On Cherednik and Nazarov-Sklyanin large N limit construction for double elliptic integrable system
aa r X i v : . [ m a t h - ph ] F e b On Cherednik and Nazarov-Sklyanin large N limit constructionfor double elliptic integrable system
A. Grekov A. Zotov Abstract
The infinite number of particles limit in the dual to elliptic Ruijsenaars model (coordinate trigono-metric degeneration of quantum double elliptic model) is proposed using the Nazarov-Sklyanin ap-proach. For this purpose we describe double-elliptization of the Cherednik construction. Namely,we derive explicit expression in terms of the Cherednik operators, which reduces to the generatingfunction of Dell commuting Hamiltonians on the space of symmetric functions. Although the doubleelliptic Cherednik operators do not commute, they can be used for construction of the N → ∞ limit. Contents p = 0 ) model 114 Matrix resolvent 135 N → ∞ limit 15 θ ω ( uZ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.3 Inverse limit of P θ ω ( uV γ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.4 Inverse limit of quantum Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 X [ n ] , Y [ n ] , α [ n ] , β [ n ]
248 Explicit form of the Dell Nazarov-Sklyanin Hamiltonians to the first order in ω Physics Department, Stony Brook University, USA; National Research University Higher School of Economics, RussianFederation; e-mail: [email protected]. Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow 119991, Russia; e-mail:[email protected]. Explicit check of the commutativity of the first few Hamiltonians to the first power in ω I − , I n ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.2 Evaluating [ I , I ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
10 Discussion 3011 Appendix A: Cherednik construction in
GL(2) case 30
12 Appendix B: Elliptic function notations 3313 Appendix C: Helpful identities 3414 Appendix D: Commutation relations for Q ∗ m , Q n List of main notations: q j , j = 1 , ..., N – positions of particles; x j = e q j – exponents of positions of particles; ∂ i = ∂ x i , so that ∂ q i = x i ∂ i ; γ i = q − x i ∂ i (1.4); ω – the elliptic modular parameter, controlling the ellipticity in momentum; p = e πıτ – the modular parameter, controlling the ellipticity in coordinates; q = e ~ – exponent of the Planck constant; t = e η – exponent of the coupling constant; u – the spectral parameter; z – the second spectral parameter; A x,p - the space of operators, generated by { x , .., x N , q x ∂ , ..., q x N ∂ N } ;: : - normal ordering on A x,p , moving all shift of operators in each monomial to the right;ˆ O ( u ) – the generating function of operators ˆ O n from [14];ˆ H ( u ) – generating function of quantum Dell Hamiltonians ˆ H n = ˆ O − ˆ O n (1.1);P θ ω ( uA )( q, t ) = P n ∈ Z ω n − n ( − u ) n A [ n ] ( q, t ), for any operator A ( q, t ) with A [ n ] ( q, t ) = A ( q n , t n ) (1.16); Q Ni = k B i = B ...B N , i.e. all products of non-commuting operators B i are left ordered products; D N ( u ) – the generating function of Dell Hamiltonians for p = 0 case (1.3); C i ( t, q ) – the trigonometric Cherednik operators (1.6); R ij ( t ) – R -operators (1.7);Λ N – the space of symmetric functions of N variables: x , ..., x N ;Λ ( k ) N ⊂ C [ x , ..., x N ] the subspace of polynomials symmetric in the variables x k +1 , ..., x N ; σ ij – an element of permutation group S N generated by permutation of variables x i and x j ; Z i – Nazarov-Sklyanin operators (1.9). Introduction and summary
We discuss the double elliptic (Dell) integrable model being a generalization of the Calogero-Ruijsenaarsfamily of many-body systems [9, 20] to elliptic dependence on the particles momenta. There are twoversions for this type of models. The first one was introduced and extensively studied by A. Mironov andA. Morozov [6]. Its derivation was based on the requirement for the model to be self-dual with respectto the Ruijsenaars (or action-angle or p-q) duality [19]. The Hamiltonians are rather complicated.They are given in terms of higher genus theta functions, and the period matrix depends on dynamicalvariables. At the same time the eigenfunctions for these Hamiltonians possesses natural symmetricproperties and can be constructed explicitly [2, 4]. Another version of the Dell model was suggestedby P. Koroteev and Sh. Shakirov in [14]. It is close to the classical model introduced previously byH.W. Braden and T.J. Hollowood [5], though precise relation between them needs further elucidation.The generating function of quantum Hamiltonians in this version are given by a relatively simpleexpression, where both modular parameters (for elliptic dependence on momenta and coordinate) arefree constants. Another feature of the Koroteev-Shakirov formulation is that it admits some algebraicconstructions, which are widely known for the Calogero-Ruijsenaars family of integrable systems. Inparticular, it was shown in our previous paper [12] that the generating function of Hamiltonians hasdeterminant representation, and the classical L -operator satisfies the Manakov equation instead of thestandard Lax representation. For both formulations the commutativity of the Hamiltonains has notbeing proved yet, but verified numerically. To find possible relation between two formulations of theDell model is an interesting open problem.In this paper we deal with the Koroteev-Shakirov formulation, and our study is based on theassumption that the following Hamiltonians indeed commute:ˆ H n = ˆ O − ˆ O n , (1.1)where ˆ O n are defined through ˆ O ( u ) = X n ,...,n N ∈ Z ω P i n i − ni ( − u ) P i n i N Y i 3n our previous paper [12] different variants of determinant representations for (1.1)-(1.2) wereproposed. Here we extend another set of algebraic constructions to the double-elliptic case (1.1). Ourfinal goal is to describe the large N limit for the Dell model. This limit is widely known for the Calogero-Moser and the Ruijsenaars-Schneider models [1, 16, 22, 17, 18, 8] including their spin generalizations[3]. The purpose of the paper is to describe N → ∞ limit of the Dell ( p = 0) model by introducingdouble-elliptic version of the Dunkl-Cherednik approach [10] and by applying the Nazarov-Sklyaninconstruction for N → ∞ limit, which was originally elaborated for the trigonometric Ruijsenaars-Schneider model [16]. For the latter model there exists a set of N commuting operators (the Cherednikoperators) C i ( t, q ) = t i − R i,i +1 ( t ) ... R iN ( t ) γ i R ,i ( t ) − ... R i − ,i ( t ) − , (1.6)acting on C [ x , ..., x N ], where the R -operators are of the form: R ij ( t ) = x i − tx j x i − x j + ( t − x j x i − x j σ ij , (1.7)and σ ij permutes the variables x i and x j . The commutativity of the Macdonald-Ruijsenaars operators(1.5) for different values of spectral parameter u follows from the commutativity of (1.6) and thefollowing relation between D N ( u ) (cid:12)(cid:12)(cid:12) ω =0 (1.5) and the Cherednik