Higher level q -oscillator representations for U q ( C (1) n ), U q ( C (2) (n+1)) and U q ( B (1) (0,n))
aa r X i v : . [ m a t h . QA ] J u l HIGHER LEVEL q -OSCILLATOR REPRESENTATIONS FOR U q ( C (1) n ) , U q ( C (2) ( n + 1)) AND U q ( B (1) (0 , n )) JAE-HOON KWON AND MASATO OKADO
Abstract.
We introduce higher level q -oscillator representations for the quantum affine(super)algebras of type C (1) n , C (2) ( n + 1) and B (1) (0 , n ). They are constructed from thefusion procedure from the fundamental q -oscillator representations obtained through thestudies of the tetrahedron equation. We prove that they are irreducible for type C (1) n and C (2) ( n + 1), and give their characters. Introduction
Let g be an affine Lie algebra and U q ( g ) the Drinfeld-Jimbo quantum group (withoutderivation) associated to it. For a node r of the Dynkin diagram of g except 0 and a positiveinteger s there exists a family of finite-dimensional U q ( g )-modules W r,s called Kirillov-Reshetikhin modules. They have distinguished properties. One of them is the existence ofcrystal bases in Kashiwara’s sense (see [1, 5, 18] and references therein). B (1) n ◦ ❯❯❯❯❯❯ ◦ n − ◦ / / ◦ n ◦ ✐✐✐✐✐✐ B (1) (0 , n ) ◦ / / ◦ n − ◦ / / • n D (1) n ◦ ❚❚❚❚❚❚ ◦ n − ❥❥❥❥❥❥ ◦ n − ◦◦ ❥❥❥❥❥❥ ◦ n ❚❚❚❚❚❚ C (1) n ◦ / / ◦ n − ◦ o o ◦ n D (2) n +1 ◦ o o ◦ n − ◦ / / ◦ n C (2) ( n + 1) • o o ◦ n − ◦ / / • n Table 1.
Dynkin diagrams of ( g , g )Consider the affine Lie algebras g = B (1) n , D (1) n , D (2) n +1 , whose Dynkin diagrams are given inthe left side of Table 1. The Kirillov-Reshetikhin modules corresponding to the node n andthe integer 1 have a simple structure. Let V be a two dimensional vector space. The actionof U q ( g ) on W n, has an easy description on V ⊗ n . It is irreducible when g = B (1) n , D (2) n +1 , butfor g = D (1) n it decomposes into two components; V ⊗ n = W n, ⊕ W n − , . For a quantum J.-H.K. is supported by the National Research Foundation of Korea(NRF) grant funded by the Koreagovernment(MSIT) (No. 2019R1A2C108483311 and 2020R1A5A1016126). M.O. is supported by Grants-in-Aid for Scientific Research No. 19K03426. This work was partly supported by Osaka City UniversityAdvanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and TheoreticalPhysics JPMXP0619217849). group U q ( g ) we can consider the quantum R matrix. We introduce a spectral parameter x to the representation W n, , and denote the associated representation by W n, ( x ). Let ∆ bethe coproduct and ∆ op its opposite. Then the quantum R matrix R ( x/y ) is defined as anintertwiner of ∆ and ∆ op , namely, linear operator satisfying R ( x/y )∆( u ) = ∆ op ( u ) R ( x/y )for any u ∈ U q ( g ) on W n, ( x ) ⊗ W n, ( y ). ( R is found to depend only on x/y .)In [15], Kuniba and Sergeev initiated an attempt to obtain quantum R matrices fromthe solution to the tetrahedron equation, three dimensional analogue of the Yang-Baxterequation. Let L be a solution of the tetrahedron equation. It is a linear operator on F ⊗ V ⊗ V where F is an infinite-dimensional vector space spanned by { | m i | m ∈ Z ≥ } . By composingthis L n times and applying suitable boundary vectors in F and F ∗ , they obtained linearoperators on ( V ⊗ n ) ⊗ ( V ⊗ n ) satisfying the Yang-Baxter equation. The commuting symmetryalgebras were found to be U q ( B (1) n ) , U q ( D (1) n ) or U q ( D (2) n +1 ). The reason they had variationswas that there were two choices of boundary vectors in each F and F ∗ corresponding to theshapes of the Dynkin diagrams at each end.To the tetrahedron equation, there is yet another solution R , which is a linear operatoron F ⊗ . In [13], Kuniba and the second author performed the same scheme to R andconstructed linear operators on ( F ⊗ n ) ⊗ ( F ⊗ n ). For the symmetry algebra this time, theyfound U q ( C (1) n ) , U q ( D (2) n +1 ) and U q ( A (2)2 n ). They called these representations on W = F ⊗ n q -oscillator ones. To be precise, for type C (1) n there are two irreducible components W + , W − ,so one can think W of either W + or W − . By construction, the q -oscillator representation W is a bosonic analogue of W n, , and it is natural to ask whether we have a higher level q -oscillator representation corresponding to W n,s for s ≥
1. However, there is a difficultyin understanding W since they do not have a suitable classical limit ( q →
1) for type D (2) n +1 and A (2)2 n .In this paper, we first resolve this difficulty by considering W for these two types as q -oscillator representations over quantum affine superalgebras g given in the right side ofTable 1 by using the twistor on quantum covering groups [4]. The filled nodes in the Dynkindiagrams signify anisotropic odd simple roots. If they were not filled, the third Dynkindiagram would be D (2) n +1 and the first one A (2) † n , where the latter is the same diagram as A (2)2 n but the opposite labeling of nodes. We then investigate the quantum R matrices for W ( x ) ⊗ W ( y ) and apply the fusion construction. As a result, we obtain a higher levelrepresentation W ( s ) for any s ∈ Z > and each U q ( C (1) n ) , U q ( C (2) ( n + 1)) and U q ( B (1) (0 , n )).Our main purpose in this paper is to prove the irreducibility of W ( s ) and compute itscharacter for U q ( C (1) n ) and U q ( C (2) ( n + 1)). We investigate the crystal base of W ( s ) in detailto show this. We further prove that W ( s ) is classically irreducible, that is, irreducible asa module over the subalgebra generated by e i , f i , k i for i = 0. Rather surprisingly, thiscoincides with the fact that the corresponding W n,s is classically irreducible. We also giveconjectures on the irreducibility of W ( s ) and its character formula for B (1) (0 , n ).We would like to remark that the correspondence between W n,s and W ( s ) as representa-tions of finite-dimensional simple Lie (super)algebras after a classical limit, appears in thecontext of super duality [3]. The theory of super duality is an equivalence between certain parabolic Bernstein-Gelfand-Gelfand categories of classical Lie (super)algebras of infinite-rank. As a special case, this yields an equivalence between the categories for G ∞ and G ∞ ,where ( G ∞ , G ∞ ) = ( B ∞ , B (0 , ∞ )) , ( D ∞ , C ∞ ). Their Dynkin diagrams are given in Table 2.Let G n and G n denote the subalgebras of G ∞ and G ∞ of finite rank n , respectively. Let V ∞ be a given integrable highest weight G ∞ -module. Under this equivalence, it corresponds toan irreducible highest weight G ∞ -module, say W ∞ , called an oscillator representation. Byapplying a truncation functor to V ∞ and W ∞ , we also obtain irreducible modules V n and W n of G n and G n , respectively. B ∞ ◦ ◦ ◦ / / ◦ B (0 , ∞ ) ◦ ◦ ◦ / / • D ∞ ◦ ❥❥❥❥❥❥ ◦ ◦ ◦ ◦ ❚❚❚❚❚❚ C ∞ ◦ ◦ ◦ o o ◦ Table 2.
Dynkin diagrams of ( G ∞ , G ∞ )Let ( g , g ) be one of the pairs of affine Lie (super)algebras ( B (1) n , B (1) (0 , n )), ( D (1) n , C (1) n ),( D (2) n +1 , C (2) ( n + 1)) in Table 1. Let G n and G n be the subalgebra of g and g correspondingto I \ { } , respectively. Assume that g = D (1) n , D (2) n +1 . Now we see that if V n is the classicallimit of a classically irreducible Kirillov-Reshetikhin U q ( g )-module, then W n correspondsto the classical limit of a higher level q -oscillator U q ( g )-module in Theorems 5.1 and 5.20.The character formula in Conjecture 5.22 is based on this observation in case of ( g , g ) =( B (1) n , B (1) (0 , n )), which is true for s = 2. We strongly expect that there is a quantum affineanalogue of super duality which relates the category of finite-dimensional U q ( g )-modules anda suitable category of infinite-dimensional U q ( g )-modules including the q -oscillator modules,and hence explains the correspondence in this paper.The paper is organized as follows: In Section 2, we briefly review the notion of quantumsuperalgebras. In Section 3, we construct a level one q -oscillator representation W of U q ( g )and study some of its properties including the crystal base. In Section 4, we introducethe quantum R matrix on W ( x ) ⊗ W ( y ) and apply fusion construction to define W ( s ) .In Section 5, we prove the irreducibility of W ( s ) and give its character formula when g = C (1) n , C (2) ( n + 1). A conjecture when g = B (1) (0 , n ) is also given. In Appendix A, we explainhow to construct a level one q -oscillator representation of U q ( g ) when g = C (2) ( n + 1) and B (1) (0 , n ) from the one for D (2) n +1 and A (2) † n in [13], respectively, by using the quantumcovering groups and twistor [4]. In Appendices B and C, we construct the quantum R matrix on W ( x ) ⊗ W ( y ) for U q ( g ) from the one in [13]. Acknowledgements
Part of this work was done while the first author was visiting OsakaCity University. He would like to thank Department of Mathematics in OCU for its supportand hospitality. The second author would like to thank Atsuo Kuniba for the collaboration[13] on which this work is based.
JAE-HOON KWON AND MASATO OKADO Quantum superalgebras
Variant of q -integer. Throughout the paper, we let q be an indeterminate. Following[4], we introduce variants of q -integer, q -factorial and q -binomial coefficient. Let ǫ = ± m ∈ Z ≥ , we set [ m ] q,ǫ = ( ǫq ) m − q − m ǫq − q − . For m ∈ Z ≥ , set [ m ] q,ǫ ! = [ m ] q,ǫ [ m − q,ǫ · · · [1] q,ǫ ( m ≥ , [0] q,ǫ ! = 1 . For integers m, n such that 0 ≤ n ≤ m , we define " mn q,ǫ = [ m ] q,ǫ ![ n ] q,ǫ ![ m − n ] q,ǫ ! . They all belong to Z [ q, q − ]. Let A be the subring of Q ( q ) consisting of rational functionswithout a pole at q = 0. Then we have[ m ] q,ǫ ∈ q − m (1 + qA ) , [ m ] q,ǫ ! ∈ q − m ( m − / (1 + A ) , " mn q,ǫ ∈ q − n ( m − n ) (1 + qA ) . We simply write [ m ] = [ m ] q, , [ m ]! = [ m ] q, ! and " mn = " mn q, .2.2. Quantum (super)algebra U q ( sl ) and U q ( osp | ) . The quantum (super)algebras U q ( sl ) ( ǫ = 1) and U q ( osp | ) ( ǫ = −
1) are defined as a Q ( q )-algebra generated by e, f, k ± satisfying the following relations: kk − = k − k = 1 , kek − = q e, kf k − = q − f, ef − ǫf e = k − k − q − q − . Set e ( m ) = e m / [ m ] q,ǫ ! and f ( m ) = f m / [ m ] q,ǫ !. We will use the following formula. Proposition 2.1. e ( m ) f ( n ) = X j ≥ ǫ mn − j ( j +1) / [ j ] q,ǫ ! f ( n − j ) j − Y l =0 ( ǫq ) j − m − n − l k − q − j + m + n + l k − q − q − ! e ( m − j ) . Proof.
