Representations of Degenerate Affine Hecke Algebra of Type C_n Under the Etingof-Freund-Ma Functor
aa r X i v : . [ m a t h . R T ] F e b REPRESENTATIONS OF DEGENERATE AFFINE HECKEALGEBRA OF TYPE C n UNDER THE ETINGOF-FREUND-MAFUNCTOR
YUE ZHAO
Abstract.
We compute the image of a polynomial GL N -module under the Etingof-Freund-Ma functor [EFM09]. We give a combinatorial description of the image interms of standard tableaux on a collection of skew shapes and analyze weights of theimage in terms of contents. Contents
1. Introduction 22. Definitions and notations 32.1. Root system of type C n C n C n Y -semisimple degenerate affine Hecke algebra representations 43. Etingof-Freund-Ma Functor 44. GL -module 55. Invariant space 65.1. Definition of the invariant space 75.2. Computation of the ( t , µ ) invariant space 75.3. A basis of invariant space and standard tableaux 95.4. One skew shape 105.5. Skew shapes and standard tableaux 126. Y - semisimplicity 136.1. Action of Y Y -semisimple representations 188. Combinatorial moves 208.1. Moves among standard tableaux 208.2. Correspondence between algebraic actions and combinatorial moves 219. Irreducible representations 229.1. The image F n,p,µ ( V ξ ) is irreducible 229.2. Irreducible representation associated to a skew shape ϕ ξn,p,µ F n,p,µ ( V ξ ) 2310.2. Combinatorial description of irreducible representations in M Y -semisimple representations 36 Date : February 15, 2021.
References 371.
Introduction
Schur-Weyl duality connects polynomial representations of GL N and representa-tions of the symmetric group S n . Let V = C N denote the vector representation of GL N . Then V ⊗ n has a GL N -action. Let S n be the symmetric group on n indices.The tensor V ⊗ n has a natural right S n -action. By the Schur-Weyl duality, we havethe decomposition V ⊗ n = M | λ | = n V λ ⊠ S λ , where n ≥ N , λ is a partition of n with at most N rows, S λ runs through all irre-ducible representations S n and V λ is the irreducible GL N -module with highest weight λ . Moreover, the actions of Jucys-Murphy elements are diagonalizable.In [AS98],Arakawa and Suzuki constructed a functor from the category of U ( gl N )-modules tothe category of representations of the degenerate affine Hecke algebra of type A n .In [CEE09], Calaque, Enriquez and Etingof generalized this functor to the categoryof representations of degenerate double affine Hecke algebra of type A n . Etingof,Freund and Ma [EFM09] extended the construction to the category of representa-tions of degenerate affine and double affine Hecke algebra of type BC n by consideringthe classical symmetric pair ( gl N , gl p × gl N − p ). As a quantization of the functors byEtingof-Freund-Ma, Jordan and Ma in [JM11] constructed functors from the categoryof U q ( gl N )-modules to the category of representations of affine Hecke algebra of type C n and from the category of quantum D -modules to the category of representationsof the double affine Hecke algebra of type C ∨ C n . The construction in [JM11] used thetheory of quantum symmetric pair ( U q ( gl N ) , B σ ) where B σ is a coideal subalgebra.This is a quantum analogue of the classical symmetric pair.On the other hand, in [Ree97], Reeder did the classification of irreducible represen-tations of affine Hecke algebra of type C with equal parameters. In [Kat09], Katoindexed and analyzed the weights of representations of affine Hecke algebra of type C n . In [Ma08], Ma analyzed the image of principal series modules under the Etingof-Freund-Ma functor. Moreover, the combinatorial description of Young diagrams isused to describe irreducible representations of the symmetric group and Hecke al-gebra of type A with standard tableaux on the Young diagram indexing the bases.Similarly, the skew shape and standard tableaux on it describes the irreducible rep-resentation of the affine Hecke algebra of type A . Moreover, in [SV05], Suzuki andVazirani introduced a description of some irreducible representations of the doubleaffine Hecke algebra of type A by periodic skew Young diagrams and periodic stan-dard tableaux on it. In [Ram03], Ram introduced the chambers and local regionsand described the representations of the affine Hecke algebra. In [Dau12], Daughertyintroduce the combinatorial description of representations of degenerate extendedtwo-boundary Hecke algebra. In [DR18], Daugherty and Ram gave a Schur-Weylduality approach to the affine Hecke algebra of type C n .This paper focuses on the representations of the degenerate affine Hecke algebra oftype C n and gives a combinatorial description which is similar to the combinato-rial description in [Dau12] and [DR18] but is via a different structure, the Etingof-Freund-Ma functor. This paper is arranged as follow: Section 2-4 are about the EPRESENTATIONS OF DEGENERATE AHA OF TYPE C n Etingof-Freund-Ma functor, the degenerate affine Hecke algebra of type C n and GL N -modules. In section 5 and section 6, we compute the underlying vector space of theimage of Etingof-Freund-Ma functor and the Y -actions. In section 7 and section8, we talk about intertwining operators and define combinatorial moves. Section 9concerns the irreducibility of the image. In section 10, we talk about how to recovera GL N -module from a representation of degenerate affine Hecke algebra of type C n . Acknowledgments.
I would like to thank Monica Vazirani for her guidance andhelpful discussions, Arun Ram for helpful comments on my first draft and for the sug-gestion on presenting the Y -action, Jos´e Simental Rodr´ıguez for detailed feedbackand helpful discussions. 2. Definitions and notations
Root system of type C n . Let h ∗ be a finite-dimensional real vector space withbasis { ǫ i | i = 1 , · · · , n } and a positive definite symmetric bilinear form ( · , · ) such that( ǫ i , ǫ j ) = δ ij . Let R n be an irreducible root system of type C n with R n = { ǫ i + ǫ j | i, j = 1 , · · · , n } ∪ { ǫ i − ǫ j | i, j = 1 , · · · , n and i = j } , and the positive roots are R n + = { ǫ i + ǫ j | i, j = 1 , · · · , n } ∪ { ǫ i − ǫ j | ≤ i < j ≤ n } . For any root α , the coroot is α ∨ = 2 α ( α, α ) . Let Q be the root lattice and Q ∨ bethe coroot lattice. Let α i = ǫ i − ǫ i +1 , for i = 1 , · · · , n − α n = 2 ǫ n . Then thecollection of simple roots are Π n = { α i | i = 1 , · · · , n } . For each simple root α i , define the reflection s i := s α i , s α i ( λ ) = λ − ( λ, α ∨ i ) α i , where λ ∈ h ∗ . Then the finite Weyl group W of type C n is generated by thegenerators s , · · · , s n − , s n with the relations s i = 1 , for i = 1 , · · · , n, (1) s i s i +1 s i = s i +1 s i s i +1 , for i = 1 , · · · , n − , (2) s n − s n s n − s n = s n s n − s n s n − , (3) s i s j = s j s i , for | i − j | > . (4)2.2. Affine Weyl group of type C n . Let W = W ⋉ Q ∨ . For any ι ∈ h ∗ , where ι = ι ǫ + · · · + ι n ǫ n and ι k ∈ Z , let y ι = y ι · · · y ι n n and the action of w ∈ W by w.y ι = y w ( ι ) .Let W = W ⋉ Q ∨ and the affine Weyl group of type C n is generatedby s , · · · , s n − , s n and Y i , for i = 1 , · · · , n with the following additional relations to(1)-(4), s i Y j = Y j s i , for j = i, i + 1 , (5) Y i Y j = Y j Y i , (6) s i Y i s i = Y i +1 , for i = 1 , · · · , n − , (7) s n Y n s n = Y n − . (8) YUE ZHAO
Definition of degenerate affine Hecke algebra of type C n . Let κ and κ be two parameters. The trigonometric degenerate affine Hecke algebra H n ( κ , κ ) isan algebra generated over C by s , · · · , s n − , γ n , where we take γ n = s n , and y , · · · , y n with relations (1)-(6) and the following relations s i y i − y i +1 s i = κ , for i = 1 , · · · , n − , (9) γ n y n + y n γ n = κ . (10)2.4. Y -semisimple degenerate affine Hecke algebra representations. Now letdefine what we mean by Y -semisimple. Let Y = C [ y , · · · , y n ] be the commutativesubalgebra of the degenerate affine Hecke algebra H n ( κ , κ ). Let L be a represen-tation of H n ( κ , κ ). For a function ζ : { , · · · , n } → C , let ζ i denote ζ ( i ) and ζ = [ ζ , · · · , ζ n ]. Define the simultaneous generalized eigenspace as L genζ = { v ∈ L | ( y i − ζ i ) k v = 0 for some k ≫ i = 1 , · · · , n } . Since the polynomial algebra Y is commutative, the restriction of L on Y decomposesto a sum of simultaneous generalized eigenspace, i.e. L = ⊕ ζ L genζ . Similarly, definethe simultaneous eigenspace L ζ = { v ∈ L | y i v = ζ i v for all i = 1 , · · · , n } . Definition 2.1.
