DIRAC COHOMOLOGY FOR DEGENERATE AFFINE HECKE-CLIFFORD ALGEBRAS

In this paper, we study the Dirac cohomology theory on a class of algebraic structures. The main examples of this algebraic structure are the degenerate affine Hecke-Clifford algebra of type An-1 by Nazarov and of classical types by Khongsap-Wang. The algebraic structure contains a remarkable subalgebra, which usually refers to Sergeev algebra for type An-1. We define an analogue of the Dirac operator for those algebraic structures. A main result is to relate the central characters of modules of those algebras with the central characters of modules of the Sergeev algebra via the Dirac cohomology. The action of the Dirac operator on certain modules is also computed. Results in this paper could be viewed as a projective version of the Dirac cohomology of the degenerate affine Hecke algebra.


Introduction
Throughout this paper, we work over the ground field C. Let W be a Weyl group. It is well known that W admits a non-trivial central extension where W is a distinguished double cover of W . The projective representations of W are linear representations of W which do not factor through W . Those representations over C have been has been known for a long time from the work of Schur,Morris,Read,Stembridge,and others [Mo1], [Mo2], [Re], [Sc], [St].
The degenerate affine Hecke-Clifford algebra for type A n−1 (see Definition 4.2) was introduced by Nazarov [Na] to study Young's symmetrizers of the projective representations of S n . The degenerate affine Hecke-Clifford algebra for other classical types was later constructed by Khongsap-Wang [WK]. Those algebras could be viewed as the projective counterpart of the degenerate affine Hecke algebra of Lusztig.
The purpose of this paper is to establish Dirac cohomology theory for those classes of algebras. We first single out the algebraic structure (see Section 3) that is necessary to prove several important results for the Dirac cohomology, and then we show that the degenerate affine Hecke-Clifford algebras considered in [Na] and [WK] satisfy that algebraic structure. Our approach is an analogue of the one recently developed for degenerate affine Hecke algebras by Barbasch-Ciubotaru-Trapa [BCT] (also see a recent extension by Ciubotaru [Ci2]).
In more detail, let H W be the associative algebra with certain important properties (see Definitions 3.1 and 3.3). The algebra H W contains a remarkable subalgebra, namely Seg(W ) (see again Definition 3.1), which is is the same as the Sergeev algebra when W is of type A n−1 .) The Dirac type element in H W is defined as an analogue of the one in [BCT] and has some nice properties. In specific examples of H W in Section 4, the Dirac type element can be viewed as the square root of a certain Casmir type element (Theorem 4.23).
For an H W -module (π, X), the Dirac cohomology is defined as H D (X) = ker π(D)/(ker π(D) ∩ im π(D)), which is a Seg(W )-module. One of our main results (Theorem 3.5) says that if X is irreducible and H D (X) is nonzero, then any irreducible Seg(W )-module in H D (X) determines the central character of X. This is an analogue to a statement for Harish-Chandra modules called Vogan's conjecture [HP]. A key step in the proof of Theorem 3.5 is to establish a canonical algebra homomorphism from the center of H W to the center of Seg(W ) (Theorem 3.4). In the case of the degenerate affine Hecke-Clifford algebra of type A n−1 , this homomorphism is shown to map onto the even elements of the center of Seg(W ) via the study of the Dirac cohomology on some modules (Corollary 7.21). The homomorphism indeed agrees with another natural map arising from the Jucys-Murphy type elements (see more detail in Remark 7.22), and hence the property of surjectivity has already been covered in the result of [Ru]. For a Dirac cohomology in other settings (see, for example, [HP]), one may apply the Dirac operator and Dirac cohomology developed in this paper to study the representation theory of H W . More precisely, the action of the Dirac operator provides information about the Seg(W )-module structure and central characters of some H W -modules (see Corollary 4.24 and Theorem 4.25).
We provide evidences that the Dirac cohomology can be useful in the representation theory by computing the action of the Dirac operators in several cases. In Section 5, we consider some basic modules for all classical types and show that the Dirac operator acts identically to zero on those modules. Those modules for type A n−1 were constructed and studied by Hill-Kujawa-Sussan [HKS]. In Section 7, we go further for type A n−1 and compute the action of the Dirac type element D on more interesting modules. We show that the Dirac cohomology of those examples does not vanish, and this indeed coincides with the expectation from the case of the degenerate affine Hecke algebra in [BCT]. While some computations can also be done for other classical types, the picture is more complete for type A n−1 to date. This paper is organized as follows. In Section 2, we review some properties of superalgebras. In Section 3, we define a certain algebraic structure H W and develop the Dirac cohomology theory for H W . We provide examples of H W in Section 4 and compute the square of the Dirac operator. In Section 5 and Section 7, we consider the Dirac cohomology for some particular modules. In Section 6, we review properties of Sergeev algebra which is needed for the computation of Section 7.
Acknowledgment. The author would like to thank Dan Ciubotaru and Peter Trapa for the suggestion of this topic and many useful discussions. He also thanks Professor Weiqiang Wang for his interest in the work and pointing out the reference [Wa]. The author would also like to thank the referees for useful suggestions and comments, and also thank one of the referees for pointing out the reference [Ru].

Notation for modules
In this paper, all the algebras are associative with a unit over C. Let A be an algebra. An A-module is denoted (π, X) or simply X, where X is a vector space and π is the map defining the action of A on X. For a ∈ A and x ∈ X, the action of a on x is written by π(a)x or a.x.
Let B be a subalgebra of A. Define Ind A B to be the induction functor, i.e., where Y is a B-module. The left adjoint functor of Ind A B is the restriction functor denoted Res A B .

