Non-stationary Ruijsenaars functions for $\kappa=t^{-1/N}$ and intertwining operators of Ding-Iohara-Miki algebra

We construct the non-stationary Ruijsenaars functions (affine analogue of the Macdonald functions) in the special case $\kappa=t^{-1/N}$, using the intertwining operators of the Ding-Iohara-Miki algebra (DIM algebra) associated with $N$-fold Fock tensor spaces. By the $S$-duality of the intertwiners, another expression is obtained for the non-stationary Ruijsenaars functions with $\kappa=t^{-1/N}$, which can be regarded as a natural elliptic lift of the asymptotic Macdonald functions to the multivariate elliptic hypergeometric series. We also investigate some properties of the vertex operator of the DIM algebra appearing in the present algebraic framework; an integral operator which commutes with the elliptic Ruijsenaars operator, and the degeneration of the vertex operators to the Virasoro primary fields in the conformal limit $q \rightarrow 1$.


Introduction
The non-stationary Ruijsenaars function f gl N (x, p|s, κ|q, t) introduced by one of the authors in [1] is by definition given in the form of the Nekrasov partition function, Here x = (x 1 , . . . , x N ), s = (s 1 , . . . , s N ), and N ∈ Z ≥1 . As for the detail, see Definition 3.21. Our goal in the present paper is to establish the transformation formula stated in Theorem 3.30 below for the non-stationary Ruijsenaars function f gl N in the special case κ = t −1/N , by using the S-duality of the intertwining operators of the Ding-Iohara-Miki (DIM) algebra. Define the elliptic shifted product Θ(a; q, p) n as the ratio of the Ruijsenaars elliptic gamma function Γ(a; q, p) by Θ(a; q, p) n := Γ(q n a; q, p) Γ(a; q, p) , Γ(a; q, p) := (qp/a; q, p) ∞ (a; q, p) ∞ .
Theorem (Theorem 3.41). We have . (1.9) Remark that setting κ = t −1 in (1.9), we recover (1.5) for N = 1. To prove Theorem 3.30 and Theorem 3.41, we use the technique of the topological vertex operator. This consists of the DIM algebra, the trivalent vertex operators, and the web diagrams encoding the structure of the Fock tensor spaces which the DIM algebra is acting on.
To fix a good starting point, we need to recall some facts about the asymptotically free eigenfunctions f gl N (x, s|q, t) for the Macdonald q-difference operator. See [4,5,6] and Appendix A as to the basic facts. (q a>k (θ ia −θ ja ) ts j /s i ; q) θ ik (q a>k (θ ia −θ ja ) qs j /s i ; q) θ ik × N k=2 1≤i≤j<k (q −θ jk + a>k (θ ia −θ ja ) qs j /ts i ; q) θ ik (q −θ jk + a>k (θ ia −θ ja ) s j /s i ; q) θ ik . (1.11) Note that the f gl N enjoys the eigenvalue equation (Fact A.2), the analytic property (Fact A.6), the bispectral duality (Fact A. 3), and the Poincaré duality (Fact A.4). We have a "web diagrammatic description" of the combinatorial structure (Definition 3.12), based on the DIM algebra. In other words, the f gl N has an interpretation as a certain Nekrasov partition function associated with the web diagram. See [7,8].
The DIM algebra has two kinds of intertwining operators, Φ : F (0,1) ⊗ F (1,0) → F (1,1) and Φ * : F (1,1) → F (1,0) ⊗ F (0,1) among the triple of Fock spaces, introduced in [9]. As for the definition of the modules F (1,M ) , F (0,1) and the intertwiners, see Fact 2.5, 2.4, and 2.6. By using physics terminology, we refer to the modules F (0,1) as the preferred directions. The matrix elements of these intertwiners are identical to the refined topological vertex. We express the composition Φ * • Φ by the cross diagram in FIGURE 1, left. Compose these operators reticulately as in FIGURE 1, right. Specialize spectral parameters in a certain manner. Attach empty diagrams to all the external edges. Then we have the Macdonald function f gl N as thus constructed matrix element (See Fact 3.13).
Suppose we consider the trace in the preferred or vertical direction, instead of the matrix element as above. See FIGURE 2 below. We will prove that the p-trace associated with the web on a cylinder gives us the non-stationary Ruijsenaars function f gl N for the special parameter κ = t −1/N for N ≥ 2, and for generic κ for N = 1. Thanks to the S-duality, we can flip the diagram, obtaining the picture as in FIGURE 3. Finally, the trace in the horizontal direction can be calculated by the standard technique, thereby giving the elliptic lift f ellip N of the asymptotically free eigenfunction f gl N for the Macdonald q-difference operator. In Section 3 are given our proofs of Theorems 3.30 and 3.41 which go along this idea. Some explanations are in order, concerning the physical background and recent related works on the non-stationary systems, including the non-stationary Heun, Lamé, and elliptic Calogero-Sutherland equations. These non-stationary equations have been extensively studied based on the perturbative approach by Atai and Langmann in [10]. Recently in [11], they obtained an integral formula for the eigenfunctions for the non-stationary elliptic Calogero-Sutherland equation for some special choices of the parameter in the "time derivative" term, using the kernel function technique.