If the restriction of L on Y decomposes to a sum of simultaneouseigenspaces, i.e. L = ⊕ ζ L ζ , then call L is Y -semisimple. The function ζ is called aweight and L ζ is the weight space of weight ζ . Etingof-Freund-Ma Functor
We recall the definition of the Etingof-Freund-Ma functor F n,p,µ in [EFM09]. Let N be a positive number and V be the vector representation of gl N . Let p, q be positiveintegers such that N = p + q . Let t = gl p × gl q and t be the subalgebra in t consistingof all the traceless elements in t . Let χ is a character defined on t as(11) χ ( (cid:20) S T (cid:21) ) = q · tr ( S ) − p · tr ( T ) , where S ∈ gl p and T ∈ gl q . For a given µ ∈ C , define a functor F n,p,µ from thecategory of gl N -modules to the category of representations of degenerate affine Heckealgebra H n (1 , p − q − µN ) F n,p,µ ( M ) = ( M ⊗ V ⊗ n ) t ,µ , where the ( t , µ )-invariant corresponds A.v = µχ ( A ) v , for all A ∈ t .Let M be the 0-th tensor factor. Let V i be the i -th tensor factor with V i = V being thevector representation for i = 1 , · · · , n . In [JM11], the action of the degenerate affineHecke algebra H n (1 , p − q − µN ) is the quasi classical limit of the action of the affineHecke algebra H n ( q, q σ , q ( p − q − τ ) ) generated by T , · · · , T n − , T n and Y ± , · · · , Y ± n . Inthe following figures, V i is the vector representation for i = 1 , · · · , n . In [JM11], theaction of T i for i = 1 , · · · , n − τ V i ,V i +1 ◦ R i,i +1 , where the flip operator τ V i ,V i +1 : V i ⊗ V i +1 → V i +1 ⊗ V i is defined by v i ⊗ v i +1 v i +1 ⊗ v i and R i,i +1 is the R matrix acting on V i ⊗ V i +1 , EPRESENTATIONS OF DEGENERATE AHA OF TYPE C n T i = · · · · · · MM V V V i V i V i +1 V i +1 V n V n Let T i = s i e ~ s i / . Proposition 39 in [Jor09] and section 10.7 of [JM11] computed theaction of s i , i.e. s i acts on F n,p,µ ( M ) by exchanging the i -th and ( i + 1)-th tensorfactors.The action of T n was defined as follows T n = · · · MM V V V V V n V n J V where the matrix J V is a right-handed numerical solution of the reflection equation R ( J V ) R ( J V ) = ( J V ) R ( J V ) R in section 7 of [JM11]. Section 10.7 of [JM11]compute the quasi classical limit of T n . Then γ n acts on F n,p,µ ( M ) by multiplyingthe n -th tensor factor by J = diag ( I p , − I q ).The action of Y was define by q nN + µ ( q − p ) − N R − ◦ τ V,M ◦ R − ◦ τ M,V . Y q nN + µ ( q − p ) − N · · · MM V V V V V n V n Let Y = e y ~ . By Proposition 10.13 in [JM11],(12) y = − X s,t ( E ts ) ⊗ ( E st ) + nN + µ ( q − p )2 − N , where E ts is the N × N matrix with the ( s, t ) entry being 1 and other entries being 0and ( E ts ) i means E ts acting on the i -th tensor factor. Let s k,l denote the transposition( k, l ) ∈ S n and γ k ∈ W denote the action multiplying the k -th factor by J . In[EFM09], the action of y is given by(13) − X s | t ( E ts ) ⊗ ( E st ) + p − q − µN γ + 12 X l> s ,l + 12 X l =1 s ,l γ γ l , where P s | t = P ps =1 P nt = p +1 + P pt =1 P ns = p +1 . In section 6.1, we show that the compu-tation via equation (13) agrees with equation (12). By the relation y k = s k − y k − s k − − s k − , we could compute the action of y k for k = 1 , · · · , n .4. GL -module We consider images of polynomial GL N -modules under Etingof-Freund-Ma functor.Recall the facts about polynomial GL N -modules. Let M be a polynomial GL N -module and H ⊂ GL N be the collection of invertible diagonal matrices. Let v ∈ M satisfy x.v = x λ · · · x λ N N v, YUE ZHAO for any x = diag ( x , · · · , x N ) ∈ H . Then v is a weight vector of H -weight λ =( λ , · · · , λ N ). The subspace M ( λ ) = { v ∈ M | x.v = x λ · · · x λ N N v, x ∈ H } is called the weight space of weight λ . Then the polynomial GL N -module M is adirect sum of weight spaces M = M M ( λ ) . Let B ⊂ GL N be the collection of all invertible upper triangular matrices. Let v ∈ M be a generator of M . If v satisfies x.v = c ( x ) v for some function c ( x ) and any x ∈ B ,then v is called a highest weight vector. If M has the unique highest weight vectorup to a scalar of the highest weight ξ , then M is a highest weight module with thehighest weight ξ and let us denote M by V ξ . A GL N -module M is irreducible ifand only if M is a highest weight GL N -module. Furthermore, two highest weight GL N -modules are isomorphic if and only if they have the same highest weight. Let ξ = P Ni =1 ξ i ǫ i satisfying ξ ≥ ξ ≥ · · · ≥ ξ N and ξ i ∈ Z for i = 1 , · · · , N . Then ξ is an integral dominant weight of GL N . Let P + denote the collection of all integraldominant weights and P + ≥ denote the collection of all integral dominant weights ξ = P Ni =1 ξ i ǫ i with ξ i ∈ N , for i = 1 , · · · , N . Then the highest weight modules withhighest weights ξ ∈ P + ≥ are all the irreducible polynomial GL N -modules. Let M bea rational GL N -module. Then M = det m ⊗ N for some m ∈ Z and a polynomial GL N -module N . Then the highest weight modules with integral dominant highestweights are all the irreducible rational GL N -modules.The collection P + ≥ has a one-to-one correspondence with the collection of partitionswith at most N parts and thus the one-to-one correspondence with Young diagramswith at most N rows. For the ease of writing, for each irreducible polynomial GL N -module V ξ with highest weight ξ ∈ P + ≥ , let us denote the corresponding partition( ξ , · · · , ξ N ) and Young diagram also by ξ . Moreover, define | ξ | = P Ni =1 ξ i for ξ ∈ P + .For a highest weight GL N -module V ξ , ξ ∈ P + ≥ , with weight space decomposition V ξ = L V ξ ( λ ), the character of V ξ χ V ξ = X λ dim ( V ξ ( λ )) x λ · · · x λ N N is the Schur polynomial s ξ ( x , · · · , x N ) of shape ξ .By Pieri’s rule, s ξ e = X ν s ν , where ν ∈ P + ≥ runs through all the shapes obtained by adding a cell to some row of ξ .Observe that e = s ξ , where ξ = (1), is the character of the vector representation V of GL N . This fact indicates how the tensor product of an irreducible polynomial GL N -module and vector representation decomposes into a sum of irreducible polynomial GL N -modules. 5. Invariant space
In this section, we compute the underlying vector space F n,p,µ ( V ξ ) = ( M ⊗ V ⊗ n ) t ,µ by finding a special basis of it and then index the basis elements by a collection ofstandard tableaux. EPRESENTATIONS OF DEGENERATE AHA OF TYPE C n Definition of the invariant space.
Let M be a GL N -module, then M has a gl N -module structure. For any X ∈ gl N and v ∈ M , X.v = ddt ( e tX .v ) t =0 . Recall the notations, K = GL p × GL q , Lie ( K ) = t and t ⊂ t which is the collectionof traceless matrices in t . Proposition 5.1.
The underlying vector space is invariant under tensoring powersof the determinant representation, i.e. ( det m ⊗ M ⊗ V ⊗ n ) t ,µ ∼ = ( M ⊗ V ⊗ n ) t ,µ , forany m ∈ C .Proof. Take any element from ( det m ⊗ M ⊗ V ⊗ n ) t ,µ , we can denote it by ⊗ w , where w ∈ M ⊗ V ⊗ n . According to the definition of invariant space( det m ⊗ M ⊗ V ⊗ n ) t ,µ = { ⊗ w | A. ( ⊗ w ) = µχ ( A )( ⊗ w ), for any A ∈ t } . Compute the action of A ∈ t A. = ddt ( e tA . ) t =0 = ddt ( det m ( e tA )) t =0 . = ddt ( e m · tr ( tA ) ) t =0 . = 0 , since tr ( A ) = 0. Then it follows A. ( ⊗ w ) = ( A. ) ⊗ w + ⊗ ( A.w )= ⊗ ( A.w ) . Hence ( det m ⊗ M ⊗ V ⊗ n ) t ,µ = { ⊗ w | ⊗ ( A.w ) = µχ ( A )( ⊗ w ), for any A ∈ t }∼ = { w | A.w = µχ ( A ) w , for any A ∈ t } =( M ⊗ V ⊗ n ) t ,µ . (cid:3) Remark 5.2.
For an irreducible rational GL N -module M , we could write M = det m ⊗ V ξ for some integer m and some highest weight module V ξ with the high-est weight ξ ∈ P + ≥ such that ξ N = 0 . Then ( M ⊗ V ⊗ n ) t ,µ = ( V ξ ⊗ V ⊗ n ) t ,µ . So it isenough to consider highest weight module V ξ with highest weight ξ ∈ P + ≥ such that ξ N = 0 , which is associated to partitions ξ of length at most N − . Computation of the ( t , µ ) invariant space. YUE ZHAO
Proposition 5.3.
The ( t , µ ) invariant space F n,p,µ ( V ξ ) = ( V ξ ⊗ V ⊗ n ) t ,µ , for µ ∈ C and ξ ∈ P + ≥ . ( V ξ ⊗ V ⊗ n ) t ,µ ∼ = Hom t ( µχ , Res gl N t V ξ ⊗ V ⊗ n ) ∼ = Hom t ( θ , Res gl N t V ξ ⊗ V ⊗ n ) , where θ is a one-dimensional t -module and θ = ( µq + | ξ | + nN ) tr gl p + ( − µp + | ξ | + nN ) tr gl q . Proof.
The ( t , µ ) invariant space F n,p,µ ( V ξ ) = ( V ξ ⊗ V ⊗ n ) t ,µ is defined to be thesubspace { v ∈ V ξ ⊗ V ⊗ n | Av = µχ ( A ) v for any A ∈ t } . To compute this subspace, we lift it to a t invariant space. Let ψ the one-dimensional t -module such that ( V ξ ⊗ V ⊗ n ) t ,µ =( Res gl N t ( V ξ ⊗ V ⊗ n ) ⊗ ψ ) t . Let t = t ⊕ C { I N } . For any P ∈ t , there is a unique decomposition P = A + B suchthat A ∈ t and B = bI N for some b ∈ C . So the t -invariant corresponds to { v ∈ V ξ ⊗ V ⊗ n | P v + ψ ( P ) v = 0 } . Then P v + ψ ( P ) v = Av + Bv + ψ ( P ) v = 0. And B = bI N acts by the scalar b ( | ξ | + n ) = ( | ξ | + n ) tr ( B ) N . Also, we have χ ( P ) = χ ( A ) + χ ( B ) = χ ( A ), since χ ( B ) = qbp − pbq = 0. So { v ∈ V ξ ⊗ V ⊗ n | P v + ψ ( P ) v = 0 } = { v ∈ V ξ ⊗ V ⊗ n | Av = µχ ( A ) v } . For any P ∈ t with P = (cid:20) S T (cid:21) where S ∈ gl p and T ∈ gl q , we have ψ ( P ) = − µχ ( A ) − | ξ | + nN tr ( B )= − µχ ( P ) − | ξ | + nN tr ( P )= ( − µq − | ξ | + nN ) tr gl p ( S ) + ( µp − | ξ | + nN ) tr gl q ( T ) . Hence it follows that the one dimensional t -module θ = ( µq + | ξ | + nN ) tr gl p + ( − µp + | ξ | + nN ) tr gl q . (cid:3) EPRESENTATIONS OF DEGENERATE AHA OF TYPE C n Remark 5.4.
The ( t , θ ) invariant space above is equivalent to the following K in-variant space. ( V ξ ⊗ V ⊗ n ) t ,µ ∼ = Hom t ( µχ , Res gl N t V ξ ⊗ V ⊗ n ) ∼ = Hom t ( θ , Res gl N t V ξ ⊗ V ⊗ n ) ∼ = Hom K ( det a ⊠ det b , V ξ ⊠ V ⊗ n ) , where a = µq + | ξ | + nN and b = − µp + | ξ | + nN . A basis of invariant space and standard tableaux.
The characters of irreducible polynomial GL N -modules are Schur functions. So wecould consider the restriction of V ξ ⊗ V ⊗ n by exploring Schur functions. Recall thefollowing fact of Schur functions. Proposition 5.5.
Let s ν ( x , · · · , x p , z p +1 , · · · , z N ) be the character of V ν , then s ν ( x , · · · , x p , z p +1 , · · · , z N ) = Σ c νω ,ω s ω ( x , · · · , x p ) s ω ( z p +1 , · · · , z N ) , where ω is a highest weight of GL p and ω is a highest weight of GL q , c νω ,ω is theLittlewood-Richardson coefficient. The Littelwood-Richardson coefficient c νω ,ω is the multiplicity of the K -module V ω ⊠ V ω in the restriction of GL N -module V ν . Let V ξ ⊗ V ⊗ n = L ν m ν V ν as GL N -modules, where ν ∈ P + ≥ and m ν ∈ N is the multiplicity of V ν in V ξ . Then the ( t , µ )invariant space F n,p,µ ( V ξ ) = Hom K ( det a ⊠ det b , Res GL N K V ξ ⊗ V ⊗ n )(14) = M ν m ν Hom K ( det a ⊠ det b , Res GL N K V ν ) . (15)Since ν ∈ P + ≥ , to guarantee Hom K ( det a ⊠ det b , Res GL N K V ν ) = 0 for each ν in (13), itsuffices to consider a, b ∈ N , otherwise F n,p,µ ( V ξ ) = ( V ξ ⊗ V ⊗ n ) t ,µ = 0. Our goal isto compute the ν such that the multiplicity of det a ⊠ det b in the K restriction of the GL N -module V ν is nonzero. To do this, we need Okada’s theorem [Oka98]. Theorem 5.6.
