) for all i, j.
2 Proof. Note that (βi , βi ) = 2βi = 2, and hence

βj βi = ?βi βj + (βj , βi ) = ?βi βj + Thus, we have

2(βj , βi ) 2 β = ?βi si (βj ). (βi , βi ) i

√ βj Φ(si ) = ? ?1βj βi ti √ √ = ? ?1ti βj βi = ?1ti βi si (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 Yamaguchi 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 CSn 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 a?ne Hecke-Cli?ord algebras of type D and B, and establish some basic properties. The degenerate a?ne Hecke-Cli?ord algebra associated to the symmetric group Sn was introduced earlier by Nazarov under the terminology of the a?ne Sergeev algebra [Naz].

3.1. The algebra Hc of type An?1 . W De?nition 3.1. [Naz] Let u ∈ C, and W = WAn?1 = Sn be the Weyl group of type An?1 . The degenerate a?ne Hecke-Cli?ord algebra of type An?1 , denoted by Hc or Hc n?1 , is the algebra generated by x1 , . . . , xn , c1 , . . . , cn , W A and Sn subject to the relation (2.3) and the following relations: xi xj = xj xi σci = cσi σ (?i, j) (i = j) (1 ≤ i ≤ n, σ ∈ Sn ) (3.1) (3.2) (3.3) (3.4) (3.5)

xi ci = ?ci xi , xi cj = cj xi xi+1 si ? si xi = u(1 ? ci+1 ci ) xj si = si xj

(j = i, i + 1)

Remark 3.2. Alternatively, we may view u as a formal parameter and the algebra Hc as a C(u)-algebra. Similar remarks apply to various algebras W introduced in this paper. Our convention c2 = 1 di?ers from Nazarov’s i which sets c2 = ?1. i 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 → f σ for σ ∈ Sn , f ∈ C[x1 , . . . , xn ]. Proposition 3.3. Let W = WAn?1 . Given f ∈ C[x1 , . . . , xn ] and 1 ≤ i ≤ n ? 1, the following identity holds in Hc : W si f = f si si + u ci ci+1 f ? f si ci ci+1 f ? f si +u . xi+1 ? xi xi+1 + xi

A 1 It is understood here and in similar expressions below that g(x) = g(x) · 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 de?nition of Hc , we have that si xk = xk si for any k if j j W j = i, i + 1. So it su?ces to check the identity for f = xk xl . We will i i+1 proceed by induction.

THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS

9

First, consider f = xk , i.e. l = 0. For k = 1, this follows from (3.4). Now i assume that the statement is true for k. Then si xk+1 i = xk si i+1 (ci ci+1 xk ? xk ci ci+1 ) (xk ? xk ) i i+1 i i+1 +u +u xi+1 ? xi xi+1 + xi xi

= xk (xi+1 si ? u(1 ? ci+1 ci )) i+1 +u

(ci ci+1 xk ? xk ci ci+1 ) (xk ? xk ) i i+1 i i+1 xi + u xi xi+1 ? xi xi+1 + xi

= xk+1 si + u i+1

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 monomial f = xk xl . The case l = 0 is established above. Assume the formula i i+1 is true for f = xk xl . Then using si xi+1 = xi si + u(1 + ci+1 ci ), we compute i i+1 that si xk xl+1 = i i+1 xl xk si + u i i+1 (xk xl ? xl xk ) i i+1 i i+1 xi+1 ? xi (ci ci+1 xk xl ? xl xk ci ci+1 ) i i+1 i i+1 xi+1 + xi · xi+1

(xk+1 ? xk+1 ) (ci ci+1 xk+1 ? xk+1 ci ci+1 ) i+1 i i+1 i +u , xi+1 ? xi xi+1 + xi

+u

= xl xk (xi si + u(1 + ci+1 ci )) i i+1 +u (ci ci+1 xk xl+1 + xl xk+1 ci ci+1 ) (xk xl+1 ? xl xk+1 ) i i+1 i i+1 i i+1 i i+1 +u xi+1 ? xi xi+1 + xi (xk xl+1 ? xl+1 xk ) i i+1 i+1 i xi+1 ? xi

= xl+1 xk si + u i+1 i +u

(ci ci+1 xk xl+1 ? xl+1 xk ci ci+1 ) i+1 i i+1 i . xi+1 + xi This completes the proof of the proposition. The algebra Hc contains C[h? ], Cn , and CW as subalgebras. We shall W denote xα = xa1 · · · xan for α = (a1 , . . . , an ) ∈ Zn , c? = c?1 · · · c?n for ? = n + n 1 1 (?1 , . . . , ?n ) ∈ Zn . 2 Below we give a new proof of the PBW basis theorem for Hc (which W has been established by di?erent methods in [Naz, Kle]), using in e?ect the Hc induced Hc -module IndWW 1 from the trivial W -module 1. This induced W module is of independent interest. This approach will then be used for type D and B. Theorem 3.4. Let W = WAn?1 . The multiplication of subalgebras C[h? ], Cn , and CW induces a vector space isomorphism C[h? ] ? Cn ? CW ?→ Hc . W
?