S-duality
Note that no explicit equations have been obtained for the non-stationary Ruijsenaars function f gl N unfortunately. So the authors have been lead to use the representation theories of the DIM algebra, to bypass the troublesome situation without any eigenvalue equations or associated kernel functions. One may find, nevertheless, the resulting formula in Theorem 3.30 can be regarded as a q-difference analogue of Atai and Langmann's integral formula in [11], suggesting the existence of q-analogue of the kernel functions.
In this occasion, we address some other related problems in the representation theory of the DIM algebra. First, we derive an integral operator I(s 1 /s 0 , . . . , s N /s 0 ) introduced in [12,13,14] from the intertwining operators of the DIM algebra. As for the definition of I(s 1 /s 0 , . . . , s N /s 0 ), see Definition 4.5. In [12,13,14], it was conjectured that the integral operator I(s 1 /s 0 , . . . , s N /s 0 ) commutes with Macdonald's difference operator. A proof has been given in [5]. We provide yet another perspective based on the vertex operator. We also discuss the elliptic analogue of the integral operator. Our elliptic integral operator can be constructed by taking the loop of intertwining operators. It commutes with the Ruijsenaars operator.
Secondly, we study the conformal limit q → 1 of the 2N -valent intertwining operator V(z), which is a main object in the authors' previous paper [7], restricting ourselves to the simplest nontrivial case N = 2. The operator V(z) is defined by the relations with the q-W N algebra, the q-Vir algebra for N = 2. We will derive the well-known relation of the Virasoro primary fields from those defining relations in the limit q → 1. In [7], a formula for the matrix elements of V(z) with respect to generalized Macdonald functions (Fact 2.21) are proved. The result Theorem 5.25 and the matrix elements formula for V(z) prove that the generalized Macdonald functions are reduced to Alba, Fateev Litvinov and Tarnopolskii's basis [15]. Hence, the matrix element formula for V(z) provides us with another proof of the 4-dimensional AGT correspondence [16].
This paper is organized as follows. Basic facts with respect to the intertwining operators of the DIM algebra are summarized in Section 2. In Section 3, we reproduce the non-stationary Ruijsenaars functions from the intertwiners on the cylindric web diagrams. We also prove Theorem 3.30 by using the S-duality. In Section 4, we derive the integral operator I(s 1 /s 0 , . . . , s N /s 0 ) and prove the commutativity with the Macdonald q-difference operator. The elliptic extension of the integral operator is also discussed. We take the conformal limit q → 1 in Section 5. In Appendix A, some facts with respect to Macdonald functions f gl N are explained. In Appendix B, we describe the construction of the non-stationary Ruijsenaars functions from the affine screening operators in the case of general κ. In Appendix C are given some straightforward but cumbersome details needed for our proofs.