For any two rectangular shapes ( a p ) and ( b q ) , where a and b arenonnegative integers and p ≤ q , then s a p · s b q = X c ν ( a p )( b q ) s ν , where c ν ( a p )( b q ) = 1 when ν satisfies the condition ν i + ν p + q − i +1 = a + b, i = 1 , · · · , p (16) ν p ≥ max ( a, b )(17) ν i = b, i = p + 1 , · · · , q (18) and c ν ( a p )( b q ) = 0 otherwise. Corollary 5.7.
Now we have the following fact, the ( t , µ ) invariant space F n,p,µ ( V ξ ) = ( V ξ ⊗ V ⊗ n ) t ,µ (19) = M ν Hom GL N ( V ν , V ξ ⊗ V ⊗ n ) , (20) where ν ∈ P + ≥ runs through all partitions satisfying (16)-(18). Moreover, by Pieri’s rule, the vector space
Hom GL N ( V ν , V ξ ⊗ V ⊗ n ) has a basisindexed by standard tableaux T such that the shape of T is ν/ξ and the dimensionof this vector space m ν = dimHom GL N ( V ν , V ξ ⊗ V ⊗ n )equals the number of standard tableaux T with the shape of T being ν/ξ . If m ν = 0,then ξ ⊂ ν and | ν | = | ξ | + n . Theorem 5.8.
The ( t , µ ) invariant space F n,p,µ ( V ξ ) = ( V ξ ⊗ V ⊗ n ) t ,µ has a one toone correspondence to the set of standard tableaux T such that the shape of T is ν/ξ for ν ∈ P + ≥ with | ν | = | ξ | + n , ν runs through all the partitions satisfying (16)-(18)and ξ ⊂ ν . Let us consider the following example of ( t , µ ) invariant space. Example 5.9.
Let M = V ξ be a GL -module, ξ = 2 ǫ + ǫ , n = 3 , p = 1 and µ = 0 .Then ( a p ) = (2 ) and ( b q ) = (2 ) .By Okada’s theorem, we could compute the shapes ν such that the invariant spaceis nonzero. (2 ) × (2 ) = + + Then a basis of the invariant space could be indexed by standard tableaux on skewshapes obtained by the shapes above skewed by ξ . In this example, we obtain an invariant space of dimensions. One skew shape.
In this subsection, we associate a skew shape ϕ ξn,p,µ to theimage F n,p,µ ( V ξ ) under Etingof-Freund-Ma functor. Let ξ = P Ni =1 ξ i ǫ i ∈ P + ≥ . Thecorresponding Young diagram ξ = ( ξ , · · · , ξ N ). The first q rows of ξ forms a Youngdiagram denoted by ξ (1) and the last p rows of ξ forms a Young diagram denoted by ξ (2) . The parameter µ gives a pair of rectangles ( a p ) and ( b q ) denoting the K -module det a ⊠ det b , where a = µq + | ξ | + nN and b = − µp + | ξ | + nN .Suppose p ≤ q . Placing the northwestern corner the rectangle ( a p ) next to the north-eastern corner of the rectangle ( b q ) forms a Young diagram β . Delete the Youngdiagram ξ (1) from northwestern corner of β . Let ξ (2) denote the skew shape obtainedby rotating ξ (2) by π . Delete the rotated ξ (2) from the southeastern corner of β , i.e. EPRESENTATIONS OF DEGENERATE AHA OF TYPE C n the skew shape ϕ ξn,p,µ is defined by ϕ ξn,p,µ = ν/ξ (1) , where ν i = a + b − ξ N − i +1 for i = 1 , · · · , p and ν i = b for i = p + 1 , · · · , q . ( a p ) ( b q ) ξ (1) ξ (2) ξ = P Ni =1 ξ i ǫ i ∈ P + ≥ q ξ (1) ξ (2) max ( a, b ) b aq pϕ ξn,p,µ Let ϕ = ϕ ξn,p,µ . If a cell ( i, j ) of the skew shape ϕ satisfy ( i +1 , j ) / ∈ ϕ and ( i, j +1) / ∈ ϕ ,then call ( i, j ) a corner of ϕ . Define γ -move on a skew shape ϕ : delete a corner ( i, j ) ∈ ϕ such that j > max ( a, b ) and 1 ≤ i ≤ p , and add the cell ( p + q − i + 1 , a + b − j + 1).Denote the γ -move by ϕ → ϕ ′ where ϕ ′ = ϕ \ ( i, j ) ∪ ( p + q − i +1 , a + b − j +1). Note thatfor a given ϕ , the γ -move stops when there is no cell ( i, j ) such that j > max ( a, b ).Given the skew shape ϕ ξn,p,µ , a collection D ( ϕ ξn,p,µ ) of skew shapes consists of ϕ ξn,p,µ and all the skew shapes obtained by applying γ -moves on ϕ ξn,p,µ for finitely manytimes. The shape ϕ ξn,p,µ is called the minimal shape of the representation F n,p,µ ( V ξ ). ϕ ξn,p,µ Continue Example 5.9, the representation F , , ( V ) is index by the following skewshape ϕ .The collection D ( ϕ ) of skew shapes is obtained as follows:5.5. Skew shapes and standard tableaux.
For the ease of description, let us usethe following definition of skew shapes and standard tableaux. Given a partition ξ = ( ξ , · · · , ξ l ), the corresponding Young diagram ξ is a subset of Z , consisting of( i, j ) such that 1 ≤ i ≤ l and 1 ≤ j ≤ ξ i . Let ν = ( ν , · · · , ν l ) and ξ = ( ξ , · · · , ξ l )such that ν i ≥ ξ i for 1 ≤ i ≤ l , then for the corresponding Young diagrams ξ ⊂ ν holds. A skew shape ν/ξ is the subset ν \ ξ of Z . For example, let ν = (7 , , , , , ξ = (5 , , , , , ν and ξ and the skew shape ν/ξ arethe following subsets of Z . ν = { ( i, j ) | ≤ i ≤ , ≤ j ≤ ν i } ,ξ = { ( i, j ) | ≤ i ≤ , ≤ j ≤ ξ i } and ν/ξ = { (1 , , (1 , , (2 , , (3 , , (3 , , (3 , , (4 , } . Define a tableau T on n -indices { , · · · , n } to be an injective map TT : { , · · · , n } → Z k ( i ( k ) , j ( k ))where i and j being two maps from { , · · · , n } to Z and the image Im ( T ) of T being a skew shape. The image Im ( T ) is also called the shape of the tableaux T .Let cont T be a map cont T : { , · · · , n } → Z k j ( k ) − i ( k ) , call cont T ( k ) is the content of k in the tableau T . If T − ( i + 1 , j ) > T − ( i, j ) and T − ( i, j + 1) > T − ( i, j ) hold for each cell ( i, j ) ∈ Im ( T ), then call T is a standardtableau.Let T ab ( ϕ ξn,p,µ ) = { T | T is a standard tableau and Im ( T ) ∈ D ( ϕ ξn,p,µ ) } . The invariant space F n,p,µ ( V ξ ) = ( V ξ ⊗ V ⊗ n ) t ,µ has a basis indexed by a collection ofstandard tableaux on the skew shapes in D ( ϕ ξn,p,µ ), i.e. all the tableaux in T ab ( ϕ ξn,p,µ ).Let v T denote the basis vector indexed by T ∈ T ab ( ϕ ξn,p,µ ). Then as a vector space F n,p,µ ( V ξ ) = ( V ξ ⊗ V ⊗ n ) t ,µ = span C { v T | T ∈ T ab ( ϕ ξn,p,µ ) } . EPRESENTATIONS OF DEGENERATE AHA OF TYPE C n Y - semisimplicity Action of Y . In this subsection let us computer the Y -actions on the invariantspace F n,p,µ ( V ξ ) = ( V ξ ⊗ V ⊗ n ) t ,µ . In [Jor09], Jordan computed the action of y andused the fact that Etingof-Freund-Ma functor is a trigonometric degeneration of thequantum case. Now let us review the computation and conduct it in the degeneratecase. Let us use the following notations in [EFM09] for sums X s,t = N X s =1 N X p =1 (21) X s | t = p X s =1 N X t = p +1 + p X t =1 N X s = p +1 (22) X st = p X s =1 p X t =1 + N X s = p +1 N X t = p +1 (23)It is easy to observe that the sum of (22) and (23) equals (21).Review the definition of y on the ( t , µ )-invariant space F n,p,µ ( V ξ ) = ( V ξ ⊗ V ⊗ n ) t ,µ in [EFM09], y = − X s | t ( E ts ) ⊗ ( E st ) + p − q − µN γ + 12 X l> s ,l + 12 X l =1 s ,l γ γ l . Compute the last two terms of y , we have12 X l> s ,l + 12 X l =1 s ,l γ γ l = 12 X l> X s,t ( E ts ) ⊗ ( E st ) l + 12 X l> X s,t ( E ts J ) ⊗ ( E st J ) l = X l> X st ( E ts ) ⊗ ( E st ) l = X st ( E ts ) ( X l> ⊗ ( E st ) l )= X st ( E ts ) (∆ ( n ) ( E st ) − ( E st ) − ( E st ) )The last step follows the fact that P l> ⊗ ( E st ) l = ∆ ( n ) ( E st ) − ( E st ) − ( E st ) , where∆ denotes the comultiplication of Lie algebra gl N and ∆ ( n ) ( E st ) = P nl =0 ( E st ) l .Applying the fact that y preserves on the ( t , µ )-invariant space F n,p,µ ( V ξ ) = ( V ξ ⊗ V ⊗ n ) t ,µ , the computation of the last two terms of y above continues as follows. X st ( E ts ) (∆ ( n ) ( E st ) − ( E st ) − ( E st ) )= p X s =1 ( µq + | ξ | + nN )( E ss ) + N X s = p +1 ( − µp + | ξ | + nN )( E ss ) − p X s =1 p ( E ss ) − N X s = p +1 q ( E ss ) − X st ( E ts ) ⊗ ( E st ) =( µq − p + | ξ | + nN ) p X s =1 ( E ss ) + ( − µp − q + | ξ | + nN ) N X s = p +1 ( E ss ) − X st ( E st ) ⊗ ( E ts ) Combining other terms in the definition of y , y = − X s,t ( E ts ) ⊗ ( E st ) + p − q − µN γ + ( µq − p + | ξ | + nN ) p X s =1 ( E ss ) + ( − µp − q + | ξ | + nN ) N X s = p +1 ( E ss ) = − X s,t ( E ts ) ⊗ ( E st ) + ( µq − p + | ξ | + nN + p − q − µN p X s =1 ( E ss ) + ( − µp − q + | ξ | + nN − p − q − µN N X s = p +1 ( E ss ) = − X s,t ( E ts ) ⊗ ( E st ) + ( | ξ | + nN + µq − µp − N N X s =1 ( E ss ) = − X s,t ( E ts ) ⊗ ( E st ) + | ξ | + nN + µq − µp − N , Remark 6.1.
Since the action in [JM11] was define on F n,p,µ ( M ) for M is a D -module, there is a difference between equation (12) and the above result. If we inputa D -module instead of V ξ , the above result will be the same with equation (12). Moreover, the action of y k for k > Proposition 6.2.