The U q ( sl ) ( ǫ = 1) case is derived easily from (1.1.23) of [12]. The U q ( osp | )( ǫ = −
1) case can be shown by induction. (cid:3)
Quantum affine (super)algebras U q ( C (1) n ) , U q ( C (2) ( n + 1)) , U q ( B (1) (0 , n )) . Set I = { , , . . . , n } . In this paper, we consider the following three Cartan data ( a ij ) i,j ∈ I , or Dynkindiagrams (cf. [9]), and ( d i ) i ∈ I such that d i a ij = d j a ji for i, j ∈ I . • C (1) n : ◦ / / ◦ n − ◦ o o ◦ n ( a ij ) i,j ∈ I = − − −
1. . . − − − ( d i ) i ∈ I = (2 , , . . . , , • C (2) ( n + 1): • o o ◦ n − ◦ / / • n ( a ij ) i,j ∈ I = − − −
1. . . − − − ( d i ) i ∈ I = (cid:0) , , . . . , , (cid:1) • B (1) (0 , n ): ◦ / / ◦ n − ◦ / / • n ( a ij ) i,j ∈ I = − − −
1. . . − − − ( d i ) i ∈ I = (2 , , . . . , , ) . Let d = min { d i | i ∈ I } . For i ∈ I , let q i = q d i , and let p ( i ) = 0 , p ( i ) ≡ d i (mod 2). Set[ m ] i = [ m ] q i , ( − p ( i ) , [ m ] i ! = [ m ] q i , ( − p ( i ) ! , (cid:20) mk (cid:21) i = (cid:20) mk (cid:21) q i , ( − p ( i ) , for 0 ≤ k ≤ m and i ∈ I .For a Cartan datum X = C (1) n , C (2) ( n + 1) , B (1) (0 , n ), the quantum affine (super)algebra U q ( X ) is defined to be the Q ( q d )-algebra generated by k ± i , e i , f i ( i ∈ I ) with the followingrelations: k i k j = k j k i , k i e j k − i = q a ij i e j , k i f j k − i = q − a ij i f j ,e i f j − ( − p ( i ) p ( j ) f j e i = δ ij k i − k − i q i − q − i , JAE-HOON KWON AND MASATO OKADO1 − a ij X m =0 ( − m + p ( i ) m ( m − / mp ( i ) p ( j ) e (1 − a ij − m ) i e j e ( m ) i = 0 ( i = j ) , − a ij X m =0 ( − m + p ( i ) m ( m − / mp ( i ) p ( j ) f (1 − a ij − m ) i f j f ( m ) i = 0 ( i = j ) , where e ( m ) i = e mi [ m ] i ! , f ( m ) i = f mi [ m ] i ! . We define the automorphism τ of U q ( X ) for X = C (1) n , C (2) ( n + 1) by τ ( k i ) = k − n − i , τ ( e i ) = f n − i , τ ( f i ) = e n − i , if X = C (1) n , (2.1) τ ( k i ) = k − n − i , τ ( e i ) = ( − δ in f n − i , τ ( f i ) = ( − δ i e n − i , if X = C (2) ( n + 1)(2.2)for i ∈ I and the anti-automorphism η of U q ( X ) by η ( k i ) = k i η ( e i ) = ( − δ i + δ in q − i k − i f i η ( f i ) = ( − δ i + δ in q − i k i e i if X = C (1) n , B (1) (0 , n ) , η ( k i ) = k i η ( e i ) = ( − δ in q − i k − i f i η ( f i ) = ( − δ in q − i k i e i if X = C (2) ( n + 1)for i ∈ I . Both τ and η are involutions.When X = C (2) ( n + 1) , B (1) (0 , n ), let U q ( X ) σ = U q ( X ) ⊕ U q ( X ) σ be the semidirect product of U q ( X ) and the group algebra generated by σ , where(2.3) σ = 1 , σk i = k i σ, σe i = ( − p ( i ) e i σ, σf i = ( − p ( i ) f i σ ( i ∈ I ) .τ and η are extended to U q ( X ) σ by τ ( σ ) = η ( σ ) = σ .The algebras U q ( C (1) n ) , U q ( C (2) ( n + 1)) σ , U q ( B (1) (0 , n )) σ have a Hopf algebra structure.In particular, the coproduct ∆ is given by∆( k i ) = k i ⊗ k i , ∆( σ ) = σ ⊗ σ, ∆( e i ) = e i ⊗ σ p ( i ) δ i k − i + σ p ( i ) δ in ⊗ e i , ∆( f i ) = f i ⊗ σ p ( i ) δ i + σ p ( i ) δ in k i ⊗ f i (2.4)for i ∈ I . 3. Level one q -oscillator representation Let W be an infinite-dimensional vector space over Q ( q d ) defined by W = M m Q ( q d ) | m i , where | m i is a basis vector parametrized by m = ( m , . . . , m n ) ∈ Z n ≥ . Let | m | = P nj =1 m j ,and let e j be the j -th standard vector in Z n for 1 ≤ j ≤ n . In this section, we introduce theso-called q -oscillator representation of level one for each algebra.3.1. Type C (1) n . U q ( C (1) n ) -module W ± . Consider the quantum affine algebra U q ( C (1) n ). Let U q ( C n )and U q ( A n − ) be the subalgebras generated by k i , e i , f i for i ∈ I \ { } and i ∈ I \ { , n } ,respectively. Proposition 3.1.
For a non-zero x ∈ Q ( q ) , the space W admits a U q ( C (1) n ) -module structuregiven as follows: e | m i = xq − [ m + 1][ m + 2][2] | m + 2 e i ,f | m i = − x − q [2] | m − e i ,k | m i = q m +1 | m i ,e j | m i = [ m j +1 + 1] | m − e j + e j +1 i ,f j | m i = [ m j + 1] | m + e j − e j +1 i ,k j | m i = q − m j + m j +1 | m i ,e n | m i = − q [2] | m − e n i ,f n | m i = q − [ m n + 1][ m n + 2][2] | m + 2 e n i ,k n | m i = q − m n − | m i , where ≤ j ≤ n − . Here we understand the vector on the right-hand side is zero whenany of its components does not belong to Z ≥ . Remark 3.2.
For | m i ∈ W , set τ ( | m i ) = | m n , . . . , m i , and extend linearly to any vectorof W . Then, when x = 1 we have the following symmetry τ ( u | m i ) = τ ( u ) τ ( | m i ) , for u ∈ U q ( C (1) n ). Here the automorphism τ on U q ( C (1) n ) is given in (2.1). Remark 3.3.
This representation originally appeared in [13, Proposition 3]. The pre-sentation above is obtained from the one in [13] by applying the basis change | m i new = ( q [2]) | m | / Q ni =1 [ m i ]! | m i old and the automorphism of U q ( C (1) n ) sending f
7→ − f , e n
7→ − e n , k i
7→ − k i for i = 0 , n with the other generators fixed.We assume that ε denotes + or − . Set ς ( ε ) = 0 and 1, when ε = + and − , respectively.For m ∈ Z ≥ , let sgn( m ) be + and − if m is even and odd, respectively.Define the subspace W ε of W by W ε = M sgn( | m | )= ε Q ( q ) | m i . JAE-HOON KWON AND MASATO OKADO
Proposition 3.4.
For a non-zero x ∈ Q ( q ) , W ε is an irreducible U q ( C (1) n ) -module. We denote this module by W ε ( x ), and call it a (level one) q -oscillator representation. Wesimply write W ε = W ε (1) as a U q ( C (1) n )-module.Let s λ ( x , . . . , x n ) denote the Schur polynomial in x , . . . , x n corresponding to a partition λ . Then as a U q ( A n − )-module, we havech W + = X l ∈ Z ≥ s ( l ) ( x , . . . , x n ) = 1 Q ni =1 (1 − x i ) , ch W − = X l ∈ Z ≥ s ( l ) ( x , . . . , x n ) = Q ni =1 (1 + x i ) − Q ni =1 (1 − x i ) . Here the weight lattice of U q ( C (1) n ) is identified with the Z -lattice spanned by e i for 1 ≤ i ≤ n ,and hence the variable x i corresponds to the weight of e i .3.1.2. Classical limit.
Let A be the localization of Z [ q, q − ] at [2] = q + q − . Let W ε ( x ) A = X sgn( | m | )= ε A | m i . Then W ε ( x ) A is invariant under e i , f i , k i and { k i } := k i − k − i q i − q − i for i ∈ I \ { } . Let W ε ( x ) = W ε ( x ) A ⊗ A C , where C is an A -module such that f ( q ) · c = f (1) c for f ( q ) ∈ A and c ∈ C .Let E i , F i and H i be the C -linear endomorphisms on W ε ( x ) induced from e i , f i and { k i } for i ∈ I \ { } . We can check that they satisfy the defining relations for the universalenveloping algebra U ( C n ) of type C n (cf. [7, Chapter 5]). Hence W ε ( x ) becomes a U ( C n )-module. Lemma 3.5.
The space W ε ( x ) is isomorphic to the irreducible highest weight U ( C n ) -modulewith highest weight − ( + ς ( ε )) ̟ n , where ̟ n is the n -th fundamental weight for C n .Proof. It is clear that E i ( | i ⊗
1) = 0 for all i ∈ I \ { } . Since H n ( | i ⊗
1) = (cid:18) k n − k − n q n − q − n | i (cid:19) ⊗ (cid:18) − q + q − | i (cid:19) ⊗ − | i ⊗ , and H i ( | i ⊗
1) = 0 for 1 ≤ i ≤ n − W + ( x ) is a highest weight U ( C n )-module with highestweight − ̟ n . It follows from the actions of E i for i ∈ I \ { } that any submodule of W + ( x )contains | i ⊗
1. This implies that W + ( x ) is irreducible. The proof for W − ( x ) is similar. (cid:3) Polarization.
Define a symmetric bilinear form on W ε by( | m i , | m ′ i ) = δ m , m ′ q − P ni =1 m i ( m i − Q ni =1 [ m i ]! , (3.1)for | m i , | m ′ i with m = ( m , . . . , m n ). Note that ( | m i , | m i ) ∈ qA . Lemma 3.6.
The bilinear form in (3.1) is a polarization on W ε , that is, ( uv, v ′ ) = ( v, η ( u ) v ′ ) , for u ∈ U q ( C (1) n ) and v, v ′ ∈ W ε .Proof. It suffices to show when u is one of the generators. If u = k i , it is trivial. Let usshow that(3.2) ( e i | m i , | m ′ i ) = ( | m i , η ( e i ) | m ′ i ) , for i ∈ I and | m i , | m ′ i ∈ W ε . The proof for f i is almost identical since (3.1) is symmetric. Case 1 . Suppose that 1 ≤ i ≤ n −
1. We may assume m ′ = m − e i + e i +1 . The right-handside is( | m i , η ( e i ) | m − e i + e i +1 i ) = ( | m i , q − i k − i f i | m − e i + e i +1 i ) = [ m i ] q − m i − m i +1 ( | m i , | m i ) , and the left-hand side is( e i | m i , | m − e i + e i +1 i ) = [ m i +1 + 1]( | m − e i + e i +1 i , | m − e i + e i +1 i )= q A [ m i +1 + 1][ m i − m i +1 + 1]! Q j = i,i +1 [ m j ]!= q m i − m i +1 − [ m i ]( | m i , | m i ) , since A = − X j = i,i +1 m j ( m j − −
12 ( m i − m i − −
12 ( m i +1 + 1) m i +1 = − X ≤ j ≤ n m j ( m j −
1) + m i − m i +1 − . Hence (3.2) holds.