Superalgebras and supermodules
A super vector space V is a Z 2 -graded vector space V = V 0 ⊕ V 1 . A super vector subspace W of V is a subspace of V such that W = (W ∩ V 0 ) ⊕ (W ∩ V 1 ). We say an element a in V 0 (resp. V 1 ) has even (resp. odd) degree, denoted deg(v) = 0 (resp. deg(v) = 1).
A superalgebra A is an algebra with a super vector space structure A = A 0 ⊕A 1 and A i A j ⊆ A i+j for i, j ∈ Z 2 . A subalgebra C of a superalgebra A is said to be a supersubalgebra of For superalgebras A and B, a superalgebra homomorphism from A to B is an For superalgebras A and B, the super tensor product of A and B, denoted A ⊗B, is a superalgebra isomorphic to A ⊗ B as vector spaces with the multiplication determined by: where a, a ∈ A and b, b ∈ B are homogeneous elements.
Let A be a superalgebra. An A-supermodule X is an A-module with a super vector space structure X = X 0 ⊕ X 1 and the property that For an A-supermodule X = X 0 ⊕ X 1 , define a map δ : Let Mod sup (A) be the category of A-supermodules. The morphisms in the category Mod sup (A) are the even homomorphisms between A-supermodules. Let Π : Mod sup (A) → Mod sup (A) be a parity change functor. That means for an A-supermodule, Π(M ) and M are isomorphic as A-modules, but have opposite Z 2 -grading.

Relations between irreducible supermodules and irreducible
modules Let A = A 0 ⊕ A 1 be a superalgebra. Given an irreducible A-module (π, Y ), we construct a supermodule as follows. Let (π, Y ) be an irreducible A-module such that Y is identified with Y as vector spaces and the A-action on Y is determined for any homogenous element a ∈ A and for v ∈ Y by Lemma 2.1. Let Y be an irreducible A-module. Let X Y = Y ⊕ Y be an Asupermodule with the supermodule structure described above. Then (1) X Y is an irreducible A-supermodule if and only if Y and Y are non-isomorphic as A-modules.
(2) If Y and Y are isomorphic as A-modules, then there is a supermodule structure on Y .
Proof. For (1), we first prove that if X Y is an irreducible A-supermodule, then Y and Y are not isomorphic as A-modules. Suppose instead there exists an Amodule isomorphism f : Y → Y , and we will derive a contradiction. Recall that Y is identified with Y as vector spaces and thus there exists a natural vector space isomorphism θ : Y → Y such that (−1) deg(a) π(a)θ = θπ(a) for any homogenous a ∈ A. Then θ • f satisfies the property that for any homogenous element a ∈ A, Then the map (θ • f ) 2 is an A-module automorphism of Y . Thus, by Schur's lemma and a suitable normalization, we may assume (θ • f ) 2 is an identity map. Then as vector spaces Then it is straightforward to verify that Ker 0 ⊕ Ker 1 ⊂ X Y gives a proper supersubmodule of X Y . We now prove that if Y and Y are not isomorphic as A-modules, X Y is an irreducible A-supermodule. Suppose instead that there exists a proper supersubmodule M of X Y and we will get a contradiction. Let One can check that f is an A-module isomorphism and so this gives a contradiction.
We now consider (2). By (1), X Y is not an irreducible A-supermodule. Let X be an irreducible supersubmodule of X Y . Then by the construction of X Y , X is isomorphic to Y = Y as A-modules. Then this gives a supermodule structure on Y .
We can also start with an irreducible A-supermodule and decompose it into irreducible A-module(s).
Lemma 2.2. Let X be an irreducible A-supermodule. Let δ be a linear automorphism on X such that δ(v) = (−1) i v for v ∈ X i (i = 0, 1). If X is not an irreducible A-module, then there exists an irreducible A-submodule Y of X such that (1) δ(Y ) is also an A-submodule of X and δ(Y ) = Y ; and (2) Y and δ(Y ) are non-isomorphic A-modules; and Proof.
Lemma 2.3. Let X and X be irreducible A-supermodules. If X and X are isomorphic as A-modules, then X and X are isomorphic, up to applying the functor Π, as A-supermodules.
Proof. Suppose X and X are also irreducible A-modules. Then X 0 , X 1 , X 0 , X 1 are irreducible A 0 -modules. Then either X 0 = X 0 or X 0 = X 1 as A 0 -modules. Then either X ∼ = X or X ∼ = Π(X ) as A-supermodules. Suppose X is not an irreducible A-module. Let X = Y ⊕ δ(Y ) and X = Y ⊕ δ(Y ) be the decomposition of X into A-modules as in Lemma 2.2. Without loss of generality, we may assume Y = Y as A-modules. Let f : Y → Y be an A-module isomorphism. Then f also induces an A-module isomorphism f : δ(Y ) → δ(Y ) such that f = δ •f •δ. Then one can show that the map f ⊕f is an A-supermodule isomorphism by checking that the map preserves grading. In particular, we also have Π(X) = X as A-supermodules in this case.
Let Irr(A) (resp. Irr sup (A)) be the set of irreducible A-modules (resp. irreducible A-supermodules). Let ∼ be the equivalence relation on Irr(A): Y ∼ Y if and only if Y = Y or Y = Y . Let ∼ Π be the equivalence relation on Irr sup (A): X ∼ Π X if and only if X = X or X = Π(X ).
Proposition 2.4. There is a natural bijection Proof. Lemmas 2.1 and 2.3 define a map from Irr(A)/∼ to Irr sup (A)/∼ Π . Lemma 2.2 defines a map in the opposite direction. The two maps are inverse to each other by Lemma 2.3.

