Duality for Knizhnik-Zamolodchikov and Dynamical Operators
aa r X i v : . [ m a t h . QA ] F e b DUALITY FOR KNIZHNIK-ZAMOLODCHIKOV AND DYNAMICALOPERATORS
V. TARASOV ◦ AND F. UVAROV ⋆ ◦ ⋆ Department of Mathematical Sciences, Indiana University – Purdue UniversityIndianapolis402 North Blackford St, Indianapolis, IN 46202-3216, USA ◦ St. Petersburg Branch of Steklov Mathematical InstituteFontanka 27, St. Petersburg, 191023, Russia
Abstract.
We consider the Knizhnik-Zamolodchikov and Dynamical operators, both dif-ferential and difference, in the context of the ( gl k , gl n )-duality for the space of polynomialsin kn anticommuting variables. We show that the Knizhnik-Zamolodchikov and Dynamicaloperators naturally exchange under the duality. Introduction
The Knizhnik-Zamolodchikov (KZ) operators is a family of pairwise commuting differen-tial operators acting on U ( gl k ) ⊗ n -valued functions. They play an important role in confor-mal field theory, representation theory, and they are closely related to the famous GaudinHamiltonians. The difference analogue of the KZ operators is the quantum Knizhnik-Zamolodchikov (qKZ) operators. There are rational, trigonometric, and elliptic versionsof the KZ and qKZ operators. For a review and references see, for example, [2].There exist other families of commuting differential or difference operators called theDynamical Differential (DD) or Dynamical Difference (qDD) operators, respectively. Thereare rational and trigonometric versions of the DD and qDD operators as well. It is knownthat the rational DD operators commute with the rational KZ operators, the trigonometricDD operators commute with the rational qKZ operators, and the rational qDD operatorscommute with the trigonometric KZ operators, see [4–6]. The qDD operators appear as theaction of the dynamical Weyl groups [3]. The DD operators are also known as the Casimirconnection, see [8, 9].Together with the KZ, DD, qKZ, and qDD operators associated with U ( gl k ) ⊗ n , we willsimultaneously consider similar operators associated with U ( gl n ) ⊗ k interchanging k and n .Let P kn be the space of polynomials in kn commuting variables. There are natural actionsof U ( gl k ) ⊗ n and U ( gl n ) ⊗ k on the space P kn , and those actions commute. This manifeststhe ( gl k , gl n )-duality, see, for example, [1]. Consider the images of the KZ, DD, qKZ, qDDoperators associated with U ( gl k ) ⊗ n and U ( gl n ) ⊗ k under the corresponding actions on P kn . Itwas proved in [7] that the images of the rational (trigonometric) KZ operators associated with U ( gl n ) ⊗ k coincide with the images of the rational (trigonometric) DD operators associatedwith U ( gl k ) ⊗ n . Similarly, the images of the rational qKZ operators associated with U ( gl n ) ⊗ k ,up to an action of a central element in gl n , coincide with the images of the rational qDDoperators associated with U ( gl k ) ⊗ n . In this paper we obtain a similar duality for the case of ◦ E-mail: [email protected], [email protected], supported in part by Simons Foundation grant 430235. ⋆ E-mail: fi[email protected].
