Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 028, 34 pages Polynomial Relations for q-Characters via the ODE/IM Correspondence Juanjuan SUN Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo 153-8914, Japan E-mail: sunjuan@ms.u-tokyo.ac.jp Received January 08, 2012, in final form May 10, 2012; Published online May 15, 2012 http://dx.doi.org/10.3842/SIGMA.2012.028 Abstract. Let Uq(b) be the Borel subalgebra of a quantum affine algebra of type X (1) n (X = A,B,C,D). Guided by the ODE/IM correspondence in quantum integrable models, we propose conjectural polynomial relations among the q-characters of certain representations of Uq(b). Key words: Borel subalgebra; q-character; Baxter’s Q-operator; ODE/IM correspondence 2010 Mathematics Subject Classification: 81R10; 17B37; 81R50 1 Introduction Let Uq(g) be a quantum affine algebra of type X (1) n (X = A,B,C,D), and let Uq(b) be its Borel subalgebra. In this paper we shall consider the problem of finding polynomial relations satisfied by the q-characters of the fundamental modules in the sense of [20] and related modules. This problem is intimately related with that of functional equations for Baxter’s Q-operators in quantum integrable models. In order to motivate the present study let us review this connection. In quantum integrable systems, one is interested in the spectra of a commutative family of transfer matrices. The latter are constructed from the universal R matrix of a quantum affine algebra, by taking the trace of the first component over some finite-dimensional representation called ‘auxiliary space’. When the auxiliary spaces are the Kirillov–Reshetikhin (KR) modules, the corresponding transfer matrices satisfy an important family of polynomial identities known as the T -system [24, 26]. (For a recent survey on this topic, see [25].) Subsequently the T - system has been formulated and proved [19, 28] as identities of q-characters. It has been shown further [21] that the T -system is actually the defining relations of the Grothendieck ring, which is a polynomial ring [17], of finite-dimensional modules of quantum affine algebras. Notice that, from the construction by trace, the q-characters and the transfer matrices are both ring homomorphisms defined on the Grothendieck ring. Since the q-characters are injective [16, 17], identities for q-characters imply the same identities for transfer matrices. Baxter’s Q-operators were first introduced in the study of the 8-vertex model [1]. Since then they have been recognized as a key tool in classical and quantum integrable systems, and there is now a vast literature on this subject. In the seminal paper [4], Bazhanov, Lukyanov and Zamolodchikov revealed that the Q-operators can also be obtained from the universal R matrix, provided the auxiliary space is chosen to be a (generically infinite-dimensional) representation of the Borel subalgebra. The work [4] for Uq(ŝl2) has been extended by several authors [2, 3, 5, 23, 29] to higher rank and supersymmetric cases. In view of the results mentioned above, it is natural to ask whether one can find polynomial relations, analogous to the T -system, for the q-characters of Uq(b) (see definitions in [20]) (and 2 J. Sun hence for the Q-operators as well). The goal of this paper is to propose candidates of such identities. Our idea is to use the so-called ODE/IM correspondence which relates the eigenvalues of Q-operators and certain ordinary linear differential equations. The reader is referred e.g. to the review [12] on this topic. In [11], the correspondence is discussed for general non-twisted affine Lie algebras using scalar (pseudo-)differential operators. In this paper we reformulate their results in terms of first-order systems. The general setting (to be explained in Section 3) is as follows. Let Lg denote the Langlands dual Lie algebra of g. Consider the following Lg-valued first- order linear differential operator L = d dx − ` x + n∑ i=1 ei + ( xMh ∨ − E ) e0. (1.1) Here e0, . . . , en are the Chevalley generators of Lg, ` is a generic element of the Cartan subalgebra Lh ⊂ Lg, h∨ is the dual Coxeter number of g, and M > 0, E ∈ C are parameters. For each fundamental representation V (a) of Lg, there is a basis {χ(a)J (x,E)} of V (a)-valued solutions to the equation Lχ = 0 characterized by the behavior χ(a)J (x,E) = u (a) J x λ(a)J (1 +O(x)), x→ 0, where λ(a)J , u (a) J are the eigenvalues of ` and the corresponding eigenvectors. Hence this basis is labeled by weight vectors of V (a). There is also a canonical solution ψ(a)(x,E) which has the fastest decay at x→ +∞. Due to the special choice (1.1) of L, the canonical solutions are shown to satisfy a system of relations similar to the Plu¨cker relations, called the ψ-system in [11] (see Subsection 3.2 below). Introduce the connection coefficients Q(a)J (E) ∈ C by ψ(a)(x,E) = ∑ J Q(a)J (E)χ (a) J (x,E). Then the ψ-system implies a set of polynomial relations among the connection coefficients Q(a)J (E). We expect the following to be true: (1) For each connection coefficient Q(a)J (E), there is associated a formal power series Q (a) J,z. Up to a simple multiplicative factor, the latter is the q-character of an irreducible highest `-weight module of Uq (b). In particular, for the highest or lowest weights of V (a) the corresponding modules are the fundamental modules [20] of Uq(b). (2) With the identification Q(a)J (E)←→ Q (a) J,z, E ←→ z, the polynomial relations for Q(a)J (E) implied by the ψ-system hold true also for Q (a) J,z. We adopt (1), (2) as our working hypothesis for finding relations among the q-characters. A few remarks are in order here. We consider separately the cases related to the spin repre- sentation (i.e., (g, a) = (C(1)n , n), (D (1) n , n − 1), (D (1) n , n)) and use the letter R (a) ε,z to denote the counterpart of Q(a)J,z. We expect that Q (a) J,z or R (a) ε,z corresponds to the q-character of an irre- ducible highest `-weight module of Uq(b) whose highest `-weight is given by formulas (4.16), (4.19)–(4.20), (4.23)–(4.24). (In the case C(1)n , there are problems with this interpretation, as Polynomial Relations for q-Characters via the ODE/IM Correspondence 3 explained in Remark 5.9.) Also the relations for the Q(a)J,z’s or R (a) ε,z ’s are not exactly the same as those for Q(a)J (E)’s, but it is necessary to fine-tune the coefficients by some power functions independent of z. The details will be given in Section 5 below. That it is natural to consider all Q(a)J,z corresponding to general weights of V (a) is a viewpoint suggested by the work [29] for type A algebras. Now let us come to the content of the present work. As a first step, we give explicit candidates for Q(1)i,z associated with each weight of the vector representation V (1). This is done by taking suitable limits of the known q-characters of KR modules given by tableaux sums. It is known [20] that for the highest and lowest weights this procedure indeed gives the irreducible q-characters. As the next step, we define the formal seriesQ(a)J,z for a general node a of the Dynkin diagram. We define them by Casorati determinants whose entries are Q(1)i,z with suitable shifts of parameters. We then give candidates of polynomial relations among the Q(a)J,z expected from the ψ-system. In the cases related to spin representations, however, we do not have explicit candidates the R(a)ε,z ’s in general, so we only write down the candidates for the relations. The resulting polynomial relations are given in Proposition 5.1, Theorem 5.2 (type A(1)n ), Conjecture 5.3 (type B (1) n ), Conjectures 5.5–5.8 (type C(1)n ) and Conjectures 5.10, 5.11 (type D (1) n ), respectively. For the type A algebras, one can check that Q(1)i,z (not necessarily the highest or lowest ones) are indeed irreducible q-characters of modules given in [23]. The polynomial relations corresponding to the ψ-system can be summarized as a single identity det ( Q(1)ν,q−2µ+n+2zx −µ+ν ν )n+1 µ,ν=1 = 1, (1.2) where xν = eν , {ν} n+1 ν=1 being an orthonormal basis related to the simple roots by αν = ν − ν+1. This relation (as an identity for Q-operators) has been known to experts under the name ‘Wronskian identity’. We shall give a direct proof that (1.2) is satisfied for q-characters in Appendix B. For the other types of algebras, the situation is less satisfactory. At the moment we do not know the irreducibility of modules corresponding to the Q(1)i,z given by our procedure (except those corresponding to the highest or lowest vectors). For the spin representations of C(1)n and D(1)n , explicit formulas for the R (a) ε,z ’s are missing. More seriously, we have not been able to prove the proposed identities for q-characters by computational methods. Instead, we support our working hypothesis by performing the following checks: 1. g = B(1)2 , proof by hand, 2. g = B(1)3 , computer check, 3. g = D(1)4 , computer check, 4. g = B(1)n , C (1) n , D (1) n , proof in the limit to ordinary characters. The main results of the present paper consist in formulating the conjectured relations, and in performing the checks mentioned above. The text is organized as follows. In Section 2, we prepare basic definitions concerning the Borel subalgebra Uq(b) of a quantum affine algebra Uq(g), and collect necessary facts about their representations. In Section 3, we give an account of the ψ-system in the ODE/IM correspondence and indicate how to derive them using the formulation by first-order systems. We note that in [11] the ψ-system for algebras other than A type is mentioned as conjectures. In Section 4, we introduce the series Q(1)i,z as limits of the q-characters of KR modules. We also define these series for the other nodes of the Dynkin diagram. Section 5 is devoted to the proposals for polynomial 4 J. Sun relations. By comparing with the relations for the connection coefficients, we write down the relations for each type of algebras A(1)n , B (1) n , C (1) n and D (1) n . In Section 6, we give a summary of our work. The text is followed by four appendices. Appendix A gives a list of realizations of the dual Lie algebras Lg. In Appendix B we give a proof of the Wronskian identity for type A(1)n . In Appendix C, we prove the identities for type B(1)n in the limit to ordinary characters. In Appendix D the same is done for type C(1)n and D (1) n . 2 Preliminaries In this section we introduce our notation on quantum affine algebras and their Borel subalgebras, and collect necessary facts that will be used later. Throughout this paper, we assume that q is a nonzero complex number which is not a root of unity. 2.1 Quantum Borel algebras Let g be an affine Lie algebra associated with a generalized Cartan matrix C = (cij)0≤i,j≤n of non-twisted type. Let D = diag(d0, . . . , dn) be the unique diagonal matrix such that DC is symmetric and d0 = 1. Set I = {1, 2, . . . , n}, and let g˚ denote the simple Lie algebra with the Cartan matrix (cij)i,j∈I . Let {αi}i∈I , {α∨i }i∈I and {ωi}i∈I be the simple roots, simple coroots and the fundamental weights of g˚, respectively. We set P = ⊕i∈IZωi, Q = ⊕i∈IZαi. Set qi = qdi . We shall use the standard notation [k]i = qki − q −k i qi − q −1 i , [n]i! = n∏ k=1 [k]i, [ n k ] i = [n]i! [k]i![n− k]i! . The quantum affine algebra Uq(g) is the C-algebra defined by generators Ei, Fi, K±1i (i = 0, . . . , n) and the relations KiK −1 i = 1 = K −1 i Ki, KiKj = KjKi, KiEjK −1 i = q cij i Ej , KiFjK −1 i = q −cij i Fj , [Ei, Fj ] = δij Ki −K −1 i qi − q −1 i , 1−cij∑ r=0 [ 1− cij r ] i (−1)rEriEjE 1−cij−r i = 0, i 6= j, 1−cij∑ r=0 [ 1− cij r ] i (−1)rF ri FjF 1−cij−r i = 0, i 6= j. We do not write the formulas defining the Hopf structure on Uq(g) since we are not going to use them. As is well known [6, 13], Uq(g) is isomorphic to the C-algebra with generators x±i,r (i ∈ I, r ∈ Z), k±1i (i ∈ I), hi,r (i ∈ I, r ∈ Z\{0}) and central elements c ±1/2, with the following defining relations kik −1 i = 1 = k −1 i ki, c 1/2c−1/2 = 1 = c−1/2c1/2, kikj = kjki, kihj,r = hj,rki, kix ± j,rk −1 i = q ±cij i x ± j,r, [hi,r, x ± j,s] = ± 1 r [rcij ]ic ∓|r|/2x±j,r+s, x±i,r+1x ± j,s − q ±cij i x ± j,sx ± i,r+1 = q ±cij i x ± i,rx ± j,s+1 − x ± j,s+1x ± i,r, Polynomial Relations for q-Characters via the ODE/IM Correspondence 5 [x+i,r, x − j,s] = δij c(r−s)/2φ+i,r+s − c −(r−s)/2φ−i,r+s qi − q −1 i , ∑ pi∈S1−cij 1−cij∑ k=0 (−1)k [ 1− cij k ] i x±i,rpi(1) · · ·x ± i,rpi(k) x±j,sx ± i,rpi(k+1) · · ·x±i,rpi(1−cij) = 0, i 6= j, for all integers rj , where Sm is the symmetric group on m letters, and the φ±i,r are given by ∞∑ r=0 φ±i,±ru ±r = k±1i exp ( ± ( qi − q −1 i ) ∞∑ s=1 hi,±su ±s ) . By definition, the Borel subalgebra Uq(b) is the Hopf subalgebra of Uq(g) generated by Ei, K±1i (0 ≤ i ≤ n). It is known [22] that Uq(b) is isomorphic to the algebra with generators Ei, K±1i (0 ≤ i ≤ n) and the defining relations KiKj = KjKi, KiEjK −1 i = q cij i Ej , 1−cij∑ r=0 [ 1− cij r ] i (−1)rEriEjE 1−cij−r i = 0, i 6= j. The algebra Uq(b) contains x+i,m, x − i,r, k ±1 i and φ + i,r, where i ∈ I, m ≥ 0 and r > 0. 2.2 Category O of Uq(b) In this subsection we recall basics about Uq(b)-modules in category O. For more details see [20]. Denote by t the subalgebra of Uq(b) generated by {k±1i }i∈I . For a Uq(b)-module V and λ ∈ P , set Vλ = { v ∈ V | kiv = q λ(α∨i ) i v, ∀ i ∈ I } . When Vλ 6= 0, it is called the weight space of weight λ. A Uq(b)-module V is said to be of type 1 if V = ⊕λ∈PVλ. A series of complex numbers Ψ = (Ψi,r)i∈I,r∈Z≥0 is called an `-weight if Ψi,0 6= 0 for all i ∈ I. We denote by t∗` the set of `-weights. For a Uq(b)-module V and Ψ ∈ t ∗ ` , the subspace V (Ψ) = { v ∈ V | ∃ p > 0, ∀ i ∈ I, ∀m ≥ 0, (φ+i,m −Ψi,m) pv = 0 } is called the `-weight space of `-weight Ψ. A Uq(b)-module V is said to be a highest `-weight module of highest `-weight Ψ ∈ t∗` if there exists a vector v ∈ V such that V = Uq(b) v and Eiv = 0, i ∈ I, φ + i,rv = Ψi,rv, i ∈ I, r ≥ 0. For each Ψ ∈ t∗` there exists a unique simple Uq(b)-module of highest `-weight Ψ. We denote it by L(Ψ). A highest `-weight module is of type 1 if its highest `-weight Ψ satisfies Ψi,0 = q pi i for some pi ∈ Z, i ∈ I. (2.1) For any non-zero complex numbers ci ∈ C×, the map Ei 7→ Ei, Ki 7→ ciKi (i = 0, . . . , n) gives rise to an automorphism of Uq(b). After twisting by such an automorphism, any highest `-weight module can be brought to one satisfying the condition (2.1). We denote by t∗`,P the set of `-weights satisfying (2.1). Set D(λ) = λ−Q+, Q+ = ∑ i∈I Z≥0αi. A Uq(b)-module V of type 1 is said to be an object in category O if 6 J. Sun 1. for all λ ∈ P we have dimVλ <∞, 2. there exist a finite number of elements λ1, . . . , λs ∈ P such that the weights of V are contained in ⋃ j=1,...,s D(λj). In what follows we shall identify Ψ ∈ t∗` with their generating series, Ψ = (Ψ1(u), . . . ,Ψn(u)), Ψi(u) = ∑ r≥0 Ψi,ru r. Simple objects in category O are classified by the following theorem. Theorem 2.1 ([20]). Suppose that Ψ ∈ t∗`,P . Then the simple module L(Ψ) is an object in category O if and only if Ψi(u) is a rational function of u for any i ∈ I. In particular, for i ∈ I and z ∈ C×, the simple modules L±i,z = L(Ψ) defined by the highest `-weight Ψj(u) = { (1− zu)±1, j = i, 1, j 6= i, are objects in category O. These modules are called the fundamental representations [20]. It is known [7, 8] that finite-dimensional simple Uq(g)-modules remain simple when restricted to Uq(b). According to the classification of the former [9, 10], the simple module L(Ψ) is finite- dimensional if its highest `-weight has the form Ψi(u) = q degPi i Pi(q −1 i u) Pi(qiu) , ∀ i ∈ I, where Pi(u) is a polynomial such that Pi(0) = 1. In the case where Pj(u) = {( 1− q−m+1i zu )( 1− q−m+3i zu ) · · · ( 1− qm−1i zu ) , j = i, 1, j 6= i, with some i ∈ I, m ∈ Z>0 and z ∈ C×, the module L(Ψ) is called a Kirillov–Reshetikhin (KR) module. We denote it by W (i)m,z. 2.3 Characters and q-characters We recall the definition of q-characters (see [20]) and characters of representations of Uq(b). Let Zt ∗ `,P denote the set of maps t∗`,P → Z. For Ψ ∈ t ∗ `,P , define [Ψ] ∈ Z t∗`,P by [Ψ](Ψ′) = δΨ,Ψ′ . For a Uq(b)-module V of type 1 in category O, its q-character χq(V ) is defined as an element of Zt ∗ `,P , χq(V ) = ∑ Ψ∈t∗`,P dimV (Ψ) · [Ψ]. Similarly let ZP denote the set of maps P → Z, and define eλ ∈ ZP by eλ(µ) = δλ,µ. The ordinary character χ(V ) is an element of ZP , χ(V ) = ∑ λ∈P dimVλ · e λ. We have a natural map $ : Zt ∗ `,P → ZP which sends [Ψ] to eλ such that Ψi,0 = q λ(α∨i ) i . Under $ the q-character specializes to the ordinary character, χ(V ) = $ (χq(V )) . Polynomial Relations for q-Characters via the ODE/IM Correspondence 7 3 ψ-system In this section, we reformulate the ψ-systems given in [11]. 