Central characters of supermodules
For a superalgebra A, let Z(A) be the center of A. Note that Z(A) is a supersubalgebra of A. Recall that Z(A) 0 is the set of even elements in Z(A).
Proposition 2.5. Let X be an irreducible A-supermodule. For z ∈ Z(A) 0 , z acts on X by the multiplication of a scalar.
Proof. If X is an irreducible A-module, then the statement follows from (ordinary) Schur's lemma (for this case). If X is not an irreducible A-module, then we can decompose X = Y ⊕ δ(Y ) as A-modules as in Lemma 2.2. Then z acts on the two modules Y and δ(Y ) by scalars, denoted λ and λ respectively. Then for v ∈ Y , Note that δ(v + δ(v)) = v + δ(v) and so v + δ(v) ∈ X 0 , and similarly v − δ(v) ∈ X 1 . Then since z is of even degree, λ = λ .
By Proposition 2.5, we can define the following: Definition 2.6. Let A be a superalgebra. Let (π, X) be an irreducible A-supermodule. Define the central character χ π to be the map from Z(A) 0 to C such that χ π (z) is the scalar of z acting on X.
The central character defined above is only for even elements in the center of a superalgebra. However, the central character indeed determines the action of odd elements in the center in the following sense: Proposition 2.7. Let z ∈ Z(A) 1 . Let X be an irreducible A-supermodule. If X is also an irreducible A-module, then z acts by zero on X. If X is not an irreducible A-module, then z acts on the two irreducible A-submodules of X by two distinct scalars √ λ and − √ λ, where λ is the scalar that z 2 ∈ Z(A) 0 acts on X.
Proof. For (1), suppose X is an irreducible A-module. Then by Schur's Lemma, z acts on X by a scalar denoted by λ. Meanwhile by Lemmas 2.1 and 2.3, X = X as A-modules. This implies z also acts by −λ on X as z is an odd element. Hence λ = 0. Now suppose X is not an irreducible A-module. Then z 2 is an even element in the center and hence acts by a scalar, denoted λ. Then z acts on the irreducible A-submodules of X by scalars √ λ and − √ λ.

H W and a Dirac type element in H W
Fix a real reflection group W . Let V be a representation of W . Fix a W -invariant inner product on V . Let {a 1 , . . . , a n } be an orthogonal basis for V .
Definition 3.1. An associative algebra H W = H W (V ) is said to have property ( * ) if it satisfies the following properties. First H W is an algebra generated by symbols f w (w ∈ W ), c i (i = 1, . . . , n) and a i (i = 1, . . . , n) such that the map from C[W ] to H W sending w to f w is an injection and the algebra has a natural basis of elements having the form a k1 1 · · · a kn n c 1 1 · · · c n n f w (k 1 , . . . , k n non-negative integers, w ∈ W , i = 0 or 1). Again we shall write w for f w for simplicity. Let Seg(W ) be the subalgebra of H W generated by all w ∈ W and c i (i = 1, . . . , n). Furthermore, the generators of H W satisfy the following relations: Here w(a i ) is the action of w on V . Furthermore, we identify the linear space spanned by c i with V via the map a i → c i and hence there is a natural action of W on c i , and w(c i ) represents such action of w on c i . Indeed, the algebra generated by the those c i is isomorphic to the Clifford algebra on the vector space V , and the subalgebra Seg(W ) is the smash product of the Clifford algebra and the group algebra of W . H W has a superalgebra structure with deg(c i ) = 1, deg(a i ) = deg(w) = 0 (i = 1, . . . , n and w ∈ W ).
In the rest of this section, H W denotes an algebra satisfying the property ( * ). Define a Dirac type element D in H W : (3.6) The following two properties will be used several times: Lemma 3.2.
(1) wD = Dw for any w ∈ W ; (2) c i D = −Dc i for any i. Proof.
(1) follows from the fact that {a i } forms an orthogonal basis and property (3.1).
Definition 3.3. The algebra H W with the property ( * ) is said to satisfy the property ( * * ) if for any h ∈ H W such that h supercommutes with elements in Seg(W ), D 2 h − hD 2 = 0.
In the next section, we shall give examples which satisfy the algebraic structure in Definitions 3.1 and 3.3. From now on, assume that H W satisfies the properties ( * ) and ( * * ).

Relation between central characters for H W and Seg(W )
Let d : A relation between Z(H W ) 0 and Z(Seg(W )) 0 is the following: Then ζ is an algebra homomorphism.
Our main result in this paper is the following, which says the central character of an H W -supermodule X is determined by the central characters of irreducible Seg(W )-supermodules in the Dirac cohomology H D (X). Here H D (X) is defined in the theorem.
Since wD = Dw and c i D = −Dc i by Lemma 3.2, ker π(D) and ker π(D) ∩ im π(D) are invariant under the action of Seg(W ). Thus H D (X) has a natural Seg(W )-module structure from the H W -module structure. The proofs of Theorems 3.4 and 3.5 are given at the end of the next subsection. Theorem 3.5 directly follows from Theorem 3.4. Readers who only want to know how Theorem 3.4 implies Theorem 3.5 may jump to the end of the next subsection.

Proof of Theorems 3.4 and 3.5
The proofs of the theorems basically follow from the ideas of proofs in [HP,Chap. 3] and [BCT,Sect. 4]. We provide some technical details for this specific case.
Let S ≤j (V ) be the vector space of polynomials of x 1 , . . . , x n with degree less than or equal to j. Let H j W be the vector space spanned by elements of the form In the following lemmas, one can see that ker d , im d , ker d, (ker d ∩ im d) Seg(W ) and so on are supersubspaces by using the fact that D is an homogenous element. Lemma 3.6. As supersubspaces of B, Here C is regarded as the C-subalgebra of B generated by 1.
Proof. Note that any element in B can be uniquely written as a linear combination of elements of the form pb for any i, j, one can see that the action of d is determined by . In order to apply the known cohomology of the Koszul complex, we identify B with C[x 1 , . . . , x n ] ⊗ ∧ • C n as vector spaces, where ∧ • C n is the exterior algebra, via the linear isomorphism η from C[x 1 , . . . , x n ] ⊗ ∧ • C n to B determined by where {e 1 , . . . , e n } is the standard basis of C n . Then, by the above description of the action of d , the map η −1 • d • η is a multiple of the differential map in the standard Koszul resolution. Then the result follows from the well known cohomology of the Koszul resolution.
Proposition 3.7. As supersubspaces of H W , Proof. By the property ( * ) of H W , a m1 1 a m2 2 · · · a mn n c 1 1 · · · c n n w (m i ∈ Z ≥0 , i ∈ {0, 1} and w ∈ W ) form a basis for H W . Then b m1 1 b m2 2 · · · b mn n c 1 1 · · · c n n w (m i ∈ Z ≥0 , i ∈ {0, 1} and w ∈ W ) also form a basis for H W . Then as linear vector spaces, we may identify H W with B ⊗ Seg(W ) via the following map: under the above identification. Then by Lemma 3.6, one has For any subspace H of H W , define H Seg(W ) to be the set of all elements supercommuting with elements in Seg(W ). If we view Seg(W ) as a subalgebra of H W , we could similarly define H Seg(W ) for any subspace H of H W . Proposition 3.7 implies the following: Lemma 3.9. As supersubspaces of H W , Proof. It is clear that Z(Seg(W )) and (ker d ∩ im d) Seg(W ) are subspaces of the space (ker d) Seg(W ) and thus (ker d ∩ im d) Seg(W ) ⊕ Z(Seg(W )) ⊂ (ker d) Seg(W ) . We will prove another inclusion by induction on the degree of filtration of an element and so the statement is clearly true. Now assume By the uniqueness of the element h 0 , h 0 supercommutes with any element in Seg(W ). This implies h 0 is also a representative of h 0 . Furthermore, h 0 supercommutes with elements in Seg(W ) and d(h 0 ) ∈ (H i W ) Seg(W ) . By the property ( * * ), We also proved in the beginning that zz ∈ ker d and thus zz ∈ (ker d∩im d) Note that z is in Z(Seg(W )) 0 since the decomposition in Lemma 3.9 is between super vector spaces. Hence we have a map ζ : It remains to prove that ζ is an algebra homomorphism. To see that ζ is an algebra map, let . This completes the proof.
Proof of Theorem 3.5. By our hypothesis, there exists a non-zero element v ∈ H D (X) such that v is in the isotypic component U of H D (X). Let v be a representative of v in ker π(D). Now by Theorem 3.4 for any z Since we choose v = 0, we can only have χ π (z) = χ σ (ζ(z)) = χ σ (z). This completes the proof.