DIM algebra, intertwiners, and Mukadé operators
We briefly recall the definition of the Ding-Iohara-Miki (DIM) algebra [18,19], its intertwining operators and the S-duality formula for the intertwining operators. Let q and t be generic complex parameters with |q|, |t −1 | < 1.
Definition 2.1. The DIM algebra, which we denote by U = U q,t , is a unital associative algebra generated by the currents x ± (z) = n∈Z x ± n z −n , ψ ± (z) = ±n∈Z ≥0 ψ ± n z −n and the central elements c ±1/2 . The defining relations are 18]). The Drinfeld coproduct gives rise to a bialgebra structure. Further, U has a Hopf algebra structure. We omit the counit and the antipode.
A U -module is called of level-(n, m) if the central elements act as c = (t/q) n/2 and (ψ + 0 /ψ − 0 ) 1/2 = (q/t) m/2 . In this paper, we use two kinds of U -modules. The first one is a free field representation with the following boson. Let H be the Heisenberg algebra generated by {a n |n ∈ Z} with the commutation relation [a n , a m ] = n 1 − q |n| 1 − t |n| δ n+m,0 . (2.11) Let |0 and 0| be the highest weight vectors defined by a n |0 = 0 (n ≥ 0) and 0| a n = 0 (n ≤ 0), respectively. Denote by F (resp. F * ) the Fock space generated from the highest weight vector |0 (resp. 0|). The bilinear form F * ⊗ F → C is defined by setting 0|0 = 1.
1 − t n n a n z −n , (2.12) 1 − t n n q −n/2 t n/2 a n z −n , (2.13) (1 − t n q −n )q n/4 t −n/4 a n z −n , (2.14) 20]). Let u be an indeterminate and M be an integer. The algebra homomorphism ρ u : U → End(F) defined by We denote by F (1,M ) u the Fock space endowed with the level (1, M )-module structure. The dual space F * can also be endowed with the right U -module structure through ρ u . Then it is denoted by F (1,M ) * u . The ρ u is called the horizontal representation. Next, we consider the level (0, 1)-module. Let F (0,1) be the vector space spanned by the vectors |λ with λ ∈ P. Define ( λ| |λ ∈ P) to be the dual basis such that λ|µ = δ λ,µ .
Fact 2.5 ( [21,22]). Let u be an indeterminate. The following action gives the level (0, 1)-module structure to F (0,1) : Here, We denote this module by F (0,1) u . This is called the vertical representation or the preferred direction. By using the two representations, the trivalent intertwiners Φ, Φ * of the DIM algebra were introduced in [9]. Similarly, there exists a unique linear operator such that ( 0| ⊗ ∅|)Φ * |0 = 1 and It is known that these intertwining operators can be realized as follows.
The symbol : · · · : means the usual normal ordering product. Similarly, Φ * λ is of the form 1 n 1 1 − q n q −n/2 t n/2 a −n u n exp ∞ n=1 1 n q n 1 − q n q −n/2 t n/2 a n u −n , (2.45) ξ λ (u) = : Throughout the paper, the case M = 0 is concerned. The intertwining operators Φ and Φ * are expressed as the trivalent diagrams in FIGURE 4. It is known that their matrix elements coincide with the Iqbal, Kozcaz and Vafa's or Awata and Kanno's refined topological vertecies [9,23,24], and the vertical representation corresponds to the preferred direction. We prepare the formula for the normal ordering of the intertwiners, in which the Nekrasov factor appears.
(2.47) Fact 2.11 ([9]). Put γ = (t/q) 1/2 . We have Furthermore, we introduce the following operators for convenience, which is expressed by the cross diagram in FIGURE 5. Let N ∈ Z ≥1 . The main objects in the previous paper [7] are the following 2N -valent intertwiners, which we called the Mukadé operators after the shape of diagrams obtained by connecting the trivalent diagrams. ("Mukade" means centipedes in Japanese. See FIGURE 6.) as the vacuum expectation value 0| · · · |0 of the operator (2.54) with respect to the level (1, 0) representation. Furthermore, we define the normalized operator Here, we have set |∅ = |∅ ⊗ · · · ⊗ |∅ .
Definition 2.14. Define the operators . (2.59) We prepare some notations to treat the N -fold Fock tensor spaces.
In [7], the S-duality formula for the matrix elements of T V and T H is proved. Recall that the matrix elements of T H with respect to the basis (|λ (1) ⊗ · · · ⊗ |λ (N ) ) can be easily calculated by operator products. On the other hand, the basis on F (N,0) u which corresponds to |λ (1) ⊗ · · · ⊗ |λ (N ) is defied as the eigenfunctions of the operator X (1) 0 given as follows. (2.66) Here, Let Λ be the ring of symmetric functions, and p n be the power sum symmetric function of degree n. Then the map gives the isomorphism as graded vector spaces between F and Λ. If N = 1, the operator X (1) 0 is essentially the same as Macdonald's difference operator under this isomorphism [25]. Therefore, its eigenfunctions can be identified with the ordinary Macdonald functions. In the case of general N , the eigenfunctions of X (1) 0 can be viewed as a generalization of Macdonald functions. Their existence theorem is given in terms of the following generalized dominance partial ordering.
Definition 2.19. We write λ ≥ L µ (resp. λ ≥ R µ) if and only if |λ| = |µ| and Let us prepare the notation for the vectors corresponding to the monomial symmetric functions.
. .] be the element in the Heisenberg algebra H such that m λ (a −n ) |0 coincides with the monomial symmetric function under the identification (2.70). m λ (a −n ) is the abbreviation for m λ (a −1 , a −2 , . . .). Note that we often substitute a n or another boson for a −n .
We state the existence theorem of the generalized Macdonald functions.
Similarly, there exists a unique vector The eigenvalues ǫ λ and ǫ * λ are of the forms 26]). It follows that The following is the S-duality formula for changing the preferred directions. See also FIGURE 6.
Theorem 2.24 is essentially proved in [7]. See Appendix C.2 as to the appearance of the factor (−1) |λ|+|µ| . For the explicit form of the matrix elements, see Fact 3.35.