3.1 Lg-connection From now on, let g be an affine Lie algebra of type X(1)n (X = A,B,C,D), and let Lg denote its Langlands dual algebra. Let h∨ be the dual Coxeter number of g (see Table 1). Table 1. g A(1)n B (1) n C (1) n D (1) n Lg A(1)n A (2) 2n−1 D (2) n+1 D (1) n h∨ n+ 1 2n− 1 n+ 1 2n− 2 dimV (1) n+ 1 2n 2n+ 2 2n Denote by ej , fj , hj (0 ≤ j ≤ n) the Chevalley generators of Lg. We set e = ∑n j=1 ej . Fix also an element ` ∈ Lh from the Cartan subalgebra of Lg, and let ζ ∈ C×. We consider the following Lg-valued connection (cf. e.g. [15]): L = d dx − ` x + e+ p(x,E)ζe0, (3.1) p(x,E) = xMh ∨ − E M > 0, E ∈ C. (3.2) Take an element hρ ∈ Lh such that [hρ, ej ] = ej (1 ≤ j ≤ n) and [hρ, e0] = −(h∨ − 1)e0. The choice (3.2) ensures the following symmetry property for L = L(x,E; ζ): ωk(hρ−1)L ( x,E; e2piikζ ) ω−khρ = L ( ωkx,ΩkE; ζ ) , (3.3) where k ∈ C and ω = e 2pii h∨(M+1) , Ω = ωh ∨M . (3.4) On any finite-dimensional Lg-module V , (3.1) defines a first-order system of differential equa- tions L(x,E; 1)φ(x,E) = 0. Quite generally, for a Lg-module V , we denote by Vk the Lg-module obtained by twisting V by the automorphism ej 7→ exp(2piikδj,0)ej , fj 7→ exp(−2piikδj,0)fj . The operator L(x,E; e2piik) represents the action of L(x,E; 1) on Vk. For a V -valued solution φ(x,E) we set φk(x,E) = ω −khρφ ( ωkx,ΩkE ) . (3.5) Then the symmetry (3.3) implies L(x,E; e2piik)φk(x,E) = 0. With each node a of the Dynkin diagram of Lg˚ is associated a fundamental module V (a) of Lg. We summarize our convention about them and some facts which will be used later. We leave the proofs to the readers (see Subsection 5.3 for an example). Remark 3.1. In [11], scalar (pseudo-)differential equations are considered. Using the realization of Lg given in Appendix A and rewriting the equation Lφ = 0 for the highest component of φ, one obtains the formulas [11, (3.18)–(3.21)] (for simplicity we have taken K = 1 there). 8 J. Sun The module V (1) is called the vector representation of Lg. Its explicit realization is given in Appendix A. If k is an integer, it is obvious that Vk = V for any V . The vector representation for g = C(1)n has the additional property V (1)1 2 ' V (1), g = C(1)n . (3.6) For general a, we distinguish the following two cases, (NS) : (g, a) 6= (C(1)n , n), (D (1) n , n− 1), (D (1) n , n), (S) : (g, a) = (C(1)n , n), (D (1) n , n− 1), (D (1) n , n). We shall refer to them as the non-spin case and the spin case, respectively. Here and after we set t = { 2, g = C(1)n , 1, otherwise. In the non-spin case, we have V (a) = a∧ V (1)a−1 2t for (NS). (3.7) In the case g = B(1)n , we have in addition V (a)1 2 ' V (2n−a), g = B(1)n , a = 1, . . . , n, (3.8) where V (a) for a > n stands for the right-hand side of (3.7). In the case (g, a) = (C(1)n , n), V (n) is the spin representation of the subalgebra Lg˚ = o(2n+1) ⊆ D(2)n+1. Likewise, in the cases (g, a) = (D (1) n , n − 1) or (D (1) n , n), V (n−1) and V (n) are the spin representations of the subalgebra Lg˚ = o(2n) ⊆ D(1)n . Let us consider the solutions at the irregular singularity x = ∞. It is convenient to use a gauge transformed form of L, xMhρLx−Mhρ = d dx + ΛxM − `+Mhρ x − Ee0x −M(hρ−1), (3.9) where Λ = e+ e0. Let µ(a) be the eigenvalue of Λ on V (a) which has the largest real part. This eigenvalue is multiplicity free, and is given explicitly as follows: µ(a) =    sin piath∨ sin pith∨ for (NS), 1 2 sin pith∨ for (S). Let u(a) be an eigenvector of Λ corresponding to µ(a). From the representation (3.9) it follows that there is a unique V (a)-valued solution ψ(a)(x,E) which satisfies the following in a sector containing the positive real axis x > 0: ψ(a)(x,E) = e− µ(a) M+1x M+1 x−Mhρ ( u(a) + o(1) ) (x→∞). We call ψ(a)(x,E) the canonical solution. In view of the relation (3.7) and the formula for µ(a) given above, we have ψ(a) = ψ(1) −a−12t ∧ψ(1) −a−32t ∧ · · · ∧ψ(1)a−1 2t for (NS). (3.10) Here ψ(i)k = ω −khρψ(i)(ωkx,ΩkE) as defined by (3.5). Polynomial Relations for q-Characters via the ODE/IM Correspondence 9 3.2 ψ-system In this subsection we state the ψ-system for the canonical solutions ψ(a)(x,E) introduced above. Let us consider them case by case. g = A(1)n (Lg = A (1) n ). For a = 1, . . . , n, we have the embedding of Lg-modules ι : 2∧ V (a)1 2 ↪→ V (a−1) ⊗ V (a+1), (3.11) where V (0) = V (n+1) = C. The explicit expression of the embedding is given in Subsection 5.1. On the space V (a−1) ⊗ V (a+1), the functions φ = ι ( ψ(a) − 12 ∧ ψ(a)1 2 ) and φ = ψ(a−1) ⊗ ψ(a+1) both satisfy the equation Lφ = 0 and have the behavior φ(x,E) = O ( exp ( − 2µ(a) cos pih∨ M + 1 xM+1 )) , x→∞ in a sector containing x > 0. Since such a solution is unique upto a constant multiple, we conclude that (after adjusting the constant multiple) ι ( ψ(a) − 12 ∧ψ(a)1 2 ) = ψ(a−1) ⊗ψ(a+1), a = 1, . . . , n. (3.12) In particular, denoting by {uj} n+1 j=1 the standard basis of V (1) we have ψ(n+1) = u1 ∧ · · · ∧ un+1. (3.13) We call the relations (3.12), (3.13) the ψ-system for A(1)n . The ψ-system for the other types can be deduced by the same argument, using the relevant embeddings of representations. We obtain relations of the following form. g = B(1)n (Lg = A (2) 2n−1). ι ( ψ(a) − 12 ∧ψ(a)1 2 ) = ψ(a−1) ⊗ψ(a+1), a = 1, . . . , n− 1, (3.14) ι ( ψ(n) − 14 ∧ψ(n)1 4 ) = ψ(n−1) − 14 ⊗ψ(n−1)1 4 . (3.15) Here ι stands for the embedding of Lg (3.11) or 2∧ V (n)1 