The action of y k , for k = 1 , · · · , n , on the invariant space ( V ξ ⊗ V ⊗ n ) t ,µ is computed by y k = − X s,t (∆ ( k − E ts ) (0 ,k ) ⊗ ( E st ) k + | ξ | + nN + µq − µp − N , where ( E ts ) (0 ,k ) denotes the tensor product ( V ξ ⊗ V ⊗ ( k − ) and hence ∆ ( k − E ts actingon ( E ts ) (0 ,k ) .Proof. We verified the action of y above. Suppose the statement is true for y i , i < k .Let compute the action of y k . By the relation s k − y k − − y k s k − = κ = 1 and the EPRESENTATIONS OF DEGENERATE AHA OF TYPE C n inductive hypothesis, it follows y k = s k − y k − s k − − s k − = − X s,t,j,l (∆ ( k − E ts ) (0 ,k − ⊗ ( E tl E st E js ) k − ⊗ ( E lt E sj ) k − X s,t ( E ts ) k − ⊗ ( E st ) k + | ξ | + nN + µq − µp − N − X s,t,j (∆ ( k − E ts ) (0 ,k − ⊗ ( E jj ) k − ⊗ ( E st ) k − X s,t ( E ts ) k − ⊗ ( E st ) k + | ξ | + nN + µq − µp − N P j ( E jj ) k − = ( I N ) k − . The above computation continues= − X s,t (∆ ( k − E ts ) (0 ,k − ⊗ ( I N ) k − ⊗ ( E st ) k − X s,t ( E ts ) k − ⊗ ( E st ) k + | ξ | + nN + µq − µp − N − X s,t (∆ ( k − E ts ) (0 ,k ) ⊗ ( E st ) k + | ξ | + nN + µq − µp − N . (cid:3) The Lie algebra gl N has a basis { E ts | ≤ s, t ≤ N } with the dual basis { E st } with respect to the Killing form. Let C denote the Casimir element of U ( gl N ), then C = P s,t E ts E st . The following computation follows∆( C ) = X s,t ∆( E ts )∆( E st )= X s,t ( E ts ⊗ ⊗ E ts )( E st ⊗ ⊗ E st )=( X s,t E ts E st ) ⊗ ⊗ ( X s,t E ts E st ) + 2 X s,t E ts ⊗ E st . Thus X s,t E ts ⊗ E st = ∆( C ) − C ⊗ − ⊗ C . Weights and contents.
In [Ram04], Ram talked about the standard tableauxand representations of affine Hecke algebra of type C and analyzed the weights interms of boxes. Now let us analyze the weights of F n,p,µ ( V ξ ) in terms of contents. Insection 5, we obtain a basis of the ( t , µ )-invariant space F n,p,µ ( V ξ ) = ( V ξ ⊗ V ⊗ n ) t ,µ indexed by T ab ( ϕ ξn,p,µ ), i.e. standard tableaux on a family of skew shapes ν/ξ where ν are obtained by Okada’s theorem. The action of y k on the basis element indexedby standard tableau T is by a scalar. Moreover, this scalar is computed in terms ofthe content of the box fixed by k . Theorem 6.3.
Let v T denote the basis element of the invariant space indexed bystandard tableau T . Then v T is an eigenvector of y k and the eigenvalue is computedas − cont T ( k ) + s , where s = | ξ | + nN + µq − µp − N .Proof. Let us T ∈ T ab ( ϕ ξn,p,µ ). Since T is a standard tableau, then T corresponds toa sequence ( ν ( k ) ) k = nk =0 of Young diagrams, where ν (0) = ξ,ν (1) = ξ ∪ T ( { } ) ,ν (2) = ξ ∪ T ( { , } ) , · · · ν ( n ) = ξ ∪ T ( { , , · · · , n } ) , where T ( { , · · · , k } ) is the collection of cells filled by numbers 1 , · · · , k , i.e. the Youngdiagram ν ( k ) is formed by adding the cells filled by numbers 1 , · · · , k to the Youngdiagram ξ . So it follows, for k = 1 , · · · , n , v T ∈ ( V ξ ⊗ V ⊗ k )[ ν ( k ) ] ⊗ V ⊗ ( n − k ) , where ( V ξ ⊗ V ⊗ k )[ V ν ( k ) ] denotes the V ν k -isotopic component of the tensor product V ξ ⊗ V ⊗ k . By the previous subsection 6.1, it follows that the term P s,t (∆ ( k − ( E ts )) (0 ,k ) ⊗ ( E st ) k acts on v T by C (0 ,k +1) − C (0 ,k ) ⊗ k − (0 ,k ) ⊗ C k . Moreover, the Casimir element acts on the highest weight module V ν by the scalar h ν, ν + 2 ρ i , where the weight 2 ρ = P Ni =1 ( N − i + 1) ǫ i . So for each k such that1 ≤ k ≤ N , C (0 ,k +1) acts on V ν ( k ) by the scalar h ν ( k ) , ν ( k ) + 2 ρ i , C (0 ,k ) acts on V ν ( k − by the scalar h ν ( k − , ν ( k − + 2 ρ i and C k acts on V by the scalar h ǫ, ǫ + 2 ρ i = N ,namely C (0 ,k +1) − C (0 ,k ) ⊗ k − (0 ,k ) ⊗ C k h ν ( k ) , ν ( k ) + 2 ρ i − h ν ( k − , ν ( k − + 2 ρ i − h ǫ, ǫ + 2 ρ i ) . Let T ( k ) be the cell ( i ( k ) , j ( k )), then ν ( k ) i ( k ) = j ( k ) = ν ( k − i ( k ) + 1 and ν ( k ) i = ν ( k − i , for i = i ( k ). 12 ( h ν ( k ) , ν ( k ) + 2 ρ i − h ν ( k − , ν ( k − + 2 ρ i − h ǫ, ǫ + 2 ρ i )= 12 (( j ( k ) + N − i ( k ) + 1)( j ( k )) − ( j ( k ) + N − i ( k ))( j ( k ) − − N )= j ( k ) − i ( k ) . Then the statement follows. (cid:3)
EPRESENTATIONS OF DEGENERATE AHA OF TYPE C n Theorem 6.4.
Let F n,p,µ ( V ξ ) denote the image of the irreducible GL N -module V ξ ,for some ξ ∈ P + , under Etingof-Freund-Ma functor. Then F n,p,µ ( V ξ ) has a basisindexed tableaux in T ab ( ϕ ξn,p,µ ) , i.e. { v T | T ∈ T ab ( ϕ ξn,p,µ ) } . This basis is a weightbasis with each basis vector v T is a weight vector of weight ζ T = − cont T + s . So F n,p,µ ( V ξ ) is a Y -semisimple representation of H n (1 , p − q − µN ) . Moreover, it isobvious different standard tableaux give different weights. Hence each weight space isone dimensional. Intertwining operators
Definition of intertwining operators.Definition 7.1.
For i = 1 , · · · , n − , define the intertwining operators φ i = [ s i , y i ] , and for γ n , define φ n = [ γ n , y n ] . Proposition 7.2.
The intertwining operators φ i satisfy the braid relations φ i φ i +1 φ i = φ i +1 φ i φ i +1 , i = 1 , · · · , n − ,φ i φ j = φ j φ i , | i − j | > ,φ n − φ n φ n − φ n = φ n φ n − φ n φ n − . Since the operators φ i ’s satisfy the same braid relations with s i ’s and γ n , it makessense to define the following. Definition 7.3.
Let W denote the finite Weyl group of type C n , for each w ∈ W ,it has a reduced expression w = s i s i . . . s i m , l ( w ) = m , here we take the convention s n = γ n . Define φ w = φ i φ i . . . φ i m . Properties of intertwining operators.
Some computations on intertwining operators:(1) φ i = s i ( y i − y i +1 ) − φ n = 2 γ n y n − κ .(2) φ i = (1 − y i + y i +1 )(1 + y i − y i +1 ), φ n = ( κ − y n )( κ + 2 y n ). Definition 7.4.
Define the actions of W on weight ζ = [ ζ , · · · , ζ n ] : for an arbitrary w ∈ W , the action of w is w.ζ = ζ ◦ w − , where we take ζ − k = − ζ k . Theorem 7.5.
Let L be a Y -semisimple module and L ζ denote the weight space ofweight ζ , then φ w L ζ ⊂ L w.ζ . Proof.
It suffices to show the statement is true for each operator φ i .Case 1. When 1 ≤ i ≤ n −
1. We have the following facts that y i φ i = φ i y i +1 ,y i +1 φ i = φ i y i , and y j φ i = φ i y j , j = i or i + 1 . Case 2. Consider φ n . We have facts that y n φ n = − φ n y n ,y j φ n = φ n y j , j = n. (cid:3) Remark 7.6.
Since each weight space of F n,p,µ ( V ξ ) is one dimensional, so the actionof φ i is either or an isomorphism. Lemma 7.7. If ζ i − ζ i +1 = ± for some i ∈ { , , · · · , n − } , then φ i v ζ = 0 , where v ζ is the weight vector of the weight ζ .Proof. Suppose that φ i v ζ = 0. Then φ i v ζ = 0. By the computation above φ i =(1 − y i + y i +1 )(1 + y i − y i +1 ). Then φ i v ζ = (1 − ζ i + ζ i +1 )(1 + ζ i − ζ i +1 ) v ζ = 0. Thenwe have that ζ i − ζ i +1 = ± (cid:3) Similarly, we have the following fact.
Lemma 7.8. If ζ n = ± κ , then φ n v ζ = 0 , where v ζ is the weight vector of the weight ζ .Proof. Suppose that φ n v ζ = 0. Then φ n v ζ = 0. By the computation above φ n =( κ − y n )( κ + 2 y n ). Then φ n v ζ = φ n = ( κ − ζ n )( κ + 2 ζ n ) v ζ = 0. Then we havethat ζ ( n ) = ± κ . (cid:3) Properties of irreducible Y -semisimple representations. Let L be anirreducible Y -semisimple representation of H n ( κ , κ ). Let ζ = [ ζ , · · · , ζ n ] is a weight L . Theorem 7.9. If ζ i = ζ i +1 for some ≤ i ≤ n − , then L ζ = 0 .Proof. Let ζ be a weight such that ζ i = ζ i +1 . Suppose there exists a nonzero element v ∈ L ζ . Consider the vector s i v . Since φ i = s i ( y i − y i +1 ) − y i +1 − y i ) s i + 1, wehave φ i v = − v . ( y i − y i +1 ) s i v =(1 − φ i ) v =2 v = 0 . And act again by y i − y i +1 , ( y i − y i +1 ) s i v =2( y i − y i +1 ) v = 0 . This means s i v belongs to the generalized eigenspace of y i − y i +1 and does not belongto the eigenspace of y i − y i +1 , which contradicts Y -semisimplicity. (cid:3) Theorem 7.10.
Let κ = 0 . If ζ n = 0 , then L ζ = 0 . EPRESENTATIONS OF DEGENERATE AHA OF TYPE C n Proof.
Let ζ be a weight such that ζ n = 0. Suppose there exists a nonzero element v ∈ L ζ . Consider the vector γ n v . Since φ n = 2 γ n y n − κ = − y n γ n + κ , we have φ n v = − κ v . 2 y n γ n v =( κ − φ n ) v =2 κ v = 0 . Act again by y n , we have 2 y n γ n v =2 κ y n v = 0 . his means s i v belongs to the generalized eigenspace of y n and does not belong to theeigenspace of y n , which contradicts Y -semisimplicity. (cid:3) Remark 7.11.