Case 2 . Suppose that i = n . We may assume m ′ = m − e n . The right-hand side is( | m i , η ( e n ) | m − e n i ) = ( | m i , − q − n k − n f n | m − e n i ) = − q m n − [ m n − m n ][2] ( | m i , | m i ) , and the left-hand side is( e n | m i , | m − e n i ) = − q [2] ( | m − e n i , | m − e n i ) = − q [2] q B [ m n − Q j = n [ m j ]! ( | m i , | m i )= − q m n − [ m n − m n ][2] ( | m i , | m i ) , since B = − X j = n m j ( m j − −
12 ( m n − m n −
3) = − X ≤ j ≤ n m j ( m j −
1) + 2 m n − . Hence (3.2) holds.
Case 3 . Suppose that i = 0. We have to show ( e v, v ′ ) = ( v, − q − k − f v ′ ). By Remark3.2 and the property ( τ ( | m i ) , τ ( | m ′ i )) = ( | m i , | m ′ i ), it is equivalent to ( f n τ ( v ) , τ ( v ′ )) =( τ ( v ) , − q − k n e n τ ( v ′ )). However, it is equivalent to the one proved in Case 1 . (cid:3) Crystal base.
Let M be a U q ( C (1) n )-module. For 1 ≤ j ≤ n −
1, we assume that e j and f j are locally nilpotent on M , and define e e j , e f j to be the usual lower crystal operators[12]. For i = 0 , n , we introduce new operators e e i and e f i as follows: Case 1 . Let u ∈ M be a weight vector such that e n u = 0 and k n u = q − ln u for some l > u k := q k ( k +2 l − n f ( k ) n u ( k ≥ . Then we define(3.4) e f n u k = u k +1 , e e n u k +1 = u k ( k ≥ . Case 2 . Let u ∈ M be a weight vector such that f u = 0 and k u = q l u for some l > u k := q k ( k +2 l − e ( k )0 u ( k ≥ . Then we define(3.6) e e u k = u k +1 , e f u k +1 = u k ( k ≥ . Remark 3.7.
The definitions of e e i and e f i ( i = 0 , n ) are based on the idea that(3.7) ( e f kn u, e f kn u ) ∈ qA ( e e k u ′ , e e k u ′ ) ∈ qA ( k ≥ , for u, u ′ ∈ W ε such that e n u = 0 and f u ′ = 0 (use Proposition 2.1).Let A be the subring of Q ( q ) consisting of functions which are regular at q = 0. Wedefine A -lattice L ε of W ε and a Q -basis B ε of L ε /q L ε by L ε = M sgn( m )= ε A | m i , B ε = { | m i (mod q L ) | sgn( m ) = ε } . It is clear from (3.1) that ( L ε , L ε ) ⊂ A , and B ε is an orthonormal basis of L ε /q L ε withrespect to ( , ) | q =0 . Proposition 3.8.
The pair ( L ε , B ε ) is a crystal base of W ε , that is, (1) L ε is invariant under e e i and e f i for i ∈ I , (2) e e i B ε ⊂ B ε ∪ { } and e f i B ε ⊂ B ε ∪ { } for i ∈ I , where we have e f i | m i ≡ | m + 2 e n i if i = n, | m + e i − e i +1 i if m i +1 ≥ and ≤ i ≤ n − , | m − e i if m ≥ and i = 0 , otherwise , (mod q L ε ) . Proof.
It is enough to prove (2).
Case 1 . Suppose that 1 ≤ i ≤ n −
1. Let | m i = | m , . . . , m n i ∈ L ε be given with m i +1 ≥
1. Since e i | m − m i e i + m i e i +1 i = 0, we have e f m i i | m − m i e i + m i e i +1 i = f m i i [ m i ]! | m − m i e i + m i e i +1 i = | m i , and hence e f i | m i = e f m i +1 i | m − m i e i + m i e i +1 i = | m + e i − e i +1 i . Case 2 . Suppose that i = n . First, suppose that m n is even. Since e n | m − m n e n i = 0and k n | m − m n e n i = q − | m − m n e n i , we have e f mn n | m − m n e n i = q ( mn ) f mn n (cid:2) m n (cid:3) n ! | m − m n e n i = (1 + q ) − mn q ( mn ) [ m n ]! (cid:2) m n (cid:3) n ! | m i , and hence e f n | m i = (1 + q ) mn q − ( mn ) (cid:2) m n (cid:3) n ![ m n ]! e f mn +1 n | m − m n e n i = (1 + q ) mn q − ( mn ) (cid:2) m n (cid:3) n ![ m n ]! (1 + q ) − mn − q ( mn +1 ) [ m n + 2]! (cid:2) m n + 1 (cid:3) n ! | m + 2 e n i = (1 + q ) − q ( mn +1 ) − ( mn ) [ m n + 2][ m n + 1] (cid:2) m n + 1 (cid:3) n | m + 2 e n i≡ | m + 2 e n i (mod q L ε ) , since q ( mn +1 ) − ( mn ) [ m n + 2][ m n + 1] (cid:2) m n + 1 (cid:3) n = q m n +1 [ m n + 2][ m n + 1] (cid:2) m n + 1 (cid:3) n ∈ (1 + qA ) . Next, suppose that m n is odd. Since e n | m − ( m n − e n i = 0 and k n | m − ( m n − e n i = q − | m − ( m n − e n i , we have e f mn − n | m − ( m n − e n i = q ( mn − )( mn +32 ) f mn − n (cid:2) m n − (cid:3) n ! | m − ( m n − e n i = (1 + q ) − mn − q ( mn − )( mn +32 ) [ m n ]! (cid:2) m n − (cid:3) n ! | m i , and hence e f n | m i = (1 + q ) mn − q − ( mn − )( mn +32 ) (cid:2) m n − (cid:3) n ![ m n ]! e f mn +12 n | m − ( m n − e n i = (1 + q ) mn − q − ( mn − )( mn +32 ) (cid:2) m n − (cid:3) n ![ m n ]! (1 + q ) − mn +12 q ( mn +12 )( mn +52 ) [ m n + 2]! (cid:2) m n +12 (cid:3) n ! | m + 2 e n i = (1 + q ) − q ( mn +12 )( mn +52 ) − ( mn − )( mn +32 ) [ m n + 2][ m n + 1] (cid:2) m n +12 (cid:3) n | m + 2 e n i≡ | m + 2 e n i (mod q L ε ) , since q ( mn +12 )( mn +52 ) − ( mn − )( mn +32 ) [ m n + 2][ m n + 1] (cid:2) m n +12 (cid:3) n = q m n +1 [ m n + 2][ m n + 1] (cid:2) m n +12 (cid:3) n ∈ (1 + qA ) . Case 3 . Suppose that i = 0. We can prove this case by the same arguments as in Case 2 by using the automorphism τ (2.1). (cid:3) Type C (2) ( n + 1) . U q ( C (2) ( n +1) -module W . Consider the quantum affine superalgebra of type C (2) ( n +1). Let U q ( B (0 , n )) and U q ( A n − ) be the subalgebras of U q ( C (2) ( n +1)) generated by k i , e i , f i for i ∈ I \ { } and i ∈ I \ { , n } , respectively. We also write U q ( B (0 , n )) = U q ( osp | n ),where osp | n is the orthosymplectic Lie superalgebra corresponding to the Dynkin diagram: ◦ n − ◦ / / • n Proposition 3.9.
For a non-zero x ∈ Q ( q ) , the space W admits an irreducible U q ( C (2) ( n +1)) σ -module structure given as follows: e | m i = xq − [ m + 1] | m + e i ,f | m i = x − q | m − e i ,k | m i = q m + | m i ,e j | m i = [ m j +1 + 1] | m − e j + e j +1 i ,f j | m i = [ m j + 1] | m + e j − e j +1 i ,k j | m i = q − m j + m j +1 | m i ,e n | m i = − q | m − e n i ,f n | m i = q − [ m n + 1] | m + e n i ,k n | m i = q − m n − | m i ,σ | m i = ( − | m | | m i , where ≤ j ≤ n − . We denote this module by W ( x ) and call it a (level one) q -oscillator representation. Wesimply write W = W (1) as a U q ( C (2) ( n + 1))-module. Note that as a U q ( A n − )-module, wehave ch W = X l ∈ Z ≥ s ( l ) ( x , . . . , x n ) = 1 Q ni =1 (1 − x i ) . Remark 3.10.
When x = 1 we also have the following symmetry τ ( u | m i ) = τ ( u ) τ ( | m i ) , for u ∈ U q ( C (2) ( n + 1)) (cf. Remark 3.2). Here the automorphism τ on U q ( C (2) ( n + 1)) isgiven in (2.2).3.2.2. Classical limit.
Let(3.8) W ( x ) A = X m A | m i , W ( x ) = W ( x ) A ⊗ A C , where A = Z [ q , q − ] and C is an A -module such that f ( q ) · c = f (1) c for f ( q ) ∈ A and c ∈ C . One can check directly that W ( x ) A is invariant under e i , f i , and { k i } for i ∈ I \ { } , andthe induced operators E i , F i , and H i on W ( x ), respectively, satisfy the defining relations of U ( osp | n ). Lemma 3.11.
The space W ( x ) is isomorphic to the irreducible highest weight U ( osp | n ) -module with highest weight − ̟ n , where ̟ n is the n -th fundamental weight for osp | n .Proof. We have H n ( | i ⊗
1) = (cid:18) k n − k − n q − q − | i (cid:19) ⊗ q − − q q − q − | i ! ⊗ −| i ⊗ , and H i ( | i ⊗
1) = 0 for 1 ≤ i ≤ n −
1. By the same argument as in Lemma 3.5, W ( x ) is anirreducible highest weight U q ( osp | n )-module with highest weight − ̟ n . (cid:3) Polarization.
Define a symmetric bilinear form on W by (3.1). Lemma 3.12.
The bilinear form in (3.1) is a polarization on W , that is, ( uv, v ′ ) = ( v, η ( u ) v ′ ) , for u ∈ U q ( C (2) ( n + 1)) and v, v ′ ∈ W .Proof. Let us show ( e n | m i , | m ′ i ) = ( | m i , η ( e n ) | m ′ i ) for | m i , | m ′ i ∈ W only. The proof for e i (1 ≤ i ≤ n −
1) is identical to Lemma 3.1, and the proof for e is obtained by using τ .We may assume m ′ = m − e n . The right-hand side is( | m i , η ( e n ) | m − e n i ) = ( | m i , − q − n k − n f n | m − e n i ) = − q m n − n [ m n ]( | m i , | m i ) , and the left-hand side is( e n | m i , | m − e n i ) = − q ( | m − e n i , | m − e n i )= − q q − P m i ( m i − Q ni =1 [ m i ]! [ m n ] q m n − = − q m n − [ m n ]( | m i , | m i ) . Hence the equality holds. (cid:3)
Crystal base.