V. TARASOV AND F. UVAROV U ( gl k ) ⊗ n - and U ( gl n ) ⊗ k -actions on the space of polynomials in kn anticommuting variables,see Theorem 4.4.The duality for the rational and trigonometric KZ and DD operators is proved in a straight-forward way. To prove the duality for the rational qKZ and qDD operators, we study theeigenvalues of the rational R -matrix and compare them to the eigenvalues of the operator B h i ( t ), which is used in the construction of the qDD operators.The ( gl k , gl n )-duality for classical integrable models related to Gaudin Hamiltonians andthe actions of gl k and gl n on the space of polynomials in anticommuting variables was studiedin [10, Section 3.3]. The result of [10] resembles what one can expect for Bethe algebras ofGaudin models discussed in our work. We will study those Bethe algebras in an upcomingpaper.The paper is organized as follows. In Section 2, we introduce necessary notations. InSection 3, we define the KZ, DD, qKZ, and qDD operators. In Section 4, we formulate andprove the main result. 2. Basic notation
Let e ab , a, b = 1 , . . . , k , be the standard basis of the Lie algebra gl k : [ e ab , e cd ] = δ bc e ad − δ ad e cb . We take the Cartan subalgebra h ⊂ gl k spanned by e , . . . , e kk , and the nilpotentsubalgebras n + and n − spanned by the elements e ab for a < b and a > b respectively. Wehave standard Gauss decomposition gl k = n + ⊕ h ⊕ n − .Let ε , . . . , ε k be the basis of h ∗ dual to e , . . . , e kk : h ε a , e bb i = δ ab . We identify h ∗ with C k mapping l ε + · · · + l k ε k to ( l , . . . , l k ). The root vectors of gl k are e ab for a = b , thecorresponding root being equal to α ab = ε a − ε b The roots α ab for a < b are positive. Thesimple roots are α , . . . , α k − : α a = ε a − ε a +1 . Denote by ρ the half-sum of positive roots.We choose the standard invariant bilinear form ( , ) on gl k : ( e ab , e cd ) = δ ad δ bc . It defines anisomorphism h → h ∗ The induced bilinear form on h ∗ is ( ε a , ε b ) = δ ab .For a gl k -module W and a weight l ∈ h ∗ let W [ l ] be the weight subspace of W of weight l For any l = ( l , . . . , l k ) with l a > l a +1 we denote by V l the irreducible gl k -module withhighest weight l . Also, for any m ∈ Z > we write V m instead of V (1 m ) , where (1 m ) =(1 , . . . , | {z } k , , . . . , V = C is the trivial gl k -module, V = C k with the natural actionof gl k , and V m is the m -th antisymmetric power of V The element I = k P a,b =1 e ab e ba is central in U ( gl k ), on the irreducible gl k -module V l it acts asmultiplication by ( l , l + 2 ρ ) .Consider the algebra X k = V • C k . We identify it with the ring of all polynomials inanticommuting variables x , . . . , x k . In particular, x i = 0 for all i = 1 , . . . , k .Define the left derivations ∂ , . . . , ∂ k as follows: if g ( x ) = x b . . . x b m for some m , where b s = a for any s , then ∂ a g ( x ) = 0 , ∂ a ( x a g ( x )) = g ( x ) . The operators of left multiplication by x , . . . , x k and the left derivations ∂ , . . . , ∂ k make thespace X k into the irreducible representation of the Clifford algebra Cliff k . UALITY FOR KNIZHNIK-ZAMOLODCHIKOV AND DYNAMICAL OPERATORS 3