When κ = 0 , it is possible for an irreducible Y -semisimple module L to contain a nonzero weight space L ζ with ζ n = 0 . In this case, γ n v ∈ C v . Otherwise,the vector v + γ n v generalizes a nonzero proper submodule of L , which contradicts theirreducibility. Lemma 7.12.
For any arbitrary w ∈ W , the intertwining operator φ w = w Π α ij ∈ R ( w ) ( y i − y j ) + X x Let ζ be a weight of L such that L ζ = 0 . Let v be a nonzero weightvector in L ζ . Then the set { φ w v | w ∈ W } spans the irreducible representation L .Proof. We need to show w.v lies in the span of { φ w v | w ∈ S n ⋉ ( Z / Z ) n } for anyarbitrary w ∈ S n ⋉ ( Z / Z ) n . We prove by induction on the length of w . When thelength of w is zero, the statement is trivial. Now assume for w with l ( w ) < k , thestatement holds, i.e. w.v can be expressed by a linear combination of elements in { φ w v | w ∈ W } . Set w is of length k and w = s i · · · s i k . Then by Lemma 7.12, wehave φ w · v = Π α ij ∈ R ( w ) ( ζ i − ζ j ) · w · v +Σ x Let ζ be a weight such that L ζ = 0 . Let w = 1 ∈ W such that w.ζ = ζ . Then φ w v = 0 .Proof. Let w = s i · · · s i k . since w.ζ = ζ , there is 1 ≤ p ≤ k such that s i · · · s i p =( hm ) where ζ h = ζ m . Consider φ i p − · · · φ i φ w v = Π ≤ j ≤ p (1 − ζ i j + ζ i j +1 )(1 + ζ i j − ζ i j +1 ) φ u v . It follows φ u v = 0 and hence φ w v = 0. (cid:3) Corollary 7.15. Let ζ be a weight such that L ζ = 0 . Then it follows dim ( L ζ ) = 1 . Proposition 7.16. (1) Let v be a nonzero weight vector of weight ζ such that | ζ i − ζ i +1 | = 1 . Then φ i v = 0 . (2) Let v be a nonzero weight vector of weight ζ such that ζ n = ± κ . Then φ n v = 0 . Remark 7.17. Some similar results also happen in degenerate affine Hecke algebraof type A n − . Let H n (1) be the degenerate affine Hecke algebra generated by s i ( i =1 , · · · n − and y i ( i = 1 · · · n ) with the following relations: s i = 1 , i = 1 , · · · , n − ,s i s j = s j s i , | i − j | > ,s i s i +1 s i = s i +1 s i s i +1 , i = 1 , · · · , n − ,y i y j = y j y i ,s i y i − y i +1 s i = 1 ,s i y j = y j s i , j = i, i + 1 . There is the same definition of Y -semisimple representation. And for any Y -semisimplerepresentation M , if a weight ζ with ζ i = ζ i +1 , then M ζ = 0 .Furthermore, we still could define the intertwining operator φ = s i y i − y i s i , then wewill also have φ i = (1 − y i + y i +1 )(1 + y i − y i +1 ) . This also implies the fact that if φ i v ζ = 0 then we have ζ i − ζ i +1 = ± . For the double affine Hecke algebra of type A , [SV05] explored similar properties in details. Combinatorial moves Moves among standard tableaux. Let T ab ( ϕ ξn,p,µ ) denote the collection of standard tableaux indexing the basis of F ( V ξ )in section 5. We define a set of moves m , · · · , m n on T ab ( ϕ ξn,p,µ ) ⊔ { } as follows.The move m i for i − , · · · , n − m i ( T ) = ( T ′ , T ′ is a standard tableau , otherwise,where T ′ ( k ) = T ( s i ( k )). The move m n is defined to be m n · T = ( , i ( n ) ≤ max ( p, q ) and j ( n ) ≤ max ( a, b ) T ′′ , otherwise,where T ′′ ( j ) = T ( j ) for each j = n and T ′′ ( n ) = ( N − i ( n ) + 1 , a + b − j ( n ) + 1). Remark 8.1. There is a easy observation. For any shape ϕ ′ ∈ D ( ϕ ξn,p,µ ) and any i ≤ min ( p, q ) , the sum of the column number of the last cell of the i -th row and thecolumn number of the last cell of the ( N − i + 1) -th row equal a + b . So T ′′ ( n ) =( N − i ( n ) + 1 , a + b − j ( n ) + 1) means that the m n -move takes the cell filled by n tothe end of the ( N − i ( n ) + 1) -th row. here m be the column number of the last cell of the ( N − i ( n ) + 1)-th row of Im ( T ). EPRESENTATIONS OF DEGENERATE AHA OF TYPE C n Correspondence between algebraic actions and combinatorial moves. Let v T denote the basis vector indexed by T ∈ T ab ( ϕ ξn,p,µ ) and ζ T denote the weightof v T , i.e. ζ T = − cont T + s . Proposition 8.2. (1) For i = 1 , · · · , n − , if m i ( T ) = holds, then m i ( T ) ∈ T ab ( ϕ ξn,p,µ ) and the common eigenbasis vector v m i ( T ) is of weight ζ m i ( T ) = s i .ζ T . (2) If m n ( T ) = , then m n ( T ) ∈ T ab ( ϕ ξn,p,µ ) and the common eigenbasis vector v m n ( T ) is of weight ζ m i ( T ) = γ n .ζ T Proof. First, for i = 1 , · · · , n − 1, if m i ( T ) = , then by the definition of the move m i , T ∈ T ab ( ϕ ξn,p,µ ) and we want to show ζ m i ( T ) = s i .ζ T .Then let us consider the case when w = γ . In this case w moves the box filled by n in the i -th row of tableau T to the end of the ( N − i + 1)-th row. So the only box inthe new tableau γ.T with a different position comparing with the tableau T is the boxfilled by n . Thus the only difference in the new weight associated to γ.T comparingwith ζ T is the eigenvalue of y n . Let ( i , j ) denote the coordinates of the box filled by n in the tableau T . Then the coordinates of the box filled by n in the new tableau γ.T is ( N − i + 1 , µ ( q − p ) + 2 | ξ | + nN − j + 1). Then the eigenvalue of y n in the new weight ζ γ.T associated to γ.T is j − i − | ξ | + nN + N µ ( p − q )2 . So the new weight equals γ.ζ T . (cid:3) Proposition 8.3. If w.T = 0 for some w ∈ W , then φ w v T = 0 .Proof. It is enough to verify the statement when w is the transposition s i or γ n .First, consider the case when w = s i , i = 1 , · · · , n − 1. Suppose φ i v T = 0 for some1 ≤ i ≤ n − φ i v T = 0 and φ i = (1 − y i + y i +1 )(1 + y i − y i +1 ). Then ζ T ( i ) − ζ T ( i + 1) = ± 1. In this case the contents of boxes filled by i and i + 1 differ by1 and hence the two boxes are adjacent and in the same row or in the same column.We have s i .T = 0 in this case. This contradicts the condition. So we have φ i v T = 0.Second, consider the case when w = γ n . Suppose φ n v T = 0 which implies theeigenvalue of y n is ± κ . Since φ n v T = 0 in this case and φ n = ( κ − y n )( κ + 2 y n ).Then the box filled by n is either ( p, µq + | ξ | + nN ) or ( q, − µp + | ξ | + nN ). But by thedefinition of action of γ n on the tableau T , we have in both cases that γ n .T = 0. Thiscontradicts the condition. Hence we have that φ n v T = 0. (cid:3) Remark 8.4. (1) If m i ( T ) = , then φ i v T = cv m i ( T ) up to a nonzero scalar c ∈ C for i = 1 , · · · , n . (2) If m i ( T ) = , then φ i v T = 0 for i = 1 , · · · , n . Example 8.5. In example 5.9, the action of intertwining operators are as follows.The diagonals give the eigenvalue of y i ’s. m m m m m m m 52 12 - 52 12 - 52 12 - 52 12 - 52 12 - 52 12 - - - - - - - m m m m m m Let k be the filling of the cell ( q, b ),we could compute that the eigenvalue of y k is − κ . Similarly, let k be the filling of the cell ( p, a ), it follows the eigenvalue of y k is κ . Furthermore, κ = p − q − a + b .9. Irreducible representations The image F n,p,µ ( V ξ ) is irreducible.Lemma 9.1. Let ϕ and ϕ be two skew shapes in D ( ϕ ) with ϕ → ϕ . Then thereexist standard tableaux T and T with Im ( T ) = ϕ and Im ( T ) = ϕ such that γ n ( T ) = T .Proof. The ϕ → ϕ implies that ϕ is obtained by moving a corner ( i, ϕ i ) of ϕ tothe end of the ( N − i + 1)-th row of ϕ . Since ( i, ϕ ) is a corner of ϕ , there exists astandard tableau T such that ( i, ϕ ) is filled by n . Applying the γ n move to T , let T = γ n ( T ). Then T is a standard tableau with Im ( T ) = ϕ . (cid:3) We show in the following the representation of degenerate affine Hecke algebraobtained through Etingof-Freund-Ma functor is irreducible. Theorem 9.2. The image F n,p,µ ( V ξ ) of a finite dimensional irreducible gl N -module V ξ under the Etingof-Freund-Ma functor is irreducible.Proof. A basis of F n,p,µ ( V ξ ) is indexed by T = { T | T is a standard tableau and Im ( T ) ∈ D ( ϕ ξn,p,µ ) } . It’s obvious to see that the underlying vector space of F n,p,µ ( V ξ ) is isomorphic to span C { v T | T ∈ T } . Let N be a submodule of F n,p,µ ( V ξ ). Then N contains at leastone weight vector of F n,p,µ ( V ξ ). Let v T be a weight vector associated to the tableau T ∈ T ab ( ϕ ξn,p,µ ) and the submodule N contains v T .We show in the following we could get every other weight vector from an arbitraryweight vector v T . We could consider the actions of signed permutations on standardtableaux since the actions of signed permutations on standard tableaux are compat-ible with the actions of intertwining operators on weight vectors. EPRESENTATIONS OF DEGENERATE AHA OF TYPE C n Case 1. For any the standard tableau T ′ with the same shape of the tableau T , thereexists w ∈ S n such that T ′ = w.T . Equivalently v T ′ = cφ ω v T where c ∈ C is nonzero.Case 2. For standard T and T with Im ( T ) → Im ( T ), combining Lemma 8.4 andCase 1, it follows T = ω ( T ) for some ω ∈ W ( BC n ) and hence v T = cφ ω v T where c ∈ C is nonzero.Furthermore, consider two