Commutative Families of the Elliptic Macdonald Operator

In the paper [J. Math. Phys. 50 (2009), 095215, 42 pages, arXiv:0904.2291], Feigin, Hashizume, Hoshino, Shiraishi, and Yanagida constructed two families of commuting operators which contain the Macdonald operator (commutative families of the Macdonald operator). They used the Ding-Iohara-Miki algebra and the trigonometric Feigin-Odesskii algebra. In the previous paper [arXiv:1301.4912], the present author constructed the elliptic Ding-Iohara-Miki algebra and the free field realization of the elliptic Macdonald operator. In this paper, we show that by using the elliptic Ding-Iohara-Miki algebra and the elliptic Feigin-Odesskii algebra, we can construct commutative families of the elliptic Macdonald operator. In Appendix, we will show a relation between the elliptic Macdonald operator and its kernel function by the free field realization.

Notations. In this paper, we use the following symbols.
Organization of this paper.
In section 1, we review the trigonometric case treated in the paper [1]. In section 2, first we recall related materials of the elliptic Ding-Iohara algebra and the free field realization of the elliptic Macdonald operator. Second we show that using the elliptic Ding-Iohara algebra and the elliptic Feigin-Odesskii algebra, we can obtain commutative families of the elliptic Macdonald operator.
In Appendix, by the free field realization of the elliptic Macdonald operator, we show a functional equation of the elliptic kernel function.

Trigonometric case
In this section, we review the construction of the commutative families of the Macdonald operator by Feigin, Hashizume, Hoshino, Shiraishi, Yanagida [1].
Set the q-shift operator as T q,x f (x) := f (qx). We define the Macdonald operator H N (q, t) (N ∈ Z >0 ) as In the following [f (z)] 1 denotes the constant term of f (z) in z.
We also have the dual version of the proposition 1.3. Let 0| be the dual vacuum vector which satisfies the condition 0|a n = 0 (n < 0) and define the dual boson Fock space F * as a right B module : F * := span{ 0|a λ : λ ∈ P} (a λ := a λ 1 · · · a ℓ(λ) ).
Then the kernel function Π(q, t)(

Trigonometric Feigin-Odesskii algebra A
In this subsection, we review basic facts of the trigonometric Feigin-Odesskii algebra [1].
Remark 1.7. The definition of the trigonometric Feigin-Odesskii algebra A above is a reduced version of the paper [1]. For instance, there would be a question why the function ε n (q; x) appears. For more detail of the trigonometric Feigin-Odesskii algebra A, see [1].
In the paper [1], the following fact is shown. (1.17) Here [f (z 1 , · · · , z n )] 1 denotes the constant term of f (z 1 , · · · , z n ) in z 1 , · · · , z n , and we extend the map O linearly. Proof . To prove the proposition, for f ∈ A m and g ∈ A n , we show Then from the relation we have the following : The relation shows that the operator-valued function This follows from the fact that the constant term is invariant under the action of the symmetric group. In addition for a symmetric function f (x 1 , · · · , x N ), we have Hence we have From them we have the following.
The trigonometric Feigin-Odesskii algebra A is commutative by means of the star product * , therefore we have the following corollary.
Let V be a C-vector space and T : V → V be a C-linear operator. Then for a subset W ⊂ V , the symbol T | W denotes the restriction of T on W . For a subset M ⊂ End C (V ), we use the symbol M| W := {T | W : T ∈ M} (W ⊂ V ). (2) The space M| Cφ N (x)|0 consists of commuting q-difference operators which contains the Macdonald operator H N (q, t) (commutative family of the Macdonald operator H N (q, t)).
Proof . (1) This statement follows from the commutativity of A and the compatibility of the star product * and the map O.
(2) Due to the free field realization of the Macdonald operator H N (q, t), the operator and (1) in the corollary 1.11, the restriction M| Cφ N (x)|0 is a space of commuting qdifference operators which contains the Macdonald operator H N (q, t).
The Macdonald operator H N (q −1 , t −1 ) is reproduced from the operator ξ(z). By this fact, we can construct another commutative family of the Macdonald operator.
We extend the map O ′ linearly.
Proof . From the relation ω(x, y) = ω ′ (y, x), for f ∈ A m , g ∈ A n we have the following.
Since the relation holds for any f ∈ A m , g ∈ A n , we have * ′ = * .
We can check the map O ′ and the star product * ′ are compatible in the similar way of the proof of the proposition 1.10. Furthermore by the lemma 1.13 as * ′ = * , we have the following corollary. (2) The space M ′ | Cφ N (x)|0 consists of commuting q-difference operators which contains the Macdonald operator H N (q −1 , t −1 ) (commutative family of the Macdonald operator H N (q −1 , t −1 )).  Proof . This proposition follows from the existence of the Macdonald symmetric functions. That is, elements of the commutative families are simultaneously diagonalized by the Macdonald symmetric functions.
From the proposition 1.15, commutative families

Elliptic case
In this section, we are going to construct a commutative family of the elliptic Macdonald operator by using the elliptic Ding-Iohara algebra and the elliptic Feigin-Odesskii algebra.