10

TA KHONGSAP AND WEIQIANG WANG

Equivalently, {xα c? w|α ∈ Zn , ? ∈ Zn , w ∈ W } forms a linear basis for Hc + 2 W (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 de?ning relations of Hc , we can W verify that the algebra Hc acts on IND by letting xi and ci act by left W multiplication, and si ∈ Sn act by si .(f c? ) = f si csi ? + u ci ci+1 f ? f si ci ci+1 f ? f si +u xi+1 ? xi xi+1 + xi c? .

For α = (a1 , . . . , an ), we denote |α| = a1 + · · · + an . De?ne a Lexicographic 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′ and aj = a′ for each j < i. i j Note that the algebra Hc is spanned by the elements of the form xα c? w. W It remains to show that these elements are linearly independent. Suppose that S := aα?w xα c? w = 0 for a ?nite sum over α, ?, w and that some coe?cient aα?w = 0; we ?x one such ?. Now consider the action S on an element of the form xN1 xN2 · · · xNn for N1 ? N2 ? · · · ? Nn ? 0. n 1 2 ? Let w be such that (xN1 xN2 · · · xNn )w is maximal among all possible w with ? n 1 2 aα?w = 0 for some α. Let α be the largest element among all α with ? aα?w = 0. Then among all monomials in S(xN1 xN2 · · · xNn ), the monomial n ? 1 2 ? ? xα (xN1 xN2 · · · xNn )w c? appears as a maximal term with coe?cient ±aα?w . ? ? n 1 2 It follows from S = 0 that aα?w = 0. This is a contradiction, and hence the ? ? elements xα c? w are linearly independent. Remark 3.5. By the PBW Theorem 3.4, the Hc -module IND introduced W in the above proof can be identi?ed with the Hc -module induced from the W trivial CW -module. The same remark applies below to type D and B. 3.2. The algebra Hc of type Dn . Let W = WDn be the Weyl group of W type Dn . It is generated by s1 , . . . , sn , subject to the following relations: s2 = 1 i si si+1 si = si+1 si si+1 si sj = sj si si sn = sn si (i ≤ n ? 1) (3.6) (3.7) (3.8) (3.9) (3.10) (i ≤ n ? 2)

(|i ? j| > 1, i, j = n) (i = n ? 2) s2 n = 1.

sn?2 sn sn?2 = sn sn?2 sn ,

In particular, Sn is generated by s1 , . . . , sn?1 subject to the relations (3.6– 3.8) above. De?nition 3.6. Let u ∈ C, and let W = WDn . The degenerate a?ne HeckeCli?ord algebra of type Dn , denoted by Hc or Hc n , is the algebra generated W D 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: sn cn = ?cn?1 sn sn ci = ci sn (i = n ? 1, n) (i = n ? 1, n). (1 ≤ i ≤ n);

sn xn + xn?1 sn = ?u(1 + cn?1 cn ) s n xi = xi s n

(3.11)

Proposition 3.7. The algebra Hc n admits anti-involutions τ1 , τ2 de?ned by D τ2 : s i → s i , τ1 : s i → s i , cj → ?cj , cj → cj , xj → xj , xj → xj , (1 ≤ i ≤ n).

Also, the algebra Hc n admits an involution σ which ?xes all generators D si , xi , ci except the following 4 generators: σ : sn → sn?1 , sn?1 → sn , xn → ?xn , cn → ?cn .

Proof. We leave the easy veri?cations on τ1 , τ2 to the reader. It remains to check that σ preserves the de?ning 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+1 si ? si xi ) = = σ(xn sn?1 ? sn?1 xn?1 ) = = Also, σ preserves (3.11) since σ(sn xn + xn?1 sn ) = ?sn?1 xn + xn?1 sn?1 = ?u(1 ? cn?1 cn ) = σ(?u(1 + cn?1 cn )). xi+1 si ? si xi u(1 ? ci+1 ci ) = σ(u(1 ? ci+1 ci )); ?xn sn ? sn xn?1 u(1 + cn cn?1 ) = σ(u(1 ? cn cn?1 )).

Hence, σ is an automorphism of Hc n . Clearly σ 2 = 1. D

The natural action of Sn on C[h? ] = C[x1 , . . . , xn ] is extended to an action of WDn by letting xsn = ?xn?1 , n xsn = ?xn , n?1 xsn = xi i (i = n ? 1, n).

Proposition 3.8. Let W = WDn , 1 ≤ i ≤ n ? 1, and f ∈ C[x1 , . . . , xn ]. Then the following identities hold in Hc : W ci ci+1 f ? f si ci ci+1 f ? f si +u , (1) si f = f si si + u xi+1 ? xi xi+1 + xi sn sn c f ?f cn?1 cn f ? f n?1 cn (2) sn f = f sn sn ? u +u . xn + xn?1 xn ? xn?1 Proof. Formula (1) has been established by induction as in type An?1 . Formula (2) can be veri?ed by a similar induction.