arbitrary standard tableaux T and T in T . Let T be astandard tableaux of shape ϕ . There is a path ϕ → ϕ → · · · → Im ( T ) and hence v T = c φ ω v T . (cid:3) Irreducible representation associated to a skew shape ϕ ξn,p,µ . Define arepresentation L ϕ ξn,p,µ of H n (1 , p − q − µN ) as follows. Let the underlying vectorspace be span C { w T | T ∈ T } . The action of H n (1 , p − q − µN ) is defined by y k w T = ( − cont T ( k ) + s ) w T , (24) s i w T = (1 − cont T ( i ) + cont T ( i + 1)) w s i ( T ) cont T ( i ) − cont T ( i + 1) + 1 cont T ( i ) − cont T ( i + 1) w T , (25) γ n w T = ( p − q − µN − cont T ( n )) w γ n ( T ) cont T ( n ) + ( p − q − µN ) 12 cont T ( n ) w T . (26) Theorem 9.3. The representation F n,p,µ ( V ξ ) is isomorphic to L ϕ ξn,p,µ .Proof. Fix a T ∈ T . Define a map f : F n,p,µ ( V ξ ) → L ϕ ξn,p,µ by f ( v T ) = w t and f ( φ i v T ) = (1 − cont T ( i ) + cont T ( i + 1)) w s i ( T ) . (cid:3) Combinatorial description In this section, we first discuss some properties of a representation of the degenerateaffine Hecke algebra H n (1 , κ ) obtained via the Etingof-Freund-Ma functor, where κ = p − q − µN , and then we show that any representation satisfying these propertiesis the image of some irreducible polynomial representation of GL N via the Etingof-Freund-Ma functor.10.1. Some facts of F n,p,µ ( V ξ ) . Let F = F n,p,µ ( V ξ ) be a representation H n (1 , p − q − µN ) obtained through Etingof-Freund-Ma functor and ζ = ( ζ , · · · , ζ n ) be weightof F such that F ζ = 0. For i = 1 , · · · , n , if there is an increasing sequence i = i
For i = 1 , · · · , n , if | ζ i | ≤ | κ | , then ζ i is fixed, i.e. there is an increasingsequence i = i < i < · · · < i m ≤ n such that | ζ i k − ζ k +1 | = 1 for k = 0 , · · · , m − and ζ i m = ± κ . Property 2. The parameter κ is an integer. If κ is even, then all ζ i ’s, for i =1 , · · · , n , are integers. If κ is odd, then all ζ i ’s, for i = 1 , · · · , n are half integers. Recall that the the cell ( p, a ) in ϕ ξn,p,µ gives the eigenvalue κ and that the cell( q, b ) gives the eigenvalue − κ . Then Property 2 follows.In [Ram04], Ram explored the facts of weights of a semisimple affine Hecke algebrarepresentation. Now let us explore facts of weights in the degenerate case. Let L be an irreducible and Y -semisimple representation of H n (1 , κ ) satisfying Property1 and Property 2 above and ζ be a weight such that L ζ = 0. Then ζ satisfies thefollowing property. Proposition 10.1. If there exist ≤ i < j ≤ n such that ζ i = ζ j , then there exist i < k < j such that ζ k = ζ i + 1 and i < k < j such that ζ k = ζ i − .Proof. Let ζ be a weight such that L ζ = 0. Suppose there exist 1 ≤ i < j ≤ n suchthat ζ i = ζ j and there is no i < k < j such that ζ k = ζ i . We proof by induction on j − i .First, if j − i = 1, then ζ i = ζ i +1 which contradicts theorem 7.9.Second, if j − i = 2, by Theorem 7.9 and Lemma 7.7, it follows ζ i +1 = ζ i ± ζ i +1 ± v be a nonzero weight vector of weight ζ . Proposition 7.16 implies φ i v = φ i +1 v =0. Combining the definition of the intertwining operators, it follows s i v = ∓ v and s i +1 v = ± v and hence ± v = s i s i +1 s i v = s i +1 s i s i +1 v = ∓ v, which is a contradiction.So the base case of the induction is j − i = 3. If ζ i = ζ i +1 ± ζ j − = ζ j ± 1. Lemma7.7 implies the existence of a weight satisfying the condition in the second case whichis a contradiction. So it hold | ζ i − ζ i +1 | = 1 and | ζ j − − ζ j | = 1. If ζ i = ζ i +1 + 1and ζ j − = ζ j + 1, then k = j − k = i + 1. Similarly, if ζ i = ζ i +1 − ζ j − = ζ j − 1, then k = i + 1 and k = j − 1. If ζ i = ζ i +1 ± ζ j − = ζ j ∓ 1, then ζ i +1 = ζ i +2 which contradicts theorem 7.9.Suppose the statement is true for all i − j < m , consider the case j − i = m .Case1. If | ζ i − ζ i +1 | 6 = 1 or | ζ j − − ζ j | 6 = 1 and let v be a nonzero weight vector of weight ζ , then φ i v or φ j − v will be a nonzero weight vector of weight s i ζ or respectively s j − ζ with s i ζ or s j − ζ has ζ i +1 = ζ j or respectively ζ i = ζ j − . Then the k and k exist bythe inductive hypothesis.Case 2. If ζ i = ζ i +1 ± ζ j − = ζ j ∓ 1, this implies ζ i +1 = ζ j − , the statement stillholds by inductive hypothesis.Case 3. If ζ i = ζ i +1 + 1 and ζ j − = ζ j + 1, then k = j − k = i + 1.Case 4. If ζ i = ζ i +1 − ζ j − = ζ j − 1, then k = i + 1 and k = j − (cid:3) Next let us explore another fact of L . Lemma 10.2. Let ζ = [ ζ , · · · , ζ n ] be a weight of L such that L ζ = 0 and ζ satisfies ζ i > | κ | for i = k, · · · , n . Then there is weight ζ ′ = [ ζ , · · · , ζ k − , − ζ n , − ζ n − , · · · , − ζ k +1 , − ζ k ] such that L ζ ′ = 0 .Proof. Let v be a nonzero weight vector of ζ . Acting on v by h = φ n ( φ n − φ n ) · · · ( φ k φ k +1 · · · φ n ) , the vector hv ∈ L ζ ′ and hv = 0 by Lemma 7.7 and Lemma 7.8. (cid:3) Definition 10.3. Let ζ = [ ζ , · · · , ζ n ] be a weight of L such that L ζ = 0 and ζ satisfiesthe condition: if a coordinate ζ i > , then ζ i is fixed, i.e. there exists an increasingsequence i = i < i < · · · < i m ≤ n such that | ζ i k − ζ i k +1 | = 1 and ζ i m = ± κ . Thenwe call ζ is a minimal weight of L . EPRESENTATIONS OF DEGENERATE AHA OF TYPE C n Proposition 10.4. There exists at least one minimal weight ζ = [ ζ , · · · , ζ n ] of L such that L ζ = 0 .Proof. Let ζ be any weight such that L ζ = 0. If 0 < ζ i ≤ | κ | , then ζ i is fixed since L satisfies Property 1. So it suffices to consider the coordinate ζ i > | κ | . We wantto show that starting with any weight ζ such that L ζ = 0, there is an algorithm toobtain a weight ζ ′ such that L ζ ′ = 0 and ζ ′ satisfies the condition: if a coordinate ζ ′ i > 0, then ζ ′ i is fixed.Suppose { ζ r , ζ r , · · · , ζ r l } is the collection of all the coordinates such that ζ r i > | κ | and ζ r i is not fixed, for 1 ≤ r < r < · · · < r l ≤ n . Let v be a nonzero weight vectorof weight ζ . We start with the rightmost coordinate ζ r l in this collection. If r l = n ,there are only the following two cases.Case 1. There exists an increasing sequence r l + 1 = j < j < · · · < j l ≤ n such that | ζ j k +1 − ζ j k | = 1 and ζ j l = ± κ . Then | ζ r l − ζ r l +1 | 6 = 1, otherwise there is an increasingsequence r l = j − < j < j < · · · < j l ≤ n such that | ζ j k +1 − ζ j k | = 1 and ζ j l = ± κ .So φ r l v is a nonzero vector of weight ζ (1) = s r l ζ .Case 2. If ζ r l +1 < − | κ | , then | ζ r l − ζ r l +1 | > φ r l v is a nonzero weightvector of weight ζ (1) = s r l ζ .Then we consider ζ (1) r l +1 and we are in the same situation. Hence we repeat this processfor ( n − r l ) times and obtain a nonzero weight vector ( φ n − · · · φ r l +1 φ r l ) v of weight ζ ( n − r l ) = ( s n − · · · s r l +1 s r l ) ζ . Next, we deal with the second rightmost coordinate ζ r l − = ζ ( n − r l ) r l − in the collectionabove and repeat the process above for ( n − − r l − ) times. We obtain a nonzeroweight vector ( φ n − · · · φ r l − +1 φ r l − )( φ n − · · · φ r l +1 φ r l ) v of weight ζ (2 n − − r l − − r l ) = ( s n − · · · s r l − +1 s r l − )( s n − · · · , s r l +1 s r l ) ζ . Next, we continue to deal with other coordinates in the collection in the order of ζ r l − , ζ r l − , · · · , ζ r and repeat the process for ( n − k − r k ) times for the coordinate ζ r k for k = 1 , · · · , l . We obtain a nonzero weight vector( φ n − l · · · φ r +1 φ r )( φ n − l +1 · · · φ r +1 φ r ) · · · ( φ n − · · · φ r l +1 φ r l ) v of weight ζ ( ln − l ( l − / − r − r ···− r l ) = ( s n − l · · · s r +1 s r )( s n − l +1 · · · s r +1 s r ) · · · ( s n − · · · s r l +1 s r l ) ζ . The weight ζ ( ln − l ( l − / − r − r ···− r l ) satisfies the condition that ζ ( ln − l ( l − / − r − r ···− r l ) i > | κ | i = n − l + 1 , · · · , n . Moreover, for i = 1 , · · · , n − l , it follows either ζ ( ln − l ( l − / − r − r ···− r l ) i < ζ ( ln − l ( l − / − r − r ···− r l ) i is fixed. Combining Lemma 10.2, there is aweight ζ ′ = γ n ( s n − γ n ) · · · ( s n − l +1 · · · s n − γ n ) ζ ( ln − l ( l − / − r − r ···− r l ) such that L ζ ( ln − l ( l − / − r − r ···− rl ) = 0 and satisfying the condition: if ζ ( ln − l ( l − / − r − r ···− r l ) i > , then ζ ( ln − l ( l − / − r − r ···− r l ) i is fixed for any i = 1 , · · · , n . (cid:3) Remark 10.5. Lemma 10.2 and Proposition 10.4 indicate that for any weight ζ suchthat L ζ = 0 and a nonzero v ∈ L ζ , there is a nonzero weight vector φ ω v ∈ L ζ ′ suchthat ζ ′ satisfies the condition in Proposition 10.4. Example 10.6. Let ζ = [ − , , , , , − , and v ∈ L is a nonzero weight vectorof weight ζ . Locate the collection of all the coordinates which are positive and notfixed: { ζ = 4 , ζ = 5 , ζ = 6 } , i.e. there are three coordinates with r = 3 , r = 4 and r = 5 . We deal with these