Proofs of Main Theorems
3.1. Non-stationary Ruijsenaars Functions and Intertwining Operators. In [1], an operator formula is given for the non-stationary Ruijsenaars functions by using the affine screening currents [28,29,30]. In this subsection, we show that the affine screening currents can be reproduced from the intertwiners of the DIM algebra in the special case of κ = t −1 , giving an expression of the nonstationary Ruijsenaars functions in terms of the Mukadé operators. To help the interested readers, the operator product formulas for the affine screenings given in [1] are reproduced in Appendix B. Figure 6. The S-duality formula.
These operators A(z) and A * (z) appear in the following decomposition of the intertwiners Φ λ (z) and Φ * λ (z). Proposition 3.2. For a partition λ = (m 1 , . . . , m l ), Let N ≥ 2 in this subsection. The case N = 1 will be considered in Subsection 3.4. Define the screening currents as follows.
Here, α 0 , α 1 , . . . , α N −1 are regarded as the classical part of the real simple roots of the affine Lie algebra gl N .
Remark 3.5. Note that these screening currents essentially coincide with those in [28,29,30] when Proposition 3.6. We have (3.10) and for N ≥ 3, 3 we obtain Let us introduce the following vertex operator. 4 These screening currents and φ 0 (z) can be obtained by a specialization of the Mukadé operators. Firstly, we consider the non-affine case and derive the Macdonald functions from specialized Mukadé operators to fix our starting point for making the p-traces (FIGURE 1). .

(3.27)
When we construct T V , we need to compose many Φ cr. 's producing a big summation running over the set of the partitions in P N −1 . By giving a certain condition to the spectral parameters attached to the internal edges, we have the "restricted operator" T V i (z). Then, one finds that all the internal partitions are allowed to run over the one row diagrams satisfying certain interlacing conditions among them.
Fact 3.11 (Appendix A in [7]). We have We call T V i the "screened vertex operator". From these screened vertex operators, we can construct the Macdonald functions.
It is known that f gl N (x; s|q, t) is an eigenfunction of Macdonald's difference operator [4,5,6]. For some basic facts about f gl N (x; s|q, t), see Appendix A. This function can be reproduced as follows.
Fact 3.13 (Appendix A of [7]). It follows that In Appendix A in [7], (3.31) was proved up to proportionality. We can easily calculate the proportional constant by taking the constant term of s i 's and using q-binomial theorem.
Remark 3.14. By using the bispectral duality proved in [5], the right hand side in (3.31) can be rewritten as We can also obtain this equation by applying the S-duality formula for the intertwiners (Theorem 2.24) to the left hand side in (3.31) and using Fact 3.13 again.
The formula (3.31) should be understood as the equation as formal power series in s i+1 /s i and x i+1 /x i (i = 1, . . . , N −1). By Fact A.6, we can also treat the variables x i and s i as complex numbers. We will give an affine analogue of the above facts. Since analyticity of the non-stationary Ruijsenaars functions has not been clarified, we treat x i 's and s i 's as indeterminates in the affine case.
Let p be an indeterminate, and consider the following "loop operator" obtained by the loop of the Mukadé operator. (See FIGURE 7.) Definition 3.16. Set the shifted screening currents The screening currents S i (z) are the realization of the operator in Appendix B in the case of κ = t −1 . In Fact 3.11, we expressed T V i by composition of screening currents S (i) (z) and φ 0 (z). In the affine case, we compose the screening currents as follow.
The operator T loop i can be expressed as follows. This is an affine analogue of Fact 3.11.
For the proof, we prepare two lemmas.
The proof is given in Section C.1.
Proof of Proposition 3.18. First, the Nekrasov factors N k=1 N µ (k−1) ,µ (k) (t δ k,i ) arise from the normal ordering product of the operator 1≤k≤N Φ cr. x k ; (See Fact 2.11). In general, for partitions ν and ρ, we N ν,ρ (1) = 0 (resp. N ν,ρ (t) = 0) if and only if ν ⊂ ρ (resp.ν ⊂ ρ). Here, we putν = (ν 2 , ν 3 . . .) for ν = (ν 1 , ν 2 , ν 3 . . .). Thus the partitions µ (k) in (3.33) are restricted by the cyclic interlacing conditions Therefore, the µ (k) 's can be expressed by the single partition (3.45) By using this λ, the partitions µ (k) 's can be written as Recalling Proposition 3.2 and Definition 3.4, we have 1≤k≤N Φ cr. x k ; For a ∈ Q, ⌊a⌋ is defined to be the integer n satisfying n ≤ a < n + 1. Lemma 3.19 and the equation (3.49) Furthermore, by using the shifted screening current S k (z)'s, the operator part in (3.47) can be rewritten as This proposition says that the vertex operators T loop i (x, p; s) can be identified with the screened vertex operators which are used to construct the non-stationary Ruijsenaars function in [1], though in our case, κ should be specialized to t −1/N . (See Appendix B.) This motivates us to state the affine analogue of Fact 3.13, that is, to construct the non-stationary Ruijsenaars function as the matrix element of the composition of T loop i (x, p; s)'s. In order to state the claim, we introduce the non-stationary Ruijsenaars function.
We cyclically identify x i+N = x i and put Then, we obtain the following theorem. (See also FIGURE 8.) Theorem 3. 22. Let (3.53) Then we obtain Here, we set Proof. Proposition 3.18 gives (3.60) Therefore, Fact B.4 shows that (3.57) is equal to Finally, it can be easily shown that the non-stationary Ruijsenaars function in (3.61) coincides with Remark 3.23. The LHS in this theorem can be rewritten by the trace of the operators T H . For an operator A ∈ End((F (0,1) ) ⊗N ), set the formal power series Then it is clear that Here, It is clear that the function f ellip N (x, s|q, t, p) is reduced to the ordinary Macdonald function, i.e., We give a realization of this elliptic lift by taking the trace of the Mukadé operators.
Note that the trace tr p d − certainly does not depend on bases.
Our main purpose in this subsection is to compute the trace of the following operator. See also FIGURE 9. We state the key property of the p-trace.

72)
where the product with respect to l can be either finite or infinite if it converges. Then, it follows that In particular, if the operators satisfy then we can rewrite the result by the elliptic gamma functions: Proof. Let A ± i (z) be the operators of the forms Since tr(p d · 1) = 1 (p;p) N ∞ , by repeating the calculation above, it can shown that for any m ∈ Z >0 , This completes the proof.