operators (1.6): D N ( u ) (cid:12)(cid:12)(cid:12) ω =0 = N Y i =1 (1 − uC i ) (cid:12)(cid:12)(cid:12) Λ N , (1.8)where Λ N is the space of symmetric functions in variables x , ..., x N .The generating function (1.5) is the one considered in [16], where the authors derived N → ∞ limitof the quantum Ruijsenaars-Schneider (or the Macdonald-Ruijsenaars) Hamiltonians. Let us recallmain steps of the Nazarov-Sklyanin construction since our paper is organized as a straightforwardgeneralization of their results to the Dell ( p = 0) case (1.3). First, one needs to express the generatingfunction (1.5) through the covariant Cherednik operators: Z i = N Y k = i x i − tx k x i − x k γ i + N X j = i ( t − x i x i − x j N Y k = i,j x j − tx k x j − x k γ j σ ij , (1.9) U i = ( t − Y j = i x i − tx j x i − x j γ i , (1.10)which satisfy the property σZ i σ − = Z σ ( i ) , σU i σ − = U σ ( i ) , σ ∈ S N , (1.11)where in the l.h.s. σ acts by permutation of variables x k . Then the generating function of theMacdonald-Ruijsenaars Hamiltonians (1.5) is represented in the form D N ( tu ) D N ( u ) − (cid:12)(cid:12)(cid:12) ω =0 = 1 − u N X i =1 U i − uZ i (cid:12)(cid:12)(cid:12) Λ N , (1.12)The next step is to construct the inverse limits for the operators U i and Z i , where the inverse limit isthe limit of the sequence Λ ← Λ ← ... (1.13) To match notations of [16] one should change u to − u . π N : Λ → Λ N , (1.14)sending the standard basis elements p n from Λ to the power sum symmetric polynomials: π N ( p n ) = N X i =1 x ni . (1.15)Finally, using (1.12) one gets the inverse limit for D N ( tu ) D N ( u ) − (cid:12)(cid:12)(cid:12) ω =0 .Our strategy is to extend the above formulae to the double-elliptic ( p = 0) case. Throughout thepaper we use the following convenient notation. For any operator A ( q, t ) setP θ ω ( uA )( q, t ) = X n ∈ Z ω n − n ( − u ) n A ( q n , t n ) = X n ∈ Z ω n − n ( − u ) n A [ n ] ( q, t ) , (1.16)i.e. notation A [ n ] ( q, t ) = A ( q n , t n ) is also used. In particular, A [1] = A . Outline of the paper and summary of results The paper is organized as follows.In Section 2 we introduce the double-elliptic ( p = 0) version of the Cherednik operators (1.6),acting on the space C [ x , ..., x N ]:P θ ω ( uC i ) = X n ∈ Z ω n − n ( − u ) n t n ( i − R i,i +1 ( t n ) ...R iN ( t n ) γ ni R ,i ( t n ) − ... R i − ,i ( t n ) − , (1.17)where R ij ( t ) is given by (1.7), and u is a spectral parameter. These operators do not commute witheach other. However, we prove the following relation between (1.17) and D N ( u ) (1.3): D N ( u ) = N Y i =1 P θ ω ( uC i ) (cid:12)(cid:12)(cid:12) Λ N = P θ ω ( uC ) ... P θ ω ( uC N ) (cid:12)(cid:12)(cid:12) Λ N . (1.18)It is the Dell ( p = 0) version of the relation (1.8). The order of operators in the above product isimportant. In what follows a product of non-commuting operators is understood as it is given in ther.h.s of (1.18). It is also mentioned in the list of notations.In Section 3 , using the covariant version of the Cherednik operators (1.9)P θ ω ( uZ i ) = X n ∈ Z ω n − n ( − u ) n " Y k = i x i − t n x k x i − x k γ ni + X j = i ( t n − x i x i − x j Y k = i,j x j − t n x k x j − x k γ nj σ ij (1.19)and the auxiliary covariant operatorsP θ ω ( uU i ) = X n ∈ Z ω n − n ( − u ) n ( t n − Y k = i x i − t n x k x i − x k γ ni (1.20)we prove the following analogue of (1.12): I N ( u ) := D N ( ut ) D N ( u ) − = 1 + N X i =1 P θ ω ( uU i ) 1P θ ω ( uZ i ) (cid:12)(cid:12)(cid:12) Λ N . (1.21)5n Section 4 the matrix resolvent of the construction is presented. Namely, consider N × N matrix Z with elements Z ii = (cid:16) Y l = i x i − tx l x i − x l (cid:17) γ i (1.22) Z ij = ( t − x j x i − x j (cid:16) Y l = i,j x i − tx l x i − x l (cid:17) γ j for i = j . (1.23)It is the Lax matrix of the trigonometric quantum Ruijsenaars-Schneider model. Together with thecolumn vector E = (1.24)and the row vector P θ ω ( u U ) = (cid:2) P θ ω ( uU ) ... P θ ω ( uU N ) (cid:3) (1.25)it provides the generating function of the Dell Hamiltonians (with p = 0) in the following way: I N ( u ) := D N ( ut ) D N ( u ) − = 1 + P θ ω ( u U )P θ ω ( u Z ) − E (cid:12)(cid:12)(cid:12) Λ N . (1.26)In Sections 5 and we describe the generalization of the Nazarov-Sklyanin N → ∞ limit con-struction for the Dell Hamiltonians (with p = 0) and the covariant Cherednik operators.Extend the homomorphism (1.14) to the space Λ[ w ] of polynomials in a formal variable w withcoefficients in Λ in the following way: τ N ( p n ) = π N ( p n ) ,τ N ( w ) = t N . (1.27)Let I ( u ) be the operator Λ → Λ[ w ], satisfying I N ( u ) π N = τ N I ( u ) . (1.28)Then the main result of these two Sections is as follows. The operator I ( u ) = θ ω ( u ) θ ω ( uw ) I ( u )does not depend on w , thus mapping the space Λ to itself. It has the form: I ( u ) = θ ω ( u ) h θ ω ( u ) + P θ ω ( uY β ) − P θ ω ( uY α )P θ ω ( uXα ) − P θ ω ( uXβ ) i − , (1.29)where the operators α [ n ] , β [ n ] , X [ n ] , Y [ n ] are defined through (5.6), (5.9), (5.10), (5.21), (5.22), (5.23),(5.24), (5.25), (6.2), (6.4).In Section 7 the expressions for the operators α [ n ] , β [ n ] , X [ n ] , Y [ n ] are derived in a more explicitform. These operators yields the generating function of the N → ∞ Hamiltonians. We prove, thatthese Hamiltonians commute as soon as the Dell Hamiltonians commute . Let us again stress that the commutativity of the Dell Hamiltonians (1.1) is a hypothesis, which was verified numer-ically. Section 8 we write down the explicit form of the first few non-trivial N → ∞ Hamiltonians tothe first power in ω .The generating function equals: I ( u ) = 1 − u + ω ( u − u − )1 − u − uJ ( u ) + ω K ( u ) + O ( ω ) , (1.30)where J ( u ) and K ( u ) are given by (8.2) and (8.3) respectively. The formulae for the first and thesecond Hamiltonians up to the first order in ω is given in (8.19) and (8.21) together with the notations(8.22)-(8.29). In the limit ω = 0, our answer (1.30) reproduces the Nazarov-Sklyanin result [16]: I ( u ) = 1 − u − u − uJ ( u ) . (1.31)In Section 9 we also verify directly that the first and the second Hamiltonians commute with eachother up to the first order in ω . Let C [ x , ..., x N ] be the space of polynomials in N variables x , ..., x N . As in (1.7) denote by σ ij theoperators acting on C [ x , ..., x N ] by interchanging the variables (particles positions) x i ↔ x j and havingthe following commutation relations with operators from A x,p – the space of operators generated by { x , ..., x N , q x ∂ , ..., q x N ∂ N } : σ ij x j = x i σ ij ,σ ij q x j ∂ j = q x i ∂ i σ ij . (2.1) By definition introduce the Dell Cherednik operators (for p = 0) acting on C [ x , ..., x N ] as follows :P θ ω ( uC i ) = X n ∈ Z ω n − n ( − u ) n t n ( i − R i,i +1 ( t n ) ...R iN ( t n ) γ ni R ,i ( t n ) − ... R i − ,i ( t n ) − , (2.2)where operators R ij ( t ) are given by (1.7).The main theorem of this Section is as follows. Theorem 2.1. D N ( u ) = N Y i =1 P θ ω ( uC i ) (cid:12)(cid:12)(cid:12) Λ N = P θ ω ( uC ) ... P θ ω ( uC N ) (cid:12)(cid:12)(cid:12) Λ N , (2.3) where Λ N – is the space of symmetric functions of N variables x , ..., x N . The ordering in the r.h.s of(2.3) is important since the operators P θ ω ( uC i ) do not commute . To prove it we need two lemmas. The first one is analogous to the Lemma 2.3 from [16]. Lemma 2.1. For k = 1 , ..., N − let C ( k )1 ,... , C ( k ) N − k be the Cherednik operators, acting on the space C [ x k +1 , ..., x N ] instead of C [ x , ..., x N ] . Then N Y i = k +1 P θ ω ( uC i ) (cid:12)(cid:12)(cid:12) Λ N = N − k Y i =1 P θ ω ( ut k C ( k ) i ) (cid:12)(cid:12)(cid:12) Λ N . (2.4) Here the notation (1.16) is used. So that, in the above definition C i are the ordinary Cherednik operators (1.6). Let us remark that while P θ ω ( uC i ) (for N > 2) indeed do not commute on the space C [ x , ..., x N ], numericalcalculations show that they do commute on a small subspace of C [ x , ..., x N ] spanned by monomials x a x a ...x a N N with a k ∈ { , } . We hope to clarify this phenomenon in our future works. roof: First, by the downward induction on k = N, N − , ..., , N Y i = k +1 P θ ω ( uC i ) (cid:12)(cid:12)(cid:12) Λ N = N Y i = k +1 P θ ω ( ut i − R i,i +1 ... R iN γ i ) (cid:12)(cid:12)(cid:12) Λ N . (2.5)The base case of induction k = N is trivial. Assuming the statement (2.5) holds true for k we need toprove it for k − 1, i.e. N Y i = k P θ ω ( uC i ) (cid:12)(cid:12)(cid:12) Λ N = P θ ω ( uC k ) N Y i = k +1 P θ ω ( ut i − R i,i +1 ... R iN γ i ) (cid:12)(cid:12)(cid:12) Λ N . (2.6)Notice that for any n ∈ Z the factors R ,k ( t n ) − , ..., R k − ,k ( t n ) − appearing in P θ ω ( uC k ) commutewith all expressions R i,i +1 ( t m ) , ..., R iN ( t m ) and q mx i ∂ i (for any m ∈ Z and i = k + 1 , ..., N ) appearingin the product in the r.h.s. of (2.6). They also act trivially on Λ N . Therefore, we can remove them: N Y i = k P θ ω ( uC i ) (cid:12)(cid:12)(cid:12) Λ N = P θ ω ( ut k − R k,k +1 ... R kN γ k ) N Y i = k +1 P θ ω ( ut i − R i,i +1 ... R iN γ i ) (cid:12)(cid:12)(cid:12) Λ N . (2.7)Hence, we proved the desired statement: N Y i = k +1 P θ ω ( uC i ) (cid:12)(cid:12)(cid:12) Λ N = N Y i = k +1 P θ ω ( ut i − R i,i +1 ... R iN γ i ) (cid:12)(cid:12)(cid:12) Λ N . (2.8)In particular, for k = 0 we have N Y i =1 P θ ω ( uC i ) (cid:12)(cid:12)(cid:12) Λ N = N Y i =1 P θ ω ( ut i − R i,i +1 ... R iN γ i ) (cid:12)(cid:12)(cid:12) Λ N . (2.9)Applying this result to the set of operators C ( k )1 , ..., C ( k ) N − k yields the following answer for the product: N − k Y i =1 P θ ω ( ut k C ( k ) i ) (cid:12)(cid:12)(cid:12) Λ N = N − k Y i =1 P θ ω ( ut i + k − R i + k,i + k +1 ... R i + k,N γ i + k ) (cid:12)(cid:12)(cid:12) Λ N . (2.10)The product in the r.h.s. of (2.10) equals N Y i = k +1 P θ ω ( ut i − R i,i +1 ... R iN γ i ) (cid:12)(cid:12)(cid:12) Λ N (2.11)by just renaming the indices from i + k to i . This is what we need since we have already proved thatthe action of this operator on Λ N coincides with that of the product N Y i = k +1 P θ ω ( uC i ) (cid:12)(cid:12)(cid:12) Λ N . (cid:4) (2.12) Introduce the following notations: for i, j = 1 , ..., N and i = j denote A [ n ] ij = A ij ( t n ) = x i − t n x j x i − x j and B [ n ] ij = B ij ( t n ) = ( t n − x j x i − x j , (2.13)8o that R ij = A ij ( t ) + B ij ( t ) σ ij . (2.14)The Dell ( p = 0) version of the Nazarov-Sklyanin operators Z i (1.9) is as follows:P θ ω ( uZ i ) = X n ∈ Z ω n − n ( − u ) n " Y k = i x i − t n x k x i − x k γ ni + X j = i ( t n − x i x i − x j Y k = i,j x j − t n x k x j − x k γ nj σ ij , (2.15)or, in the notations (2.13):P θ ω ( uZ i ) = X n ∈ Z ω n − n ( − u ) n " Y k = i A [ n ] ik γ ni + X j = i B [ n ] ij Y k = i,j A [ n ] jk γ nj σ ij . (2.16)Similarly to the operators Z i (1.9) they make up a covariant set with respect to the symmetric group S N , acting by the permutation of variables: σ − P θ ω ( uZ i ) σ = P θ ω ( uZ σ ( i ) ) , σ ∈ S N . (2.17)Let us now introduce one more convenient notation, which we need to formulate the next lemma. For k = 1 , .., N − ( k ) N ⊂ C [ x , ..., x N ] the subspace of polynomials symmetric in the variables x k +1 , ..., x N . Then Λ N ⊂ Λ (1) N ⊂ ... ⊂ Λ ( N − N = C [ x , ..., x N ] . (2.18)The second lemma we need for the proof of the Theorem 2.1 is as follows. Lemma 2.2. P θ ω ( uC ) (cid:12)(cid:12)(cid:12) Λ (1) N = P θ ω ( uZ ) (cid:12)(cid:12)(cid:12) Λ (1) N . (2.19) Proof: Consider each term in the sum over n ∈ Z separately. The statement then reduces to C ( q n , t n ) (cid:12)(cid:12)(cid:12) Λ (1) N = Z ( q n , t n ) (cid:12)(cid:12)(cid:12) Λ (1) N . (2.20)The latter directly follows from the Proposition 2.4 in [16], which reads C ( q, t ) (cid:12)(cid:12)(cid:12) Λ (1) N = Z ( q, t ) (cid:12)(cid:12)(cid:12) Λ (1) N . (cid:4) (2.21) Let us prove the main theorem of this Section. Proof: The proof is by induction on the number of variables (particles). For a single particle thestatement is true. Assume it is true for N − N Y i =1 P θ ω ( uC i ) (cid:12)(cid:12)(cid:12) Λ N = P θ ω ( uC ) N Y i =2 P θ ω ( uC i ) (cid:12)(cid:12)(cid:12) Λ N = P θ ω ( uC ) N − Y i =1 P θ ω ( utC (1) i ) (cid:12)(cid:12)(cid:12) Λ N (2.22)due to Lemma 2.1 (2.4) for k = 1. By the induction assumptionP θ ω ( uC ) N − Y i =1 P θ ω ( utC (1) i ) (cid:12)(cid:12)(cid:12) Λ N = P θ ω ( uC ) D N − ( tu | x , ..., x N ) (cid:12)(cid:12)(cid:12) Λ N . (2.23)9he operator D N − ( tu | x , ..., x N ) maps the space Λ N to Λ (1) N . Therefore, using Lemma 2.2 (2.19) wehave P θ ω ( uC ) D N − ( tu | x , ..., x N ) (cid:12)(cid:12)(cid:12) Λ N = P θ ω ( uZ ) D N − ( tu | x , ..., x N ) (cid:12)(cid:12)(cid:12) Λ N . (2.24)Hence, we must prove the following relation: D N ( u | x , x , ..., x N ) (cid:12)(cid:12)(cid:12) Λ N = P θ ω ( uZ ) D N − ( tu | x , ..., x N ) (cid:12)(cid:12)(cid:12) Λ N . (2.25)Let us verify it by direct calculation. Write down both parts of (2.25) explicitly: X n ,...,n N ∈ Z ω P i n i − ni ( − u ) P i n i N Y i Let us assume that the Dell Hamiltonians (1.1) commute with each other [ H N ( u ) , H N ( v )] = [ D − N, D N ( u ) , D − N, D N ( v )] = 0 (3.2) for any u and v . Then [ I N ( u ) , I N ( v )] = 0 . (3.3) Proof: Consider the ratios of the Shakirov-Koroteev Hamiltonians (1.1): H N ( tu ) H N ( u ) − . (3.4)Obviously, this expression commutes with itself for different values of u :[ H N ( tu ) H N ( u ) − , H N ( tv ) H N ( v ) − ] = 0 . (3.5)On the other hand, we have H N ( tu ) H N ( u ) − = D − N, D N ( ut ) D N ( u ) − D N, = D − N, I N ( u ) D N, , (3.6)i.e. the new generating function I N ( u ) is conjugated to the function generating the commuting set ofoperators. Therefore, the Hamiltonians produced by (3.1) also commute with each other. (cid:4) Following [16] define the operators U , ..., U N : U i ( q, t ) = ( t − Y k = i x i − tx k x i − x k γ i . (3.7)They also form