coordinates from right to left. First, we deal with therightmost coordinate ζ = 6 in this collection and apply the process for ( n − r ) = 2 times. We obtain a nonzero weight vector ( φ n − · · · φ r ) v = ( φ φ ) v of weight ζ ( n − r ) = ζ (2) = ( s s ) ζ = [ − , , , , − , , . Then we work on with the coordinate ζ = ζ (2)4 = 5 and apply the process for ( n − − r ) times. We obtain a nonzero weight vector ( φ n − · · · φ r )( φ n − · · · φ r ) v = ( φ φ )( φ φ ) v of weight ζ (2 n − − r − r ) = ζ (4) = ( s s ) ζ (2) = ( s s )( s s ) ζ = [ − , , , − , , , . Finally, we deal with the coordinate ζ = ζ (4)3 = 4 and apply the process for n − − r times. We obtain a nonzero weight vector ( φ n − · · · φ r )( φ n − · · · φ r )( φ n − · · · φ r ) v = ( φ φ )( φ φ )( φ φ ) v of weight ζ (3 n − − r − r − r ) = ζ (6) = ( s s ) ζ (4) = [ − , , − , , , , . Now the weight ζ (6) satisfies the condition in Lemma 10.2 with ζ (6) i > | κ | for i =5 , , . Moreover, for each i = 1 , · · · , , either ζ (6) i < or that ζ (6) i is fixed.Applying Lemma 10.2, we obtain a nonzero weight vector φ ( φ φ )( φ φ φ )( φ φ )( φ φ )( φ φ ) v of weight ζ ′ = γ ( s γ )( s s γ ) ζ (6) = [ − , , − , , − , − , − . Example 10.7. Let ζ = [0 , , − , , − , , and v ∈ L is a nonzero weight vectorof weight ζ . There are three coordinates ζ = 4 , ζ = 6 and ζ = 5 satisfying thecondition that i = 2 , , , there is no increasing sequence i < i < · · · < i l ≤ n suchthat | ζ i k +1 − ζ i k | = 1 and | ζ i l | = ± κ . Starting with the coordinate with maximal index i = 6 and applying the intertwining operators, it follows [0 , , − , , − , , 1] [0 , , − , , − , , 5] [0 , , − , − , , , 5] [0 , − , − , , , , s s s s s s and by Lemma 10.2 [0 , − , − , , , , 5] [0 , − , − , , − , , 6] [0 , − , − , , − , − , 4] [0 , − , − , , − , − , − s s γ s γ γ EPRESENTATIONS OF DEGENERATE AHA OF TYPE C n Let ζ ′ = [0 , − , − , , − , − , − . Then there is a nonzero weight vector φ ( φ φ )( φ φ φ )( φ φ φ )( φ φ ) φ v ∈ L ζ ′ . Remark 10.8. For any minimal weight ζ of F = F n,p,µ ( V ξ ) such that F ζ = 0 , let T ζ be the corresponding standard tableau. Then Im ( T ζ ) is the minimal shape ϕ ξn,p,µ of F n,p,µ ( V ξ ) . Before introducing the third property of F n,p,µ ( V ξ ), we need the following definitionand lemma. Definition 10.9. Let ζ = [ ζ , · · · , ζ n ] be a weight. If a coordinate ζ i , i = 1 , , · · · , n ,satisfies the condition that there is no i < k ≤ n such that ζ k = ζ i ± , then thecoordinate ζ i is a corner of ζ . Remark 10.10. Let ζ = [ ζ , · · · , ζ n ] and T ζ is the corresponding standard tableau.For i = 1 , · · · , n , ζ i is a corner of ζ if and only if T ( i ) is a southeastern corner of Im ( T ζ ) . Example 10.11. Let ζ = [0 , − , − , , − , − , − . Then ζ = − , ζ = 1 , ζ = − and ζ = − are corners of ζ . The corresponding standard tableau T ζ has southeasterncorners , , and . Lemma 10.12. Let L be an irreducible and Y -semisimple representation of H n (1 , κ ) satisfying Property 1. Let ζ be a minimal weight of L such that L ζ = 0 . For i =1 , · · · , n , if the coordinate ζ i is a corner of ζ , then ζ i = ± κ or ζ < − | κ | .Proof. First, since L satisfies Property 1, if | ζ i | < | κ | , then ζ i is fixed, i.e. there isan increasing sequence i = i < i < · · · < i m ≤ n such that | ζ i k − ζ k +1 | = 1 for k = 0 , · · · , m − ζ i m = ± κ . This contradicts the fact that ζ i is a corner of ζ .Second, suppose ζ i > | κ | . Since ζ is a minimal weight, ζ i if fixed, which againcontradicts the fact that ζ i is a corner. (cid:3) Now we introduce the third property of F n,p,µ ( V ξ ). Property 3. Let ζ be a minimal weight such that F ζ = 0 . If ζ k is the rightmostcoordinate equal to | κ | and ζ r is the rightmost coordinate equal to − | κ | , then at leastone of these two coordinates is not a corner.Proof. Let T ζ be the corresponding standard tableau of weight ζ . Since ζ is a minimalweight, the shape Im ( T ζ ) is the minimal shape ϕ = ϕ ξn,p,µ . So it suffices to show thatit is impossible for T ζ to have T ζ ( k ) and T ζ ( r ) at southeastern corners simultaneously,equivalently, it is impossible for ϕ to have a southeast corner at eigenvalue κ and asoutheastern corner at eigenvalue − κ simultaneously. Let p ≤ q , a = µq + | ξ | + nN and b = − µp + | ξ | + nN . Suppose ϕ simultaneously has a southeast corner at eigenvalue κ and a southeasterncorner at eigenvalue − κ , then p < q and a > b follow. In this case, ϕ has cell ( p, a )at eigenvalue − | κ | and cell ( q, b ) at eigenvalue | κ | . Furthermore, the fact that cell( p, a ) is a southeastern corner indicates ξ (2)1 = ξ q +1 = b . The fact that cell ( q, b ) ∈ ϕ indicates ξ (1) q = ξ q < b . This contradicts ξ ∈ P + ≥ . (cid:3) Combinatorial description of irreducible representations in M . Let M ( H n (1 , κ )) be collection of Y -semisimple representations of H n (1 , κ ) satisfyingProperties 1-3. In this subsection, we show that any irreducible representation in M ( H n (1 , κ )) is isomorphic to the image F n,p,µ ( V ξ ) for a tuple of n, p, µ and some ξ ∈ P + ≥ .Let L ∈ M ( H n (1 , κ )) be irreducible and ζ be a minimal weight such that L ζ = 0.Recall, if ζ i ≥ 0, then there is an increasing sequence k < · · · < k m such that ζ k i +1 = ζ k i ± ζ k m = ± κ . The weight ζ gives a standard tableau T ζ suchthat ζ k = − cont T ζ ( k ) + s for some fixed number s where s − κ is an integer. Let Im ( T ζ ) = ν/β such that β < ν and β ℓ ( ν ) < ν ℓ ( ν ) . Let us explore in different casesdepending on corners. According to Lemma 10.11, if ζ i is a corner of ζ , for some i = 1 , · · · , n , then ζ i = ± κ or ζ i < − | κ | . For any minimal ζ , there is at least onecorner of ζ . Let the coordinate ζ r be the corner of ζ such that i ( r ) is the maximal of { i ( i ) | ζ i is corner of ζ } and the coordinate ζ r is the corner of ζ such that i ( r ) is thesecond largest number in { i ( i ) | ζ i is corner of ζ } if ζ r exists. It is obvious ζ r < ζ r .There are the following cases. If ζ r = | κ | , then ζ r < − | κ | or ζ r doesn’t exist.By Lemma 10.11, if ζ r = | κ | and ζ r = − | κ | , then ζ violates Property 3. When ζ r = − | κ | , ζ r < − | κ | or there is no ζ r . When ζ r < − | κ | , ζ r < − | κ | or ζ r doesn’texist. So let us discuss in five cases. Case 1. The corner ζ r = | κ | and the corner ζ r < − | κ | .Denote T ζ ( r ) = ( i , j ) and T ζ ( r ) = ( i , ν i ). Let j = i + s + | κ | . In this case, settwo rectangles ( a p ) = (( ν − j ) i )and ( b q ) = (( ν − j ) i ) . Claim 10.13. Following the setting above, the number ν i − j − j ≥ .Proof. Since ζ r is a corner, there exists a weight ˜ ζ such that L ˜ ζ = 0, Im ( T ˜ ζ ) = Im ( T ζ )and T ˜ ζ ( n ) = ( i , ν i ), where T ˜ ζ denotes the standard tableau given by the weight ˜ ζ .Let v be a nonzero weight vector of weight ˜ ζ . Since ˜ ζ n = ± κ , it follows that φ n v isa nonzero weight vector of weight γ n ˜ ζ . Moreover, the standard tableau T γ n ˜ ζ given by γ n ˜ ζ n satisfies that Im ( T γ n ˜ ζ )) = Im ( T ζ ) \ { ( i , ν i ) } ∪ { ( i + 1 , j + j − ν i + 1) } since ( γ n ˜ ζ ) n = − ˜ ζ n . Lemma 10.1 implies T γ n ˜ ζ is a standard tableau and hence Im ( T γ n ˜ ζ ) is a skew shape. This fact forces j + j − ν i + 1 ≤ ν i − j − j ≥ . (cid:3) EPRESENTATIONS OF DEGENERATE AHA OF TYPE C n Set ξ (1) = ( ξ (1)1 , · · · , ξ (1) i ) with ξ (1) k = β k + ν − j − j , for k = 1 , · · · , i and ξ (2) = ( ξ (2)1 , · · · , ξ (2) i ) with ξ (2) k = ν − ν i − k +1 , for k = 1 , · · · , i . Furthermore, set ξ = ( ξ , · · · , ξ i + i ) with ξ k = ξ (1) k , for k = 1 , · · · , i and ξ k = ξ (2) k − i , for k = i + 1 , · · · , i + i . Remark 10.14. Claim 10.13 implies the following two facts. (1) It follows ν − j − j ≥ . (2) The inequality ν − ν i = ξ (2)1 ≤ ξ (1) i = β i + ν − j − j holds and hence ξ isa well-defined Young diagram. Example 10.15. Continue Example 10.7. An irreducible representation L in M ( H (1 , − ,we start with a minimal weight ζ = [0 , − , − , , − , − , − and the standard tableauof ζ . The corners of ζ are ζ = − , ζ = 1 , ζ = − and ζ = − . Furthermore, ζ r = ζ = 1 and ζ r = ζ = − i j i j − − s = − ν = (5 , , , β = (3 , ν = 5 i = 4 , j = 1 i = 3 , j = 2 Place the southeastern corner of (( ν − j ) i ) at the cell ( i , j ) and northeastern cornerof ( ν − j ) i at the cell (1 , ν ) . The gray part on the left forms ξ (1) and the gray parton the right forms ξ (2) . i j ν − − a p ) = (4 )( b q ) = (3 ) ξ (1) = (5 , , , ξ (2) = (2 , , Furthermore, we obtain other parameters of Etingof-Freund-Ma functor as N = p + q = 7 , p = 3 and µ = a − bN = . ξ = (5 , , , , , , ξ (1) ξ (2) ξ (1) ξ (2) b aq p Case 2. The corner ζ r = − | κ | and the corner ζ r < − | κ | .Denote T ζ ( r ) = ( i , j ) and T ζ ( r ) = ( i , ν i ). Let j = i + s − | κ | . In this case, settwo rectangles ( a p ) = (( ν − j ) i )and ( b q ) = (( ν − j ) i ) . We have a similar claim to that in Case 1. Claim 10.16. Following the setting above, the number ν i − j − j ≥ . The proof is the same with that in Case 1.Similarly, let ξ (1) = ( ξ (1)1 , · · · , ξ (1) i ) with ξ (1) k = β k + ν − j − j , for k = 1 , · · · , i and ξ (2) = ( ξ (2)1 , · · · , ξ (2) i ) with ξ (2) k = ν − ν i − k +1 , for k = 1 , · · · , i . Furthermore, set ξ = ( ξ , · · · , ξ i + i ) with ξ k = ξ (1) k , for k = 1 , · · · , i and ξ k = ξ (2) k − i , for k = i + 1 , · · · , i + i . Example 10.17. Let L be an irreducible representation in M ( H (1 , − with aminimal weight ζ = [ − , , , − , − , − , − and the standard tableau of ζ . Thecorners of ζ are ζ = − , ζ = − and ζ = − . Furthermore, ζ r = ζ = − and ζ r = ζ = − − − i j i j s = − ν = (5 , , β = (4 , , i = 3 , j = 3 i = 2 , j = 0 Place the southeastern corner of ( b q ) at the cell ( i , j ) and northeastern corner of ( a p ) at the cell (1 , ν ) . The gray part on the left forms ξ (1) and the gray part on theright forms ξ (2) . EPRESENTATIONS OF DEGENERATE AHA OF TYPE C n i j ν − − 11 ( a p ) = (2 )( b q ) = (5 ) ξ (1) = (6 , , ξ (2) = (1 , Furthermore, we obtain other parameters of Etingof-Freund-Ma functor as N = q + p = 5 , q = 3 and µ = b − aN = . ξ = (6 , , , , ξ (1) ξ (2) b aq pξ (1) ξ (2) Case 3. The corner ζ r = | κ | and the corner ζ r doesn’t exist. Let j = s + | κ | .Then the cell (0 , j ) on the diagonal of eigenvalue − | κ | . We explore the following intwo subcases. Case 3a. j ≥ 1. Set two rectangles ( a p ) = ( j )and ( b q ) = ( ν ℓ ( ν )+11 ) . Moreover, ξ = ( ξ , · · · , ξ ℓ ( ν ) ) with ξ = ν + j and ξ k = β k − . Example 10.18. Let L be an irreducible representation in M ( H (1 , − with aminimal weight ζ = [ − , , , , , , such that L ζ = 0 . There is only one corner ζ = 1 . So ζ r = ζ = 1 = | κ | . The standard tableau of ζ is as follows. − 142 3 75 6 j s = 1 ν = (3 , , β = (2 , , ℓ ( ν ) = 3 j = 2 The two rectangles are ( a p ) = (2 ) and ( b q ) = (3 ) . Place the southeastern corner of ( b q ) at T ζ ( r ) = T ζ (7) and the northwestern corner of ( a p ) at the cell (0 , ν + 1) . Thegray area forms ξ . j ν + 10 1 − 142 3 75 6 ( a p ) = (2 )( b q ) = (3 ) ξ = (5 , , , , Furthermore, we obtain other parameters of Etingof-Freund-Ma functor as N = p + q = 5 , p = 1 and µ = a − bN = − . ξ = (5 , , , , ξb aq p Case 3b. j ≤ 0. Set two rectangles( a p ) = (1 )and ( b q ) = (( ν − j + 1) ℓ ( ν )+1 ) . Moreover, ξ = ( ξ , · · · , ξ ℓ ( ν ) ) with ξ = ν − j + 2 and ξ k = β k − − j + 1. Example 10.19. Let L be an irreducible representation in M ( H (1 , − with aminimal weight ζ = [0 , − , − , , , , such that L ζ = 0 . There is only one corner ζ = 1 . So ζ r = ζ = 1 = | κ | . The standard tableau of ζ is as follows. − j 14 2375 6 s = − ν = (2 , , , β = (1 , , , ℓ ( ν ) = 4 j = 0 The two rectangles are ( a p ) = (1 ) and ( b q ) = (3 ) . Place the southeastern corner of ( b q ) at T ζ ( r ) = T ζ (7) and the northwestern corner of ( a p ) at the cell (0 , ν + 1) . Thegray area forms ξ . EPRESENTATIONS OF DEGENERATE AHA OF TYPE C n − j ν + 1 14 2375 6 ( a p ) = (1 )( b q ) = (3 ) ξ = (4 , , , , , Furthermore, we obtain other parameters of Etingof-Freund-Ma functor as N = p + q = 6 , p = 1 and µ = a − bN = − . ξ = (4 , , , , , ξ b aq p Case 4. The corner ζ r = − | κ | and there is no corner ζ r . Set j = s − | κ | . Thenthe cell (0 , j ) is on the diagonal of eigenvalue | κ | . Let us discuss in two subcases. Case 4a. When j ≥ 1. Set two rectangles( a p ) = ( j )and ( b q ) = ( ν ℓ ( ν )+11 ) . Moreover, ξ = ( ξ , · · · , ξ ℓ ( ν ) ) with ξ = ν + j and ξ k = β k − . Example 10.20. Let L be an irreducible representation in M ( H (1 , − with aminimal weight ζ = [4 , , , − , , , − such that L ζ = 0 . There is only one corner ζ = − . So ζ r = ζ = − − | κ | . The standard tableau of ζ is as follows. − j s = 3 ν = (6 , β = (5 , ℓ ( ν ) = 2 j = 2 The two rectangles are ( a p ) = (2 ) and ( b q ) = (6 ) . Place the southeastern corner of ( b q ) at T ζ ( r ) = T ζ (7) and the northwestern corner of ( a p ) at cell (0 , ν + 1) = (0 , .The gray area forms ξ . − j ν + 10 ( a p ) = (2 )( b q ) = (6 ) ξ = (8 , , , Furthermore, we obtain other parameters of Etingof-Freund-Ma functor as N = q + p = 4 , q = 3 and µ = b − aN = 1 . ξ = (8 , , , ξb aq p Case 4b. When j ≤ 0. Set two rectangles( a p ) = (1 )and ( b q ) = (( ν − j + 1) ℓ ( ν )+1 ) . Moreover, ξ = ( ξ , · · · , ξ ℓ ( ν ) ) with ξ = ν − j + 2 and ξ k = β k − − j + 1. Example 10.21. Let L be an irreducible representation in M ( H (1 , − with aminimal weight ζ = [0 , − , , , − , , − such that L ζ = 0 . There is only one corner ζ = − . So ζ r = ζ = − − | κ | . The standard tableau of ζ is as follows. j − 14 23 756 s = 1 ν = (4 , β = (1 , ℓ ( ν ) = 2 j = 0 The two rectangles are ( a p ) = (1 ) and ( b q ) = (5 ) . Place the southeastern corner of ( b q ) at T ζ ( r ) = T ζ (7) and the northwestern corner of ( a p ) at the cell (0 , ν + 1) . Thegray area forms ξ . j ν + 1 − 14 23 756 ( a p ) = (1 )( b q ) = (5 ) ξ = (6 , , , Furthermore, we obtain other parameters of Etingof-Freund-Ma functor as N = q + p = 4 , q = 3 and µ = b − aN = 1 . ξ = (6 , , , ξ b aq p EPRESENTATIONS OF DEGENERATE AHA OF TYPE C n Case 5. The corner ζ r < − | κ | . Let j = ν ℓ ( ν )+ | κ | + ζ r and j = ν ℓ ( ν ) − | κ | + ζ r .Set two rectangles ( a p ) = (( ν − j ) ℓ ( ν ) )and ( b q ) = (( ν − j ) ℓ ( ν ) ) . Claim 10.22. According to the setting above, the number ν ℓ ( ν ) − j − j ≥ Proof. There exist a weight ˜ ζ such that L ˜ ζ = 0, Im ( T ˜ ζ ) = Im ( T ζ ) and T ˜ ζ ( n ) =( ℓ ( ν ) , ν ℓ ( ν ) ). Let v be a nonzero weight vector of weight ˜ ζ . Since ζ r < − | κ | , weobtain a nonzero weight vector φ n v of weight γ n ˜ ζ . Moreover, Im ( T γ n ˜ ζ ) = Im ( T ζ ) \ { ( ℓ ( ν ) , ν ℓ ( ν ) ) } ∪ { ( ℓ ( ν ) + 1 , ℓ ( ν ) − ν ℓ ( ν ) + 2 s + 1) } . Since Im ( T γ n ˜ ζ ) is a skew shape, it follows 2 ℓ ( ν ) − ν ℓ ( ν ) + 2 s + 1 ≤ 1. Applying j = ν ℓ ( ν ) + | κ | + ζ r and j = ν ℓ ( ν ) − | κ | + ζ r , the statement ν ℓ ( ν ) − j − j ≥ (cid:3) Set ξ (1) = ( ξ (1)1 , · · · , ξ (1) ℓ ( ν ) ) with ξ (1) k = β k + ν − j − j for k = 1 , · · · , ℓ ( ν ), ξ (2) = ( ξ (2)1 , · · · , ξ (2) ℓ ( ν ) ) with ξ (2) k = ν − ν ℓ ( ν ) − k +1 for k = 1 , · · · , ℓ ( ν ) and ξ = ( ξ , · · · , ξ ℓ ( ν ) ) with ξ k = ξ (1) k for k = 1 , · · · , ℓ ( ν ) and ξ k = ξ (2) k − ℓ ( ν ) for k = ℓ ( ν ) + 1 , · · · , ℓ ( ν ). Remark 10.23. Claim 10.22 implies the following two facts. (1) It follows ν − j − j ≥ . (2) The inequality ν − ν ℓ ( ν ) = ξ (2)1 ≤ ξ (1) ℓ ( ν ) = ν − j − j holds and hence ξ is awell-defined Young diagram. Example 10.24. Let L be an irreducible representation in M ( H (1 , − with aminimal weight ζ = [ − , − , − , − , − , − , − such that L ζ = 0 . The corners of ζ are ζ = − , ζ = − and ζ = − . So ζ r = ζ = − . The standard tableau of ζ isas follows. − − s = − ν = (4 , , β = (2 , , ℓ ( ν ) = 3 The two rectangles ( a p ) = (3 ) and ( b q ) = (5 ) follow. Place the northeastern cornerof ( a p ) = (3 ) at the cell (1 , ν ) and the southeastern corner of ( b q ) = (5 ) at the cell ( ℓ ( ν ) , ℓ ( ν ) + | κ | + s . The gray area on the left forms ξ (1) and the gray area on theright forms ξ (2) . − ℓ ( ν ) ℓ ( ν ) + s + | κ | ν ( a p ) = (3 )( b q ) = (5 ) ξ (1) = (6 , , ξ (2) = (2 , , So the three shapes ( a p ) , ( b q ) and ξ are set as follows. The other parameters ofEtingof-Freund-Ma functor are set as N = 6 , p = 3 and µ = 1 / . ξ (1) ξ (2) b aq p ξ (1) ξ (2) ξ = (6 , , , , , Remark 10.25. When we fix the number n , for different input ( ξ, N, p, µ ) , we couldactually get isomorphic H n -modules. Consider the following example of representa-tions of H (1 , − .Let ξ = (3 , , , N = 4 , p = 1 and µ = − .In this case, a = µq + | ξ | + nN = 2 and b = − µp + | ξ | + nN = 3 . Then the image F = F , , − ( V ξ ) is an H (1 , − -module with the following minimal shape ϕ ξ , , − =(5 , , / (3 , , . ξb aq p Then the basis is indexed by the standard tableaux on the skew shapes: (5 , , /ξ , (4 , , , /ξ and (3 , , , /ξ . There is a minimal weight ζ = [ , − , − ] such that F ζ = 0 . Now let us recover a functor F n,p ′ ,µ ′ such that F n,p ′ ,µ ′ ( V ξ ′ ) is an H (1 , − -module with a minimal weight ζ = [ , − , − ] . According to Case 1, ( a ′ p ′ ) = (3 ) , ( b ′ q ′ ) = (3 ) , ξ ′ = (4 , , and µ ′ = 0 . ξ ′ b ′ a ′ q ′ p ′ − Other Y -semisimple representations. The image of the Etingof-Freund-Ma functor does not exhaust all the Y -semisimple representations. The follow-ing are two examples of Y -semisimple H n (1 , κ ) representation which are not in M ( H n (1 , κ )). Example 10.26. 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