The Lie algebra gl k acts on the space X k by the rule: e ab · p = x a ∂ b p for any p ∈ X k .Denote the obtained gl k -module by V • , then(2.1) V • = k M l =0 V l , the submodule V l being spanned by the homogeneous polynomials of degree l . A highestweight vector of the submodule V l is x x . . . x l .The gl k -action on X k naturaly extends to a U ( gl k ) ⊗ n -action on ( X k ) ⊗ n . For any g ∈ U ( gl k ),set g ( i ) = 1 ⊗ · · · ⊗ g i − th ⊗ · · · ⊗ ∈ U ( gl k ) ⊗ n . We consider U ( gl k ) as the diagonal subalgebraof U ( gl k ) ⊗ n , that is, the embedding U ( gl k ) ֒ → U ( gl k ) ⊗ n is given by the n-fold coproduct: x x (1) + · · · + x ( n ) for any x ∈ gl k . This corresponds to the standard gl k -module structureon ( X k ) ⊗ n as the tensor product of gl k -modules.Let Ω = k P a,b =1 e ab ⊗ e ba be the Casimir tensor, and letΩ + = 12 k X a =1 e aa ⊗ e aa + X a
Fix a nonzero complex number κ . Consider differential operators ∇ z , . . . , ∇ z n and b ∇ z , . . . , b ∇ z n with coefficients in U ( gl k ) ⊗ n depending on complex variables z , . . . , z n and λ , . . . , λ k : ∇ z i ( z ; λ ) = κ ∂∂z i − k X a =1 λ a ( e aa ) ( i ) − n X j =1 ,j = i Ω ( ij ) z i − z j b ∇ z i ( z ; λ ) = κz i ∂∂z i − k X a =1 ( λ a − e aa e aa ) ( i ) − n X j =1 ,j = i z i Ω +( ij ) + z j Ω − ( ij ) z i − z j The differential operators ∇ z , . . . , ∇ z n (resp. b ∇ z , . . . , b ∇ z n ) are called the rational (resp. trigonometric ) Knizhnik-Zamolodchikov (KZ) operators .Introduce differential operators D λ , . . . , D λ k and b D λ , . . . , b D λ k with coefficients in U ( gl k ) ⊗ n depending on complex variables z , . . . , z n and λ , . . . , λ k : D λ a ( z ; λ ) = κ ∂∂λ a − n X i =1 z i ( e aa ) ( i ) − k X b =1 ,b = a e ab e ba − e aa λ a − λ b . b D λ a ( z ; λ ) = κλ a ∂∂λ a + e aa − n X i =1 z i ( e aa ) ( i ) − k X b =1 X i The differential operators D λ , . . . , D λ k (resp. b D λ , . . . , b D λ k ) are called the rational (resp. trigonometric ) differential dynamical (DD) operators , see [4, 5].For any a, b = 1 , . . . , k , a = b , introduce the series B ab ( t ) depending on a complex variable t : B ab = 1 + ∞ X s =1 e sba e sab s ! s Y j =1 ( t − e aa + e bb − j ) − . The action of this series is well defined on any finite-dimensional gl k -module W giving anEnd( W )-valued rational function of t .Denote λ bc = λ b − λ c . Consider the products X , . . . , X k depending on complex variables z , . . . , z n , and λ , . . . , λ k : X a ( z ; λ ) = ( B ak ( λ ak ) . . . B a,a +1 ( λ a,a +1 )) − n Y i =1 ( z − e aa i ) ( i ) B a ( λ a − κ ) . . . B a − ,a ( λ a − ,a − κ ) . The products X , . . . , X k act on any n -fold tensor product W ⊗· · ·⊗ W n of finite-dimensional gl k -modules.Denote by T u a difference operator acting on a function f ( u ) by( T u f )( u ) = f ( u + κ ) . Introduce difference operators Q λ , . . . , Q λ k : Q λ a ( z ; λ ) = X a ( z ; λ ) T λ a . The operators Q λ , . . . , Q λ k are called the ( rational ) difference dynamical (qDD) operators .[6]For any finite-dimensional irreducible gl k -modules V and W , there is a distinguishedrational function R V W ( t ) of t with values in End( V ⊗ W ) called the rational R -matrix. It isuniquely determined by the gl k -invariance(3.1) [ R V W ( t ) , g ⊗ ⊗ g ] = 0 for any g ∈ gl k , the commutation relations(3.2) R V W ( t )( te ab ⊗ k X c =1 e ac ⊗ e cb ) = ( te ab ⊗ k X c =1 e cb ⊗ e ac ) R V W ( t ) , and the normalization condition(3.3) R V W ( t ) v ⊗ w = v ⊗ w, where v and w are the highest weight vectors of V and W respectively.Denote z ij = z i − z j and R ij ( t ) = ( R W i W j ( t )) ( ij ) . Consider the products K , . . . , K n depending on complex variables z , . . . , z n and λ , . . . , λ k : K i ( z ; λ ) = ( R in ( z in ) . . . R i,i +1 ( z i,i +1 )) − k Y a =1 ( λ − e aa a ) ( i ) R i ( z i − κ ) . . . R i − ,i ( z i − ,i − κ ) . The products K , . . . , K n act on any n -fold tensor product W ⊗ · · · ⊗ W n of gl k -modules.Introduce difference operators Z z , . . . , Z z n : Z z i ( z ; λ ) = K i ( z ; λ ) T z i . The operators Z z , . . . , Z z n are called ( rational ) quantized Knizhnik-Zamolodchikov (qKZ)operators. UALITY FOR KNIZHNIK-ZAMOLODCHIKOV AND DYNAMICAL OPERATORS 5 It is known that the introduced operators combine into three commutative families, see[4–6] for more references. Theorem 3.1. The operators ∇ z , . . . , ∇ z k , D λ , . . . , D λ k pairwise commute. Theorem 3.2. The operators b ∇ z , . . . , b ∇ z n , Q λ , . . . , Q λ k pairwise commute. Theorem 3.3. The operators b D λ , . . . , b D λ k , Z z , . . . , Z z n pairwise commute. The ( gl k , gl n )-duality Consider the ring P kn of polynomials in kn anticommuting variables x ai , a = 1 , . . . , k , i =1 , . . . , n . As a vector space, P kn is isomorphic to ( X k ) ⊗ n , the isomorphism ϕ : ( X k ) ⊗ n → P kn being given by(4.1) ϕ ( p ⊗ · · · ⊗ p n )( x , . . . , x kn ) = p ( x , . . . , x k ) p ( x , . . . , x k ) . . . p n ( x n , . . . , x kn ) , and to ( X n ) ⊗ k , the isomorphism ϕ : ( X n ) ⊗ k → P kn being given by(4.2) ϕ ( p ⊗ · · · ⊗ p k )( x , . . . , x kn ) = p ( x , . . . , x n ) p ( x , . . . , x n ) . . . p k ( x k , . . . , x kn ) . We transfer the gl k -action on ( X k ) ⊗ n to P kn using the isomorphism ϕ . Similarly, wetransfer the gl n -action on ( X n ) ⊗ k to P kn using the isomorphism ϕ .We will write superscripts h k i and h n i to distinguish objects related to the Lie algebras gl k and gl n , respectively. For example, e h k i ab , a, b = 1 , . . . , k , denote the generators of gl k , and e h n i ij , i, j = 1 , . . . , n , denote the generators of gl n . Then P kn is isomorphic to ( V h k i• ) ⊗ n as a gl k -module by (4.1), and it is isomorphic to ( V h n i• ) ⊗ k as a gl n -module by (4.2).It is easy to check that gl k - and gl n -actions on P kn commute. Therefore, P kn is a gl k ⊕ gl n -module. We have the following theorem, see for example [1]: Theorem 4.1. For any partition l , denote its transpose by l ′ . The gl k ⊕ gl n -module P kn hasthe decomposition: P kn = M l =( l ,...,l k ) ,l n. V h k i l ⊗ V h n i l ′ . Fix vectors l = ( l , . . . , l n ) ∈ Z n > and m = ( m , . . . , m k ) ∈ Z k > such that P ni =1 l i = P ka =1 m a . Let Z kn [ l , m ] = { ( d ai ) a =1 ,...,ki =1 ,...,n ∈ { , } kn | k X a =1 d ai = l i , n X i =1 d ai = m a } . Denote by P kn [ l , m ] ⊂ P kn the span of all monomials x d = x d . . . x d k k . . . x d n n . . . x d kn kn suchthat d = ( d ai ) ∈ Z kn [ l , m ]. Then by (2.1), the maps ϕ and ϕ induce the isomorpfisms of therespective weight subspaces ( V h k i l ⊗ · · · ⊗ V h k i l n )[ m , . . . , m k ] and ( V h n i m ⊗ · · · ⊗ V h n i m k )[ l , . . . , l n ]with the space P kn [ l , m ].There is another description of the isomorphisms ϕ and ϕ .For any a = ( a , . . . , a r ), i = ( i , . . . , i s ), such that 1 a < · · · < a r k , 1 i < · · ·