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TA KHONGSAP AND WEIQIANG WANG

3.3. The algebra Hc of type Bn . Let W = WBn be the Weyl group of W type Bn , which is generated by s1 , . . . , sn , subject to the de?ning relation for Sn on s1 , . . . , sn?1 and the following additional relations: si sn = sn si (sn?1 sn )4 = 1, s2 = 1. n (1 ≤ i ≤ n ? 2) (3.12) (3.13)

We note that the simple re?ections s1 , . . . , sn belongs to two di?erent conjugacy classes in WBn , with s1 , . . . , sn?1 in one and sn in the other. De?nition 3.9. Let u, v ∈ C, and let W = WBn . The degenerate a?ne Hecke-Cli?ord algebra of type Bn , denoted by Hc or Hc n , is the algebra W B 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: sn ci = ci sn (i = n) √ sn xn + xn sn = ? 2 v s n xi = xi s n sn cn = ?cn sn

(i = n).

√ The factor 2 above is inserted for the convenience later in relation to the spin a?ne Hecke algebras. When it is necessary to indicate u, v, we will write Hc (u, v) for Hc . For any a ∈ C\{0}, we have an isomorphism of W W superalgebras ψ : Hc (au, av) → Hc (u, v) given by dilations xi → axi for W W 1 ≤ i ≤ n, while ?xing each si , ci . The action of Sn on C[h? ] = C[x1 , . . . , xn ] can be extended to an action of WBn by letting xsn = ?xn , n xsn = xi , i (i = n). Proposition 3.10. Let W = WBn . Given f ∈ C[x1 , . . . , xn ] and 1 ≤ i ≤ n ? 1, the following identities hold in Hc : W ci ci+1 f ? f si ci ci+1 f ? f si +u , (1) si f = f si si + u xi+1 ? xi xi+1 + xi √ f ? f sn (2) sn f = f sn sn ? 2v . 2xn Proof. The proof is similar to type A and D, and will be omitted. 3.4. PBW basis for Hc . Note that Hc contains C[h? ], Cn , CW as subalW W gebras. We have the following PBW basis theorem for Hc . W Theorem 3.11. Let W = WDn or W = WBn . The multiplication of subalgebras C[h? ], Cn , and CW induces a vector space isomorphism Equivalently, the elements {xα c? w|α ∈ Zn , ? ∈ Zn , w ∈ W } form a linear + 2 basis for Hc (called a PBW basis). W C[h? ] ? Cn ? CW ?→ Hc . W

THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS

13

Proof. For W = WDn , we can verify by a direct lengthy computation that the Hc n?1 -action on IND = C[x1 , . . . , xn ]?Cn (see the proof of Theorem 3.4) A naturally extends to an action of Hc n , where (compare Proposition 3.8) sn D acts by sn .(f c? ) = f sn csn ? ? u cn?1 cn f ? f sn cn?1 cn f ? f sn ?u xn + xn?1 xn ? xn?1 c? .

Similarly, for W = WBn , the Hc n?1 -action on IND extends to an action of A Hc n , where (compare Proposition 3.10) sn acts by B sn .(f c? ) = f sn csn ? ? √ 2v f ? f sn ? c. 2xn

It is easy to show that, for either W , the elements xα c? w span Hc . It W remains to show that they are linearly independent. We shall treat the WBn 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 modi?cation as follows. The w = ((η1 , . . . , ηn ), σ) ∈ WBn = ? {±1}n ? Sn may now not be unique, but the σ and the α are uniquely ? determined. Then, by the same argument on the vanishing of a maximal ? term, we obtain that w aα?w xα (xN1 xN2 · · · xNn )w = 0, and hence, n ? ? ? 1 2 aα?w (?1) ? ?
(η1 ,...,ηn ) Pn
i=1

ηi Ni

= 0.

By choosing N1 , . . . , Nn with di?erent parities (2n choices) and solving the 2n linear equations, we see that all aα?w = 0. This can also be seen more ? ? explicitly by induction on n. By choosing Nn to be even and odd, we deduce Pn?1 that for a ?xed ηn , (η1 ,...,ηn?1 )∈{±1}n?1 aα?w (?1) i=1 ηi Ni = 0, which is the ? ? equation for (n ? 1) xi ’s and the induction applies. 3.5. The even center for Hc . The even center of a superalgebra A, deW 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(Hc ) W of Hc is isomorphic to C[x2 , . . . , x2 ]W . n 1 W Proof. We ?rst show that every W -invariant polynomial f in x2 , . . . , x2 is n 1 central in Hc . Indeed, f commutes with each ci by (3.2) and clearly f W commutes with each xi . By Proposition 3.8 for type Dn or Proposition 3.10 for type Bn , si f = f si for each i. Since Hc is generated by ci , xi and si for W all i, f is central in Hc and C[x2 , . . . , x2 ]W ? Z(Hc ). n 1 W W On the other hand, take an even central element C = aα,?,w xα c? w in c . We claim that w = 1 whenever a HW α,?,w = 0. Otherwise, let 1 = w0 ∈ W be maximal with respect to the Bruhat ordering in W such that aα,?,w0 = 0. Then xw0 = xi for some i. By Proposition 3.8 for type Dn or Proposition 3.10 i for type Bn , x2 C ? Cx2 is equal to α,? aα,?,w0 xα (x2 ? (xw0 )2 )c? w0 plus a i i i i