Examples of H W and their Dirac cohomology theory
Let W be a classical Weyl group and let R = R(W ) be the root system associated to W . Let k : R → C be a function such that k(α 1 ) = k(α 2 ) if α 1 = w(α 2 ) for some w ∈ W . We shall write k α for k(α). For any α ∈ R, let s α be the simple reflection associated to α.
Let e 1 , . . . , e n be the standard basis of R n . Let , be the inner product on R n such that e i , e j = δ ij .
4.1. Type A n−1 Notation 4.1. Set W = W (A n−1 ) to be the Weyl group of type A n−1 . The root system R(A n−1 ) of type A n−1 is the set Fix a set R + of positive roots We usually write α > 0 for α ∈ R + (A n−1 ) and write α < 0 for −α ∈ R + (A n−1 ). The set of simple roots ∆ is Since there is only one W -orbit for R(A n−1 ), we simply write k for k α for any α ∈ R(A n−1 ). For i = j, let Thus α ij is always a positive root. For a root α ∈ R(A n−1 ), let s α be the corresponding simple reflection in W (A n−1 ). For simplicity, set s ij = s αij .
Definition 4.2 ( [Na]).The degenerate affine Hecke-Clifford algebra for type A n−1 , denoted H Cl W (An−1) , is the associative algebra with a unit generated by the symbols given by w → f w is an algebra injection; ( for all i, j with |i − j| > 1. We later simply write w for f w . The algebra has a superalgebra structure with deg(c i ) = 1, deg(w) = 0 for w ∈ W (A n−1 ), and deg(x i ) = 0.
Let s α = s α c α . For later convenience, we also set s ij = s αij = s αij c αij , The notations y i and x i will be used to define the Dirac type element in H Cl W (An−1) and are inspired by the setting in the degenerate affine Hecke algebra in [BCT]. (1) c i y j = −y j c i for any i, j; The above lemma is elementary. We skip the proof. We shall use the natural permutation of W (A n−1 ) on the set {1, . . . , n} below.
Lemma 4.5. Let w ∈ W (A n−1 ). Then In particular, for α > 0, When l(w) = 1, w = s α for some α ∈ ∆. We consider three cases. When e i , α = 0, it is easy to see s α y i s α − y i = 0. Now consider the case e i , α = 1. In this case, we have For e i , α = −1, by using s α s α s α = − s α and the computation in the case e i , α = 1, we have s α y i+1 s α = y i + √ 2k s α .
We now use an induction on l(w). Assume l(w) = k for some k > 1. Write w = s α w for some simple reflection s α and w ∈ W (A n−1 ) with l(w ) = k − 1. Set = 1 if α, w(e i ) = 0 and = 0 otherwise. Then This proves the first assertion. The second assertion follows from the first one with the equation that Proof.
By Lemma 4.5, the term √ 2 2 k =i s i,k y j +y j k =i s i,k +y i l =j s l,j + l =j s l,j y i is in Seg(W (A n−1 )). This completes the proof. (1) Proof. For (1), it suffices to show when w = s α for some α ∈ ∆. Fix an i. By the definition of x i , it suffices to show s α y i s α = y sα(i) . We consider two cases. In the case that e i , α = 0, s α (α i,j ) > 0 for any j = i. Then s α s i,j s α = s i,sα(j) for any j = i. Thus, the last equality in Lemma 4.5 becomes In the case that e i , α = 0, let k = i − 1 or i + 1 such that α = α i,k . Then, by Lemmas 4.4(4) and 4.5, For (2), it is straightforward from Lemma 4.4 and y i = x i c i .
Remark 4.8. The subalgebra of H Cl W (An−1) generated by the elements y i and s i,j is the degenerate spin affine Hecke algebra of type A n−1 defined in [Wan,Sect. 3.3]. (Other classical types for the degenerate spin affine Hecke algebra are established in [WK,Sect. 4].) The degenerate spin affine Hecke algebra can be regarded as a more elementary analogue of the degenerate affine Hecke algebra, and the notions of y i can be regarded as the Drinfield presentation [Dr] under the analogue.