a covariant set with respect to the action of the symmetric group S N by the permutationsof variables x , ...x N , i.e. σ − P θ ω ( uU i ) σ = P θ ω ( uU σ ( i ) ) σ ∈ S N . (3.8)The double elliptic generalization of the Nazarov-Sklyanin construction is based on the following result. Theorem 3.1. D N ( ut ) D N ( u ) − = 1 + N X i =1 P θ ω ( uU i ) 1P θ ω ( uZ i ) (cid:12)(cid:12)(cid:12) Λ N . (3.9)11 roof: The proof is again analogous to the one from [16]. Multiplying both parts of (3.9) by D N ( u )one obtains D N ( ut ) − D N ( u ) = N X i =1 P θ ω ( uU i ) 1P θ ω ( uZ i ) D N ( u ) (cid:12)(cid:12)(cid:12) Λ N . (3.10)Because of the covariance property of both U i and Z i it is equal to D N ( ut ) − D N ( u ) = N X i =1 σ i P θ ω ( uU ) 1P θ ω ( uZ ) D N ( u ) (cid:12)(cid:12)(cid:12) Λ N . (3.11)Next, due to Lemmas (2.2) and (2.1) (for k = 1) and the Theorem 2.1 from the previous Section (see(2.3)) we have the following chain of equalities:P θ ω ( uU ) 1P θ ω ( uZ ) D N ( u | x , ..., x N ) (cid:12)(cid:12)(cid:12) Λ N = P θ ω ( uU ) 1P θ ω ( uZ ) N Y i =1 P θ ω ( uC i ) (cid:12)(cid:12)(cid:12) Λ N = (3.12)= P θ ω ( uU ) N Y i =2 P θ ω ( uC i ) (cid:12)(cid:12)(cid:12) Λ N = P θ ω ( uU ) N Y i =1 P θ ω ( tuC (1) i ) (cid:12)(cid:12)(cid:12) Λ N = (3.13)= P θ ω ( uU ) D N ( tu | x , ..., x N ) (cid:12)(cid:12)(cid:12) Λ N , (3.14)i.e. we need to prove that D N ( ut ) − D N ( u ) = N X i =1 σ i P θ ω ( uU ) D N ( tu | x , ..., x N ) (cid:12)(cid:12)(cid:12) Λ N . (3.15)The l.h.s. of (3.15) equals X n ,...,n N ∈ Z ω P i n i − ni ( − u ) P i n i ( t P i n i − Y ≤ a
L, M ] , L, M ∈ Mat N , (4.1)where L is the quantum (i.e operator valued) Lax matrix, and M is the quantum accompany (or M -)matrix. In the classical limit ~ → L = { H, L } = [ L, M ],which integrals of motion (the Hamiltonians) are given by tr( L k ). In quantum case tr( L k ) are no moreconserved since matrix elements of L -matrix do not commute. However, if the zero sum condition X i M ij = X j M ij = 0 (4.2)13olds true, then the total sums of the Lax matrix powers H k = X i,j ( L k ) ij (4.3)are conserved operators [23], i.e. [ H k , H ] = 0 . (4.4)In this Section we perform a kind of the above construction applicable for the Dell model. Let f ∈ Λ (1) N .Consider the column vector F = fσ ( f )... σ N ( f ) (4.5)Next, define the set of operator valued N × N matrices Z [ n ] with matrix elements Z [ n ] ij , acting on C [ x , ..., x N ] as follows: Z [ n ] ii = (cid:16) Y l = i A [ n ] il (cid:17) γ ni , (4.6) Z [ n ] ij = B [ n ] ij (cid:16) Y l = i,j A [ n ] jl (cid:17) γ nj for i = j . (4.7)The previously defined operators Z [ n ] i then take the form: Z [ n ] i = Z [ n ] ii + X j = i Z [ n ] ij σ ij . (4.8)From (4.8) we conclude that Z [ n ]1 fσ ( Z [ n ]1 f )... σ N ( Z [ n ]1 f ) = Z [ n ]1 fZ [ n ]2 σ ( f )... Z [ n ] N σ N ( f ) = Z [ n ] F . (4.9)Indeed, for the first component it holds due to (4.8) for i = 1, while for the rest components we have: Z [ n ] i σ i ( f ) = Z [ n ] ii σ i ( f ) + Z [ n ] i σ i σ i ( f ) + X j =1 ,i Z [ n ] ij σ ij σ i ( f ) = (4.10)= Z [ n ] ii σ i ( f ) + Z [ n ] i ( f ) + X j =1 ,i Z [ n ] ij σ j ( f ) = Z [ n ] i ( f ) + X j =1 Z [ n ] ij σ j ( f ) , (4.11)as it should be. Since for any n the column Z [ n ] F has the same form as F with only f being replacedby Z [ n ]1 ( f ), the following equality holds: P θ ω ( uZ ) − f P θ ω ( uZ ) − σ ( f )...P θ ω ( uZ N ) − σ N ( f ) = P θ ω ( u Z ) − F . (4.12)14he inverse operator is understood as the power series expansion in ω . For example, the first two termsare of the form:P θ ω ( u Z ) − = (1 − u Z [1] ) − − ω (1 − u Z [1] ) − ( u Z [2] − u − Z [ − )(1 − u Z [1] ) − + O ( ω ) . (4.13)Notice that the matrix P θ ω ( u Z ) appeared in our previous paper [12] as the one, whose determinant givesthe generating function D N ( u ). Actually, Z = L RS is the Lax matrix of the quantum trigonometricRuijsenaars-Schneider model.If f ∈ Λ N , so that f = σ ( f ) = ... = σ N ( f ) , (4.14)then F = E f , (4.15)where E is the column vector with all elements equal to 1 (1.24). Thus, we get the following statement. Corollary 4.1. Let P θ ω ( u U ) = (cid:2) P θ ω ( uU ) ... P θ ω ( uU N ) (cid:3) . (4.16) Then D N ( ut ) D N ( u ) − = 1 + P θ ω ( u U ) P θ ω ( u Z ) − E (cid:12)(cid:12)(cid:12) Λ N . (4.17) N → ∞ limit The goal of this Section is to develop the N → ∞ constructions for the described Hamiltonians.Namely, we need to represent them as operators on the space Λ – the inverse limit of the sequence ofΛ N . The starting point is the formula (3.9). As we will show, it can be rewritten in the form: I N ( u ) = 1 + P θ ω ( uV γ )P θ ω ( uZ ) − (cid:12)(cid:12)(cid:12) Λ N , (5.1)where V = ( t − N X i =1 (cid:16) Y ≤ l ≤ Nl = i A il (cid:17) σ i . Next, we find the inverse limit for the operator P θ ω ( uZ ). As it does not preserve the space Λ N , itslimit will map the space Λ to Λ[ v ] - space of polynomials in the formal variable v with coefficients inΛ. Λ[ v ] (more precisely v Λ[ v ]) could be understood as an auxiliary space in the terminology of spinchains. The same can be done for P θ ω ( uV γ ). Its limit actually will be the operator Λ[ v ] → Λ[ w ], foryet another formal variable w . And thus the inverse limit I ( u ) of the generating function I N ( u ) will beconstructed. However its coefficients will be operators Λ → Λ[ w ]. The dependence on w will be theneliminated by the renormalization of the generating function I ( u ). So that finally, its coefficient willbecome just operators acting on Λ. The role of the parameter w is explained in the end of section 5(5.52 - 5.56). First, let us introduce some standard notations for symmetric functions. In this Section we use thenotations defined in the Section ”1.Symmetric functions” from [16], so we recommend to read it first.We only briefly recall them here. Let Λ be the inverse limit of the sequenceΛ ← Λ ← ... (5.2)15n the category of graded algebras. Let us introduce the standard basis in Λ. For the Young diagram λ , the power sum symmetric functions are defined as follows: p λ = p λ p λ ...p λ ℓ ( λ ) . (5.3)Under the canonical homomorphism π N (acting from Λ to Λ N ) p n maps to π N : p n → p n ( x , . . . , x N ) = N X i =1 x ni . (5.4)Let us introduce the scalar product h , i on Λ. For any two partitions λ and µ h p λ , p µ i = k λ δ λµ where k λ = 1 k k ! 