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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 = aα,? xα c? . Since xi C = Cxi for each i, then (3.2) forces C to be in C[x1 , . . . , xn ]. Now by (3.2) and ci C = Cci for each i we have that C ∈ C[x2 , . . . , x2 ]. n 1 Since si C = Csi for each i, we then deduce from Proposition 3.8 for type Dn or Proposition 3.10 for type Bn that C ∈ C[x2 , . . . , x2 ]W . n 1 This completes the proof of the proposition. 3.6. The intertwiners in Hc . In this subsection, we will de?ne the interW twiners in the degenerate a?ne Hecke-Cli?ord algebras Hc . W The following intertwiners φi ∈ Hc (with u = 1) for W = WAn?1 were W introduced by Nazarov [Naz] (also cf. [Kle]), where 1 ≤ i ≤ n ? 1: A direct computation using (3.4) provides another equivalent formula for φi : We de?ne the intertwiners φi ∈ Hc for W = WDn (1 ≤ i ≤ n) by the W same formula (3.14) for 1 ≤ i ≤ n ? 1 and in addition by letting We de?ne the intertwiners φi ∈ Hc for W = WBn (1 ≤ i ≤ n) by the W same formula (3.14) for 1 ≤ i ≤ n ? 1 and in addition by letting √ (3.16) φn ≡ φB = 2x2 sn + 2vxn . n n 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) φ2 = 2u2 (x2 + x2 ) ? (x2 ? x2 )2 (1 ≤ i ≤ n ? 1, ?W ); i i+1 i i+1 i (2) φ2 = 2u2 (x2 + x2 ) ? (x2 ? x2 )2 , for type Dn ; n n n n?1 n?1 (3) φ2 = 4x4 ? 2v 2 x2 , for type Bn ; n n n (4) φi f = f si φi (?f ∈ C[x1 , . . . , xn ], ?i, ?W ); (5) φi cj = csi φi (1 ≤ j ≤ n, ?i, ?W ); j (6) φi φj φi · · · = φj φi φj · · ·.
mij mij

φi = (x2 ? x2 )si ? u(xi+1 + xi ) ? u(xi+1 ? xi )ci ci+1 . i+1 i

(3.14)

φi = si (x2 ? x2 ) + u(xi+1 + xi ) + u(xi+1 ? xi )ci ci+1 . i i+1

φn ≡ φD = (x2 ? x2 )sn + u(xn ? xn?1 ) ? u(xn + xn?1 )cn?1 cn . (3.15) n n n?1

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 de?ned in Proposition 3.7. Part (3) and (5) follow by a direct veri?cation. 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 de?ned 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 mij

si sj si · · · = sj si sj · · ·, (denoting this element by w). Let IND be the subalgebra of Hc generated W by C[x1 , . . . , xn ] and Cn . Denote by ≤ the Bruhat ordering on W . Then we can write φi φj φi · · · = f w + pu,w u
u<w

for some f ∈ C[x1 , . . . , xn ], and pu,w ∈ IND. We may rewrite φi φj φi · · · = f w +
′ ru,w φu u<w

where φu := φa φb · · · for any subword u = sa sb · · · of w = si sj si · · · , ′ and ru,w is in some suitable localization of IND with the central element 2 2 2 1≤k≤n xk 1≤i<j≤n (xi ? xj ) ∈ 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 φj φi φj · · · = f w +
′′ ru,w φu u<w

with the same coe?cient of w as for φi φj φi · · ·, according to Lemma 3.14. The di?erence ? := (φi φj φi · · · ? φj φi φj · · ·) is of the form ?=
u<w

ru,w φu

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

ru,w φu p =
u<w

ru,w pu φu .

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: φ0 φ0 φ0 · · · = φ0 φ0 φ0 · · · j i j i j i
mij mij

where φ0 denotes the specialization φi |u=0 of φ at u = 0 (or rather φB |v=0 n i when i = n in the type Bn case.)

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Proof. Let W = WBn . For 1 ≤ i ≤ n ? 1, mi,i+1 = 3. So we have φ0 φ0 φ0 = (x2 ? x2 )si (x2 ? x2 )si+1 (x2 ? x2 )si i i+1 i i+1 i i+2 i+1 i+1 i = (x2 ? x2 )(x2 ? x2 )(x2 ? x2 )si+1 si si+1 i+2 i+1 i+2 i i+1 i = (x2 ? x2 )(x2 ? x2 )(x2 ? x2 )si si+1 si i+1 i i+2 i i+2 i+1