Let M be a U q ( C (2) ( n + 1))-module. For 1 ≤ j ≤ n −
1, we assumethat e j and f j are locally nilpotent on M , and define e e j , e f j to be the usual lower crystaloperators. For i = 0 , n , we consider the operators e e i and e f i defined in the same way as in U q ( C (1) n ) (3.3)–(3.6), which also satisfy (3.7).Let A be the subring of Q ( q ) consisting of functions which are regular at q = 0. Wedefine the A -lattice L of W and a Q -basis B of L /q L by L = M m A | m i , B = { | m i (mod q L ) } . (3.9)It is clear from (3.1) that ( L , L ) ⊂ A , and B is an orthonormal basis of L /q L with respectto ( , ) | q =0 . Proposition 3.13.
The pair ( L , B ) is a crystal base of W in the sense of Proposition 3.8,where e f i | m i ≡ | m + e n i if i = n, | m + e i − e i +1 i if m i +1 ≥ and ≤ i ≤ n − , | m − e i if m ≥ and i = 0 , otherwise , (mod q L ) . Proof.
It suffices to prove (2) when i = 0 , n since the other cases are proved in Proposition3.8. Let us prove the case of e f n only. Recall that [ m ] n = [ m ] q , − for m ∈ Z ≥ .Let | m i be given. Since e n | m − m n e n i = 0 and k n | m − m n e n i = q − | m − m n e n i , wehave e f m n n | m − m n e n i = q mn ( mn +1)2 n f m n n [ m n ] q n , − ! | m − m n e n i = q mn ( mn − n [ m n ]![ m n ] qn, − ! | m i , and hence e f n | m i = q − mn ( mn − n [ m n ] q n , − ![ m n ]! e f m n +1 n | m − m n e n i = q − mn ( mn − n [ m n ] q n , − ![ m n ]! q mn ( mn +1)2 n [ m n + 1]![ m n + 1] qn, − ! | m + e n i≡ q m n n [ m n + 1][ m n + 1] qn, − | m + e n i = | m + e n i (mod q L ) . (cid:3) Type B (1) (0 , n ) . U q ( B (1) (0 , n )) -module W . Consider the quantum affine superalgebra of type B (1) (0 , n ).Let U q ( B (0 , n )) (or U q ( osp | n )) and U q ( A n − ) be the subalgebras of U q ( B (1) (0 , n )) gener-ated by k i , e i , f i for i ∈ I \ { } and i ∈ I \ { , n } , respectively. Proposition 3.14.
For a non-zero x ∈ Q ( q ) , the space W admits an irreducible U q ( B (1) (0 , n )) σ -module structure given as follows: e | m i = xq − [ m + 1][ m + 2][2] | m + 2 e i ,f | m i = − x − q [2] | m − e i ,k | m i = q m +1 | m i ,e j | m i = [ m j +1 + 1] | m − e j + e j +1 i ,f j | m i = [ m j + 1] | m + e j − e j +1 i ,k j | m i = q − m j + m j +1 | m i ,e n | m i = − q | m − e n i ,f n | m i = q − [ m n + 1] | m + e n i , k n | m i = q − m n − | m i ,σ | m i = ( − | m | | m i , where ≤ j ≤ n − . We also denote this module by W ( x ) and call it a (level one) q -oscillator representation.Note that the classical limit of W ( x ) as a U q ( osp | n )-module is the same as in Lemma 3.11.3.3.2. Polarization and crystal base.
Lemma 3.15.
The bilinear form in (3.1) is a polarization on W , that is, ( uv, v ′ ) = ( v, η ( u ) v ′ ) , for u ∈ U q ( B (1) (0 , n )) and v, v ′ ∈ W .Proof. All the cases are already shown in Lemmas 3.6 and 3.12 since the action of e i for0 ≤ i < n (resp. i = n ) is the same as the one for C (1) n (resp. C (2) ( n + 1)). (cid:3) We define the A -lattice L of W and a Q -basis B of L /q L as in (3.9). We also definethe operators e e i and e f i in the same way as in U q ( C (1) n ) and U q ( C (2) ( n + 1)). Proposition 3.16.
The pair ( L , B ) is a crystal base of W in the sense of Proposition 3.8,where e f i | m i ≡ | m + e n i if i = n, | m + e i − e i +1 i if m i +1 ≥ and ≤ i ≤ n − , | m − e i if m ≥ and i = 0 , otherwise , (mod q L ) . Proof.
It follows from Propositions 3.8 and 3.13. (cid:3) Quantum R -matrix and fusion construction In this section, we review the quantum R -matrix and its spectral decomposition foreach quantum affine (super)algebra and explain how to construct higher level q -oscillatorrepresentations by so-called fusion construction.Let x, y ∈ Q ( q d ) be generic, and let W be a level one q -oscillator representation of U q ( X )including W ε ( ε = ± ) for type C (1) n . The quantum R -matrix R ( x, y ) on W ( x ) ⊗ W ( y ) isdefined as a linear operator satisfying R ( x, y )∆( a ) = ∆ op ( a ) R ( x, y )for a ∈ U q ( X ), where ∆ op denotes the opposite coproduct, namely, the coproduct obtainedby interchanging the first and second components in ∆. If W ( x ) ⊗ W ( y ) is irreducible, then R ( x, y ) is unique up to a scalar function of x, y and depends only on z = x/y . Let P be thelinear operator on W ( x ) ⊗ W ( y ) such that P ( u ⊗ v ) = v ⊗ u and set ˇ R ( x, y ) = P R ( x, y ).Then ˇ R ( x, y ) maps W ( x ) ⊗ W ( y ) to W ( y ) ⊗ W ( x ). We also need to care about the difference between the coproduct (2.4) and that of [13]and Appendix A. Let ¯∆ be the coproduct of the latter. Let ς be the automorphism given by ς ( e i ) = e i k − i , ς ( f i ) = k i f i , ς ( k i ) = k i . Then we have ∆( a ) = ¯∆ op ( ς ( a )). Hence, to translatethe results in [13] and Appendix A, we replace ˇ R ( x, y ) with ˇ R ( y, x ). The component V l or V εl appearing in the spectral decomposition should be replaced with P V l . Thus, we obtainthe spectral decomposition of ˇ R ( x, y ) as follows. Note that z = x/y .For type C (1) n , we have(4.1) ˇ R ε ( x, y ) = X l ∈ Z ≥ l/ Y j =1 − q j − zz − q j − P εl where ˇ R ε ( x, y ) : W ε ( x ) ⊗ W ε ( y ) → W ε ( y ) ⊗ W ε ( x ) for ε = + , − and P εl is the projectiononto V εl .For C (2) ( n + 1), from Proposition C.4 and the spectral decomposition for U q ( D (2) n +1 ) in[13, Proposition 7], we have(4.2) ˇ R ( x, y ) = X l ∈ Z ≥ l Y j =1 − q ) j zz + ( − q ) j P l , where P l is the projection onto V l .Finally for B (1) (0 , n ), from Proposition C.4 and the spectral decomposition for U q ( A (2) † n )in Appendix B, we have(4.3) ˇ R ( x, y ) = X l ∈ Z ≥ l/ Y j =1 − q j − zz − q j − P l + X l ∈ Z ≥ ( l − / Y j =0 − q j +1 zz − q j +1 P l . Next, we explain the fusion construction. For s ≥
2, let S s denote the group of permu-tations on s letters generated by s i = ( i i + 1) for 1 ≤ i ≤ s −
1. We have U q ( X )-linearmaps ˇ R w ( x , . . . , x s ) : W ( x ) ⊗ · · · ⊗ W ( x s ) −→ W ( x w (1) ) ⊗ · · · ⊗ W ( x w ( s ) ) , for w ∈ S s and generic x , . . . , x s ∈ Q ( q ) satisfyingˇ R ( x , . . . , x s ) = id W ( x ) ⊗···⊗W ( x s ) , ˇ R s i ( x , . . . , x s ) = (cid:0) ⊗ ji +1 id W ( x j ) (cid:1) , ˇ R ww ′ ( x , . . . , x s ) = ˇ R w ′ ( x w (1) , . . . , x w ( s ) ) ˇ R w ( x , . . . , x s ) , for w, w ′ ∈ S s with ℓ ( ww ′ ) = ℓ ( w ) + ℓ ( w ′ ) where ℓ ( w ) denotes the length of w . Hence wehave a U q ( X )-linear map ˇ R s = ˇ R w ( x , . . . , x s ) with x i = q d (2 i − s − :ˇ R s : W ( q d (1 − s ) ) ⊗ . . . ⊗ W ( q d ( s − ) −→ W ( q d ( s − ) ⊗ . . . ⊗ W ( q d (1 − s ) ) . Here w is the longest element in S s and d = min { d i | i ∈ I } . Now we define a U q ( X )-module(4.4) W ( s ) = Im ˇ R s . Remark 4.1.
Let R univ be the universal R matrix for the quantum affine (super)algebra U q ( X ) [6]. Suppose that W is a finite-dimensional irreducible U q ( X )-module. Then R univ is rationally renormalizable in the sense of [11], that is, there exists c ∈ Q ( q d )(( y/x )) suchthat we have a well-defined map(4.5) cR univ : W ( x ) ⊗ W ( y ) −→ W ( y ) ⊗ W ( x ) , for x, y . Then we may apply [11, Theorem 3.12] to prove that W ( s ) is irreducible. However,the q -oscillator module W is infinite dimensional and R univ on W ( x ) ⊗W ( y ) is not rationallyrenormalizable. We expect that (4.5) still has a meaning, but do not know how to justify it.5. Higher level q -oscillator representation Type C (1) n . For s ≥ ε = ± , let W ( s ) ε denote the higher level q -oscillator modulein (4.4) corresponding to W ε . The following is the main result in this section. Theorem 5.1.
For s ≥ , W ( s ) ε is an irreducible U q ( C (1) n ) -module, which is also irreducibleas a U q ( C n ) -module. Moreover, its character is given by ch W ( s ) ε = X λ ∈ P ε ℓ ( λ ) ≤ s s λ ( x , . . . , x n ) , where P ε is the set of partitions λ = ( λ i ) i ≥ with sgn( λ i ) = ε for all i with λ i = 0 , and ℓ ( λ ) denotes the length of λ . Corollary 5.2.
The character of W ( s ) ε has a stable limit for s ≥ n as follows: ch W ( s ) ε = X λ ∈ P ε ℓ ( λ ) ≤ n s λ ( x , . . . , x n )= 1 Q ≤ i ≤ j ≤ n (1 − x i x j ) ( ε = +) . Let us construct a certain Q ( q )-basis of W (2) ε , which is compatible with the action ofˇ R ( z ), and plays an important role in the proof of Theorem 5.1. We note from (4.1) that W (2) ε = V ε = U q ( C n )( | ς ( ε ) e n i ⊗ | ς ( ε ) e n i ) , and hence it is irreducible. Moreover, we have the following character formula for W (2) ε . Proposition 5.3.
We have ch W (2) ε = ch V ε = X λ ∈ P ε ℓ ( λ ) ≤ s λ ( x , . . . , x n ) . Proof.