2 k k ! . . . (5.5)The operator conjugation with respect to this form will be indicated as ⊥ . In particular, the operatorconjugated to the multiplication by p n is just p ⊥ n = n ∂∂p n . (5.6)We use the following vertex operators Λ → Λ[ v ]: H ( v ) = 1 + h v + h v + . . . = exp (cid:16) X n ≥ p n n v n (cid:17) , (5.7) H ⊥ ( v ) = 1 + h ⊥ v + h ⊥ v + . . . = exp (cid:16) X n ≥ p ⊥ n n v n (cid:17) , (5.8) Q ( v ) = 1 + Q v + Q v + . . . = H ( v ) H ( tv ) = exp (cid:16) X n ≥ − t n n p n v n (cid:17) , (5.9) Q ∗ ( v ) = 1 + Q ∗ v + Q ∗ v + . . . = H ⊥ ( v ) H ⊥ ( qv ) = exp (cid:16) X n ≥ − q n n p ⊥ n v n (cid:17) . (5.10)From the definition of H ⊥ ( v ) we see that, it acts on p n as follows: H ⊥ ( v ) p n = v n + p n . (5.11)It can be also verified by direct calculation that π N Q ( v ) = Y i ≥ − t x i v − x i v π N . (5.12)The last thing we will need is the standard scalar product on C [ v ]: h v k , v m i = δ km . (5.13)Denote as v ◦ the operator conjugate to the multiplication by v with respect to this scalar product,extended from C [ v ] to Λ[ v ] by linearity. That is, more explicitly v ◦ : v n ( v n − if n > , n = 0 . (5.14)Operators of multiplication by any elements f ∈ Λ as well as conjugate to them f ⊥ also extends fromΛ to Λ[ v ] by C [ v ]-linearity. 16 .2 Inverse limit of P θ ω ( uZ ) We will find the inverse limit of the of the operator P θ ω ( uZ ) restriction on the subspace Λ (1) N . To dothis, one needs to extend the canonical homomorphism π N to a homomorphism π (1) N : Λ[ v ] → Λ (1) N (5.15)as follows: π (1) N : v → x , (5.16) π (1) N : p n → N X i =1 x ni . (5.17)Recall that the operator P θ ω ( uZ ) has the form:P θ ω ( uZ ) = X n ∈ Z ω n − n ( − u ) n W [ n ]1 γ [ n ]1 , (5.18)where γ [ n ]1 = γ n = q − nx ∂ , (5.19) W [ n ]1 = Y k =1 x − t n x k x − x k + X j =1 ( t n − x x − x j Y k =1 ,j x j − t n x k x j − x k σ j . (5.20)Let us find the inverse limits for each W [ n ]1 and γ [ n ]1 separately. Following the Nazarov-Sklyanin con-struction we introduce two homomorphisms ξ and η of Λ[ v ], which acts trivially on Λ, but shift v asfollows: ξ : v → q − v , (5.21) η : v → tv . (5.22)Define also γ [ n ] = ξ n Q ∗ [ n ] ( v ) , (5.23) W [ n ] = η n Q [ n ] ( v ◦ ) , (5.24) Z [ n ] = W [ n ] γ [ n ] . (5.25)Then the following theorem holds. Theorem 5.1. For any n ∈ Z the following diagram: Λ[ v ] Λ[ v ]Λ (1) N Λ (1) NZ [ n ] π (1) N π (1) N Z [ n ]1 (5.26)17 nd, consequently the following one: Λ[ v ] Λ[ v ]Λ (1) N Λ (1) N P θ ω ( uZ ) π (1) N π (1) N P θ ω ( uZ ) (5.27) is commutative.Proof: We prove the statements separately for the pairs γ [ n ]1 , γ [ n ] and W [ n ]1 , W [ n ] . Notice first, that γ [ n ] = ξ n H ⊥ ( q n v ) − H ⊥ ( v ) = H ⊥ ( v ) − ξ n H ⊥ ( v ) = γ n . (5.28)So that the statement for γ [ n ]1 , γ [ n ] just follows from the statement for γ , γ , which was proved in [16].Since H ⊥ ( v ) is the algebra homomorphism, the proof reduces to explicit verification of actions on thegenerators p n and v . We just repeat their arguments here.For v : v q − v q − x γ π N and v x q − x . π N γ For p n : p n q − n v n − v n + p n q − n x + x n + . . . + x nNγ π N and p n x n + . . . + x nN q − n x + x n + . . . + x nN . π N γ So, they are the same.Let us proceed to W [ n ] . By construction W [ n ] commutes with the multiplication by any f ∈ Λ. Atthe same time we see that W [ n ]1 commutes with the multiplication by π (1) N ( f ). So it is enough to provethat π (1) N W [ n ] = W [ n ]1 π (1) N only on the elements: 1 , v, v , ... ∈ Λ[ v ]. Consider the generating function ofthese elements 11 − uv (5.29)in some variable u . By applying π (1) N W [ n ] to it one obtains:11 − u v Q [ n ] ( u )1 − u t n v − u t n x N Y i =1 − u t n x i − u x i , W [ n ] π (1) N (5.30)where the result of the first action is found as follows: Q [ n ] ( v ◦ ) 11 − uv = ∞ X k =0 Q [ n ] k ( v ◦ ) k ∞ X m =0 u m v m = (5.31) ∞ X k =0 ∞ X m = k Q [ n ] k u m v m − k = ∞ X k =0 ∞ X s =0 Q [ n ] k u k + s v s = Q [ n ] ( u )1 − uv . (5.32)On the other hand, by applying W [ n ]1 π (1) N to the same generating function, we find11 − u v − u x − u x Y The following diagrams are commutative Λ[ v ] Λ[ w ]Λ (1) N Λ NV [ n ] π (1) N τ N V [ n ]1 (5.37) for all n ∈ Z Proof: The operator V [ n ] commute with the multiplication by any f ∈ Λ N . The operator V n actingon Λ (1) N commutes with the multiplication by any π (1) N ( f ). Therefore, it is enough to check the equality τ N V [ n ] = V [ n ]1 π (1) N only on the powers of v . Again, we will use the generating function of these powers11 − uv . 19y applying τ N V [ n ] to it, one obtains:11 − u v w n − Q [ n ] ( u ) t nN − N Y i =1 − u t n x i − u x i . V [ n ] τ N (5.38)On the other hand, by applying π (1) N V [ n ]1 to the same function, we get11 − u v − u x N X i =1 t n − − u x i Y ≤ l ≤ Nl = i x i − t n x l x i − x l . π (1) N V [ n ]1 (5.39)The results of these two actions are equal to each other. Indeed the l.h.s. and the r.h.s. are the sameas they were in [16] with t replaced by t n . The equality can be proved by considering both parts as therational function in u . The coincidence of residues at poles and asymptotic behaviour can be verifieddirectly. (cid:4) By surjectivity of π (1) N the last proposition implies, that V [ n ]1 maps Λ (1) N to Λ N . Summarizing results of the two previous subsections we come to the following statement. Theorem 5.2. The following operators (as maps Λ[ v ] → Λ N ) are equal τ N (cid:16) θ ω ( uV γ )P θ ω ( uZ ) − (cid:17) = (cid:16) θ ω ( uV γ )P θ ω ( uZ ) − (cid:17) π (1) N . (5.40) Proof: For any l ∈ N , k , ..., k l ∈ N , n, m, n , ..., n l ∈ Z , the operators V [ n ] γ [ m ] (cid:0) Z [ ~n ] (cid:1) ~k = V [ n ] γ [ m ] l Y i =1 (cid:0) Z [ n i ] (cid:1) k i (5.41)map Λ[ v ] to Λ[ w ]. It follows from the Proposition 5.1 and the Theorem 5.1 that the diagram below iscommutative: Λ[ v ] Λ[ v ] Λ[ v ] Λ[ w ]Λ (1) N Λ (1) N Λ (1) N Λ N (cid:16) Z [ ~n ] (cid:17) ~k π (1) N γ [ m ] π (1) N V [ n ] π (1) N τ N (cid:16) Z [ ~n ]1 (cid:17) ~k γ [ m ]1 V [ n ]1 where we introduced a natural notation (cid:0) Z [ ~n ]1 (cid:1) ~k = l Y i =1 (cid:0) Z [ n i ]1 (cid:1) k i . (5.42)The main statement of the Theorem then follows by expanding the left and right hand sides of (5.40)as the power series in ω at first, and then by expanding the resulting coefficients of this series in thepowers of u . (cid:4) δ the embedding of Λ to Λ[ v ] as the subspace of degree zero in v , and by ε the naturalembedding of Λ N to Λ (1) N . Then we have the commutative diagramΛ Λ[ v ]Λ N Λ (1) Nδπ N π (1) N ε (5.43)In the above notations we have I N ( u ) = 1 + P θ ω ( uV γ )P θ ω ( uZ ) − ε . (5.44)It is natural to define the inverse limit of this operator as follows: I ( u ) = 1 + P θ ω ( uV γ )P θ ω ( uZ ) − δ . (5.45)Then from the Theorem 5.2 we come to Corollary 5.1. The following diagram is commutative: Λ Λ[ w ]Λ N Λ NI ( u ) π N τ N I N ( u ) (5.46) Namely, I N ( u ) π N = τ N I ( u ) . (5.47) Proof: Indeed, one needs to use the commutativity of the diagram (5.43) and the theorem 5.2 : I N ( u ) π N = 1 + P θ ω ( uV γ )P θ ω ( uZ ) − επ N = (5.48)= 1 + P θ ω ( uV γ )P θ ω ( uZ ) − π (1) N δ = (5.49)= 1 + τ N P θ ω ( uV γ )P θ ω ( uZ ) − δ = (5.50)= τ N I ( u ) . (cid:4) (5.51)Hence, the inverse limit of the Hamiltonians is constructed.Let us normalize the operator I ( u ) in order to make it independent of the variable w . The eigenvalueof the operator I N ( u ) on the trivial eigenfunction 1 ∈ Λ N equals N Y i =1 θ ω ( ut i ) θ ω ( ut i − ) = θ ω ( ut N ) θ ω ( u ) (5.52)because of our convention for D N ( u ) ([12]) D N ( u ) = 1 Q i Consider the action of the W [ m ] operator on the power v n for n > W [ m ] v n = η m Q [ m ] ( v ◦ ) v n = η m (cid:16) Q [ m ]1 v + ... + Q [ m ] n v n (cid:17) v n (7.9)since all higher terms become zero. Hence, we get W [ m ] v n = t mn v n + t m ( n − Q [ m ]1 v n − + ... + t m Q [ m ] n − v + Q [ m ] n . (7.10)24y definition, Y [ m ] is the projection of this map to Λ, so that the image of v n is as follows: Y [ m ] v n = h , W [ m ] v n i = Q [ m ] n . (7.11)Similarly, X [ m ] is the projection on v Λ[ v ]. Hence, X [ m ] v n = W [ m ] v n − Y [ m ] v n = n − X j =0 t m ( n − j ) Q [ m ] j v n − j . (7.12)Consider γ [ m ] = ξ [ m ] Q ∗ [ m ] ( v ) . (7.13)Then it follows from the definition (6.2) that β [ m ] = γ [ m ] v = ∞ X n =1 q − nm Q ∗ [ m ] n v n , (7.14)and the action of operator γ [ m ] on v n for n > α [ m ] : α [ m ] v n = γ [ m ] v n = ∞ X j =0 q − m ( n + j ) v n + j Q ∗ [ m ] j . (cid:4) (7.15) ω In this Section we derive the Hamiltonians I and I to the first order in ω . Proposition 8.1. I ( u ) = 1 − u + ω ( u − u − )1 − u − uJ ( u ) + ω K ( u ) + O ( ω ) , (8.1) where J ( u ) is given by J ( u ) = Y [1] − uα [1] X [1] β [1] , (8.2) and K ( u ) = u − u − + u Y [2] β [2] − u − Y [ − β [ − ++ u Y [1] α [1] (cid:0) − uX [1] α [1] (cid:1) − (cid:0) u X [2] α [ − − u − X [ − α [ − (cid:1)(cid:0) − uX [1] α [1] (cid:1) − X [1] β [1] ++ u (cid:0) u Y [2] α [2] − u − Y [ − α [ − (cid:1)(cid:0) − uX [1] α [1] (cid:1) − X [1] β [1] ++ uY [1] α [1] (cid:0) − uX [1] α [1] (cid:1) − (cid:0) u X [2] β [2] − u − X [ − β [ − (cid:1) . (8.3) Proof: Starting with (6.14) and expanding every ” θ -function” P θ ω ( uA ) to the first order in ω asP θ ω ( uA ) = A [0] − uA [1] + ω (cid:0) u A [2] − u − A [ − (cid:1) + O ( ω ) (8.4)25ne obtains: I ( u ) = (cid:0) − u + ω ( u − u − ) (cid:1)" − u + ω ( u − u − ) − uY [1] β [1] + ω (cid:0) u Y [2] β [2] − u − Y [ − β [ − (cid:1) −− n − uY [1] α [1] + ω (cid:0) u Y [2] α [2] − u − Y [ − α [ − (cid:1)on − uX [1] α [1] + ω (cid:0) u X [2] α [2] − u − X [ − α [ − (cid:1)o − n − uX [1] β [1] + ω (cid:0) u X [2] β [2] − u − X [ − β [ − (cid:1)o − + O ( ω ) , (8.5)where we have used the observation that X [0]="1" , (8.6) Y [0]="0" , (8.7) α [0]="1" , (8.8) β [0]="0" . (8.9)In the zero order in ω the denominator in the above formula (the expression in square brackets) equals:1 − u − uY [1] β [1] − u Y [1] α [1] (cid:0) − uX [1] α [1] (cid:1) − X [ − β [ − , (8.10)which indeed can be rewritten as: 1 − u − uJ ( u ) . (8.11)Therefore, we have reproduced the Nazarov-Sklyanin result: I ( u )="1" − u − u − uJ ( u ) + O ( ω ) . (8.12)Gathering the terms in front of the first power of ω in the denominator we arrive at the expression(8.3) for K ( u ). (cid:4) Expanding the formula (8.1) in ω further, one="if (!window.__cfRLUnblockHandlers) return false; " obtains: I ( u )="1" − u − u − uJ ( u ) + ω ( u − u − − u − uJ ( u ) − − u − u − uJ ( u ) K ( u ) 11 − u − uJ ( u ) ) + O ( ω ) . (8.13)From their definitions it is clear that J ( u ) and K ( u ) have the following expansions in u : J ( u )="∞" X n="0" J n u n , (8.14) K ( u )="∞" X n="−" K n u n . (8.15)Hence, from (8.13) one can obtain the following expressions for the several first Hamiltonians: I n="O" ( ω ) for n < − , (8.16) I −="−" ωK − + O ( ω ) , (8.17) I="1" + ω n − K + K − − (1 + J ) K − − K − (1 + J ) − (1 + J ) o + O ( ω ) , (8.18)26r I="1" − ω n J + K + K − + J K − + K − J o + O ( ω ) . (8.19)And I="J" + ω n − K − (1 + J ) K − K (1 + J ) −− (1 + J ) K − (1 + J ) − (cid:0) (1 + J ) − J (cid:1) K − − K − (cid:0) (1 + J ) − J (cid:1) ++ K + (1 + J ) K − + K − (1 + J ) − (cid:0) (1 + J ) − J (cid:1)o + O ( ω ) , (8.20)or I="J" − ω n K + K − + K − (2 J + J − J ) + (2 J + J − J ) K − + J K − J ++ K + J K + K J + (1 + 2 J + J − J ) o + O ( ω ) . (8.21) Explicit form of J , J , K − , K , K Here we derive explicit expressions for the operators J , J , K − , K , K . The operator J has the form: J="Y" [1] β [1]="∞" X n="1" q − n Q [1] n Q ∗ [1] n , (8.22)where we have used the formulas (7.1)-(7.4), (8.2) and (8.14).Using the (DIM vertex) operator e ( z )="Q" ( z − ) Q ∗ ( q − z )="exp" (cid:16) X n ≥ − t n n z − n p n (cid:17) exp (cid:16) − X n ≥ − q − n n z n p ⊥ n (cid:17) , (8.23) J is represented as 1 + J="I" dzz e ( z ) . (8.24)The same can be done for K , K and K − . From (8.3) one obtains: − K −="1" + J [ −="I" dzz e [ − ( z ) , (8.25) − K="Y" [1] α [1] X [ − β [ − + Y [ − α [ − X [1] β [1] , (8.26) − K="Y" [1] α [1] X [ − α [ − X [1] β [1] + Y [1] α [1] X [1] α [1] X [ − β [ − + Y [ − α [ − X [1] α [1] X [1] β [1] . (8.27)With the help of (7.1)-(7.4) it is rewritten as − K="∞" X n="1" n − X j="0" ∞ X i="0" n q j − i t j − n Q [1] n − j + i Q ∗ [1] i Q [ − j Q ∗ [ − n + q i − j t n − j Q [ − n − j + i Q ∗ [ − i Q [1] j Q ∗ [1] n o , (8.28)27nd − K="∞" X n="1" n − X j="0" ∞ X i="0" n − j + i X l="0" ∞ X k="0" ( q − n + l − k t l − i Q [1] n − j + i − l + k Q ∗ [1] k Q [ − l Q ∗ [ − i Q [1] j Q ∗ [1] n ++ q − n + l − k +2 j − i t i − l Q [1] n − j + i − l + k Q ∗ [1] k Q [1] l Q ∗ [1] i Q [ − j Q ∗ [ − n ++ q n − l + k t i − l +2 n − j Q [ − n − j + i − l + k Q ∗ [ − k Q [1] l Q ∗ [1] i Q [1] j Q ∗ [1] n ) . (8.29) ω [ I − , I n ] It is well known that Macdonald polynomials have the symmetry M λ ( x | q, t )="M" λ ( x | q − , t − ) . (9.1)Hence, the Macdonald-Ruijsenaars operators with q and t inverted commute with the original ones.Therefore,="if (!window.__cfRLUnblockHandlers) return false; " using (8.17) and (8.25) one obtains:[ I − , I n ]="ω" [ J [ − , I n ] + O ( ω )="O" ( ω ) . (9.2)For I this is easily verified explicitly:[ J [1]0 , J [ − ]="=" I dww I w dzz e [1] ( z ) e [ − ( w )="I" dww I w dzz ( w − z )( w − q − t − z )( w − q − z )( w − t − z ) : e [1] ( z ) e [ − ( w ) : . (9.3)Calculating the residues one obtains:[ J [1]0 , J [ − ]="=" I dww n (1 − q )(1 − t − )(1 − qt − ) : e [1] ( qw ) e [ − ( w ) : + (1 − q − )(1 − t )(1 − q − t ) : e [1] ( tw ) e [ − ( w ) : o="0" . (9.4) [ I , I ] Due to the remarks from the previous subsection it is clear that J already commutes with all termsin I to the first order in ω , except for maybe K . Therefore, to prove[ I , I ]="O" ( ω ) (9.5)we only="if (!window.