= (x2 ? x2 )si+1 (x2 ? x2 )si (x2 ? x2 )si+1 i+2 i+1 i+1 i i+2 i+1 = φ0 φ0 φ0 . i+1 i i+1 Note that mij = 2 for j = i, i + 1; clearly, in this case, φ0 φ0 = φ0 φ0 . i j j i Noting that mn?1,n = 4, we have φ0 φ0 φ0 φ0 = 4(x2 ? x2 )sn?1 x2 sn (x2 ? x2 )sn?1 x2 sn n?1 n n?1 n n n?1 n n?1 n n = 4(x2 ? x2 )x2 (x2 ? x2 )x2 sn?1 sn sn?1 sn n n n?1 n?1 n?1 n = 4x2 sn (x2 ? x2 )sn?1 x2 sn (x2 ? x2 )sn?1 n n?1 n n n n?1 = φ0 φ0 φ0 φ0 . n n?1 n n?1

= 4x2 (x2 ? x2 )x2 (x2 ? x2 )sn sn?1 sn sn?1 n n?1 n?1 n?1 n n

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-de?ned element φw ∈ Hc given by φw = φi1 · · · φim where w = si1 · · · sim is any W reduced expression for w. These elements φw should play an important role for the representation theory of the algebras Hc . It will be very interestW ing to classify the simple modules of Hc and to ?nd a possible geometric W realization. This was carried out by Lusztig [Lu1, Lu2, Lu3] for the usual degenerate a?ne Hecke algebra case. 4. Degenerate spin affine Hecke algebras In this section we will introduce the degenerate spin a?ne Hecke algebra when W is the Weyl group of types Dn or Bn , and then establish the connections with the corresponding degenerate a?ne Hecke-Cli?ord algebras Hc . See [W1] for the type A case. W H? W 4.1. The skew-polynomial algebra. We shall denote by C[b1 , . . . , bn ] the C-algebra generated by b1 , . . . , bn subject to the relations bi bj + bj bi = 0 (i = 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 := bk1 · · · bkn for α = (k1 , . . . , kn ) ∈ Zn , and it contains a + n 1 polynomial subalgebra C[b2 , . . . , b2 ]. n 1

THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS

17

4.2. The algebra H? of type Dn . Recall that the spin Weyl group CW ? W associated to a Weyl group W is generated by t1 , . . . , tn subject to the relations as speci?ed in Example 2.1. De?nition 4.1. Let u ∈ C and let W = WDn . The degenerate spin a?ne Hecke algebra of type Dn , denoted by H? or H? n , is the algebra generated D W by C[b1 , . . . , bn ] and CW ? subject to the following relations: ti bi + bi+1 ti = u (1 ≤ i ≤ n ? 1)

ti bj = ?bj ti tn bn + bn?1 tn = u

(j = i, i + 1, 1 ≤ i ≤ n ? 1) (i = n ? 1, n).

tn bi = ?bi tn

The algebra H? is naturally a superalgebra by letting each ti and bi be W odd generators. It contains the type An?1 degenerate spin a?ne Hecke algebra H?n?1 (generated by b1 , . . . , bn , t1 , . . . , tn?1 ) as a subalgebra. A Proposition 4.2. The algebra H? n admits anti-involutions τ1 , τ2 de?ned by D τ1 : ti → ?ti , τ 2 : ti → ti , bi → bi bi → ?bi (1 ≤ i ≤ n). (1 ≤ i ≤ n);

Also, the algebra H? n admits an involution σ which swaps tn?1 and tn while D ?xing all the remaining generators ti , bi . Proof. Note that we use the same symbols τ1 , τ2 , σ to denote the (anti-) involutions for H? n and Hc n in Proposition 3.7, as those on H? n are the D D D restrictions from those on Hc n via the isomorphism in Theorem 4.4 below. D 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 . W De?nition 4.3. Let u, v ∈ C, and W = WBn . The degenerate spin a?ne Hecke algebra of type Bn , denoted by H? or H?n , is the algebra generated B W by C[b1 , . . . , bn ] and CW ? subject to the following relations: ti bi + bi+1 ti = u (1 ≤ i ≤ n ? 1)

ti bj = ?bj ti t n bn + bn t n = v

(j = i, i + 1, 1 ≤ i ≤ n ? 1) (i = n).

tn bi = ?bi tn

Sometimes, we will write H? (u, v) or H?n (u, v) for H? or H?n to indicate B W B W the dependence on the parameters u, v.

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4.4. A superalgebra isomorphism. Theorem 4.4. Let W = WDn or W = WBn . Then, (1) there exists an isomorphism of superalgebras Φ : Hc ?→Cn ? H? W W which extends the isomorphism √ : Cn ? CW ?→ Cn ? CW ? (in Φ Theorem 2.4) and sends xi ?→ ?2ci bi for each i; (2) the inverse Ψ : Cn ? H? ?→Hc extends Ψ : Cn ? CW ? ?→ Cn ? CW W W 1 (in Theorem 2.4) and sends bi ?→ √ ci xi for each i. ?2 Theorem 4.4 also holds for WAn?1 (see [W1]). Proof. We only need to show that Φ preserves the de?ning relations in Hc W which involve xi ’s. Let W = WDn . Here, we will verify two such relations below. The veri?cation of the remaining relations is simpler and will be skipped. For 1 ≤ i ≤ n ? 1, we have Φ(xi+1 si ? si xi ) = ci+1 bi+1 (ci ? ci+1 )ti ? (ci ? ci+1 )ti ci bi = u(1 ? ci+1 ci ), = (1 ? ci+1 ci )bi+1 ti + (1 ? ci+1 ci )ti bi