92)
where we put (3.93) x ′ and s ′ are the same ones in Theorem 3.22: For the proof, we prepare the following lemma.
Proof. The first vacuum expectation value can be directly calculated as The second one can be calculated by using the q-binomial theorem: Proof of Theorem 3.30. In this proof, we use the same notation as in (3.70). For the sketch of the proof, see FIGURE 3 in Introduction. By virtue of Theorem 2.24, we have where µ i = (µ (i,1) , . . . , µ (i,N ) ) ∈ P N and µ 0 = µ N . It is clear that Therefore, it follows that As a result, by Theorem 3.22 (see also Remark 3.23), we obtain

Another Expression.
In the previous subsection, we have established the relationship between f gl N and f ellip by taking traces of intertwiners. By changing the computation method to take the trace, another expression can be obtained. That is, we use the generalized Macdonald functions as a basis. We first fix the normalization of the generalized Macdonald functions |P λ , which simplifies the matrix elements of the Mukadé operators.
This normalization is based on our yet unfinished study of Conjecture 3.38 in [7]. Note, however, that we do not need the conjecture itself here.

Fact 3.35 ([7]). We have
Here By this matrix element formula, the trace of T N (u; x) can be calculated as follows.
Proposition 3.36. It follows that Proof. By Fact 3.35 and Lemma 3.31, we have Furthermore, it can be shown that Thus, we have (3.114).

3.4.
Case N = 1. In this subsection, we treat the case N = 1. This case is special in the sense that the κ parameter is not specialized. This is because in this case, the ratio of spectral parameters v/u is the free parameter, and it becomes the κ parameter.
Proof. We note the formula for the normal ordering: By Lemma 3.28, we can show that the given trace is Next, we make the loop in the vertical direction. We obtain the following lemma. and take the normal ordering (Fact 2.11).
Combining these two lemmas results in the following summation formula.

(3.127)
This gives the proof of the conjecture in [31], which claims the two different forms of the mixed Hodge polynomials of certain twisted GL(n, C)-character varieties of Riemann surfaces with g = 1.
The similar proof is given in [2,3]. Physically, this relates the partition function of the 5d N = 1 * U (1) gauge theory to that of the 6d theory with one tensor multiplet.

Integral Operator of Macdonald Functions.
We return to the non-affine case with N ≥ 2. In Fact 3.13, the ordinary Macdonald functions were constructed from the screened vertex operators. In this section, an integral operator introduced in [12,13,14] will be constructed from them. We treat the spectral parameters s = (s 1 . . . , s N ) as generic complex variables in this section. First, we rewrite the screened vertex operators (non-affine case) by the contour integrals.
Proof. It follows from the operator product formulas (Proposition 3.9) and Ramanujan's 1 ψ 1 summation formula ((5.2.1) in [17]): Proof. First, we adjust the contour of the integration to the condition |q| < |y i−1 /y i | < 1 (i = 1, . . . , k). Here, we put y 0 = x. Note that no pole affects this change. Then we have (4.7) By the deformation of the formal series The deformation (4.8) itself is not well-defined because m≥0 (q n s k+1 s i+1 ) m does not converge for arbitrary n ∈ Z. However, considering the matrix elements of the operators, we can justify the calculation (4.9). For more detail, see Remark A.2 in [7]. Fact 3.13 can be rewritten as follows. 6 This screened vertex operator corresponds to Φ (k) (x) in [7] after transformation x → t −1 x and yi → (q/t) i t −1 yi and modification of the integration contour. Actually, a more strict condition is imposed for integration contour in [7] in order to show that the screening currents commute with X (r) (z) (r = 1, . . . , N ). However, only commutativity with X (1) (z) is required to show Fact 4.4. Hence we adopt this integration contour in this paper.

Fact 4.4 ([7], Theorem 3.26). It follows that
(4.10) We introduce the following integral operator, which is essentially the same as the one in [14].