The vectors v h k i d , d ∈ Z kn [ l , m ] form a basis of the weight subspace ( V h k i l ⊗· · · ⊗ V h k i l n )[ m , . . . , m k ] . Similarly, the vectors v h n i d , d ∈ Z kn [ l , m ] form a basis of the weightsubspace ( V h n i m ⊗ · · · ⊗ V h n i m k )[ l , . . . , l n ] . Let ε ( d ) be a sign function such that x d . . . x d n n . . . x d k k . . . x d kn kn = ε ( d ) x d . Lemma 4.3. We have ϕ ( v d ) = x d , ϕ ( v d ) = ε ( d ) x d . Consider the action of KZ, qKZ, DD and qDD operators associated with the Lie algebras gl k and gl n on P kn -valued functions of z , . . . , z n and λ , . . . , λ k , treating the space P kn asa tensor product ( V h k i• ) ⊗ n of gl k -modules and as a tensor product ( V h n i• ) ⊗ k of gl n -modules.We will write F ⋍ G if the operators F and G act on the P kn -valued functions in the sameway.Denote h k i = (1 , , . . . , | {z } k ) and h n i = (1 , , . . . , | {z } n ). Theorem 4.4. For any i = 1 , ..., n and a = 1 , ..., k the following relations hold (4.5) ∇ h k i z i ( z, λ, κ ) ⋍ D h n i z i ( λ, − z, − κ ) , (4.6) ∇ h n i λ a ( λ, z, κ ) ⋍ D h k i λ a ( z, − λ, − κ ) , (4.7) b ∇ h k i z i ( z, λ, κ ) ⋍ − b D h n i z i ( − λ + h k i , z, − κ ) , (4.8) b ∇ h n i λ a ( λ, z, κ ) ⋍ − b D h k i λ a ( − z + h n i , λ, − κ ) , (4.9) Z h k i z i ( z, λ, κ ) ⋍ N h n i i ( z ) Q h n i z i ( λ, − z, − κ ) , (4.10) Z h n i λ a ( λ, z, κ ) ⋍ N h k i a ( λ ) Q h k i λ a ( z, − λ, − κ ) , where (4.11) N h n i i ( z ) = Q j
Proof . Verification of relations (4.5), (4.6), (4.7), (4.8) is straightforward. To check (4.9)and (4.10), we have to show that(4.13) R h k i ij ( t ) ⋍ C h n i ij ( t ) B h n i ij ( − t ) , (4.14) R h n i ab ( t ) ⋍ C h k i ab ( t ) B h k i ab ( − t ) , We will prove relation (4.14). Relation (4.13) can be proved similarly.Note, that both action of R h n i ab ( t ) on P kn and action of C h k i ab ( t ) B h k i ab ( − t ) on P kn involve onlythe variables x a , . . . , x an , x b , . . . , x bn . Therefore, it is sufficient to prove (4.14) for the caseof k = 2, a = 1, b = 2.The gl n - module P ,n is isomorphic to V h n i• ⊗ V h n i• . Consider the submodule V h n i m ⊗ V h n i m ⊂ P ,n . We have the following decomposition of the gl n -module:(4.15) V h n i m ⊗ V h n i m = min( m ,m ) M m =max(0 ,m + m − n ) V h n i l ( m ) . Here l ( m ) = (2 , , . . . , , , . . . , , , . . . , m times and 1 repeats m + m − m times. Denote by v m a highest weight vector of the summand V h n i l ( m ) given by formula(A.1).Define the scalar product on P ,n by the rule: h f, f i = 1, if f ∈ P ,n is a nonzeromonomial, and h f, h i = 0, if f, h ∈ P ,n are two non-proprosional monomials. Lemma 4.5. We have h v m , v m i 6 = 0 for every m . The proof is straightforward by formula (A.1). Lemma 4.6. h w , e h n i ij w i = h e h n i ji w , w i for any w , w ∈ P ,n , and i, j = 1 , . . . , n . The proof is straightforward. Corollary 4.7. If vectors w and ˜ w belong to distinct summands of decomposition (4.15) ,then h w, ˜ w i = 0 .Proof. The summands of decomposition (4.15) are eigenspaces of the operator I h n i , andthe corresponding eigenvalues are distinct. Notice that the operator I h n i is symmetric withrespect to the scalar product h· , ·i . Together with Lemma 4.6 this implies the statement. (cid:3) Denote L ij ( t ) = t ( e h n i ij ) (1) + n X k =1 ( e h n i ik ) (1) ( e h n i kj ) (2) ,M ij ( t ) = t ( e h n i ij ) (1) + n X k =1 ( e h n i kj ) (1) ( e h n i ik ) (2) ,α m ( t ) = h L m + m − m +1 ,m ( t ) · v m , v m − i , β m ( t ) = h M m + m − m +1 ,m ( t ) · v m , v m − i . Lemma 4.8. The functions α m ( t ) and β m ( t ) are nonzero, and (4.16) α m ( t ) β m ( t ) = t + 1 + m − mt − m − m . V. TARASOV AND F. UVAROV The proof is given in Appendix.Due to relation (3.1), for any m there exists a scalar function ρ m ( t ) such that R h n i ( t ) w = ρ m ( t ) w for any w ∈ V h n i l ( m ) . Lemma 4.9. It holds that (4.17) ρ m ( t ) ρ m − ( t ) = α m ( t ) β m ( t ) . Proof. Let us single out the term V l ( m − in the decomposition (4.15): V h n i m ⊗ V h n i m = V l ( m − L ˜ V . Then we can write L m + m − m +1 ,m ( t ) · v m = w + ˜ w , where w ∈ V l ( m − and˜ w ∈ ˜ V . By the definition of L m + m − m +1 ,m ( t ), the vector w has weight l ( m − w = av m − for some scalar a . By Corollary 4.7, we have α m ( t ) = h L m + m − m +1 ,m ( t ) · v m , v m − i = a h v m − , v m − i . Notice that R h n i ( t ) ˜ w ∈ ˜ V , because R -matrix R h n i ( t ) acts as a multiplication by a scalarfunction on each summand of the decomposition (4.15). Then, by Corollary 4.7, h R h n i ( t ) ˜ w, v m − i =0, and h R h n i ( t ) L m + m − m +1 ,m ( t ) · v m , v m − i = h R h n i ( t ) w, v m − i == ρ m − ( t ) a h v m − , v m − i = ρ m − ( t ) α m ( t ) . On the other hand, relation (3.2) gives h R h n i ( t ) L m + m − m +1 ,m ( t ) · v m , v m − i = h M m + m − m +1 ,m ( t ) R h n i ( t ) · v m , v m − i = ρ m ( t ) β m ( t ) . Thus we get α m ( t ) ρ m − ( t ) = ρ m ( t ) β m ( t ), which is relation (4.17). (cid:3) By formulae (4.17), (4.16),(4.18) ρ m ( t ) = m Y s =1 ρ s ( t ) ρ s − ( t ) = m − Y s =0 t + m − st − m + s . For the last equality we used that ρ = 1 by the normalization condition (3.3).Consider a decomposition of the gl -module: V h i l ⊗ · · · ⊗ V h i l n = M m | l | / V h i ( | l |− m,m ) ⊗ W h i m , where | l | = P i l i and W h i m are multiplicity spaces. Let( V h i l ⊗ · · · ⊗ V h i l n )[ m , m ] m = ( V h i l ⊗ · · · ⊗ V h i l n )[ m , m ] ∩ ( V h i ( | l |− m,m ) ⊗ W h i m ) . Lemma 4.10. It holds that (4.19) B h i ( t ) | ( V h i l ⊗···⊗ V h i ln )[ m ,m ] m = m − Y s = m t + m − st − m + s . Proof. The modules V h i l i can have only the following highest weights: (1 , , (1 , , (0 , V h i (0 , and V h i (1 , are one-dimensional and the elements e , e , and e − e act there by zero. Thus, it is enough to consider the case when V h i l i = V h i (1 , for all i , so formula (4.19) follows from [7, formula (5.13)]. (cid:3) UALITY FOR KNIZHNIK-ZAMOLODCHIKOV AND DYNAMICAL OPERATORS 9 Comparing formulas (4.18), (4.19), and (4.12), we conclude:(4.20) ρ m ( t ) = ( B h i ( − t ) C h i ( t )) | ( V h i l ⊗···⊗ V h i ln )[ m ,m ] m . Lemma 4.11. For the Casimir elements I h i and I h n i , we have (4.21) I h i − X a =1 e h i aa ⋍ − I h n i + n n X i =1 e h n i ii . The proof is straightforward.Recall that on the irreducible gl k -module V l , the element I h k i acts as a multiplication by( l , l + 2 ρ ). Then it is easy to verify that(4.22) ( I h i − X a =1 e h i aa ) | V h i ( | l |− m,m ) ⊗ W h i m = ( − I h n i + n n X i =1 e h n i ii ) | V h n i l ( m ) . Comparing formulae (4.21) and (4.22), and using the fact that the Casimir elements acton distinct irreducible modules as a multiplication by distinct scalar functions, we get thatunder isomorphisms (4.1) and (4.2) the respective images of V h i ( | l |− m,m ) ⊗ W h i m and V h n i l ( m ) in P ,n coincide. To indicate that, we will write V h i ( | l |− m,m ) ⊗ W h i m ⋍ V h n i l ( m ) .We also have ( V h i l ⊗ · · · ⊗ V h i l n )[ m , m ] ⋍ ( V h n i m ⊗ V h n i m )[ l , . . . , l n ]. Therefore ( V h n i m ⊗ V h n i m )[ l , . . . , l n ] ∩ V h n i l ( m ) ⋍ ( V h i l ⊗· · ·⊗ V h i l n )[ m , m ] m . Now we see that (4.20) gives us relationbetween actions of operators B h i ( t ), C h i ( t ), and R h n i ( t ) on one particular submodule of P ,n ,proving the theorem. Appendix A. Proof of Lemma 4.8 Let(A.1) v m = m Y i =1 