Φ(sn xn + xn?1 sn ) = (cn?1 + cn )tn cn bn + cn?1 bn?1 (cn?1 + cn )tn = ?u(1 + cn?1 cn ). = ?(1 + cn?1 cn )tn bn ? (1 + cn?1 cn )bn?1 tn

Now let W = WBn . For 1 ≤ i ≤ n ? 1, as in the proof in type Dn , we have Φ(xi+1 si ? si xi ) = u(1 ? ci+1 ci ). Moreover, we have √ √ ?2 ?2 Φ(sn xn + xn sn ) = √ cn tn cn bn + √ cn bn cn tn ?1 ?1 √ √ = 2cn tn cn bn + 2cn bn cn tn √ √ = ? 2(tn bn + bn cn ) = ? 2v, √ √ ?2 Φ(sn xj ) = √ cn tn cj bj = 2cn tn cj bj ?1 √ √ = 2cj cn tn bj = 2cj bj cn tn = Φ(xj sn ), for j = 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 W W algebra C[b1 , . . . , bn ] and the spin Weyl group algebra CW ? as subalgebras. We have the following PBW basis theorem for H? . W Theorem 4.5. Let W = WDn or W = WBn . The multiplication of the subalgebras CW ? and C[b1 , . . . , bn ] induces a vector space isomorphism C[b1 , . . . , bn ] ? CW ? ?→ H? . W Theorem 4.5 also holds for WAn?1 (see [W1]). Proof. It follows from the de?nition that H? is spanned by the elements of W the form bα σ where σ runs over a basis for CW ? and α ∈ Zn . By Theo+ rem 4.4, we have an isomorphism ψ : Cn ? H? ?→Hc . Observe that the imW W age ψ(bα σ) are linearly independent in Hc by the PBW basis Theorem 3.11 W for Hc . Hence the elements bα σ are linearly independent in H? . W W 4.6. The even center for H? . W Proposition 4.6. Let W = WDn or W = WBn . The even center of H? is W isomorphic to C[b2 , . . . , b2 ]W . n 1 Proof. By the isomorphism Φ : Hc → Cn ? H? (see Theorems 4.4) and the W W description of the center Z(Hc ) (see Proposition 3.12), we have W Thus, C[b2 , . . . , b2 ]W ? Z(H? ). n 1 W Now let C ∈ Z(H? ). Since C is even, C commutes with Cn and thus comW mutes with the algebra Cn ? H? . Then Ψ(C) ∈ Z(Hc ) = C[x2 , . . . , x2 ]W , n 1 W W and thus, C = ΦΨ(C) ∈ Φ(C[x2 , . . . , x2 ]W ) = C[b2 , . . . , b2 ]W . n n 1 1 In light of the isomorphism Theorem 4.4, the problem of classifying the simple modules of the spin a?ne Hecke algebra H? is equivalent to the W classi?cation problem for the a?ne Hecke-Cli?ord algebra Hc . It remains W to be seen whether it is more convenient to ?nd the geometric realization of H? instead of Hc . W W 4.7. The intertwiners in H? . The intertwiners Ii ∈ H? (1 ≤ i ≤ n ? 1) W W for W = WAn?1 were introduced in [W1] (with u = 1): The commutation relations in De?nition 4.1 gives us another equivalent expression for Ii : We de?ne the intertwiners Ii ∈ H? for W = WDn (1 ≤ i ≤ n) by the W same formula (4.1) for 1 ≤ i ≤ n ? 1 and in addition by letting In ≡ ID = (b2 ? b2 )tn ? u(bn ? bn?1 ). n n n?1 (4.2) Ii = ti (b2 ? b2 ) + u(bi+1 ? bi ). i i+1 Ii = (b2 ? b2 )ti ? u(bi+1 ? bi ). i+1 i (4.1) Z(Cn ? H? ) = Φ(Z(Hc )) = Φ(C[x2 , . . . , x2 ]W ) = C[b2 , . . . , b2 ]W . W 1 n 1 n W
?

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TA KHONGSAP AND WEIQIANG WANG

Also, we de?ne the intertwiners Ii ∈ H? for W = WBn (1 ≤ i ≤ n) by W the same formula (4.1) for 1 ≤ i ≤ n ? 1 and in addition by letting In ≡ IB = 2b2 tn ? vbn . n n (4.3) Proposition 4.7. The following identities hold in H? , for W = WAn?1 , W WBn , or WDn : (1) Ii bi = ?bi+1 Ii , Ii bi+1 = ?bi Ii , and Ii bj = ?bj Ii (j = i, i + 1), for 1 ≤ i ≤ n ? 1, 1 ≤ j ≤ n, and any W ; In addition, (2) In bn?1 = ?bn In , In bn = ?bn?1 In , and In bi = ?bi In (i = n ? 1, n), for type Dn ; (3) In bn = ?bn In , and In bi = ?bi In (i = n), for type Bn . Proof. (1) We ?rst prove the case when j = i: Ii bi = (b2 ? b2 )ti bi ? u(bi+1 ? bi )bi i+1 i = (b2 ? b2 )(?bi+1 ti + u) ? u(bi+1 bi ? b2 ) i+1 i i

= ?bi+1 (b2 ? b2 )ti ? u(bi+1 ? bi ) i+1 i = ?bi+1 Ii .