Write W ε = W ε ( q ± ) for short since we may consider the action of U q ( C n ) only. Let( W ε ⊗W ε ) A be the A -span of | m i⊗| m ′ i in W ε ⊗W ε . Then ( W ε ⊗W ε ) A is also invariant under e i , f i , k i and { k i } for i ∈ I \{ } . This yields its classical limit W ε ⊗ W ε := ( W ε ⊗W ε ) A ⊗ A C ,which is a U ( C n )-module. Also, we have as a U ( C n )-module W ε ⊗ W ε ∼ = W ε ⊗ W ε . By Lemma 3.5, W ε is an irreducible highest weight module. By the theory of super duality[3], it belongs to a semisimple category of U ( C n )-module which is closed under tensor product(see [16, Section 5.4] for more details, where we put m = 0 there). Hence W ε ⊗ W ε issemisimple, and the classical limit V ε , the submodule generated by ( | ς ( ε ) e n i ⊗ | ς ( ε ) e n i ) ⊗ U ( C n )-module with highest weight − (1 + 2 ς ( ε )) ̟ n . Thecharacter of V ε and hence V ε follows from [17, Theorem 6.1]. (cid:3) We construct a Q ( q )-basis of W (2) ε which is compatible with its U q ( A n − )-crystal base.For this, we find all the U q ( A n − )-highest weight vectors in W (2) ε .For l ∈ Z ≥ , let(5.1) v l = l X k =0 ( − k q k ( k − l +1) " lk − | k e n − + ( l − k ) e n i ⊗ | ( l − k ) e n − + k e n i . Lemma 5.4.
For l ∈ Z ≥ , v l is a U q ( A n − ) -highest weight vector in W (2) ε , and v l ≡ | l e n i ⊗ | l e n − i (mod q L ⊗ ε ) , where sgn( l ) = ε .Proof. It is straightforward to check that e i v l = 0 for 1 ≤ i ≤ n −
1. Next we claim that v l ∈ W (2) ε . Note that ch W ε = X l ∈ ς ( ε )+2 Z > s ( l ) ( x , . . . , x n ) , and hence(5.2) ch W ⊗ ε = (ch W ε ) = X sgn( | λ | )=+ ℓ ( λ ) ≤ m λ s λ ( x , . . . , x n ) , where for λ = ( λ , λ ), m λ = λ − λ − ς ( ε ) if λ > λ , λ = λ . Let S l be the U q ( A n − )-submodule of W ⊗ ε generated by v l . Since the character of S l is s ( l ) ( x , . . . , x n ), and the multiplicity of s ( l ) ( x , . . . , x n ) in (5.2) is one, it follows fromProposition 5.3 that S l ⊂ W (2) ε . This shows that v l ∈ W (2) ε . The lemma follows from q k ( k − l +1) " lk − ∈ q k (1 + qA ). (cid:3) One can prove more directly that v l ∈ W (2) ǫ using the following lemma. Lemma 5.5.
Set E = e (2) n − · · · e (2)1 e , where it should be understood as e when n = 2 . Thenfor l ∈ Z ≥ we have ( E e (2)1 E − e E ) ) v l = q − [2][3] ([ l + 1][ l + 2]) v l +2 . Proof.
Denote the module W ε by W ε,n to signify the rank n and let W ǫ,n be a linear subspaceof W ε,n spanned by the vectors | n − , m n − , m n i . Let π : W ε,n → W ε, be a linear mapdefined by π ( | m i ) = | m n − , m n i , where m = ( m , . . . , m n ). Then we can show by directcalculation that the following diagram commutes.( W ε,n ) ⊗ π ⊗ / / E (cid:15) (cid:15) W ⊗ ε, e (cid:15) (cid:15) ( W ε,n ) ⊗ π ⊗ / / W ⊗ ε, This fact reduces the proof of the lemma to the case of n = 2.When n = 2, one calculates e e (2)1 e v l = X k c k [ l − k + 1][ l − k + 2] × { q − l +2 k − [ k + 1][ k + 2] | k + 2 , l − k + 2 i ⊗ | l − k, k i + ( q − [ l − k + 1][ l − k + 2] + q − l − [ k − k ]) | k, l − k + 2 i ⊗ | l − k + 2 , k i + q − k [ l − k + 3][ l − k + 4] | k − , l − k + 2 i ⊗ | l − k + 4 , k i} . Here c k = ( − k q k ( k − l +1) " lk − and we have used the relation q l − k − [ l − k ] c k +1 +[ k +1] c k =0. On the other hand, we also get( e e ) v l =[2] X k c k [ l − k + 1][ l − k + 2] × { [3] q − l +2 k − [ k + 1][ k + 2] | k + 2 , l − k + 2 i ⊗ | l − k, k i + A k | k, l − k + 2 i ⊗ | l − k + 2 , k i + [3] q − k [ l − k + 3][ l − k + 4] | k − , l − k + 2 i ⊗ | l − k + 4 , k i} , where A k = q l − k q − q − { (1 + q − l − )( q [ k + 1][ l − k + 2] − [ k ][ l − k + 3]) − q − l +2 k (1 + q − )([ k + 1][ l − k + 2] − q − [ k ][ l − k + 3]) } . Combining these results, we obtain( e e (2)1 e − e e ) ) v l = [2][3] [ l + 1][ l + 2] X k c k q − k − [ l − k + 1][ l − k + 2] | k, l − k + 2 i ⊗ | l − k + 2 , k i = q − [2][3] ([ l + 1][ l + 2]) v l +2 . (cid:3) For l ∈ Z ≥ and l ′ = 2 m ∈ Z ≥ , set v l,l ′ = q m ( m +2 l +1)2 n f ( m ) n v l . Note that v l,l ′ may not be equal to e f mn v l in the sense of (3.4) since e n v l,l ′ = 0 in general. Lemma 5.6.
For l ∈ Z ≥ and l ′ ∈ Z ≥ with sgn( l ) = ε , v l,l ′ is a U q ( A n − ) -highest weightvector in W (2) ε , and v l,l ′ ≡ | l e n i ⊗ | l e n − + l ′ e n i (mod q L ⊗ ε ) . Proof.
Let us assume that l is even, and hence ε = +, since the proof for odd l is almostidentical. Since e j (1 ≤ j ≤ n −
1) commutes with f n , it is clear that v l,l ′ is a U q ( A n − )-highest weight vector in W (2) ε .Let l ′ = 2 m . For 0 ≤ c ≤ l , we have | c e n − + ( l − c ) e n i ≡ e f ⌊ l − c ⌋ n | c e n − i if c is even , e f ⌊ l − c ⌋ n | c e n − + e n i if c is odd , (mod q L + ) . Put a = ⌊ l − c ⌋ and b = ⌊ c ⌋ . Case 1 . Suppose that c is even. Let u = | c e n − i , u = | ( l − c ) e n − i . We have∆( f ( m ) n )( e f an u ⊗ e f bn u )= m X k =0 q − k ( m − k ) n f ( m − k ) n k kn ( e f an u ) ⊗ f ( k ) n ( e f bn u )= m X k =0 q − k ( m − k ) − ( +2 a ) kn f ( m − k ) n e f an u ⊗ f ( k ) n e f bn u = m X k =0 q − k ( m − k ) − ( +2 a ) k + a + b n f ( m − k ) n f ( a ) n u ⊗ f ( k ) n f ( b ) n u = m X k =0 q − k ( m − k ) − ( +2 a ) k + a + b n " m − k + aa n " k + bb n f ( m − k + a ) n u ⊗ f ( k + b ) n u = m X k =0 q − k ( m − k ) − ( +2 a ) k + a + b − ( m − k + a )22 − ( k + b )22 n " m − k + aa n " k + bb n e f m − k + an u ⊗ e f k + bn u = m X k =0 f a,b ( q ) e f m − k + an u ⊗ e f k + bn u . Multiplying q m ( m +2 l +1)2 n on both sides, we have q m ( m +2 l +1)2 n f a,b ( q ) ∈ q d (1 + qA ), where d = m ( m + 2 l + 1) − k ( m − k ) − (cid:18)
12 + 2 a (cid:19) k + a + b − ( m − k + a ) − ( k + b ) − m − k ) a − kb = 2 lm + ( m − k ) − ma − kb = 2 lm + ( m − k ) − m (cid:18) l − c (cid:19) − k (cid:16) c (cid:17) = ( m − k ) + 2 c ( m − k ) = (2 c + 1)( m − k )(5.3)since a = l − c and b = c . Case 2 . Suppose that c is odd. Let u = | c e n − + e n i , u = | ( l − c ) e n − + e n i . We have∆( f ( m ) n )( e f an u ⊗ e f bn u )= m X k =0 q − k ( m − k ) n f ( m − k ) n k kn ( e f an u ) ⊗ f ( k ) n ( e f bn u )= m X k =0 q − k ( m − k ) − ( +2 a ) n f ( m − k ) n e f ( a ) n u ⊗ f ( k ) n e f ( b ) n u = m X k =0 q − k ( m − k ) − ( +2 a ) k + a ( a +2)2 + b ( b +2)2 n f ( m − k ) n f ( a ) n u ⊗ f ( k ) n f ( b ) n u = m X k =0 q − k ( m − k ) − ( +2 a ) k + a ( a +2)2 + b ( b +2)2 n " m − k + aa n " k + bb n f ( m − k + a ) n u ⊗ f ( k + b ) n u = m X k =0 q − k ( m − k ) − ( +2 a ) k + a ( a +2)2 + b ( b +2)2 − ( m − k + a )( m − k + a +2)2 − ( k + b )( k + b +2)2 n " m − k + aa n " k + bb n e f m − k + an u ⊗ e f k + bn u = m X k =0 g a,b ( q ) e f m − k + an u ⊗ e f k + bn u . Multiplying q m ( m +2 l +1)2 n on both sides, we have q m ( m +2 l +1)2 n g a,b ( q ) ∈ q d ′ (1 + qA ), where d ′ = d − k + 2 a + 2 b − m − k + a ) − k + b )= d − k − m = 2 lm + ( m − k ) − ma − kb − k − m = 2 lm + ( m − k ) − m (cid:18) l − c − (cid:19) − k (cid:18) c − (cid:19) − k − m = ( m − k ) + 2 c ( m − k ) = (2 c + 1)( m − k )(5.4) by putting a = l − c − and b = c − . By (5.3), (5.4), and Lemma 5.4, we have q m ( m +2 l +1)2 n f ( m ) n v l ≡ | l e n i ⊗ | l e n − + 2 m e n i (mod q L ⊗ ) . (cid:3) Corollary 5.7.
The set { v l,l ′ | l ∈ Z ≥ , l ′ ∈ Z ≥ , sgn( l ) = ε } is the set of U q ( A n − ) -highest weight vectors in W (2) ε .Proof. The character of the U q ( A n − )-submodule of W ε generated by v l,l ′ is s λ ( x , . . . , x n )where λ = ( l ′ + l, l ). Hence it follows from Proposition 5.3 that there is no other U q ( A n − )-highest weight vectors in W (2) ε . (cid:3) Now we define the pair ( L (2) ε , B (2) ε ) by L (2) ǫ = X l ∈ Z ≥ sgn( l )= ε X l ∈ Z ≥ X r ≥ ≤ i ,...,i r ≤ n − A e f i . . . e f i r v l ,l , B (2) ǫ = n e f i . . . e f i r v l ,l (mod q L (2) ε ) (cid:12)(cid:12)(cid:12) l ∈ Z ≥ , sgn( l ) = ε, l ∈ Z ≥ , r ≥ , ≤ i , . . . , i r ≤ n − o \ { } . Proposition 5.8.