Remark 4.6.
In what follows, we assume |t −1 | < |q| so that the integration contour is well-defined.
Consider the N + 1 fold Fock tensor spaces   Proof. In this proof, we put s i,j := γ −i+1 t −δ i>j s j . Here δ a>b is 1 if a > b or 0 if a ≤ b. By taking the normal ordering, we have (ty j,1 /y i,2 ; q) ∞ (y j,1 /y i,2 ; q) ∞ reproduce Macdoanld functions construct the integral operator Figure 10. Operators in Proposition 4.7. .
We prove the commutativity between the integral operator I and Macdonald's difference operator. For the proof, we need to take care of the analyticity of the domain. Hence, let us define the following region. so that Here, D x (s; q, t) is the Macdonald q-difference operator: In the proof, we use the following fact.
Here, we put Proof of Proposition 4.9. A direct calculation gives and By these equations and Fact 4.10, we can show that for a function f (x 1 , . . . , x N ) ∈ O(U N |t −1 | ), D x (s; q, q/t)I(s 1 /s 0 , . . . , s N /s 0 )f (x 1 , . . . , x N ) We have poles in y k of the each term containing the difference operator T q −1 ,y k at y k = q a x k , y k = q −a tx k , y k = q a+1 x j , y k = q −a tx i (a = 0, 1, 2, . . ., i < k < j). Therefore, by the change of variable y k → qy k in the each term (note that there is no pole between y k and qy k ), we can show that (4.28) is equal to (1 − y j /y i ) · D y (s 1 , . . . , s N |q, q/t)f (y 1 , . . . y N ). (4.29) This complete the proof. Proof. Since by fixing s, the Macdonald function f gl N (x; s|q, q/t) is in O(U N |t −1 | ) (Fact A.6), we can use Theorem 4.9. Moreover, by the uniqueness of Fact A.2, we can show that f gl N (x; s|q, q/t) is an eigenfunction of I(s 1 /s 0 , . . . , s N /s 0 ): (4.31) The expansion (Proposition 4.2) makes it clear that the constant term of the LHS is 1. Hence, the eigenvalue is 1.

4.2.
Integral Operator in Elliptic Case. In this subsection, we give brief discussion on the elliptic lift of the integral operator. Namely, consider the trace at the horizontal representation of the operator in Proposition 4.7. Then we can derive the following integral operator.
θ p (y j /y i ) · f (y 1 , . . . , y N ). Γ(qy j /tx i ; q, p) Γ(qy j /x i ; q, p) . (4.33) The following trace can be viewed as the action of I ellip on the non-stationary Ruijsenaars functions. where we used the same notation in Theorem 3.22.
Proof. This can be proved similarly to the one of Proposition 4.7. By Lemma 3. (ts j /s i ; q) ∞ (qs j /s i ; q) ∞ · f ellip N (s; x|q, t, p) .
Applying Theorem 3.30, we obtain the claim.
We can obtain the commutativity between the integral operator I ellip and the Ruijsenaars operator. Here, Since the proof is quite similar to that of Theorem 4.9, we omit it. Concerning the equation with respect to the kernel function, we can use the following fact.  4.15 ([33, 34]). It follows that (4.38) Here, we put In this subsection, we have derived the integral operator I ellip and given commutativity with the Ruijsenaars operator. Unfortunately, the non-stationary Ruijsenaars functions f gl N are not the eigenfunctions of the Ruijsenaars operator. So, neither for I ellip . It is left to a future study to find an operator whose eigenfunctions are f gl N .

5.
Conformal Limit q → 1 5.1. Preparation. In this section, we will derive the relation of the Virasoro algebra and the primary field from the relation of the q-Virasoro algebra and the Mukadé operator T V . Firstly, we define the following algebra.  26]). The operator X (k) (z) is of the form Here, Λ (j) (z) is defined in Definition 2.17.
It is known that the operator V(x) exists uniquely [7]. Moreover, their matrix element formula is proved.
Fact 5.6 ( [7]). It follows that Thus, the V(x) can be realized by T V with some modifications to spectral parameters.