The proof for Ii bi+1 = ?bi Ii is similar and thus skipped. For j = i, i + 1, we have ti bj = ?bj ti , and hence Ii bj = ?bj Ii . (2) We prove only the ?rst identity. The proofs of the remaining two identities are similar and will be skipped. In bn?1 = (b2 ? b2 )tn bn?1 ? u(bn ? bn?1 )bn?1 n n?1 = ?bn (b2 ? b2 )tn ? u(bn ? bn?1 ) n n?1 = ?bn In . = (b2 ? b2 )(?bn tn + u) ? u(bn bn?1 ? b2 ) n n?1 n?1

The proof of (3) is analogous to (2), and is thus skipped. Recall the superalgebra isomorphism Φ : Hc ?→Cn ? H? de?ned in SecW W tion 4 and the elements βi ∈ Cn de?ned in Section 2. Theorem 4.8. Let W be either WAn?1 , WDn , or WBn . The isomorphism √ Φ : Hc ?→ Cn ? H? sends φi → ?2 ?1βi Ii for each i. More explicitly, Φ W W sends √ φi ?→ ? ?2(ci ? ci+1 ) ? Ii (1 ≤ i ≤ n ? 1); √ φn ?→ ? ?2(cn?1 + cn ) ? In for type Dn ; √ φn ?→ ?2 ?1cn ? In for type Bn .

THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS

21

√ √ Proof. Recall that the isomorphism Φ sends si → ? ?1βi ti , xi → ?2ci bi for each i. So, for 1 ≤ i ≤ n ? 1, we have the following Φ(φi ) = Φ (x2 ? x2 )si ? u(xi+1 + xi ) ? u(xi+1 ? xi )ci ci+1 i+1 i √ √ = ? ?2(ci ? ci+1 )(b2 ? b2 )ti ? u ?2(ci+1 bi+1 ? ci bi ) i+1 i √ ?u ?2(ci+1 bi+1 ? ci bi )ci ci+1 √ = ? ?2(ci ? ci+1 ) (b2 ? b2 )ti ? u(bi+1 ? bi ) i+1 i √ = ? ?2(ci ? ci+1 ) ? Ii .

Next for φn ∈ Hc n , we have D

We skip the computation for φn ∈ Hc n which is very similar but less B complicated. Proposition 4.9. The following identities hold in H? , for W = WAn?1 , W WBn , or WDn : (1) I2 = u2 (b2 + b2 ) ? (b2 ? b2 )2 , for 1 ≤ i ≤ n ? 1 and every type of i i+1 i i+1 i W. (2) I2 = u2 (b2 + b2 ) ? (b2 ? b2 )2 , for type Dn . n n n n?1 n?1 (3) I2 = 4b4 ? v 2 b2 , for type Bn . n n n 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 Ii Ij Ii · · · = (?1)mij +1 Ij Ii Ij · · · .
mij mij

Φ(φn ) = Φ (x2 ? x2 )sn + u(xn ? xn?1 ) ? u(xn + xn?1 )cn?1 cn n n?1 √ √ = ? ?2(cn + cn?1 )(b2 ? b2 )tn + u ?2(cn bn ? cn?1 bn?1 ) n n?1 √ ?u ?2(cn bn ? cn?1 bn?1 )cn?1 cn √ = ? ?2(cn + cn?1 ) (b2 ? b2 )tn ? u(bn ? bn?1 ) n n?1 √ = ? ?2(cn?1 + cn ) ? In .

Proof. By Theorem 2.2, we have βi βj βi · · · = (?1)mij +1 βj βi βj · · · .
mij mij

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?n?1 A can be found in [W1].

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5. Degenerate covering affine Hecke algebras In this section, the degenerate covering a?ne Hecke algebras associated to the double covers W of classical Weyl groups W are introduced. It has as its natural quotients the usual degenerate a?ne Hecke algebras HW [Dr, Lu1, Lu2] and the spin degenerate a?ne Hecke algebras H? introduced by W the authors. Recall the distinguished double cover W of a Weyl group W from Section 2.2. 5.1. The algebra H? of type An?1 . W De?nition 5.1. Let W = WAn?1 , and let u ∈ C. The degenerate covering a?ne Hecke algebra of type An?1 , denoted by H? or H?n?1 , is the algebra W A ?1 , . . . , tn?1 , subject to the relations for W ? generated by x1 , . . . , xn and z, t ? ? and the additional relations: z xi = xi z, ? ? z is central of order 2 (5.1) (5.2) (5.3) (5.4) xi xj = z xj xi (i = j) ?? ? ? ?? ti xj = z xj ti (j = i, i + 1) ? ? ?? ti xi+1 = z xi ti + u. ?? Clearly H? contains CW as a subalgebra. W 5.2. The algebra H? of type Dn . W De?nition 5.2. Let W = WDn , and let u ∈ C. The degenerate covering a?ne Hecke algebra of type Dn , denoted by H? or H? n , is the algebra W D ? ? generated by x1 , . . . , xn and z, t1 , . . . , tn , subject to the relations (5.1–5.4) ? ? and the following additional relations: ? ? tn xi = z xi tn (i = n ? 1, n) ?? 5.3. The algebra H? of type Bn . W ? ? ? tn xn = ??n?1 tn + u. x