We have (1) L (2) ε ⊂ L ⊗ ε and B (2) ε ⊂ B ⊗ ε , (2) ( L (2) ε , B (2) ε ) is a U q ( A n − ) -crystal base of W (2) ε .Proof. (1) By Proposition 3.8, L ⊗ ε is a crystal base of W ⊗ ε as a U q ( A n − )-module, henceit is invariant under e f i for 1 ≤ i ≤ n −
1. By Lemma 5.6, we have e f i . . . e f i r v l ,l ∈ L ⊗ ε andhence e f i . . . e f i r v l ,l ∈ B ⊗ ε (mod q L ⊗ ε ).(2) By definition of ( L (2) ε , B (2) ε ) and Lemma 5.6, ( L (2) ε , B (2) ε ) is a U q ( A n − )-crystal base ofthe submodule V of W (2) ǫ generated by v l ,l for l , l . On the other hand, we have V = W (2) ε by Proposition 5.3. Hence ( L (2) ε , B (2) ε ) is a U q ( A n − )-crystal base of W (2) ε . (cid:3) For | m i = | m , . . . , m n i ∈ W , let T ( m ) denote the semistandard tableau of shape ( | m | ),a single row of length | m | , with letters in { n < · · · < } such that the number of occurrencesof i is m i for 1 ≤ i ≤ n .Suppose that | m i , . . . , | m s i are given such that | m | ≤ · · · ≤ | m s | . Let λ = ( | m s | ≥· · · ≥ | m | ), which is a partition or its Young diagram, and λ π denote the Young diagramobtained by 180 ◦ -rotation of λ . We denote by T ( m , . . . , m s ) the row-semistandard tableauof shape λ π , whose j -th row from the top is equal to T ( m j ) for 1 ≤ j ≤ s . Example 5.9.
Suppose that n = 5. If | m i = | , , , , i and | m i = | , , , , i , then T ( m , m ) = . Proposition 5.10.
We have B (2) ε = n | m i ⊗ | m i (mod q L (2) ε ) (cid:12)(cid:12)(cid:12) | m | ≤ | m | , T ( m , m ) is semistandard o . Proof.
For l ∈ Z ≥ and l ∈ Z ≥ with sgn( l ) = ε , let us identify v l ,l = | l e n i⊗| l e n − + l e n i in B ⊗ ε with the pair ( l e n , l e n − + l e n ) and the connected component of v l ,l asa U q ( A n − )-crystal with the set of corresponding set of pairs ( m , m ) ′ s . Then T ( v l ,l )is the semistandard tableau of shape ( l + l , l ) π . Since e e j v l ,l = 0 for 1 ≤ j ≤ n − T ( v l ,l ) is the tableau of highest weight and the set(5.5) n T (cid:16) e f i . . . e f i r v l ,l (cid:17) (cid:12)(cid:12)(cid:12) r ≥ , ≤ i , . . . , i r ≤ n − o \ { } is equal to the set of semistandard tableau of shape ( l + l , l ) π with letters { n < · · · < } . (cid:3) Let | m i , | m i ∈ B ε be given with | m | = d and | m | = d , let P ( m , m ) denotea unique semistandard tableau of shape µ π for some partition µ , which is equivalent to | m i ⊗ | m i as an element of U q ( A n − )-crystals. Indeed, if we read the row word of T ( m )from left to right, and then apply the Schensted’s column insertion to T ( m ) in a reverse waystarting from the right-most column, then the resulting tableau is P ( m , m ). So P ( m , m )is of shape ( d ′ , d ′ ) π for some d ′ ≤ d ′ with d ′ ≤ d , d ′ ≥ d , and d ′ + d ′ = d + d . Inparticular, P ( m , m ) = T ( m , m ) if d ≤ d and | m i ⊗ | m i ∈ B (2) ε . Example 5.11.
Let | m i , | m i be as in Example 5.9. Then P ( m , m ) = . Let l ∈ Z ≥ and l ∈ Z ≥ be given with sgn( l ) = ε . Put λ = ( λ , λ ) = ( l + l , l ).Let SST ( λ π ) be the set of semistandard tableaux of shape λ π with letters in { n < · · · < } .For each T ∈ SST ( λ π ), we choose i , . . . , i r ∈ I \ { , n } such that T = T (cid:16) e f i . . . e f i r v l ,l (cid:17) (see (5.5)), and define(5.6) v T = e f i . . . e f i r v l ,l ∈ L (2) ε . By Proposition 5.10, we have a Q ( q )-basis of W (2) ε (5.7) G λ ∈ P ε ℓ ( λ ) ≤ { v T | T ∈ SST ( λ π ) } . Lemma 5.12.
For T ∈ SST ( λ π ) , we have v T = | m i ⊗ | m i + X m ′ , m ′ c m ′ , m ′ | m ′ i ⊗ | m ′ i , where P ( m , m ) = T , P ( m ′ , m ′ ) is of shape µ π with µ ⊲ λ and µ = λ , and c m ′ , m ′ ∈ qA .Here ⊲ denotes a dominance order on partitions, that is, µ > λ , and µ + µ = λ + λ .Proof. By Lemmas 5.4 and 5.6 (see also their proofs), we observe that(5.8) v l ,l = | l e n i ⊗ | l e n − + l e n i + X c x,y,z,w | x e n − + y e n i ⊗ | z e n − + w e n i , where the sum is over ( x, y, z, w ) such that(1) 0 < x ≤ l with x + z = l , (2) y ≥ z , w ≥ x with y + w = l + l ,(3) c x,y,z,w ∈ qA .We may regard | l e n i ⊗ | l e n − + l e n i as the case when ( x, y, z, w ) = (0 , l , l , l ). Then itis not difficult to see that if the shape of P ( x e n − + y e n , z e n − + w e n ) is µ π = ( µ , µ ) π ,then µ = z = l − x ≤ l and hence µ ⊲ λ , and µ = λ when x > i , . . . , i r ∈ I \ { , n } be the sequence in (5.6). By the tensor product rule of crystals,we have(5.9) e f i . . . e f i r ( | x e n − + y e n i ⊗ | z e n − + w e n i ) = X m , m c m , m | m i ⊗ | m i , where the sum is over m , m such that(1) c m , m ( q ) ∈ A such that c m , m (0) = | m i ⊗ | m i = e f i . . . e f i r ( | x e n − + y e n i ⊗ | z e n − + w e n i ),0 otherwise . (2) ν ⊲ λ and ν = λ , where ν π is the shape of P ( m , m ).Therefore, we obtain the result by (5.8) and (5.9). (cid:3) Corollary 5.13.
We have L (2) ε = L ⊗ ε ∩ W (2) ε .Proof. It is clear that L (2) ε ⊂ L ⊗ ε ∩ W (2) ε by Proposition 5.8. Conversely, suppose that v ∈ L ⊗ ε ∩ W (2) ε is given. By (5.7), we have(5.10) v = X T c T v T , for some c T ∈ Q ( q ). We may assume that all the shape of T in (5.10) is the same. Fix T with c T = 0. Let | m i ⊗ | m i be such that | m i ⊗ | m i appears in (5.10) with non-zerocoefficient, and P ( m , m ) = T . By Lemma 5.12, the coefficient of | m i ⊗ | m i is c T . Hence c T ∈ A , and v ∈ L (2) ε . (cid:3) Proof of Theorem 5.1.
Let W ⊗ ε = W (2) ε ⊕ W , where W is the complement of W (2) ε in W ⊗ ε as a U q ( A n − )-module since it is completely reducible. By Corollary 5.13, we have(5.11) L ⊗ ε = L (2) ε ⊕ M (2) , where M (2) = L ⊗ ε ∩ W is the crystal lattice of W as a U q ( A n − )-module. Then we have(5.12) ˇ R ( L ⊗ ε ) ⊂ L (2) ε , ˇ R | q =0 ( B ⊗ ε ) ⊂ B (2) ε . More generally, by (4.1) and (5.11), we have for a ∈ Z > (5.13) ˇ R ( q − a )( L ⊗ ε ) ⊂ L ⊗ ε . For each 1 ≤ i ≤ s −
1, we haveˇ R s = ˇ R s i ( · · · , q s − i − | {z } i , q s − i +1 | {z } i +1 , · · · ) ˇ R w s i ( q − s , · · · , q s − ) . We have ˇ R w s i ( q − s , · · · , q s − )( L ⊗ sε ) ⊂ L ⊗ sε by (5.13), and hence by (5.12)ˇ R s ( L ⊗ sε ) ⊂ L ⊗ i − ε ⊗ L (2) ε ⊗ L ⊗ s − i − ε , ˇ R s ( B ⊗ sε ) ⊂ B ⊗ i − ε ⊗ B (2) ε ⊗ B ⊗ s − i − ε . Therefore ˇ R s ( B ⊗ sε ) is spanned by B ( s ) ε , where B ( s ) ε = n | m i ⊗ . . . ⊗ | m s i (mod q L ⊗ sε ) (cid:12)(cid:12)(cid:12) | m j i ⊗ | m j +1 i ∈ B (2) ε (1 ≤ j ≤ s − o . By Proposition 5.10, the set n T ( m , . . . , m s ) (cid:12)(cid:12)(cid:12) | m i ⊗ . . . ⊗ | m s i ∈ B ( s ) ε o is equal to the set of semistandard tableau of shape λ π where λ = ( | m s | ≥ · · · ≥ | m | ).Hence(5.14) ch W ( s ) ε = X λ ∈ P ε ℓ ( λ ) ≤ s s λ ( x , . . . , x n ) . Let V ( s )0 be the U q ( C n )-submodule of W ( s ) ε generated | ς ( ε ) e n i ⊗ s . The classical limit V ( s )0 of V ( s )0 is a highest weight U ( C n )-module with highest weightΛ ( s ) := − s ( 12 + ς ( ε )) ̟ n . On the other hand, by [17, Theorem 6.1] the character of the irreducible highest weight U ( C n )-module with highest weight Λ ( s ) , say V (Λ ( s ) ), is also equal to (5.14). Since V (Λ ( s ) )is a quotient of V ( s )0 , we conclude thatch W ( s ) ε = ch V ( s )0 = ch V ( s )0 = ch V (Λ ( s ) ) . In particular, V ( s )0 is an irreducible U q ( C n )-module and hence W ( s ) ε = V ( s )0 is an irreducible U q ( C (1) n )-module. This completes the proof. (cid:3) Type C (2) ( n + 1) . Let us prove that W ( s ) is an irreducible U q ( C (2) ( n + 1))-module.The proof is similar to that of Theorem 5.1 for U q ( C (1) n ). So we give a sketch of the proofand leave the details to the reader.We first consider W (2) . By (4.2), we have(5.15) W (2) = V = U q ( osp | n ) | i ⊗ | i , which is an irreducible representation of U q ( osp | n ) and hence of U q ( C (1) ( n +1)). By similararguments as in Proposition 5.3, we have the following. Proposition 5.14.
We have ch W (2) = ch V = X λ ∈ P ℓ ( λ ) ≤ s λ ( x , . . . , x n ) . Lemma 5.15.
For l ∈ Z ≥ , let v l be the vector in (5.1) . Then v l is a U q ( A n − ) -highestweight vector in W (2) , and v l ≡ | l e n i ⊗ | l e n − i (mod q L ⊗ ) . Proof.
Since the actions of Chevalley generators for 1 ≤ i ≤ n − C (1) n , it follows from Lemma 5.4 that v l is a U q ( A n − )-highest weight vector. Note that(5.16) ch W ⊗ = (ch W ) = X ℓ ( λ ) ≤ m λ s λ ( x , . . . , x n ) , where m λ = λ − λ . Then we have v l ∈ W (2) by the same argument as in Lemma 5.4. (cid:3) We have an analogue of Lemma 5.5, which also proves that v l ∈ W (2) . Lemma 5.16.