4 ↪→ V (n−1) − 14 ⊗ V (n−1)1 4 , which follows from (3.11) and (3.8). g = C(1)n (Lg = D (2) n+1). ι ( ψ(a) − 14 ∧ψ(a)1 4 ) = ψ(a−1) ⊗ψ(a+1), a = 1, . . . , n− 2, (3.16) ι ( φ(n−1) ) = ψ(n) − 12 ∧ψ(n)1 2 , (3.17) ι ( φ(n) ) = ψ(n) − 14 ⊗ψ(n)1 4 . (3.18) Here we have set φ(n−1) = ψ(1) −n−24 ∧ψ(1) −n−44 ∧ · · · ∧ψ(1)n−2 4 , φ(n) = ψ(1) −n−14 ∧ψ(1) −n−34 ∧ · · · ∧ψ(1)n−1 4 , 10 J. Sun and ι stands for an analog of (3.11) or the embeddings n−1∧ V (1)n−2 4 ' V (n−1) ↪→ 2∧ V (n)1 2 , n∧ V (1)n−1 4 ↪→ V (n) − 14 ⊗ V (n)1 4 , where (3.6) is taken into account. g = D(1)n (Lg = D (1) n ). ι ( ψ(a) − 12 ∧ψ(a)1 2 ) = ψ(a−1) ⊗ψ(a+1), a = 1, . . . , n− 3, (3.19) ι ( ψ(n−2) ) = ψ(n−1) − 12 ∧ψ(n−1)1 2 , ι ( ψ(n−2) ) = ψ(n) − 12 ∧ψ(n)1 2 , (3.20) ι ( φ(n−1) ) = ψ(n−1) ⊗ψ(n). (3.21) We have set φ(n−1) = ψ(1) −n−22 ∧ψ(1) −n−42 ∧ · · · ∧ψ(1)n−2 2 and ι stands for the embedding (3.11) or n−2∧ V (1)n−3 2 ' V (n−2) ↪→ 2∧ V (n−1)1 2 , n−2∧ V (1)n−3 2 ' V (n−2) ↪→ 2∧ V (n)1 2 , n−1∧ V (1)n−2 2 ↪→ V (n−1) ⊗ V (n). 3.3 Connection coefficients Now we introduce the connection coefficients Q(a)J (E). First let us consider the vector represen- tation V (1). We choose ` generic. Set ` = diag(`1, . . . , `N ) (N = dimV (1), see Appendix A), and let uj (1 ≤ j ≤ N) be the corresponding eigenvector of `. Then there is a unique V (1)-valued solution χ(1)j (x,E) characterized by the expansion at the origin, χ(1)j (x,E) = x `juj (1 +O(x)) , x→ 0. From the symmetry (3.3) of L(x,E) we find that χ(1)j,k(x,E) = ω kλjχ(1)j (x,E), k ∈ (1/t)Z, (3.22) where we have set `− hρ = diag(λ1, . . . , λN ). (3.23) Define Q(1)j (E) ∈ C by ψ(1)(x,E) = N∑ j=1 Q(1)j (E)χ (1) j (x,E). (3.24) From (3.22) and (3.24), we have for k ∈ (1/t)Z ψ(1)k (x,E) = N∑ j=1 Q(1)j,k(E)ω kλjχ(1)j (x,E), Polynomial Relations for q-Characters via the ODE/IM Correspondence 11 where Q(1)j,k(E) = Q (1) j (Ω kE). For a sequence J = (j1, . . . , ja), introduce the notation Q(a)J (E) = det ( Q(1) jm, 2l−a−12t (E) · ω(2l−a−1)λjm/2t ) 1≤l,m≤a , χ(a)J (x,E) = χ (1) j1,− a−1 2t (x,E) ∧ · · · ∧ χ(1) ja,−a−12t (x,E). It follows from (3.10) that in the non-spin case ψ(a)(x,E) = ∑ J Q(a)J (E)χ (a) J (x,E), (3.25) where the sum is taken over all J = (j1, . . . , ja), 1 ≤ j1 < · · · < ja ≤ N . One can similarly define the connection coefficients in the spin case as well. 4 Series Q(a)J,z, R (a) ε,z It has been shown in [20] that the fundamental modules L±i,z of the Borel subalgebra arise as certain limits of the KR modules. In this section we follow the same procedure to obtain a family of formal power series Q(1)i,z associated with each weight space of the vector representation of Lg. Up to simple overall multipliers, those corresponding to the highest or lowest weights are the irreducible q-characters χq(L ± i,z). We expect that in general the Q (1) i,z are also proportional to irreducible q-characters of Uq(b). (See however Remark 5.9.) Recall in Subsection 2.3, for Ψ ∈ t∗`,P , the element [Ψ] ∈ Z t∗`,P is defined by [Ψ](Ψ′) = δΨ,Ψ′ . Below we shall use the following elements of Zt ∗ `,P : Yi,z = [(1, . . . , i-th (1− zu)−1, . . . , 1)], eωi = [(1, . . . , i-th qi , . . . , 1)], (4.1) Yi,z = [(1, . . . , i-th qi 1− q−1i zu 1− qizu , . . . , 1)] = eωiYi,qizY −1 i,q−1i z , (4.2) Ai,z = Yi,q−1i z Yi,qiz ∏ 1≤j≤n, cji=−1 Y −1j,z ∏ 1≤j≤n, cji=−2 Y −1 j,q−1j z Y −1j,qjz ∏ 1≤j≤n, cji=−3 Y −1 j,q−2j z Y −1j,z Y −1 j,q2j z . (4.3) Highest `-weights are monomials in Y±1i,z and e ±ωi . Abusing the notation, for a monomial M = [Ψ] we shall also write L(M) for L(Ψ). 4.1 The limiting procedure Let us illustrate on examples the procedure for taking the limit. Example 4.1 (g = A(1)2 ). We consider first the case g = A (1) 2 . Following [26, 27], we write 1 z = Y1,z, 2 z = Y −1 1,q2zY2,qz, 3 z = Y −1 2,q3z. Then the q-character χq ( W (1)m,z ) of the KR module W (1)m,z is presented as a sum over the tableaux ∑ k1+k2+k3=m, k1,k2,k3≥0 k1 ︷ ︸︸ ︷ 1 · · · 1 k2 ︷ ︸︸ ︷ 2 · · · 2 k3 ︷ ︸︸ ︷ 3 · · · 3 12 J. Sun where the k-th box from the right carries the parameter qm+1−2kz. This can be rewritten further as m∏ j=1 Y1,qm+1−2jz ∑ k1+k2+k3=m, k1,k2,k3≥0 k2+k3∏ j=1 A−11,qm+2−2jz k3∏ j=1 A−12,qm+3−2jz = ∑ k1+k2+k3=m, k1,k2,k3≥0 e(k1−k2)ω1e(k2−k3)ω2 × [( (1− q−mz)(1− qm−2k3+2z) (1− qm−2(k2+k3)z)(1− qm−2(k2+k3)+2z) , (1− qm−2(k3+k2)+1z)(1− qm+3z) (1− qm−2k3+1z)(1− qm−2k3+3z) )] . Let us consider the limit m→∞. There are three possibilities to obtain meaningful answers, k1 →∞, k2, k3 : finite, q −k1 → 0, k2 →∞, k1, k3 : finite, q −k2 → 0, k3 →∞, k1, k2 : finite, q −k3 → 0. Writing eαi = xi/xi+1 and defining for i = 1, 2, 3 Q(1)i,z = i−1∏ j=1 (1− xj/xi)× lim ki→∞ q−kiz→0 x−mi (x1x2x3) m 3 χq ( W (1)m,q−mz ) , we obtain the result Q(1)1,z = Y1,z ∑ k2,k3≥0 k2+k3∏ j=1 A−11,q2−2jz k3∏ j=1 A−12,q3−2jz, Q(1)2,z = Y −1 1,q2zY2,qz ∑ k3≥0 k3∏ j=1 A−12,q3−2jz, Q (1) 3,z = Y −1 2,q3z. Up to simple multipliers, they are the irreducible q-characters χq ( L−1,z ) = Q(1)1,z, χq ( L(Y−11,q2zY2,qz) ) = Q(1)2,z 1− e−α1 , χq ( L+2,q3z ) = Q(1)3,z (1− e−α1−α2)(1− e−α2) . Using the explicit construction of modules [23], it can be checked that the second one is the q- character of the simple module whose highest `-weight corresponds to the monomial Y−11,q2zY2,qz. We give two more examples. Example 4.2 (g = B(1)2 ). Q(1)1,z = Y1,z    ∑ k2,k2¯,k1¯≥0 K1¯∏ l=1 A−11,q3−2lz K2¯∏ l=1 A−12,q2−2lz K2¯∏ l=1 A−12,q3−2lz K2∏ l=1 A−11,q2−2lz + ∑ k2,k2¯,k1¯≥0 K1¯∏ l=1 A−11,q3−2lz K2¯∏ l=1 A−12,q2−2lz K2¯+1∏ l=1 A−12,q3−2lz K2+1∏ l=1 A−11,q2−2lz    , Polynomial Relations for q-Characters via