Type B n
For type B n , we modify the original definition in [WK]. More precisely, the algebra we considered in Definition 4.11 is a deformation of the algebra in [WK]. It is not hard to do a similar modification for type A n−1 . The main reason for this modification is to construct an explicit module in the next section, which cannot be done in the original definition of [WK] (by our approach). Considering the lack of existing literature for the representation theory of the degenerate affine Hecke-Clifford algebra for other classical types, such examples may be interesting and important.
Notation 4.10. Let W = W (B n ) be the Weyl group of type B n . Let the set R(B n ) of roots for type B n be The roots ±e i ±e j (i = j) are long, while the roots ±e i are short. Fix a set R + (B n ) of positive roots: The set ∆ of simple roots is For i = j > 0, define α ij as in (4.8), define α i,−j = e i + e j and define α i = e i . We also define s ij = s αi,j , s i,−j = s αi,−j and s i = s αi .
We have a natural embedding R(A n−1 ) ⊂ R(B n ). and a natural embedding W (A n−1 ) ⊂ W (B n ) (i.e., the group W (A n−1 ) being the group generated by s i,i+1 for i = 1, . . . , n − 1). given by w → f w is an algebra injection; (2) f sn c n = −c n f sn and f sn c i = c i f sn for i = n; (3) f sn x j − x j f sn = 0 for j = n ; (4) x i x j − x j x i = N Bn c j c i for i = j. We shall again simply write w for f w .
When N Bn = 0, H Cl W (Bn) (k, N Bn ) coincides with the degenerate affine Hecke-Clifford algebra of type B n in [WK,Def. 3.9].
For N Bn = 0, while x i and x j does not commute for i = j, we still have The algebra H Cl W (Bn) hence still has some nice properties such as the commutation relations with intertwining operators (but we do not need this in this paper).
For i = j > 0, define c αij as in (4.9) and define Set s i,−j = s i,−j c αi,−j . We also set s α = s i = s i c i . Since we have modified the original definition of the degenerate affine Hecke-Clifford algebra for type B n in [WK], we will give a proof for the existence of the PBW type basis.
Proof. We follow the argument in [Kl,Thm. 3.2.2]. We consider the algebra H generated by {x i }, {c i } and {s i,i+1 } n−1 i=1 ∪ {s n } subject to the relations (3), (4), (5), (6) in Definition 4.2 and the relation (2) (but not (1)) in Definition 4.18 (with a trivial replacement of notations). We resolve the minimal ambiguities according to the Bergman's diamond lemma [Be]. For example, we may consider an ordering s < c n < · · · < c 1 < x n < · · · < x 1 , where s is any simple reflection in W (B n ). This induces a semigroup ordering on x i , c i , s (i = 1, . . . , n and s runs for all simple reflections) from the length of words and the lexicographical ordering. Then one checks that Similarly, Similarly, for i > j > k, The calculation for x i (x j x k ) is similar. Other minimal ambiguities can be checked similarly.
Let I be the two-sided ideal of H generated by the relations of W (B n ) (e.g., Bn) . Let P be the subalgebra of H generated by by x i and c i . It is straightforward to check that (s 2 − 1)P = P(s , and other similar equations. Those equations can also be deduced from Lemma 4.14 and its proof below.
Lemma 4.14. wy In particular, for α > 0 Proof. The relation s n x n + x n s n = − √ 2k α implies s n y n − y n s n = − √ 2k α c n . The latter equation is also equivalent to s n y n s −1 n = y n + √ 2k α s n . The remaining proof is just similar to the case of A n−1 in the proof of Lemma 4.5.
For i > 0, define y i = x i c i . (4.10) We also define y −i = y i and y −i = y i . There is a natural permutation of W (B n ) on the set {±1, . . . , ±n}.
(2) For i = j, y i y j + y j y i ∈ Seg(W (B n )).
Proof. For (1), it suffices to check when w = s α is a simple reflection. It is the direct consequence of the expression (4.10) for y i , Lemma 4.14, and the fact that s α s α s α = − s α . For (2), using the expression (4.10), we have y i y j + y j y i = y i y j + y j y i + √ 2 2 α>0, α,ej =0 k α (y i s α + s α y i ) + α>0, α,ei =0 k α (y j s α + s α y j ) Since y i y j + y j y i = N Bn , we only need to consider and show that the middle term is in Seg(W (B n )): which is in Seg(W (B n )) by Lemma 4.14.
Proposition 4.16. The superalgebra H Cl W (Bn) satisfies the property ( * ). Proof. Let x i = −y i c i . We set W in Definition 3.1 to be W (B n ) and a i to be x i . With Lemma 4.15, one can verify relations (3.1) to (3.5) in Definition 3.1 (also see more detail for type A n−1 in Section 4.1). By Proposition 4.12, H Cl W (Bn) has a PBW type basis. These show the proposition.