Set E = e n − · · · e e , where it is understood as e when n = 2 . Then for l ≥ we have ( E e n − E − e n − E ) v l = ( − l q − / (1 + q )[2] [ l + 1] v l +1 . Lemma 5.17.
For l, m ∈ Z ≥ , let v l,m = q m ( m +4 l +3)2 n f ( m ) n v l . Then v l,m is a U q ( A n − ) -highest weight vector in W (2) , and v l,m ≡ | l e n i ⊗ | l e n − + m e n i (mod q L ⊗ ) . Proof.
Since e j for 1 ≤ j ≤ n − f n , v l,m is a U q ( A n − )-highest weightvector in W (2) ǫ .For 0 ≤ c ≤ l , put a = l − c and b = c . Let u = | c e n − i , u = | ( l − c ) e n − i . By (2.4), we have∆( f ( m ) n )( e f an u ⊗ e f bn u )= m X k =0 σ k q − k ( m − k ) n f ( m − k ) n k kn ( e f an u ) ⊗ f ( k ) n ( e f bn u )= m X k =0 σ k q − k ( m − k ) − (1+2 a ) kn f ( m − k ) n e f an u ⊗ f ( k ) n e f bn u = m X k =0 σ k q − k ( m − k ) − (1+2 a ) k + a ( a +1)2 + b ( b +1)2 n f ( m − k ) n f ( a ) n u ⊗ f ( k ) n f ( b ) n u = m X k =0 σ k q − k ( m − k ) − (1+2 a ) k + a ( a +1)2 + b ( b +1)2 n " m − k + aa n " k + bb n f ( m − k + a ) n u ⊗ f ( k + b ) n u = m X k =0 σ k q − k ( m − k ) − (1+2 a ) k + a ( a +1)2 + b ( b +1)2 − ( m − k + a )( m − k + a +1)2 − ( k + b )( k + b +1)2 n " m − k + aa n " k + bb n e f m − k + an u ⊗ e f k + bn u = m X k =0 σ k f a,b ( q ) e f m − k + an u ⊗ e f k + bn u . Multiplying q m ( m +4 l +3)2 n on both sides, it is straightforward to see that q m ( m +4 l +3)2 n f a,b ( q ) ∈ q (2 c +1)( m − k ) n (1 + q A ) . This implies that v l,m ≡ | l e n i ⊗ | l e n − + m e n i (mod q L ⊗ ). (cid:3) Now we define the pair ( L (2) , B (2) ) by L (2) = X l ,l ∈ Z ≥ X r ≥ ≤ i ,...,i r ≤ n − A e f i . . . e f i r v l ,l , B (2) = n e f i . . . e f i r v l ,l (mod q L (2) ) (cid:12)(cid:12)(cid:12) l , l ∈ Z ≥ , r ≥ , ≤ i , . . . , i r ≤ n − o \ { } . Proposition 5.18.
We have (1) L (2) ⊂ L ⊗ and B (2) ⊂ B ⊗ , (2) ( L (2) , B (2) ) is a U q ( A n − ) -crystal base of W (2) , where B (2) = n | m i ⊗ | m i (mod q L (2) ) (cid:12)(cid:12)(cid:12) | m | ≤ | m | , T ( m , m ) is semistandard o . Proof.
It follows from the same arguments as in Propositions 5.8 and 5.10. (cid:3)
Corollary 5.19.
We have L (2) = L ⊗ ∩ W (2) .Proof. By Proposition 5.18, one can check that Lemma 5.12 also holds for W (2) , whichimplies L (2) = L ⊗ ∩ W (2) . (cid:3) Theorem 5.20.
For s ≥ , W ( s ) is an irreducible U q ( C (2) ( n + 1)) -module, which is alsoirreducible as a U q ( osp | n ) -module. Moreover, its character is given by ch W ( s ) = X λ ∈ P ℓ ( λ ) ≤ s s λ ( x , . . . , x n ) . Proof of Theorem 5.20.
We may apply the same arguments as in Theorem 5.1 and the resultin [17, Theorem 6.1] by using Proposition 5.18 and Corollary 5.19. (cid:3)
Corollary 5.21.
The character of W ( s ) ε has a stable limit for s ≥ n as follows: ch W ( s ) = X λ ∈ P ℓ ( λ ) ≤ n s λ ( x , . . . , x n ) = 1 Q ≤ i ≤ n (1 − x i ) Q ≤ i 0. Generalizing the decomposition of W (2) into U q ( osp | n )-modules, we have the following conjecture on W ( s ) . Conjecture 5.22. For s ≥ , W ( s ) is an irreducible U q ( B (1) (0 , n )) -module and its characteris given by ch W ( s ) = X λ ∈ P + ℓ ( λ ) ≤ min { n,s/ } ch V (Λ ( s ) λ ) . Remark 5.23. The family of infinite-dimensional U ( osp | n )-modules V (Λ ( s ) λ ) have beenintroduced in [2] in connection with Howe duality. They are unitarizable and form a semisim-ple tensor category. The Weyl-Kac type character formula for V (Λ ( s ) λ ) can be found in [2,Theorem 6.13]. Corollary 5.24. For s ≥ n , we have ch W ( s ) = P λ ∈ P + s λ ( x , . . . , x n ) Q ≤ i ≤ n (1 − x i ) Q ≤ i The first equation follows from the fact [17, Corollary 6.6] that if λ ∈ P + with ℓ ( λ ) ≤ n , then ch V (Λ ( s ) λ ) = s λ ( x , . . . , x n ) Q ≤ i ≤ n (1 − x i ) Q ≤ i Appendix A. Twistor In this appendix, we review the twistor introduced in [4] that relate quantum groups toquantum supergroups. We use it to relate the q -oscillator representation of U q ( D (2) n +1 ) in[13] to a representation of U q ( C (2) ( n + 1)). An advantage to do so is that in the latter wecan take a classical limit q → 1. We also obtain a representation of U q ( B (1) (0 , n )) from the q -oscillator representation of U q ( A (2) † n ), where A (2) † n is the same Dynkin diagram as A (2)2 n in[8] but the labeling of nodes are opposite.A.1. The twistor of the covering quantum group. We review the covering quantumgroup and the twistor map introduced in [4]. Our notations for a Cartan datum is closerto Kac’s book [8]. Let I be the index set of the Dynkin diagram, { α i } i ∈ I the set of simpleroots, ( a ij ) i,j ∈ I the Cartan matrix. The symmetric bilinear form ( · , · ) on the weight latticeis normalized so that it satisfies d i = ( α i , α i ) / ∈ Z for any i ∈ I . It is also assumed that a ij ∈ Z if d i ≡ j ∈ I . The parity function p ( i ) taking values in { , } isconsistent with d i , namely, p ( i ) ≡ d i (mod 2). We set q i = q d i , π i = π d i .Let q, π be indeterminates and i = √− 1. For a ring R with 1, we set R π = R [ π ] / ( π − U associated to a Cartan datum is the Q π ( q, i )-algebra with generators E i , F i , K ± i , J ± i for i ∈ I subject to the following relations. J i J j = J j J i , K i K j = K j K i , J i K j = K j J i ,J i E j = π a ij E j J i , J i F j = π a ij F j J i ,K i E j = q a ij E j K i , K i F j = q a ij F j K i ,E i F j − π p ( i ) p ( j ) F j E i = δ ij J i K i − K − i π i q i − q − i , − a ij X l =0 ( − l π l ( l − p ( i ) / lp ( i ) p ( j ) (cid:20) − a ij l (cid:21) q i ,π i E − a ij − li E j E li = 0 ( i = j ) , − a ij X l =0 ( − l π l ( l − p ( i ) / lp ( i ) p ( j ) (cid:20) − a ij l (cid:21) q i ,π i F − a ij − li F j F li = 0 ( i = j ) . Remark A.1. We changed the notations from [4]. We replaced v with q , t with i , J d i i and K d i i with J i and K i .We extend U by introducing generators T i , Υ i for i ∈ I . They commute with each otherand with J i , K i . They also have the commutation relations with E i , F i as T i E j = i a ij E j T i , T i F j = i − a ij F j T i , Υ i E j = i φ ij E j Υ i , Υ i F j = i − φ ij F j Υ i where φ ij = d i a ij if i > j,d i if i = j, − p ( i ) p ( j ) if i < j. We denote this extended algebra by b U . Theorem A.2 ([4]) . There is a Q ( i ) -algebra automorphism b Ψ on b U such that E i i − d i Υ − i T i E i , F i F i Υ i , K i T i K i ,J i T i J i , T i T i , Υ i Υ i ,q i − q, π 7→ − π. A.2. Image of the twistor b Ψ . We apply the twistor b Ψ given in the previous subsectionfor the Cartan datum corresponding to B n , namely, I = { , , . . . , n } and the Cartan matrixis given by ( a ij ) = − − − − . Through it, we are to regard the q -oscillator representation W = L m Q ( q ) | m i of U q ( B n )(subalgebra of U q ( D (2) n +1 ) generated by e i , f i , k i for 1 ≤ i ≤ n ) given in Proposition 1 of[13] as a representation of U q ( osp | n ). Although we normalized the symmetric bilinear form on the weight lattice so that ( α i , α i ) ∈ Z for any i ∈ I in the previous subsection, werenormalize it so that ( α n , α n ) = to adjust it to the notations in [13]. The generators T i , Υ i are represented on W as T i | m i = i m i +1 − m i | m i (1 ≤ i < n ) i − m n | m i ( i = n ) , Υ i | m i = i − m i | m i (1 ≤ i < n ) i | m |− m n | m i ( i = n ) . Let u i ( i ∈ I , u = e, f, k ) be the generators of U q ( B n ) ( π = 1) and ¯ u i = ˆΨ( u i ) bethe image ( π = − 1) of the twistor ˆΨ. Then ¯ u i satisfy the relations for U ¯ q ( osp | n ) where¯ q = i − q . On the space W , they act as follows.