De?nition 5.3. Let W = WBn , and let u, v ∈ C. The degenerate covering a?ne Hecke algebra of type Bn , denoted by H? or H?n , is the algebra W B ? ? generated by x1 , . . . , xn and z, t1 , . . . , tn , subject to the relations (5.1–5.4) ? ? and the following additional relations: ? ? tn xi = z xi tn (i = n) ?? 5.4. PBW basis for H? . W ? ? tn xn = ??n tn + v. x ?

Proposition 5.4. Let W = WAn?1 , WDn , or WBn . Then the quotient of the covering a?ne Hecke algebra H? by the ideal z ? 1 (respectively, by W the ideal z + 1 ) is isomorphic to the usual degenerate a?ne Hecke algebras HW (respectively, the spin degenerate a?ne Hecke algebras H? ). W

THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS

23

Proof. Follows by the de?nitions in terms of generators and relations of all the algebras involved. Theorem 5.5. Let W = WAn?1 , WDn , or WBn . Then the elements xα w, ? ? n and w ∈ W , form a basis for H? (called a PBW basis). ? where α ∈ Z+ W Proof. By the de?ning relations, it is easy to see that the elements xα w ? ? form a spanning set for H? . So it remains to show that they are linearly W independent. ? ? For each element t ∈ W , denote the two preimages in W of t by {t, z t}. Now suppose that 0= aα,t xα t + bα,t z xα t. ?? ? ? ? ?

Let I + and I ? be the ideals of H? generated by z ? 1 and z + 1 respectively. W Then by Proposition 5.4, H? /I + ? HW and H? /I ? ? H? . Consider the = W = W W projections: By abuse of notation, denote the image of xα in HW by xα . Observe that ? 0 = Υ+ (aα,t xα t + bα,t xα z t) = ?? ? ?? ? (aα,t + bα,t )xα t ∈ HW . ? ? Υ+ : H? ?→ H? /I + , W W Υ? : H? ?→ H? /I ? . W W

Since it is known [Lu1] that {xα t|α ∈ Zn and t ∈ W } form a basis for + the usual degenerate a?ne Hecke algebra HW , aα,t = ?bα,t for all α and t. ? ? ? ? Similarly, denoting the image in CW ? of t by t, we have 0 = Υ? (aα,t xα t + bα,t xα z t) = ?? ? ?? ? ? (aα,t ? bα,t )xα t ∈ H? . ? ? W

? Since {xα t} is a basis for the spin degenerate a?ne Hecke algebra H? , we W have aα,t = bα,t for all α and t. Hence, aα,t = bα,t = 0, and the linear ? ? ? ? independence is proved. References
[BK] [Dr] [EG] [IY] J. Brundan, A Kleshchev, Hecke-Cli?ord superalgebras, crystals of type A2l and ? modular branching rules for Sn , Represent. Theory 5 (2001), 317–403. V. Drinfeld, Degenerate a?ne Hecke algebras and Yangians, Funct. Anal. Appl. 20 (1986), 58–60. P. Etingof and V. Ginzburg, Symplectic re?ection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243–348. S. Ihara and T. Yokonuma, On the second cohomology groups (Schur multipliers) of ?nite re?ection groups, J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 11 (1965), 155–171. A. Jones and M. Nazarov, A?ne Sergeev algebra and q-analogues of the Young symmetrizers for projective representations of the symmetric group, Proc. London Math. Soc. 78 (1999), 481–512. T. J?ze?ak, A class of projective representations of hyperoctahedral groups and o Schur Q-functions, Topics in Algebra, Banach Center Publ., 26, Part 2, PWNPolish Scienti?c Publishers, Warsaw (1990), 317–326. G. Karpilovsky, The Schur multiplier, London Math. Soc. Monagraphs, New Series 2, Oxford University Press, 1987.
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A. Kleshchev, Linear and projective representations of symmetric groups, Cambridge Tracts in Mathematics 163, Cambridge University Press, 2005. [KW] T. Khongsap and W. Wang, Hecke-Cli?ord algebras and spin Hecke algebras II: the rational double a?ne type, Preprint 2007. [Lu1] G. Lusztig, A?ne 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 re?ection 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, Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. reine angew. Math. 139 (1911), 155– 250. [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. Algebra 145 (1992), 396–453. [W1] W. Wang, Double a?ne Hecke algebras for the spin symmetric group, preprint 2006, math.RT/0608074. [W2] ———, Spin Hecke algebras of ?nite and a?ne 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)

[Kle]


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