the ODE/IM Correspondence 13 Q(1)2,z = Y −1 1,q2zY2,qz    ∑ k2¯,k1¯≥0 K1¯∏ l=1 A−11,q3−2lz K2¯∏ l=1 A−12,q2−2lz K2¯∏ l=1 A−12,q3−2lz + ∑ k2¯,k1¯≥0 K1¯∏ l=1 A−11,q3−2lz K2¯∏ l=1 A−12,q2−2lz K2¯+1∏ l=1 A−12,q3−2lz    , Q(1)2¯,z = Y1,qzY −1 2,q2z ∑ k≥0 k∏ l=1 A−11,q3−2lz, Q (1) 1¯,z = Y −1 1,q3z. Here we set K1¯ = k1¯, K2¯ = k2¯ + k1¯ and K2 = k2 + k2¯ + k1¯. We have χq(L − 1,z) = Q (1) 1,z and χq(L + 1,q3z) = Q(1)1¯,z (1− e−α1)(1− e−α1−α2)(1− e−α1−2α2) . Example 4.3 (g = C(1)2 ). Q(1)1,z = Y1,z ∑ k2,k0,k2¯,k1¯≥0 K1¯∏ l=1 A−11,q3−lz K2¯∏ l=1 A−12,q2−lz k0∏ l=1 A−1 2,q2−K2¯−2lz K2∏ l=1 A−11,q1−lz, Q(1)2,z = Y −1 1,qzY2,zY2,qz ∑ k0,k2¯,k1¯≥0 K1¯∏ l=1 A−11,q3−lz K2¯∏ l=1 A−12,q2−lz k0∏ l=1 A−1 2,q2−K2¯−2lz , Q(1)0,z = Q̂ (1) 0,z + Q̂ (1) 0¯,z, Q̂(1)0,z = Y2,qzY −1 2,q2z ∑ k2¯,k1¯≥0, K2¯:even K1¯∏ l=1 A−11,q3−lz K2¯ 2∏ l=1 A−12,q3−2lz, Q̂(1)0¯,z = x −1 2 Y −1 1,qzY1,q2zY2,zY −1 2,q3z ∑ k2¯,k1¯≥0, K2¯:odd K1¯∏ l=1 A−11,q3−lz K2¯−1 2∏ l=1 A−12,q2−2lz, Q(1)2¯,z = Y1,q2zY −1 2,q2zY −1 2,q3z ∑ k1¯≥0 k1¯∏ l=1 A−11,q3−lz, Q(1)1¯,z = Y −1 1,q3z. Here we set K1¯ = k1¯, K2¯ = k2¯ + k1¯ and K2 = k2 + 2k0 + k2¯ + k1¯. Note that the weight 0 has multiplicity 2. Correspondingly Q(1)0,z is a sum of two terms Q̂ (1) 0,z and Q̂ (1) 0¯,z. These terms cannot be separated in the process of taking the limit. Since they have highest `-weights whose ratio is not a monomial of the Ai,z’s, Q (1) 0,z cannot be an irreducible q-character. We have χq(L − 1,z) = Q (1) 1,z and χq(L + 1,q3z) = Q(1)1¯,z (1− e−α1)(1− e−α1−α2)(1− e−2α1−α2)2 . 4.2 Series Q(1)i,z In order to discuss the general case, let us prepare some notation. For g = X(1)n (X = A,B,C,D), introduce a parametrization of {αi, ωi} by orthonormal vectors {i}, and an index set J with 14 J. Sun a partial ordering ≺ as follows. A(1)n : αi = i − i+1 (1 ≤ i ≤ n), ωi = 1 + · · ·+ i − i n+ 1 n+1∑ j=1 j (1 ≤ i ≤ n), J = {1, 2, . . . , n+ 1}, 1 ≺ 2 ≺ · · · ≺ n+ 1, B(1)n : αi = i − i+1 (1 ≤ i ≤ n− 1), αn = n, ωi = 1 + · · ·+ i (1 ≤ i ≤ n− 1), ωn = 1 2 (1 + · · ·+ n), J = {1, . . . , n, n¯, . . . , 1¯}, 1 ≺ · · · ≺ n ≺ n¯ ≺ · · · ≺ 1¯, C(1)n : αi = 1 √ 2 (i − i+1) (1 ≤ i ≤ n− 1), αn = √ 2n, ωi = 1 √ 2 (1 + · · ·+ i) (1 ≤ i ≤ n), J = {1, . . . , n, 0, n¯, . . . , 1¯}, 1 ≺ · · · ≺ n ≺ 0 ≺ n¯ ≺ · · · ≺ 1¯, D(1)n : αi = i − i+1 (1 ≤ i ≤ n− 1), αn = n−1 + n, ωi = 1 + · · ·+ i (1 ≤ i ≤ n− 2), ωn−1 = 1 2 (1 + · · ·+ n−1 − n), ωn = 1 2 (1 + · · ·+ n−1 + n), J = {1, . . . , n, n¯, . . . , 1¯}, 1 ≺ · · · ≺ n− 1 ≺ n n¯ ≺ n− 1 ≺ · · · ≺ 1¯. Define also xi and fj,k for j, k ∈ J by A(1)n : xi = e i (1 ≤ i ≤ n+ 1), fj,k = 1 1− xk/xj , B(1)n : xi = e i = x−1i¯ (1 ≤ i ≤ n), fj,k = 1 + δk,j¯/xj 1− xk/xj , C(1)n : xi = e 1√ 2 i = x−1i¯ (1 ≤ i ≤ n), x0 = 1, fj,k = 1 (1− xk/xj)(1 + δk,0xk/xj) , D(1)n : xi = e i = x−1i¯ (1 ≤ i ≤ n), fj,k = 1− δk,j¯xk/xj 1− xk/xj . In the following, in the sum of the form ∑ ki,··· ,kj , unless mentioned explicitly, ki, . . . , kj run over all non-negative integers, and we use the abbreviation Kl = ∑j µ=l kµ for i ≺ l ≺ j. In the case g = C(1)n and l ≺ 0, we set Kl = ∑n j=l kj + 2k0 + ∑n j=1 kj¯ . We give below the formula for Q(1)i,z for each i ∈ J . Case A(1)n : For 1 ≤ i ≤ n+ 1, Q(1)i,z = Φi,z ∑ ki+1,...,kn+1 n∏ j=i Kj+1∏ l=1 A−1j,qj+1−2lz, where Φi,z = Y −1 i−1,qizYi,qi−1z. (4.4) Polynomial Relations for q-Characters via the ODE/IM Correspondence 15 Case B(1)n : For 1 ≤ i ≤ n, Q(1)i,z = Φi,z    ∑ ki+1,...,kn,kn¯,...,k1¯ n∏ j=i Kj+1∏ l=1 A−1j,qj+1−2lz n∏ j=1 Kj¯∏ l=1 A−1j,q2n−j−2lz + ∑ ki+1,...,kn,kn¯,...,k1¯ n∏ j=i 1+Kj+1∏ l=1 A−1j,qj+1−2lz n∏ j=1 Kj¯∏ l=1 A−1j,q2n−j−2lz    , Q(1)i¯,z = Φi¯,z ∑ ki−1,...,k1¯ i−1∏ j=1 Kj¯∏ l=1 A−1j,q2n−j−2lz, where Φi,z = Y −1 i−1,qizYi,qi−1z, Φi¯,z = Yi−1,q2n−i−1zY −1 i,q2n−i . (4.5) Case C(1)n : Q(1)i,z = Φi,z ∑ ki+1,...,kn,k0,kn¯,...,k1¯ n−1∏ j=i Kj+1∏ l=1 A−1 j,q 1+j−2l 2 z × n−1∏ j=1 Kj¯∏ l=1 A−1 j,q 2n+3−j−2l 2 z Kn¯∏ l=1 A−1 n,q n+2−2l 2 z k0∏ l=1 A−1 n,q n+2−2(Kn¯+2l) 2 z , 1 ≤ i ≤ n, Q(1)0,z = Q̂ (1) 0,z + Q̂ (1) 0¯,z, Q̂(1)0,z = Φ0,z ∑ kn¯,...,k1¯, Kn¯:even n−1∏ j=1 Kj¯∏ l=1 A−1 j,q 2n+3−j 2 −lz Kn¯ 2∏ l=1 A−1 n,q n+4 2 −2lz , Q̂(1)0¯,z = x −1 n Φ0¯,z ∑ kn¯,...,k1¯, Kn¯:odd n−1∏ j=1 Kj¯∏ l=1 A−1 j,q 2n+3−j 2 −lz Kn¯−1 2∏ l=1 A−1 n,q n+2 2 −2lz , Q(1)i¯,z = Φi¯,z ∑ ki−1,...,k1¯ i−1∏ j=1 Kj¯∏ l=1 A−1 j,q 2n+3−j 2 −lz , 1 ≤ i ≤ n, where Φi,z = Y −1 i−1,q i 2 z Y i,q i−1 2 z , Φi¯,z = Yi−1,q 2n+2−i 2 z Y−1 i,q 2n+3−i 2 z , 1 ≤ i ≤ n− 1, (4.6) Φn,z = Y −1 n−1,q n 2 z Y n,q n−2 2 z Y n,q n 2 z , Φn¯,z = Y n−1,q n+2 2 z Y−1 n,q n+2 2 z Y−1 n,q n+4 2 z , (4.7) Φ0,z = Yn,q n2 zY −1 n,q n+2 2 z , Φ0¯,z = Y −1 n−1,q n 2 z Y n−1,q n+2 2 z Y n,q n−2 2 z Y−1 n,q n+4 2 z . (4.8) Case D(1)n : Q(1)i,z = Φi,z    ∑ ki+1,...,kn,...,k1¯ n−2∏ j=1 Kj¯∏ l=1 A−1j,q2n−1−j−2lz Kn−1∏ l=1 A−1n,qn−2lz n−1∏ j=i Kj+1∏ l=1 A−1j,qj+1−2lz 16 J. Sun + ∑ ki+1,...,kn¯,...,k1¯, kn¯≥1 n−1∏ j=1 Kj¯∏ l=1 A−1j,q2n−1−j−2lz Kn¯∏ l=1 A−1n,qn−2lz n−2∏ j=i Kj+1∏ l=1 A−1j,qj+1−2lz    , 1 ≤ i ≤ n− 1, Q(1)n,z = Φn,z ∑ kn−1,...,k1¯ n−2∏ j=1 Kj¯∏ l=1 A−1j,q2n−1−j−2lz Kn−1∏ l=1 A−1n,qn−2lz, Q(1)i¯,z = Φi¯,z ∑ ki−1,...,k1¯ i−1∏ j=1 Kj¯∏ l=1 A−1j,q2n−1−l−2lz, 1 ≤ i ≤ n, where Φi,z = Y −1 i−1,qizYi,qi−1z, Φi¯,z = Yi−1,q2n−2−izY −1 i,q2n−1−iz, 1 ≤ i ≤ n− 2, (4.9) Φn−1,z = Y −1 n−2,qn−1zYn−1,qn−2zYn,qn−2z, Φn−1,z = Yn−2,qn−1zY −1 n−1,qnzY −1 n,qnz, (4.10) Φn,z = Y −1 n−1,qnzYn,qn−2z, Φn¯,z = Yn−1,qn−2zY −1 n,qnz. (4.11) Remark 4.4. As in [26, 27], it is known that the q-character of KR module W (1)m,z for B (1) n can be written in terms of tableau of elementary boxes a , a ∈ {1, . . . , n, 0, n¯, . . . , 1¯}, while