¯ e i | m i = i m i +1 [ m i ] | m − e i + e i +1 i , ¯ f i | m i = i − m i [ m i +1 ] | m + e i − e i +1 i , ¯ k i | m i = i m i − m i +1 q − m i + m i +1 | m i , ¯ e n | m i = κ i −| m | [ m n ] | m − e n i , ¯ f n | m i = i | m |− m n | m + e n i , ¯ k n | m i = i m n +1 q − m n − | m i , where 1 ≤ i < n , κ = ( q + 1) / ( q − e , ¯ f , ¯ k , we want to make W a quantum group module inSection A.1 associated to the affine Dynkin datum C (2) ( n + 1) or B (1) (0 , n ). For the formerwe set ¯ e | m i = x i m −| m | | m + e i , ¯ f | m i = x − κ i | m | +1 [ m ] | m − e i , ¯ k | m i = i − m − q m + | m i , and for the latter ¯ e | m i = x ( − | m | | m + 2 e i , ¯ f | m i = x − ( − | m | [ m ][ m − | m − e i , ¯ k | m i = − q m +1 | m i , where x is the so-called spectral parameter. We also note that the quantum parameter isstill ¯ q = i − q .To obtain the representation for the quantum parameter q , we need to we switch q to i q (¯ q to q ). Also, the relations in Section A.1 and those in Section 2.3 are different. Forthe node i that is signified as • in the Dynkin diagram, there is a relation e i f i + f i e i = k i − k − i q − q − in Section 2.3 rather than e i f i + f i e i = k i − k − i − q − q − in Section A.1. The former relation is realized by deleting κ from the action of ¯ e i or ¯ f i inthe formulas of the q -oscillator representation above. By doing so, we obtain¯ e | m i = x i m −| m | | m + e i for U q ( C (2) ( n + 1)) x ( − | m | | m + 2 e i for U q ( B (1) (1 , n )) , ¯ f | m i = x − i | m | +2 m +1 [ m ] | m − e i for U q ( C (2) ( n + 1)) x − ( − | m | +1 [ m ][ m − | m − e i for U q ( B (1) (0 , n )) , ¯ k | m i = q m + | m i for U q ( C (2) ( n + 1)) q m +1 | m i for U q ( B (1) (0 , n )) , ¯ e i | m i = ( − − m i + m i +1 +1 [ m i ] | m − e i + e i +1 i , ¯ f i | m i = ( − − m i + m i +1 +1 [ m i +1 ] | m + e i − e i +1 i , ¯ k i | m i = q − m i + m i +1 | m i , ¯ e n | m i = i −| m | +2 m n [ m n ] | m − e n i , ¯ f n | m i = i | m |− m n | m + e n i , ¯ k n | m i = q − m n − | m i , for 1 ≤ i ≤ n − U q ( C (2) ( n + 1)) (resp. U q ( B (1) (0 , n ))) in Proposition 3.9 (resp.3.14), we perform the basis change | m i to i s ( m ) q −| m | / Q nj =1 [ m j ]! | m i where s ( m ) = −| m | ( | m | +1) / − P j m j . Next we apply the algebra automorphism sending e n 7→ − e n , f n 7→ − f n andthe other generators fixed. For U q ( C (2) ( n + 1)) σ , we also apply e σe , f f σ .Accordingly, the coproduct also changes. For U q ( B (1) (0 , n )), we alternatively apply e i [2] e , f i [2] f . Appendix B. Quantum R matrix for U q ( A (2) † n )In this appendix, we consider the quantum R matrix for the q -oscillator representationof U q ( A (2) † n ) where A (2) † n is the Dynkin diagram whose nodes have the opposite labelings to A (2)2 n . Next we identify it as the one for U q ( B (1) (0 , n )).B.1. q -oscillator representation for U q ( A (2) † n ) . By A (2) † n we denote the following Dynkindiagram. ◦ / / ◦ n − ◦ / / ◦ n Although we did not deal with the q -oscillator representation for U q ( A (2) † n ) in [13], it is easyto guess from other cases given there. On the space W , the actions are given as follows. e | m i = x | m + 2 e i ,f | m i = x − [ m ][ m − | m − e i ,k | m i = − q m +1 | m i ,e i | m i = [ m i ] | m − e i + e i +1 i ,f i | m i = [ m i +1 ] | m + e i − e i +1 i ,k i | m i = q − m i +2 m i +1 | m i ,e n | m i = i κ [ m n ] | m − e n i ,f n | m i = | m + e n i ,k n | m i = i q − m n − / | m i , where 0 < i < n , κ = ( q + 1) / ( q − π x . U q ( B n )-highest weight vectors { v l | l ∈ Z ≥ } are calculated in [13, Prop 4]. We take thecoproduct (C.1) with π = 1. Lemma B.1. For x, y ∈ Q ( q ) we have (1) ( π x ⊗ π y )∆( f f (2)1 · · · f (2) n − ) v l = − [ l ][ l − ( q l − x − + q − y − ) v l − ( l ≥ , (2) ( π x ⊗ π y )∆( e n e (2) n − · · · e (2)1 e ) v = i κ [2]1 − q (( y + qx ) v − q ( y + x )∆( f n ) v ) . Define ˇ R KO ( z, q ) as in Proposition C.4 for U q ( B (1) n ). The existence of such ˇ R KO ( z, q )is essentially given in [13, Theorem 13]. Namely, although A (2) † n is not listed there, thecorresponding gause transformed quantum R matrix is S , ( z ) and the proof has been doneas the cases (i),(iv) and (v). Proposition B.2. We have the following spectral decomposition ˇ R KO ( z ) = X l ∈ Z + l/ Y j =1 z + q j − q j − z P l + X l ∈ Z + ( l − / Y j =0 z + q j +1 q j +1 z P l , where P l is the projector on the subspace generated by the U q ( B n ) -highest weight vector v l ( l ≥ . Appendix C. Quantum R matrix for U q ( C (2) ( n + 1)) and U q ( B (1) (0 , n ))In this appendix, we compare the quantum R matrix for the q -oscillator representationfor U q ( C (2) ( n + 1)) with the one for U q ( D (2) n +1 ) given in [13]. We also consider the quantum R matrix for U q ( B (1) (0 , n )) based on the results in [13].C.1. Gauge transformation. We take the following coproduct∆( k i ) = k i ⊗ k i , ∆( e i ) = 1 ⊗ e i + e i ⊗ σ − π p ( i ) k i , ∆( f i ) = f i ⊗ σ − π p ( i ) + k − i ⊗ f i , (C.1) for i ∈ I , where σ satisfies (2.3). We also take the same coproduct (C.1) for u i . Let Γ be anoperator acting on W ⊗ by(C.2) Γ | m i ⊗ | m ′ i = i P k,l ϕ kl m k m ′ l | m i ⊗ | m ′ i , for m = ( m , . . . , m n ) and m ′ = ( m ′ , . . . , m ′ n ). Here we have the constraint ϕ kl + ϕ lk = 0.Then by [20] (see also [19]), ∆ Γ ( u ) = Γ − ∆( u )Γgives another coproduct of U q ( B n ) acting on W ⊗ . Take ϕ kl to be 1 for k < l . We also set(C.3) K | m i = i c ( m ) | m i , where c ( m ) = − X k m k + X k (cid:18) k − n − (cid:19) m k . Set γ i ( m ) = −| m | + m ( i = 0 and for U q ( C (2) ( n + 1))) − | m | + 2 m ( i = 0 and for U q ( B (1) (0 , n ))) m i + m i +1 (0 < i < n ) −| m | + m n ( i = n ) ,β i ( m ) = m + n ( i = 0 and U q ( C (2) ( n + 1)))2 m + 2 n + 1 ( i = 0 and U q ( B (1) (0 , n ))) − m i + m i +1 (0 < i < n ) − m n ( i = n ) . Let α = e for U q ( C (2) ( n + 1))) , e for U q ( B (1) (0 , n ))), α i = − e i + e i +1 (0 < i < n ), and α n = − e n . Lemma C.1. The following formulas hold for m , m ′ , and i ∈ I ; (1) Γ − (1 ⊗ e i )Γ | m i ⊗ | m ′ i = i − γ i ( m ) | m i ⊗ e i | m ′ i , (2) Γ − ( e i ⊗ | m i ⊗ | m ′ i = i γ i ( m ′ ) e i | m i ⊗ | m ′ i , (3) Γ − (1 ⊗ f i )Γ | m i ⊗ | m ′ i = i γ i ( m − α i ) | m i ⊗ f i | m ′ i , (4) Γ − ( f i ⊗ | m i ⊗ | m ′ i = i − γ i ( m ′ − α i ) f i | m i ⊗ | m ′ i . Lemma C.2. The following formulas hold for m and i ∈ I ; (1) K − e i K | m i = i β i ( m ) e i | m i , (2) K − f i K | m i = i − β i ( m − α i ) f i | m i . Proposition C.3. For u i ( i ∈ I , u = e, f, k ) , we have ∆(¯ u i ) | m i ⊗ | m ′ i = i Λ i ( m + m ′ ) ( K ⊗ K ) − ∆ Γ ( u i )( K ⊗ K ) | m i ⊗ | m ′ i , on W ⊗ . Here Λ i ( m ) = m i + m i +1 − ( δ i + δ in ) | m | − nδ i ( u = e ) m i + m i +1 + ( δ i + δ in )( | m | + 1) − u = f )2 m i − m i +1 ( u = k ) , except when i = 0 and for U q ( B (1) (0 , n )) , where Λ ( m ) = m − | m | − n + 1 ( u = e )2 m − | m | − n + 3 ( u = f )0 ( u = k ) . Here we should understand m = m n +1 = 0 .Proof. It follows from Lemmas C.1 and C.2, and the following calculations. For instance,for i = n ∆(¯ e n ) | m i ⊗ | m ′ i = (1 ⊗ ¯ e n + ¯ e n ⊗ σ ¯ k n ) | m i ⊗ | m ′ i = κ ( i −| m ′ | [ m ′ n ] | m i ⊗ | m ′ − e n i + ( − | m ′ | i −| m | +2 m ′ n q − m ′ n − [ m n ] | m − e n i ⊗ | m ′ i ) , ∆ Γ ( e n ) | m i ⊗ | m ′ i = (Γ − (1 ⊗ e n )Γ + Γ − ( e n ⊗ · (1 ⊗ k n )) | m i ⊗ | m ′ i = κ ( i | m |− m n +1 [ m ′ n ] | m i ⊗ | m ′ − e n i + i −| m ′ | + m ′ n +2 q − m ′ n − [ m n ] | m − e n i ⊗ | m ′ i ) , and for i = n ∆(¯ e i ) | m i ⊗ | m ′ i = (1 ⊗ ¯ e i + ¯ e i ⊗ ¯ k i ) | m i ⊗ | m ′ i = i m ′ i +1 [ m ′ i ] | m i ⊗ | m ′ − e i + e i +1 i + i m i +1 +2 m ′ i − m ′ i +1 q − m ′ i +2 m ′ i +1 [ m i ] | m − e i + e i +1 i ⊗ | m ′ i , ∆ Γ ( e i ) | m i ⊗ | m ′ i = (Γ − (1 ⊗ e i )Γ + Γ − ( e i ⊗ · (1 ⊗ k i )) | m i ⊗ | m ′ i = i − m i − m i +1 [ m ′ i ] | m i ⊗ | m ′ − e i + e i +1 i + i m ′ i + m ′ i +1 q − m ′ i +2 m ′ i +1 [ m i ] | m − e i + e i +1 i ⊗ | m ′ i . (cid:3) For a quantum group such as U = U q ( D (2) n +1 ), U q ( A (2) † n ), U q ( C (2) ( n + 1)), U q ( B (1) (0 , n ))a quantum R matrix R ( z ) is defined, if it exists, as an intertwiner satisfyingˇ R ( z )( π x ⊗ π y )∆( u ) = ( π y ⊗ π x )∆( u ) ˇ R ( z ) , where ˇ R ( z ) = P R ( z ), P is the transposition of the tensor components and z = x/y . We alsonote that the coproduct we use here is (C.1). For U = U q ( D (2) n +1 ) or U q ( A (2) † n ), the existenceof quantum R matrices are proved in [13] or Appendix B. We denote them by ˇ R KO ( z ).Let ˇ R new ( z ) be the quantum R matrices for the quantum groups U = U q ( C (2) ( n + 1)) or U q ( B (1) (0 , n )). From the previous proposition, we have Proposition C.4. For generic x, y ∈ Q ( q ) , ˇ R new ( z ) and ˇ R KO ( z ) have the following rela-tion: ˇ R new ( z, − q ) = ( K ⊗ K ) − Γ − ˇ R KO ( z, q )Γ( K ⊗ K ) . References [1] V. Chari, D. Hernandez, Beyond Kirillov-Reshetikhin modules , Contemp. Math. 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Department of Mathematical Sciences and Research Institute of Mathematics, Seoul Na-tional University, Seoul 08826, Korea E-mail address : [email protected] Department of Mathematics, Osaka City University, Osaka 558-8585, Japan E-mail address ::