Conformal Limit
and we parametrize v ′ i and u ′ i by the parameters P ′ , P and α such that Here, we set P 1 = −P 2 = P, P ′ 1 = −P ′ 2 = P ′ . Write Q := b + b −1 .
Even if we add an arbitrary parameter to (5.12) and (5.13) to keep the degree of freedom of u ′ i and v ′ i , the results stay the same.
It is easy to show the following lemma.
Lemma 5.9. If (5.10) is satisfied, it follows that 14) The parametrization above is designed so that the following factor appears in the limit of the Nekrasov factor.
Proof. Follows by direct calculation.
Let us define bosons independent of .
Definition 5.12. Define the Heisenberg algebra a n (n ∈ Z =0 ) by the commutation relation [a n , a m ] = −b −2 nδ n+m,0 , (5.19) and we assume that It is known that the generalized Macdonald functions are reduced to the generalized Jack functions [37] in the limit → 0. They are defined as eigenfunctions of the following Hamiltonian. The existence theorem also follows similarly to the generalized Macdonald functions.
is a modified Hamiltonian of the Calogero-Sutherland model: Fact 5.14 ( [38]). There exists a unique function |J λ = |J λ (u ′ 1 , u ′ 2 ) satisfying the following two conditions: Similarly, there exists a unique function J λ | = J λ (u ′ 1 , u ′ 2 )| such that We call the eigenfunctions |J λ and J λ | the generalized Jack functions.
In fact, the generalized Jack functions are defined for general N . Also for general N , they correspond to the limit of the generalized Macdonald function. Next, we take the limit of the generator X (i) (z). In advance, we decompose the generator X (i) (z) into the q-Virasoro algebra and some Heisenberg algebra in order to obtain the relation of the Virasoro Primary fields. This decomposition can be obtained by the following linear transformation of the bosons.
Furthermore, for n ∈ Z =0 , define Proposition 5.18. For n, m ∈ Z, it follows that Proof. It follows from direct computation. Moreover, define Proposition 5.20. X (1) (z) can be decomposed as Moreover we have Proposition 5.21. The operator T (z) satisfies the defining relation of the q-Virasoro algebra: These proposition can be shown by direct calculation. (See also [35]). For representation theory of the q-Virasoro algebra, we refer the reader to [39]. Now, we consider the limit of these generators.
If P, P ′ and b are generic, the following vectors form a basis on F −µ 2 · · · |0 , λ, µ ∈ P, (5.49) Hence, we can identify F (2,0) u with the tensor product of the Verma module of the Virasoro algebra L n and the Fock space of the Heisenberg algebra β (2) n . Further, |0 can be regarded as the tensor product of the highest wight vector of the highest weight Q 2 4 − P 2 and the vacuum state of the Fock space. For simplicity, we hereafter assume that P, P ′ and b are generic so that the modules are irreducible.
By Fact 5.6, Proposition 5.11 and Fact 5.15, we can assume the following expansion. Namely, there is no pole at = 0.
We obtain the result that the operator Φ H.V. (z) corresponds to the Virasoro primary fields. Moreover, we obtain The proof is given in Section 5.3. Let us also state the following proposition.
Proposition 5.26. We have where we put Proof. By Proposition 5.29, we can show that for k > 0, m∈Z : β By using these commutation relations, we can prove (5.78) at each coefficient of z −k .
By the above lemmas, we can obtain the following relation between the Virasoro algebra L n and V 0 (w).
Proposition 5.33. For n ∈ Z, we have (5.82) Proof. By Lemma 5.31 and Lemma 5.32, we have By taking the coefficient of z −n , we get (5.82).
We have proved the relations among V 0 (w), the Heisenberg algebra β (2) n and the Virasoro algebra L n . Conversely, we can show that an operator satisfying these relations is unique up to the vacuum expectation value.
By the uniqueness of Lemma 5.34, we can prove Theorem 5.25.
Proof of Theorem 5.25. Firstly, we prepare notations of the Verma modules and the Fock space explained in Remark 5.23. Let M h 1 (resp. M h 2 ) be the Verma module of the Virasoro algebra with the highest weight vector |h 1 (resp. |h 2 ) of highest weight h 1 = Q 2 4 − P 2 (resp. h 2 = Q 2 4 − P ′2 ). Let F u be the Fock space of the Heisenberg algebra β as representation spaces of the algebra L n ⊗ β (2) n . Further, let h 2 | be the dual vector such that h 2 | L 0 = h 2 h 2 |, h 2 | L −n = 0 (n > 0), and h 2 |h 2 = 1.
Here, we put Note that there is a typo in [1], and the corresponding formula Theorem 2.13 in [1] should be read as in Fact B.4.
Appendix C. Some Combinatorial Formulas C.1. Proof of Lemma 3.19. By using factorials, we can rewrite the Nekrasov factors as follows.