__cfRLUnblockHandlers) return false; " need to verify [ J , K ]="0" . (9.6)A direct proof is too cumbersome since it involves the sixth order expressions in the operators Q k .For this reason we verify (9.6) by calculating the action of its l.h.s. on the space of the power sum28ymmetric functions (5.3)-(5.4) using computer. Explicit form of the coefficients Q k and Q ∗ k followsfrom their definitions (5.9)-(5.10): Q [1]1="(1" − t ) p , Q [1]2="(1" − t )2 p + (1 − t ) p ,Q [1]3="(1" − t )3 p + (1 − t )(1 − t )2 p p + (1 − t ) p , . . . (9.7)and Q ∗ [1]1="(1" − q ) ∂ p , Q ∗ [1]2="(1" − q ) ∂ p + (1 − q ) ∂ p ,Q ∗ [1]3="(1" − q ) ∂ p + (1 − q )(1 − q ) ∂ p ∂ p + (1 − q ) ∂ p , . . . (9.8)where the notation A [ n ] ( q, t )="A" ( q n , t n ) is again used. Plugging (9.7)-(9.8) into the definitions of J (8.22) and K (8.26), we get these operators as differential operators acting on the space of polynomials(5.3)-(5.4) of variables p , p , ... . This space has a natural grading. For a monomial p k p k p k ... thedegree in the original variables x j is equal to deg="k" + 2 k + 3 k + ... . The degree 1 subspace isspanned by p , the degree 2 – by { p , p } , the degree 3 – by { p , p p , p } and so on. In particular, thedegree of Q k polynomials is equal to k , and the action of Q ∗ k reduces the degree of a monomial by k .It is easy to see from (8.22), (8.26) that the operators J and K preserve the degree of a monomial.Therefore, in order to prove [ J , K ]="0" we need to verify [ J , K ] f="0" for any basis function f from aspan of a subspace of a given degree, i.e. for f="p" , p , p , p , p p , p , ... . Using computer calculationswe have verified [ J , K ] f="0" for all possible choices of the basis function f up to degree 5. The actionsof J and K on basis functions of degrees 1 and 2 is given below: J p="(1" − t )(1 − q − ) p , K p="(1" − t )(1 − q − )( q + t − ) p ; (9.9) J p="(cid:16)" q − q (cid:17) (cid:16) (1 − t ) p + (1 − t ) p (cid:17) ,K p="q" − t q (cid:16) p q − p q + p t q − p q t + 2 p q t + 2 p t q − p t q + (9.10)+ p t q + 2 p t q − p t q − p t q + p t q − p t q + p t q + p t q + p q t −− p qt + p q t + p q t + 3 p q t − p q t − p qt + 2 p qt − p q t + 3 p q t −− p t − p t q − p q t − p qt + 3 p q t − p q t + p q t − p q t − p q t ++ q t p + qtp + 10 p q t − p q t + p t − p t + 3 p t − p t q − p q t −− p t + p t − p t + p t − p q + p q − p t q + 4 q t p (cid:17) ,J p="2(1" − t )(1 − q − ) p − (cid:16) q − q (cid:17) (cid:16) (1 − t ) p + (1 − t ) p (cid:17) ,K p="−" t q (cid:16) p q − p q t − p q + p t − p t + p t − p q + p q + 2 q t p + (9.11)+2 qtp + p q t + p q t − p q t − p q t − p q t + p q t − p q t − p t q −− p t q − p tq + 2 p qt + p t q + 2 p t q − p t q + 2 p q t + 2 p t q − p t q −− p qt + p qt − p t − p q t − p q + 2 p q − p q t + p t − p qt + 2 p qt + 8 q t p (cid:17) . Whether Dell-DIM and Dell-DAHA exist?< p data-cf-modified-2bc2250b8b2898eea743a24e->
The Cherednik construction could be understoodmore formally from the point of view of the of Double Affine Hecke Algebras (DAHA) theory. Thespace of polynomials C [ x , ..., x N ] serves naturally as a representation of the DAHA. One can considerthe special subalgebra - spherical DAHA, which preserve the subspace of symmetric polynomials Λ N inside of C [ x , ..., x N ]. The Macdonald-Ruijsenaars operators would then represent a center in thespherical DAHA, and the corresponding Cherednik operators represent a center in the DAHA itself[10]. In the N → ∞ limit the spherical DAHA is equivalent to quantum toroidal algebra (DIM) [21].The N → ∞ limit of the Hamiltonians is thus realized as residues of the certain vertex operators inthe Fock representation of this algebra [11], [18]. We could ask a question, whether any analogues ofthese algebraic constructions exist in our case? The answer is unknown, but let us make the followingcomment. Notice that the N → ∞ limit of the covariant Cherednik operator Z is (up to shift operators)equal to the vertex operator e ( v ) in the Fock representation of the DIM algebra: Z = η Q ( v ) ξ Q ∗ ( v ) = η Q ( v ) Q ∗ ( q − v ) ξ = η e ( v ) ξ . (10.1)So it might be natural to expect that if the dual to elliptic generalization of the DIM algebra exists,the operator of the form P θ ω ( ue ( v ))might play some role in its Fock representation, with u being the evaluation parameter of this repre-sentation. Towards the Dell spin chain. In the final part of our previous paper [12] we discussed a double-elliptization of quantum R -matrix. Here we used the Cherednik operators constructed via R -operators.In some special cases (when q and t are related) these R -operators may become endomorphisms offinite-dimensional spaces, i.e. R -operators become quantum R -matrices in these representations. Inthe coordinate trigonometric case, it simply follows from the fact that operators (1.6) and consequently(1.17) preserve the space of polynomials of the fixed degree in C [ x , ..., x N ], which is finite dimensional.Following [13] in this way a correspondence between the Cherednik’s description of the Ruijsenaars-Schneider model and the spin-chain (constructed through the R -matrix) can be established. Similarprocedure can be applied to the obtained double-elliptic Cherednik operators. Then on the spin chainside it is natural to expect the Dell generalization of the spin chain. We hope to study this possibilityin our future work. 11 Appendix A: Cherednik construction in GL(2) case In this Section we consider 2-body systems, i.e. N = 2 case. In the GL(2) case the Dell-Cherednik operators have the form:P θ ω ( uC ) = X n ∈ Z ω n − n ( − u ) n (cid:16) x − t n x x − x + ( t n − x x − x σ (cid:17) q nx ∂ (A.1)P θ ω ( uC ) = X n ∈ Z ω n − n ( − u ) n q nx ∂ (cid:16) t n x − x x − x + (1 − t n ) x x − x σ (cid:17) , (A.2)30o that P θ ω ( uC )P θ ω ( uC ) = X n ,n ∈ Z ( − u ) n + n ω n − n + n − n × (A.3) × (cid:16) x − t n x x − x + ( t n − x x − x σ (cid:17) q n x ∂ q n x ∂ (cid:16) t n x − x x − x + (1 − t n ) x x − x σ (cid:17) . Restriction to Λ gives (the action of σ on Λ is trivial):P θ ω ( uC )P θ ω ( uC ) (cid:12)(cid:12)(cid:12) Λ N = (A.4)= X n ,n ∈ Z ( − u ) n + n ω n − n + n − n t n (cid:16) x − t n x x − x + ( t n − x x − x σ (cid:17) q n x ∂ q n x ∂ (cid:12)(cid:12)(cid:12) Λ N == X n ,n ∈ Z ( − u ) n + n ω n − n + n − n t n (cid:16) x − t n x x − x q n x ∂ q n x ∂ + ( t n − x x − x q n x ∂ q n x ∂ (cid:17) . Gathering terms in front of the fixed powers of the shift operators q n x ∂ q n x ∂ , one obtains:P θ ω ( uC )P θ ω ( uC ) (cid:12)(cid:12)(cid:12) Λ N = (A.5)= X n ,n ∈ Z ( − u ) n + n ω n − n + n − n (cid:16) x − t n x x − x t n + ( t n − x x − x t n (cid:17) q n x ∂ q n x ∂ , or P θ ω ( uC )P θ ω ( uC ) (cid:12)(cid:12)(cid:12) Λ N = X n ,n ∈ Z ( − u ) n + n ω n − n + n − n t n x − t n x x − x q n x ∂ q n x ∂ , (A.6)as it should be. In the elliptic case the Cherednik operators are given in terms of elliptic R -operators, which can berepresented in several different ways [7, 13]. We use the formulation of [13] since the elliptic Macdonald-Ruijsenaars were obtained in that paper. Namely, R ij = (cid:18) ϑ ( η ) ϑ ′ (0) (cid:19) φ ( q i − q j , η ) − φ ( q i − q j , z i − z j ) σ ij ! , (A.7)where z i are the spectral parameters. The function φ and the theta-function are defined by (B.3) and(B.6). The operator (A.7) satisfies the unitarity condition: R ( z − z ) R ( z − z ) = (cid:18) ϑ ( η ) ϑ ′ (0) (cid:19) ℘ ( η ) − ℘ ( z − z ) ! . (A.8)To obtain the Macdonald operators through (A.7) one should set z k = kη , so that z = − η , and ther.h.s. of (A.8) vanishes. It happens due to R | z k = kη = (cid:18) ϑ ( η ) ϑ ′ (0) (cid:19) φ ( q − q , η )(1 − σ ) , (A.9) The normalization factor ( ϑ ( η ) /ϑ ′ (0)) is not important in the Ruijsenaars-Schneider case since it affects the commonfactor only. But it becomes important when averaging in P θ ω . . For this reason below we use the limit ǫ → z = − η + ǫ asit is performed in [13].For the Cherednik operators C ( η, ~ ) = R | ( z = − η + ǫ ) e ~ ∂ , C ( η, ~ ) = e ~ ∂ R ( z = η − ǫ ) . (A.10)define the Dell-Cherednik operators asP θ ω ( uC i ) = X k ∈ Z ω k − k ( − u ) n C i ( kη, k ~ ) , (A.11)and compute H ( u ) = lim ǫ → (cid:16) ϑ ′ (0) θ ( − ǫ ) : P θ ω ( uC )P θ ω ( uC ) : (cid:17) , (A.12)where the normal ordering :: is understood as moving shift operators to the right with keeping theaction of the permutation operators. For example, : e ~ ∂ f ( x ) σ := f ( x ) e ~ ∂ σ = σ f ( x ) e ~ ∂ .Notice that we did not use this ordering in the previous subsection when considered the trigonometriccoordinate case. The reason is that in the trigonometric case we have the property R ij | Λ = 1, while inthe elliptic case R ij | Λ is a function of q ij , and the action of shift operators may cause some unwantedshifts of arguments .Let us evaluate the expression (A.12): H ( u ) = lim ǫ → " ϑ ′ (0) θ ( − ǫ ) X n ,n ∈ Z ω n n − n − n ( − u ) n + n (cid:18) ϑ ( n η ) ϑ ′ (0) (cid:19) (cid:18) ϑ ( n η ) ϑ ′ (0) (cid:19) ×× : (cid:16) φ ( q , n η ) − φ ( q , ǫ − n η ) σ (cid:17) e n ~ ∂ e n ~ ∂ (cid:16) φ ( q , n η ) − φ ( q , − ǫ + n η ) σ (cid:17) : . (A.13)The expression in the square brackets takes the following form after restriction on Λ : (cid:16) φ ( q , n η ) − φ ( q , − ǫ + n η ) σ (cid:17) | Λ = ǫφ ′ ( q , n η ) + o ( ǫ ) , (A.14)where φ ′ is a derivative of φ with respect to the second argument. Then we can evaluate the limit ǫ → H ( u ) | Λ = − X n ,n ∈ Z ω n n − n − n ( − u ) n + n (cid:18) ϑ ( n η ) ϑ ′ (0) (cid:19) (cid:18) ϑ ( n η ) ϑ ′ (0) (cid:19) ×× : (cid:16) φ ( q , n η ) − φ ( q , − n η ) σ (cid:17) e n ~ ∂ e n ~ ∂ φ ′ ( q , − n η ) : | Λ , (A.15)where we have also used the parity (B.10). By moving the permutation operator to the right and usingalso the normal ordering we get H ( u ) | Λ = − X n ,n ∈ Z ω n n − n − n ( − u ) n + n (cid:18) ϑ ( n η ) ϑ ′ (0) (cid:19) (cid:18) ϑ ( n η ) ϑ ′ (0) (cid:19) ×× (cid:16) φ ( q , n η ) φ ′ ( q , − n η ) − φ ( q , − n η ) φ ′ ( q , n η ) (cid:17) e n ~ ∂ e n ~ ∂ . (A.16) Another reason is that in fact we are performing double-elliptization of the set of operators (products of Cherednikoperators) from [13], where all the shift operators can be moved to the right due to the property e ~ ∂ i e ~ ∂ j R ij = R ij e ~ ∂ i e ~ ∂ j ,which is obviously not true for e n i ~ ∂ i e n j ~ ∂ j appearing in the Dell case. So we use the normal ordering to move the shiftoperators to the right by hands. φ ( q , n η ) φ ′ ( q , − n η ) − φ ( q , − n η ) φ ′ ( q , n η ) = φ ( q , ( n − n ) η )( ℘ ( n η ) − ℘ ( n η ))= φ ( q , ( n − n ) η ) φ ( n η, n η ) φ ( n η, − n η ) == − (cid:18) ϑ ′ (0) ϑ ( n η ) (cid:19) (cid:18) ϑ ′ (0) ϑ ( n η ) (cid:19) ϑ ( q + ( n − n ) η )) ϑ ( q ) ϑ (( n + n ) η ) ϑ ′ (0) . (A.17)Plugging it into (A.16) we get H ( u ) | Λ = X n ,n ∈ Z ω n n − n − n ϑ (( n + n ) η ) ϑ ′ (0) ( − u ) n + n ϑ ( q + ( n − n ) η )) ϑ ( q ) e n ~ ∂ e n ~ ∂ . (A.18)The latter almost coincide with the Dell operator (1.2) for N = 2. The first difference between (1.2)and (A.18) is in replacing θ p with ϑ , which is a simple modification of (1.2) (see explanation in [12]).The second difference is in presence of the factor ϑ (( n + n ) η ) /ϑ ′ (0). It is unessential since the factordepends on n + n and does not affect the definition of the coefficients ˆ O k . 12 Appendix B: Elliptic function notations We use several definitions of theta-functions. The first is the one is θ p ( x ) = X n ∈ Z p n − n ( − x ) n , (B.1)where the moduli of elliptic curve τ ∈ C , Im τ > p = e πiτ . (B.2)It was used in [14] and enters (1.2). Another theta-function is the standard Jacobi one: ϑ ( z ) = ϑ ( z | τ ) = − i X k ∈ Z ( − k e πi ( k + ) τ e πi (2 k +1) z . (B.3)The definitions (B.1) and (B.3) are easily related: θ p ( x ) = ip − x ϑ ( w | τ ) , x = e πiw . (B.4)In the trigonometric limit p → θ p ( x ) → (1 − x ) , ϑ ( w ) → − ip ( √ x − / √ x ) . (B.5)In the elliptic coordinate Dell model we also use the elliptic Kronecker function φ ( z, u ) = ϑ ′ (0) ϑ ( z + u ) ϑ ( z ) ϑ ( u ) (B.6)and the corresponding addition formulae: φ ( z, u ) ∂ v φ ( z, v ) − φ ( z, v ) ∂ u φ ( z, u ) = ( ℘ ( u ) − ℘ ( v )) φ ( z, u + v ) , (B.7) φ ( z, − q ) φ ( z, q ) = ℘ ( z ) − ℘ ( q ) , (B.8)where ℘ ( z ) is the Weierstrass ℘ -function: ℘ ( z ) = − ∂ z log ϑ ( z | τ ) + 13 ϑ ′′′ (0) ϑ ′ (0) . (B.9)Parity of the functions is as follows: ϑ ( z ) = − ϑ ( − z ) , φ ( z, u ) = − φ ( − z, − u ) , φ ′ ( z, u ) = φ ′ ( − z, − u ) , (B.10)where φ ′ ( z, u ) = ∂ u φ ( z, u ). 33 Proof of (2.28) Let us prove the identity: N Y l =2 t n l x − t n x l x − x l == t P i n i t − n N Y l =2 x − t n x l x − x l + N X j =2 t − n j ( t n j − x j x − x j N Y l =2 l = j x j − t n j x l t n l x j − t n j x l t n l x − t n x l x − x l ! . (C.1)The factors Q Nl =2 ( x − x l ) in the denominator can be cancelled out leaving us with N Y l =2 ( t n l x − t n x l ) == t P i n i (cid:16) t − n N Y l =2 ( x − t n x l ) + N X j =2 t − n j ( t n j − x j N Y l =2 l = j x j − t n j x l t n l x j − t n j x l ( t n l x − t n x l ) (cid:17) . (C.2)The l.h.s. and the r.h.s. of (C.2) are both polynomials in x of degree N − 1. To prove the equality wemust verify that their zeros and the asymptotic behaviors at x → ∞ coincide. The zeros of the l.h.s.are located at x = t n − n a x a a = 2 , ..., N . (C.3)Let us show that these points are zeros of the r.h.s. as well. Plugging (C.3) into the r.h.s. of (C.2) onesees that in the sum over j only the term with j = a survives, so the r.h.s. is equal to t P i n i t − n N Y l =2 ( t n − n a x a − t n x l )+ t − n a ( t n a − x a Y ≤ a
14 Appendix D: Commutation relations for Q ∗ m , Q n Using Q ∗ ( w ) Q ( z − ) = (1 − q wz )(1 − t wz )(1 − wz )(1 − qt wz ) Q ( z − ) Q ∗ ( w ) (D.1)and taking the coefficient in front of the power w m z − n we get:[ Q ∗ m , Q n ] = (1 − q )(1 − t ) min( m,n ) X i =1 min( m,n ) − i X j =0 ( qt ) j Q n − i − j Q ∗ m − i − j . (D.2)More generally, for s, r ∈ Z :[ Q ∗ [ s ] m , Q [ r ] n ] = (1 − q s )(1 − t r ) min( m,n ) X i =1 min( m,n ) − i X j =0 ( q s t r ) j Q [ r ] n − i − j Q ∗ [ s ] m − i − j . (D.3) Acknowledgments We are grateful to A. Grosky, M. Matushko, A. Mironov, A. Morozov, V. Rubtsov, I. Sechin, M.Bershtein, Sh. Shakirov, A. Zabrodin and Y. Zenkevich for useful comments and discussions.The work of A. Zotov was performed at the Steklov International Mathematical Center and supportedby the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2019-1614). References [1] I. Andric, A. Jevicki, H. 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