Type D n
Notation 4.17. Let W (D n ) be the Weyl group of type D n . Let the set R(D n ) of roots for type D n be R(D n ) = {±e i ± e j : 1 ≤ i < j ≤ n} ⊂ R(B n ).
Since there is only one W -orbit for R(D n ), we simply write k for k α for any α ∈ R(D n ). We shall regard W (D n ) as the subgroup of W (B n ) generated by elements s i,j and s i,−j for i, j > 0. We shall also keep using the notations in Notation 4.10.
Remark 4.19. We can explicitly write down the commutation formula from the algebra structure of H Cl W (Bn) . For example, s n−1,−n x n−1 + x n s n−1,−n = s n s n−1,n s n x n−1 + x n s n s n−1,n s n = −s n s n−1,n x n s n + s n x n−1 s n−1,n s n = s n (−s n−1,n x n + x n−1 s n−1,n )s n = s n (k(−1 + c n c n−1 ))s n = k(−1 + c n−1 c n ).
This agrees with a relation in [WK,Def. 3.6]. When N Dn = 0, H Cl W (Dn) (k, 0) is isomorphic to the degenerate affine Hecke-Clifford algebra of type D n defined in [WK,Def. 3.6]. (We remark that in [WK], their convention for c i satisfies c 2 i = 1 rather than c 2 i = −1.) We again define Again, for notational convenience, set y −i = y i .
(1) c i y j = −y j c i for any i, j; (2) s α y i s −1 α = y sα(i) ; (3) for i = j, y i y j + y j y i ∈ Seg(W (D n )).
Proof. Note that y i is defined as the one in (4.10) for type B n in Section 4.2 since we have k B α = 0 for any short root α ∈ R(B n ). Then the results can be established by Lemma 4.14 and by investigating the proof of Lemma 4.15. (4.12) Using the expressions in Section 4.1, the explicit form of the Dirac element D is as: (1) Type A n−1 and D n : (2) Type B n : In types A n−1 and D n , we consider that all the roots are long.
It is not hard to verify that ι is well-defined and is an involution. For ι(α, β) = (α , β ), one can also check that s α s β + s α s β = 0. Thus each term s α s β in the expression α>0,β>0,sα(β)>0 s α s β can be paired with another one and get canceled. This proves the expression is zero.
By Proposition 4.9, Proposition 4.16, and Proposition 4.21, H satisfies the property ( * ) and hence we can define Seg(W ) to be a subalgebra of H according to Definition 3.1.
We compute the square of the Dirac element D. This is an analogue of [BCT,Thm. 2.11]. where Moreover, H satisfies the property ( * * ).
Proof. We only do for types A n−1 and B n , and the case for type D n follows from type B n . By Lemma 4.5 and Lemma 4.14, for any α > 0, Now, by (4.13) and Lemma 4.22, We can directly verify that Ω H is in the center of H and Ω Seg(W ) is in the center of Seg(W ). Hence, H has the property ( * * ).
We obtain the following Parthasarathy-Dirac-type inequality. Examples satisfying the hypothesis of Corollary 4.24 below will be considered in Section 7 (see Proposition 7.12).
Corollary 4.24. Suppose an irreducible H-module (π, X) satisfies the property that X admits a non-degenerate positive-definite Hermitian form such that the adjoint operator of π(D) is −π(D). For any irreducible Seg(W )-module (σ, U ), Proof. Let U X be an U -isotypical component of X and let u ∈ U . The corollary follows from The conclusion of this section is a version of Theorem 3.5 in specific cases. Dn) . Let (π, X) be an irreducible supermodule of H with the central character χ π (Definition 2.6). Let D be the Dirac element in H in (4.12). Define the Dirac cohomology H D (X) as in Theorem 3.5. Then H D (X) has a natural Seg(W )-module structure. Suppose Hom Seg(W ) (U, H D (X)) = 0, for some Seg(W )-module (σ, U ). Then χ π = χ σ , where χ σ is defined as in (3.7).

Construction of some modules
In this section, we construct some modules for the degenerate affine Hecke-Clifford algebra of classical types.
In type A n−1 , we follow the construction in [HKS,Sect. 4.1], which uses a Jucys-Murphy-type element. For type B n , we use a slightly different approach. The underlying idea of the construction is to first consider a Seg(W )-module and then try to extend the action to the entire degenerate affine Hecke-Clifford algebra. However, we may not expect that this process always works, and indeed, we can only do it for certain parameters.
Type A n−1 : Let Cl n be the subalgebra of H Cl W (An−1) generated by all c i . Define St W (An−1) to be an H Cl W (An−1) -supermodule, which is identified with Cl n as vector spaces and the action of H n on St W (An−1) is determined by the following: (5.14) where 1 is the identity in Cl n and where v is any vector in Cl n and the actions of s i,j and c i , c j are the ones defined in (5.14) and (5.15). The notation St W (An−1) stands for a Steinberg-type module as it performs the role of Steinberg module in the degenerate affine Hecke algebra. It is straightforward to check that the above actions define an H Cl W (An−1) -module by verifying the defining relations of H Cl W (An−1) . Some details can be found in [HKS,Prop. 4.1.1].
Type B n : Let α be a long root in R(B n ) and let β be a short root in R(B n ). Set N Bn = 2(n−1)k 2 α + √ 2k α k β . Let Cl n be the subalgebra of H Cl W (Bn) generated by the elements c i , which is isomorphic to the Clifford algebra. Let U (n) be an irreducible supermodule of Cl n . The actions of H Cl W (Bn) on U (n) ⊗U (n) are determined by the following: The above actions are indeed well defined: Proposition 5.1. For N Bn = 2(n − 1)k 2 α + √ 2k α k β , the actions (5.16)-(5.18) above on U (n) ⊗U (n) define an H Cl W (Bn) (k, N Bn )-module.
Proof. The computation is straightforward for verifying the defining relations of H Cl W (Bn) . For example, Moreover, for i < j, note that and hence x i x j −x j x i = (2(n−1)k 2 α + √ 2k α k β )c j c i . Other relations can be verified similarly (and more easily).
Denote the above H Cl W (Bn) -module by St Bn .

Dirac cohomology
We keep using the notation in Section 5.
Proposition 5.2. Set N Bn = 2(n − 1)k 2 α + √ 2k α k β (with the notations in Section 5.1) and set N Dn = 2(n − 1)k 2 . Let H = H Cl An−1 , H Cl Bn (k, N Bn ) or H Cl Dn (k, N Dn ). Let X = St An−1 , St Bn or St Dn be an H-module defined in Section 5.1. The Dirac operator D acts identically as zero on X. In particular, H D (X) = 0.

Proof. Type
Type B n : Recall that St Bn is isomorphic to U ⊗U as vector spaces in the notation of Section 5.1. For u ⊗ v ∈ U ⊗U , Type D n : Recall that H Cl W (Dn) is a subalgebra of H Cl W (Bn) (k B , N Dn ) (see the notation of k B in Definition 4.18). The Dirac operator for H Cl W (Dn) is the same as the Dirac operator for H Cl W (Bn) (k B , N Dn ). Then the vanishing result follows from the result for type B n , which has just been proven.

Sergeev algebra
The main purpose of this section is to review several results about Sergeev algebra, which will be useful for computing the Dirac cohomology of some modules for H Cl W (An−1) in the next section. Some results can also be formulated to other types and one may refer to [WK,Sect. 2]. Starting from this section, we consider type A n−1 only and we shall usually use the notation S n for W (A n−1 ) (where S n represents the symmetric group). Write R for R(A n−1 ) and R + for R + (A n−1 ). Recall that ∆ is the set of simple roots in R.