the elemen- tary box 0 is allowed to appear at most once. When taking limit of the q-character of W (1)m,z, we have only 2n kinds of meaningful results, then we use the index set {1, . . . , n, n¯, . . . , 1¯}. By similar reason we use {1, . . . , n, 0, n¯, . . . , 1¯} as the index set of C(1)n , which looks like opposite to the usual one. 4.3 Series Q(a)J,z In this subsection we give the definition of Q(a)J,z where J ∈ J a and (g, a) 6= (C(1)n , n), (D (1) n , n−1), (D(1)n , n). We recall that q1 = q1/2 for C (1) n and q1 = q in the other cases. We fix our convention about the indices as follows. For J = (j1, . . . , ja) ∈ J a, we denote by J ∼ the underlying set {j1, . . . , ja} ⊆ J . Set further J = (ja, . . . , j1), (4.12) J∗ = (i1, . . . , ib), with J ∗ ∼ = J \J ∼ and i1 ≺ · · · ≺ ib. (4.13) We say J is increasing if j1 ≺ · · · ≺ ja. For an element J = (j1, . . . , ja) ∈ J a, we define Q (a) J,z by Q(a)J,z = det ( Q(1) jν ,q −a−1+2µ 1 z xµ−1jν )a µ,ν=1 = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ Q(1) j1,q 1−a 1 z x0j1 Q (1) j2,q 1−a 1 z x0j2 · · · Q (1) ja,q 1−a 1 z x0ja Q(1) j1,q 3−a 1 z xj1 Q (1) j2,q 3−a 1 z xj2 · · · Q (1) ja,q 3−a 1 z xja ... ... ... Q(1) j1,q a−1 1 z xa−1j1 Q (1) j2,q a−1 1 z xa−1j2 · · · Q (1) ja,q a−1 1 z xa−1ja ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ . (4.14) We set Q(0)∅,z = 1. Polynomial Relations for q-Characters via the ODE/IM Correspondence 17 Note that if the entries of J ∈ J a are not distinct (i.e. ]J ∼ < a), then Q(a)J,z = 0. Define Q¯(a)J = $ ( Q(a)J,z ) . Hence by the definition of Q(a)J,z, it is easy to get Q¯(a)J = ∏ i∈J,j∈J∗, i≺j fi,j ∏ i,j∈J, i≺j ((xj − xi)fi,j) . (4.15) For an increasing element J = (j1, . . . , ja) ∈ J a, set ΦJ,z = a∏ k=1 Φjk,qa+1−2k1 z , (4.16) where Φj,z are defined in (4.4)–(4.11) with Yi,z being given in (4.1). Let L(ΦJ,z) be the unique irreducible Uq(b)-module with the highest `-weight ΦJ,z. Except in the case g = C (1) n , we expect that the q-character of L(ΦJ,z) is given by χq(L(ΦJ,z)) = ∏ i∈J,j∈J∗ ij fj,i ∏ i,j∈J, i≺j ( −x−1i ) · Q(a)J,z. In particular, from (4.15) we expect that χ (L(ΦJ,z)) = ∏ (i,j)∈(J×J∗)∪(J∗×J), i≺j fi,j ×    1, g = A(1)n , ∏ i,¯i∈J, i≺i¯ ( 1 + x−1i ) , g = B(1)n , ∏ i,¯i∈J, i≺i¯ ( 1− x−2i ) , g = D(1)n . In the case of g = C(1)n , we expect the same to be true for Q (a) J,z if j1, . . . , ja 6= 0. As noted already in Example 4.3, the series Q(1)0,z does not correspond to an irreducible q-character. We expect rather that Q̂(1)0,z, Q̂ (1) 0¯,z correspond to irreducible q-characters. See also Remark 5.9. 4.4 Series R(n)ε,z for the spin node: case C (1) n In this subsection, we introduce another set of series R(n)ε,z for the spin node. Unlike the se- ries Q(a)J,z, in general we do not know the explicit formulas for them. We define R (n) ε,z by the q-characters of the irreducible Uq(b) module L(Mε,z), and give a rule to determine the highest `-weight Mε,z which is a monomial in C[Y±1i,z ]1≤i≤n,z∈C× . Exhibiting the n-dependence explicitly, let Pn be the weight lattice of the simple Lie algebra of type Cn. Set En = {ε = (ε1, . . . , εn) | εi = ±1, 1 ≤ i ≤ n} . We define two weight functions w1 : En −→ Pn, ε 7−→ 1 2 n∑ i=1 εii, (4.17) 18 J. Sun w2 : {J | J ∈ J a(1 ≤ a ≤ ]J ), J ∼ ⊆ J} −→ Pn, (j1, . . . , ja) 7−→ a∑ k=1 sgn(jk)jk , (4.18) where 0 = 0, and for j ∈ J we set sgn(j) =    +, j  n, 0, j = 0, −, j  n¯. Borrowing an idea from [26], we introduce monomials Mε,z in C[Y±1i,z ]1≤i≤n,z∈C× inductively as follows. Define two operators τY , τ zc by τY : Yi,z → Yi+1,z, τ z c : Yi,z → Yi,qcz. For n = 1 we set M+,z = Y1,z and M−,z = Y −1 1,q2z. In the general case we set M(++ξ),z = τ Y(M(+ξ),z), M(+−ξ),z = Y1,q n2 zτ Y(M(−ξ),z), (4.19) M(−+ξ),z = Y −1 1,q n+2 2 z τ z1 τ Y(M(+ξ),z), M(−−ξ),z = τ z 1 τ Y(M(−ξ),z), (4.20) where ξ ∈ En−2. Now let ε ∈ En with w1(ε) = 12 (∑s k=1 ϕk − ∑t k=1 ψk ) , and consider the simple module L(Mε,z). We define R (n) ε,z and R¯ (n) ε through the q-character and the character as follows. R(n)ε,z = χq (L(Mε,z)) ∏ 1≤k≤l≤t (1− xψkxψl) ∏ 1≤k≤s,1≤l≤t, ϕk>ψl ( 1− x−1ϕkxψl ) , R¯(n)ε = χ (L(Mε,z)) ∏ 1≤k≤l≤t (1− xψkxψl) ∏ 1≤k≤s,1≤l≤t, ϕk>ψl ( 1− x−1ϕkxψl ) . For the latter we have the following guess. Conjecture 4.5. For ε = (ε1, . . . , εn) ∈ En, we have χ (L(Mε,z)) = ∏ 1≤i≤n 1 1− x−2εii ∏ 1≤iψl ( 1− x−1ϕkxψl ) , R¯(n−ς)ε = χ (L(Mε,z)) ∏ 1≤kψl ( 1− x−1ϕkxψl ) . Conjecture 4.6. For ε = (ε1, . . . , εn) ∈ En,ς (ς = 0, 1), we have χ (L(Mε,z)) = ∏ 1≤i n+ 1− i. Proposition B.2. For an element J = (j1, j2, . . . , ja) ∈ J a, we have Q(a)J,z = Y −1 n+1−a,qn+1z ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∆(0)j1,q1−azx 0 j1 ∆ (0) j2,q1−az x0j2 · · · ∆ (0) ja,q1−az x0ja ∆(1)j1,q3−azxj1 ∆ (1) j2,q3−az xj2 · · · ∆ (1) ja,q3−az xja ... ... ... ∆(a−1)j1,qa−1zx a−1 j1 ∆ (a−1) j2,qa−1z xa−1j2 · · · ∆ (a−1) ja,qa−1z xa−1ja ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ . Proof. From the definition we have Q(a)J,z = a∏ k=1 Y−1n,qn−a+2kz ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∆(0)j1,q1−azx 0 j1 ∆ (0) j2,q1−az x0j2 · · · ∆ (0) ja,q1−az x0ja ∆(0)j1,q3−azxj1 ∆ (0) j2,q3−az xj2 · · · ∆ (0) ja,q3−az xja ... ... ... ∆(0)j1,qa−1zx a−1 j1 ∆ (0) j2,qa−1z xa−1j2 · · · ∆ (0) ja,qa−1z xa−1ja ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ . For k = 2, . . . , a, we subtract the (k− 1)-th row multiplied by xn+1 from the k-th row and then extract the factor A−1n,qn−a−1+2kz. Thus we get Q(a)I,z = a∏ k=1 Y−1n,qn−a+2kz a∏ k=2 A−1n,qn−a−1+2kz × ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∆(0)j1,q1−azx 