x i x i X { ε } x ε ,m +1 x ε ,m +2 ..x ε m m − m ,m + m − m with { ε } = (cid:8) ( ε , ε , . . . , ε m + m − m ) : ε i = 1 or 2, P i ε i = m − m + 2( m − m ) (cid:9) . One caneasily prove that v m is a highest weight vector of weight l ( m ) .It follows from the construction of the scalar product h· , ·i that α m ( t ) (resp. β m ( t ))equals the sum of the coefficients of those monomials presented in L m + m − m +1 ,m ( t ) · v m (resp. M m + m − m +1 ,m ( t ) · v m ) that also appear in v m − . In fact, all monomials either in L m + m − m +1 ,m ( t ) · v m or in M m + m − m +1 ,m ( t ) · v m appear in v m − as well.We will start with α m . Denote s = m + m − m + 1. Let us inspect what happens whenwe apply various terms of L sm ( t ) to v m . For the sum n P k =1 ( e sk ) (1) ( e km ) (2) , we can assume that m k < s . If k > m , then the operator ( e sk ) (1) ( e km ) (2) will send a monomial in v m to zeroif and only if this monomial does not depend on x k . That is, we look at all terms in v m corresponding to ε k − m = 1. There are C m − mm + m − m − such terms with the same contribution( − m + m + m . We leave the details of this calculation to a reader. Under the assumption m < k < s , there are m + m − m different values of k , which yield the overall contribution( − m + m + m ( m + m − m ) C m − mm + m − m − to α m ( t ). If k = m , then we have ( e sk ) (1) ( e km ) (2) · v m = ( e sm ) (1) · v m . Therefore, all C m − mm + m − m termsin v m equally contribute ( − m + m + m ( m + m − m ).Finally, the term t ( e ij ) (1) in L sm ( t ) generates the contribution t ( − m + m + m C m − mm + m − m to α m ( t ), which can be seen similarly to the case k = m considered above.Thus we obtained(A.2) ( − m + m + m α m = ( t + 1) C m − mm + m − m + ( m + m − m ) C m − mm + m − m − . The similar arguments give us(A.3) ( − m + m + m β m = ( t − C m − mm + m − m − ( m + m − m ) C m − mm + m − m − . Since ( m + m − m ) C m − mm + m − m − = ( m − m ) C m − mm + m − m and ( m + m − m ) C m − mm + m − m − = ( m − m ) C m − mm + m − m , Lemma 4.8 is proved. References [1] S-J. Cheng, W. Wang, Dualities and Representations of Lie Superalgebras , Amer. Math.Soc. (2012), 1087 - 1098.[2] P. Etingof, I. Frenkel, A. Kirillov, Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations , Math. Surveys and Monographs, vol. 58, Amer. Math. Soc.,Providence, RI, 1998.[3] P. Etingof, A. Varchenko, Dynamical Weyl Groups and Applications , Adv. Math. 167(2002), no. 1, 74-127.[4] G. Felder, Ya. Markov, V. Tarasov and A. Varchenko, Differential Equations Compatiblewith KZ Equations , Math. Phys., Analysis and Geometry 3 (2000), 139177.[5] V. Tarasov and A. Varchenko, The Dynamical Differential Equations Compatible with theRational qKZ Equations , Letters in Mathematical Physics 71(2) (2004).[6] V. Tarasov and A. Varchenko, Difference Equations Compatible with Trigonometric KZDifferential Equations , Int. Math. Res. Notices (2000), no. 15, 801829.[7] V. Tarasov, A. Varchenko, Duality for Knizhnik-Zamolodchikov and Dynamical Equa-tions , Acta Appl. Math. 73 (2002), no. 1-2, 141154.[8] V. Toledano-Laredo, The Trigonometric Casimir Connection of a Simple Lie Algebra ,Journal of Algebra, vol. 329, iss. 1, 1 March 2011, 286-327.[9] V. Toledano-Laredo, Y. Yang, The Elliptic Casimir Connection of a Simple Lie Algebra ,arXiv:1805.12261.[10] B. Vicedo, C. Young, ( gl m , gl n ) -Dualities in Gaudin Models with Irregular Singularities-Dualities in Gaudin Models with Irregular Singularities