The superalgebra C[ S n ] −
Let S n be the group generated by the elements ψ, t 1,2 , . . . , t n−1,n subject to the following relations: Then S n is a double cover of S n via the map determined by sending t αi to the transposition between i and i + 1, and ψ → 1. We also sometimes write t αi,i+1 for t i,i+1 if we want to refer to the simple root α i,i+1 . Denote by C[ S n ] the group algebra of S n with a basis labeled as e w : w ∈ S n . Define C[ S n ] − := C[ S n ]/ e ψ + 1 . We shall simply write w for the image of e w in C[ S n ] − . There is a superalgebra structure on C[ S n ] − with deg( t α ) = 1 for all α ∈ ∆.
Lemma 6.1. Given an S n -representation U and a C[ S n ] − -module U , there exists a natural C[ S n ] − -module structure on U ⊗ U characterized by where α ∈ ∆, u ∈ U , and u ∈ U .
Define an equivalence relation on Irr(C[ S n ] − ): U ∼ sgn U if and only if U = U or U = sgn ⊗U as C[ S n ] − -modules, where sgn is the sign representation of S n and the C[ S n ] − -module structure of sgn ⊗U is defined in Lemma 6.1.
Proposition 6.2. There is a natural bijection Proof. It suffices to see that the equivalence relation ∼ in Proposition 2.4 is the same as ∼ sgn . This follows from deg( t α ) = 1 for all α ∈ ∆ and definitions.
6.2. Sergeev algebra Definition 6.3. Recall that H Cl W (An−1) is defined in Definition 4.2. The Sergeev algebra, denoted Seg n , is the subalgebra of H Cl W (An−1) generated by the elements w ∈ W (A n−1 ) = S n and c i (i = 1, . . . , n). In other words, since H Cl W (An−1) satisfies the property ( * ), Seg n is the same as Seg(W An−1 ) in Definition 3.1. We shall use notations in Section 4.1 (e.g., s α , c α , s α ).
Let Cl n be the supersubalgebra of Seg n generated by c i (i = 1, . . . , n). There exists a unique, up to applying the functor Π, irreducible supermodule of Cl n . Let U (n) be a fixed choice of an irreducible supermodule of Cl n . The dimension of U (n) is 2 n/2 for n even and 2 (n+1)/2 for n odd.
The relation between subalgebras Seg n and C[ S n ] − is the following.
Proof. Define a map: One can verify that the map is an isomorphism.
For any α ∈ R + , define t α ∈ C[S n ] − such that s α maps to t α ⊗ c α under the map in the proof of Lemma 6.4.
Here is an analogue of Lemma 6.1: Lemma 6.5. Given an S n -representation U and a Seg n -module U , there exists a natural Seg n -module structure on U ⊗ U characterized by where α ∈ ∆, i = 1, . . . , n, u ∈ U , and u ∈ U .
6.3. Relation between supermodules of C[ S n ] − and Seg n Recall from [BK] (our formulation here is a bit different) a natural functor F : The Seg n -supermodule structure of X ⊗ U (n) is characterized by It is straightforward to check that the above equations define a Seg n -module. Next, define The C[ S n ] − -module structure is given by for θ ∈ Hom Cl n (U (n), Y ), ( t α .θ)(u) = (s α c α ).θ(u) (α ∈ ∆).
Proposition 6.6 ( [BK,Thm. 3.4]). The functors F and G form an adjoint pair, i.e., there is a natural isomorphism Furthermore, if n is even, G • F = Id and F • G = Id. If n is odd, G • F = Id ⊕ Π and F • G = Id ⊕ Π, where Π is defined in Section 2.2.
Let U Cl n be a Seg n -module defined by Sn] triv, where C[S n ] is regarded as the subalgebra of Seg n generated by the elements f sα for all α ∈ ∆ and triv is the trivial representation of C[S n ]. In particular, dim C U Cl n = 2 n .
We define a corresponding C[ S n ] − -module U spin as follows. If n is even, define U spin = G(U Cln ). If n is odd, by [Kl,Prop. 13.2.2] and [Kl,Thm. 22.2 An immediate consequence of Proposition 6.6 is given below.

Spectrum of the Dirac operator for type A n−1
We have seen the action of the Dirac operator on certain modules. In this section, we will go further for type A n−1 and compute the action of D on some interesting H Cl W (An−1) -modules. We shall see that Theorem 4.25 for H Cl W (An−1) has interesting consequences. We shall write H Cl n for H Cl W (An−1) for simplicity. We keep using the notations in Section 4.1 and Section 6. 7.1. Further notation for the root system of type A n−1 A partition of n is a sequence of positive integers (n 1 , . . . , n r ) such that n 1 ≥ n 2 ≥ . . . ≥ n r and n 1 + · · · + n r = n. For a partition λ = (n 1 , . . . , n r ) of n, let I λ = {1, . . . , n} \ {n 1 , n 1 + n 2 , . . . , n 1 + · · · + n r } and let Let V λ be the real span of ∆ λ in R n and let R + λ = V λ ∩ R + . 7.2. Central characters for H Cl n The center of H Cl n plays a role in the following computations. Proposition 7.1 ( [Kl,Thm. 14.3.1]). The center Z(H Cl n ) of H Cl n is the set of all symmetric polynomials in C[x 2 1 , x 2 2 , . . . , x 2 n ]. In particular, any element in Z(H Cl n ) is of even degree.
Definition 7.2. Recall that the central character χ π : Z(H Cl n ) 0 → C of an irreducible supermodule (π, X) is defined in Definition 2.6. By Proposition 7.1, we can also write χ π : Z(H Cl n ) → C. For an element γ = (a 1 , . . . , a n ) ∈ C n , define χ γ : C[x 2 1 , . . . , x 2 n ] → C such that χ γ (x 2 i ) = a i . Define χ γ to be the restriction of χ γ to Z(H Cl n ). For the central character χ π of X, there exists a unique γ ∈ C n , up to permutations of coordinates, such that χ π = χ γ . We may also say that γ is the central character of X.
An H Cl n -module (π, X) is said to be quasisimple if any element in Z(H Cl n ) acts by a scalar. In this case, γ defined as above is still called the central character of X.