0 j1 ∆ (0) j2,q1−az x0j2 · · · ∆ (0) ja,q1−az x0ja ∆(1)j1,q3−azxj1 ∆ (1) j2,q3−az xj2 · · · ∆ (1) ja,q3−az xja ... ... ... ∆(1)j1,qa−1zx a−1 j1 ∆ (1) j2,qa−1z xa−1j2 · · · ∆ (1) ja,qa−1z xa−1ja ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ . Repeating the above steps in the order b = 2, . . . , a − 1, by replacing xn+1 and A −1 n,qn−a−1+2kz with xn+2−b and A −1 n+1−b,qn−a−b+2kz (k = b+ 1, . . . , a), we get Q(a)I,z = a∏ k=1 Yn,qn−a+2kz a−1∏ b=1 a∏ k=b+1 A−1n+1−b,qn−a−b+2kz × ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∆(0)j1,q1−azx 0 j1 ∆ (0) j2,q1−az x0j2 · · · ∆ (0) ja,q1−az x0ja ∆(1)j1,q3−azxj1 ∆ (1) j2,q3−az xj2 · · · ∆ (1) ja,q3−az xja ... ... ... ∆(a−1)j1,qa−1zx a−1 j1 ∆ (a−1) j2,qa−1z xa−1j2 · · · ∆ (a−1) ja,qa−1z xa−1ja ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ . The proof is over by noting that a∏ k=1 Y−1n,qn−a+2kz a−1∏ b=1 a∏ k=b+1 A−1n+1−b,qn−a−b+2kz = Y −1 n+1−a,qn+1z.  28 J. Sun Now we give a proof of Theorem 5.2. Proof of Theorem 5.2. By Proposition B.2, for J = (1, 2, . . . , n+ 1), we have Q(n+1)J,z = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∆(0)1,q−nzx 0 1 ∆ (0) 2,q−nzx 0 2 · · · ∆ (0) n+1,q−nzx 0 n+1 ∆(1)1,q2−nzx1 ∆ (1) 2,q2−nzx2 · · · ∆ (1) n+1,q2−nzxn+1 ... ... ... ∆(n)1,qnzx n 1 ∆ (n) 2,qnzx n 2 · · · ∆ (n) n+1,qnzx n n+1 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ . By Lemma B.1, we get Q(n+1)J,z = (−1) (n+1)n/2 n∏ b=1 xn+1−bb , which implies the result.  C Proofs of identities for characters in the case B(1)n In this section, we set g = B(1)n . First we introduce some notation. For i, j ∈ J and an element J ∈ J a, we define hJ = ∏ k,k¯∈J, k≺k¯ ( x 1 2 k + x − 12 k ) , x˜i,J = ∏ k∈J, k≺i ( −x 1 2 δi,k¯−1 i ) ∏ k∈J∗, i≺k ( x 1− 12 δk,¯i k ) , gi,j = fi,jx 1− 12 δj,¯i j = −gj,i, gi,J = ∏ k∈J, k 6=i gi,k, where fi,j is defined in Subsection 4.2. By (4.15) and the above notation, one has Q¯(a)J = ∏ i∈J,j∈J∗, i≺j fi,j ∏ i,j∈J, i≺j ((xj − xi)fi,j) = X −1 J hJ ∏ i∈J,j∈J∗, i≺j fi,j , where XJ is given in (5.5). A direct computation gives ∏ i∈J∗,j∈J, i≺j fi,j = ∏ i∈J,j∈J∗, i≺j fi,j , ∏ j∈J,¯i≺j fi¯,j = ∏ j∈J,j≺i fj,i, ∏ i∈J−jk,j∈(J−jk) ∗, i≺j fi,j ∏ i∈J,j∈J∗, i≺j fi,j = ∏ i∈J, i≺jk gjk,j ∏ j∈J∗, jk≺j gjk,j x˜jk,J . Furthermore, for two elements J1 and J2 of J n and jk ∈ J2 ∼ ∩ J∗1 ∼ , we have xjk,J1 x˜jk,J2 xjk,J2 x˜jk,J1 = −x2jkxJ1x −1 J2 . Using the above formulas, the specialization of Conjecture 5.3 to the characters Q¯(a)J reduces to the following proposition. Polynomial Relations for q-Characters via the ODE/IM Correspondence 29 Proposition C.1. For two increasing elements J1 = (i1, . . . , in), J2 = (j1, . . . , jn) ∈ J n, we have x − 12 J1 hJ∗1 hJ2 − x − 12 J2 hJ∗2 hJ1 = x −1 J1 x−1J2 ∑ j∈J2∩J∗1 x 3 2 J1 x 3 2 j gj,J∗1 gj,J∗2 hJ2−jhJ∗1−j . (C.1) Proof. Generally let T1 = { j ∈ J1 ∼ ∩ J∗2 ∼ ∣ ∣ ∣ ∣ j¯ /∈ J1, j¯ ∈ J ∗ 2 or j¯ ∈ J1 ∼ , j¯ /∈ J∗2 ∼ } , T2 = { j ∈ J1 ∼ ∩ J∗2 ∼ ∣ ∣ ∣ ∣j¯ /∈ J1∼ , j¯ /∈ J∗2 ∼ } , T3 = { j ∈ J1 ∼ ∩ J∗2 ∼ ∣ ∣ ∣ ∣j¯ ∈ J1∼ ∩ J∗2 ∼ , j ≺ j¯ } , S1 = { j ∈ J2 ∼ ∩ J∗1 ∼ ∣ ∣ ∣ ∣j¯ /∈ J2∼ , j¯ ∈ J∗1 ∼ , or j¯ ∈ J2 ∼ , j¯ /∈ J∗1 ∼ } , S2 = { j ∈ J2 ∼ ∩ J∗1 ∼ ∣ ∣ ∣ ∣j¯ /∈ J2∼ , j¯ /∈ J∗1 ∼ } , S3 = { j ∈ J2 ∼ ∩ J∗1 ∼ ∣ ∣ ∣ ∣j¯ ∈ J2∼ ∩ J∗1 ∼ , j ≺ j¯ } , then we have J1 ∼ ∩ J∗2 ∼ = ⋃ 1≤i≤3 Ti ∪ T 3, J2 ∼ ∩ J∗1 ∼ = ⋃ 1≤i≤3 Si ∪S3, and ](J1 ∼ ∩ J∗2 ∼ ) = ](J2 ∼ ∩ J∗1 ∼ ). Here for a subset S ⊆ J , we set S = {¯i | i ∈ S}. Let FJ1,J2(xj) = x 3 2 j xJ1x −1 J2 hJ2−jhJ¯∗1−j¯ hJ¯∗1 hJ2 gj,J∗1 gj,J∗2 . We also introduce a function f(z) = 1 (1 + z) ∏ k∈T1 (z − xk) ∏ k∈S1 (z − xk) ∏ k∈S2 zxk − 1 z − xk ∏ k∈T3 (z − xk)(zxk − 1) ∏ k∈S3 (z − xk)(zxk − 1) , which has poles only at z = −1,∞ and xj (j ∈ J2 ∼ ∩ J∗1 ∼ = ⋃ 1≤i≤3 Si ∪ S3). A direct computation gives, for j ∈ S3 Res ( f, x−1j ) = 1 xj(1 + x2j )(1− xj) ∏ k∈T1 (1− xkxj) ∏ k∈S1 (1− xkxj) ∏ k∈S2 xk − xj 1− xkxj ∏ k∈T3 (xj − xk)(xjxk − 1) ∏ k∈S3, k 6=j (xj − xk)(xjxk − 1) = cFJ1,J2(x −1 j ), where c = xT3xS2 xS3 . Similarly we obtain Res(f, xj) = cFJ1,J2(xj), j ∈ Si, i = 1, 2, 3. On the other hand, by setting hi = x 1 2 i + x − 12 i , it is easy to see that Res (f,−1) = xT3 xS3 ( xT1 xS1 ) 1 2 ∏ k∈T1 hk ∏ k∈S1 hk ∏ k∈T3 h2k ∏ k∈S3 h2k , Res (f,∞) = − xT3xS2 xS3 = −c. 30 J. Sun Since the right-hand side of (C.1) equals hJ¯c1hJ2x − 12 J1 ∑ j∈J2 ∼ ∩J∗1 ∼ FJ1,J2(xj), it simplifies to hJ¯∗1 hJ2x − 12 J1 (−c−1)   −c+ xT3 xS3 ( xT1 xS1 ) 1 2 ∏ k∈T1 hk ∏ k∈S1 hk ∏ k∈T3 h2k ∏ k∈S3 h2k    . Now it is sufficient to prove ( xT1 xS1 ) 1 2 ∏ k∈T1 hk ∏ k∈S1 hk ∏ k∈T3 h2k ∏ k∈S3 h2k = xS2 x 1 2 J1 x 1 2 J2 hJ¯∗2 hJ1 hJ¯∗1 hJ2 which is immediate. We finish the proof.  D Proofs for identities of characters of case C(1)n and D (1) n In this section we consider g = C(1)n , D (1) n . By (4.15) and Conjecture 4.5, Conjectures 5.5–5.8 reduce to the following identities (−1) u(u+1) 2 x2σ ∏ 1≤k≤u, 1≤l≤u+1 1 (1− xσkxηl)(xσk − xηl) ∏ 1≤k≤u+1 (1− x2ηk) = ∑ (−1) s(s+1) 2 x2γ ∏ 1≤kj ( yi − y −1 j ) ∏ i,j∈J2, i r(m,n) for n > m and l(n,n) = r(n,n). From this we conclude that if n > m then R(m,n) = 0, and if n = m then R(n,n) is a constant multiple of L(n,n). The constant is shown to be 1 by setting x = 0 and using the Weyl denominator formula of type Cn, n∏ j=1 ynj · det ( y−n+j−1i − y n−j+1 i ) 1≤i,j≤n = n∏ j=1 ( 1− y2j ) ∏ 1≤i r(m,n) if m ≤ n − 2, and k(n−1,n) = r (n−1,n)  . Hence we find that R (m,n)  = 0 if m ≤ n − 2, and R(n−1,n) is a constant multiple of K(n−1,n). The constant can be found by setting x = 0 and using the Weyl denominator formula of type Dn, n∏ j=1 ynj · det ( y−n+ji +  y n−j i ) 1≤i,j≤n = (1 + ) ∏ 1≤i