Induced modules
Let us recall a construction of some H Cl n -modules in [HKS,Sect. 4], which is indeed modified from the module of type A n−1 in Section 5.1. There are also some similar constructions of H Cl n -modules in [Wa,Sect. 4]. Fix a partition λ = (n 1 , n 2 , . . . , n r ) of n. Let S λ be the subgroup of S n generated by s i,i+1 for i = {1, . . . , n} \ {n 1 , n 1 + n 2 , . . . , n 1 + · · · + n r }. It is easy to see that S λ is isomorphic to S n1 × . . . × S nr . Let H Cl λ be the supersubalgebra of H Cl n generated by all w ∈ S λ , x i (i = 1, . . . , n) and c i (i = 1, . . . , n). Let Seg λ be the supersubalgebra of H Cl λ generated by all w ∈ S λ and c i (i = 1, . . . , n). Let St λ be an H Cl λ module which is identified with Cl n as vector spaces and the action of H Cl λ is characterized by: where v is any vector in Cl n and the actions of s i,j and c i , c j are the ones defined in (5.14) and (5.15). It is straightforward to check that the above actions define an H Cl λ -module by verifying the defining relations of H Cl λ . Some details can be found in [HKS,Prop. 4 Lemma 7.3. The element x 2 i acts on St λ by a scalar (i − n k − 1)(i − n k ) where k = 0, . . . , r − 1 and i = n k + 1, . . . , n k+1 .
Define the Dirac-type element D λ in H Cl λ as: Proposition 7.4. The element D λ acts as zero on the H Cl λ -module St λ . Proof. It follows a similar computation of type A n−1 in the proof of Proposition 5.2.
To compute the Dirac cohomology of the above induced modules, we need some more information discussed in the next subsections. 7.4. S n -structure and Seg n -structure of (π λ , X λ ) We continue to fix a partition λ of n. Recall that in Definition 4.2(1), H Cl n contains C[S n ] as a subalgebra. Let (π V , V = C n ) be the S n -representation such that elements in S n permute the coordinates.
Lemma 7.5. The restriction of X λ to C[S n ] is isomorphic to Proof. Note that the restriction of St λ to C[S λ ] is isomorphic to Res It is well known that we have the following C[S n ]-isomorphism: Here the module in the right-hand side is viewed as the tensor product of two S n -representations. The isomorphism is given by Note that the space ⊕ n i=0 ∧ i V can be identified with Cl n via the map determined by e i1 ∧ · · · ∧ e ir → c i1 · · · c ir , where {e 1 , . . . , e n } is the standard basis of V = C n . Thus X λ = Ind St λ can be identified with, as vector spaces, C[S n ] ⊗ C[S λ ] triv ⊗ U Cl n via the identification in Lemma 7.5 and the above identification between ⊕ n i=1 ∧ i V and Cl n . Then if we translate the action of the subalgebra Seg n under the above identifications, then we have: We have just proven that: Lemma 7.6. As Seg n -supermodules, where the supermodule in the right hand side has the Seg n -supermodule structure described in Lemma 6.5.
Recall that F is the functor defined in Section 6.3. Proposition 7.7. As Seg n -supermodules, where (C[S n ] ⊗ C[S λ ] triv) ⊗ U spin has C[ S n ] − -supermodule described in Lemma 6.1.
Proof. By Lemma 7.6, it suffices to show By Lemma 6.7, there is a Seg n -module isomorphism f from U Cln to F (U spin ) = U spin ⊗ U (n). Then define a vector space isomorphism of Seg n -modules Using the module structure described before Lemma 7.6, one can check the linear isomorphism is Seg n -equivariant. 7.5. Hermitian form on (π λ , X λ ) We continue to fix a partition λ of n. In this subsection, we shall construct a Hermitian form on the H Cl n -module (π λ , X λ ) such that the adjoint operator of π λ (D) with respect to such form is −π λ (D). We will see this makes the computation for the Dirac cohomology H D (X) of those modules X much easier.
Recall that Seg λ is a subalgebra of H Cl λ . Lemma 7.8. There exists a Seg λ -invariant positive definite Hermitian form on St λ . Seg λ U Cl n as Seg λ -modules, it suffices to consider the case when λ = (n). Recall that U Cl n = Seg n ⊗ C[Sn] triv in Section 6.3. Define · , · : U Cl n × U Cl n → C such that for 1 ≤ i 1 < · · · < i r ≤ n and 1 ≤ j 1 < · · · < j s ≤ n,

Proof. Since Res
It is straightforward to check that , satisfies the desired properties.
Lemma 7.9. · , · defined above is a positive definite Hermitian form.
Proof. This follows from the property that · , · λ is positive definite and Hermitian.
We next compute the adjoint operator of π λ (D) with respect to · , · . We begin with some lemmas.
This completes the proof.
For each λ ∈ P n , define a S n -representation: where sgn and triv are respectively the sign and trivial representations of S λ , and λ t is the conjugate of λ. It is well known that W λ exhausts the list of irreducible representations of S n . Define Let P dist n be the set of partitions of n with distinct parts. Recall that we denote by Irr sup C[ S n ] − (resp. Irr sup Seg n ) the set of irreducible supermodules of C[ S n ] − (resp. Seg n ). Recall that the equivalence relation ∼ Π on Irr sup C[ S n ] − or Irr sup Seg n is defined in Section 2.3.
Proof. Note that for an irreducible C[ S n ] − -supermodule U , F (U ) is either an irreducible supermodule or the direct sum of two irreducible supermodules of opposite grading. Thus we could define Ψ 2 (λ) to be the unique equivalence class in Irr sup Seg n /∼ Π containing the irreducible supermodule(s) in F (U ) for a representative U ∈ Φ 1 (λ), where Φ 1 is defined in Proposition 7.15. It remains to check those two properties. Recall that F (U ) = U ⊗U (n) and that the action of Seg n on F (U ) is defined in Section 6.3. Then for u ⊗ u ∈ U ⊗ U (n), Thus for any irreducible supermodule (σ , U ) in F (U ), χ σ (Ω Seg n ) = χ σ (Ω C[ Sn] − ). Then combining this with Lemma 7.14 and Proposition 7.15, we have shown the first property.
Lemma 7.17. For a partition λ of n with distinct parts, there exists a representative U ∈ Φ 2 (λ) such that Hom Seg n (U, Res ⊇ Hom Seg n (U, F (W λ ⊗ U spin )) (by definition of W λ ).
The statement now follows from Proposition 7.16.