Elliptic Ding-Iohara algebra U (q, t, p)
The elliptic Ding-Iohara algebra is an elliptic analog of the Ding-Iohara algebra introduced by the author [3]. First we recall the definition of the elliptic Ding-Iohara algebra and its free field realization.
Definition 2.1 (Elliptic Ding-Iohara algebra U(q, t, p)). Set the structure function g p (x) as Let x ± (p; z) := n∈Z x ± n (p)z −n , ψ ± (p; z) := n∈Z ψ ± n (p)z −n be currents and γ be central, invertible element satisfying the following relations : (2.1) Here we define the delta function as δ(x) := n∈Z x n . We define the elliptic Ding-Iohara algebra U(q, t, p) to be an associative C-algebra generated by {x ± n (p)} n∈Z , {ψ ± n (p)} n∈Z and γ.
Theorem 2.2 (Free field realization of the elliptic Ding-Iohara algebra U(q, t, p)). Set an algebra of boson B a,b generated by {a n } n∈Z\{0} , {b n } n∈Z\{0} and the following relations : [a m , a n ] = m(1 − p |m| ) Let |0 be the vacuum vector which satisfies the condition a n |0 = b n |0 = 0 (n > 0) and set the boson Fock space F as a left B a,b module.
The elliptic Macdonald operator H N (q, t, p) (N ∈ Z >0 ) is defined as follows.
By the operators η(p; z), ξ(p; z) which are in the theorem 2.2, we can reproduce the elliptic Macdonald operator as follows [3].
We use the symbol φ N (p; x) := N j=1 φ(p; x j ). (1) The elliptic Macdonald operator H N (q, t, p) is reproduced by the operator η(p; z) as follows.
The operators E(p; z), F (p; z) reproduce the elliptic Macdonald operators H N (q, t, p), H N (q −1 , t −1 , p) as follows. (2.14) ) be the kernel function of the elliptic Macdonald operator defined as Then the kernel function Π(q, t, p)

Elliptic Feigin-Odesskii algebra A(p)
The elliptic Feigin-Odesskii algebra is defined in the similar way of the trigonometric case except the emergence of elliptic functions [1].
Similar to the trigonometric case, the following is shown [1].

Commutative families M(p), M ′ (p)
For the operators E(p; z), F (p; z) which are used in the theorem 2.4, we have the following proposition [3].
Proposition 2.9. (1) Operators E(p; z), F (p; z) satisfy the relation as Due to the relations operator-valued functions as are symmetric in x 1 , · · · , x N .
In the similar way of the trigonometric case, we can check the following.
Proposition 2.11. The map O p and the star product * are compatible : for f, g ∈ A(p),  A(p)). The space is a commutative algebra of boson operators.
(2) The space M(p)| Cφ N (p;x)|N consists of commuting elliptic q-difference operators which contains the elliptic Macdonald operator H N (q, t, p) (commutative family of the elliptic Macdonald operator H N (q, t, p)).
A commutative family of the elliptic Macdonald operator H N (q −1 , t −1 , p) is also constructed as follows.
We define a linear map O ′ p : A(p) → End(F α ) (α ∈ C) as follows.
We extend the map linearly.
In the same way of the trigonometric case, we have the following lemma.
Lemma 2.14. Set another star product * ′ as

(2.25)
In the elliptic Feigin-Odesskii algebra A(p), we have * ′ = * . The theorem 2.16 is the elliptic analog of the proposition 1.15. But we can't prove the theorem 2.16 in the similar way of the proof of the proposition 1.15, because we don't have an elliptic analog of the Macdonald symmetric functions. Hence we will show the theorem 2.16 in a direct way. For the proof we prepare the following lemma.
By the definition of O p , O ′ p , operators O p (ε r (q, p; z)), O ′ p (ε s (q, p; w)) are the constant terms of the following operators.
Then their functional parts take the following forms.
and its kernel function Then we have the following functional equation.
Here we denote the Macdonald operator which acts on functions of x 1 , · · · , x M by H M (q, t) x .
In the following, we will show the elliptic analog of the theorem 3.1 by the free field realization of the elliptic Macdonald operator.

Recollection : free field realization of the elliptic Macdonald operator
The elliptic Macdonald operator H N (q, t, p) (N ∈ Z >0 ) is defined as First we review the free field realization of the elliptic Macdonald operator. In the following we use the notations in section 2.1.
We use the symbol φ N (p; x) := N j=1 φ(p; x j ). The elliptic Macdonald operator H N (q, t, p) is reproduced by the operator η(p; z) as follows. (1 − t n )(qt −1 p) n (1 − q n )(1 − p n ) b n z −n n exp n>0 1 − t n (1 − q n )(1 − p n ) a n z n n . (3.7) We use the symbol φ * N (p; x) := N j=1 φ * (p; x j ). The elliptic Macdonald operator H N (q, t, p) is reproduced by the operator η(p; z) as follows.

Elliptic kernel function and its functional equation
For a partition λ, set n λ (a) := ♯{i : λ i = a}, z λ := a≥1 a n λ (a) n λ (a)! and define z λ (q, t, p), z λ (q, t, p) by We define a bilinear form •|• : F * × F → C by the following conditions.
Hence by taking the limit p → 0 the equation (3.11) reduces to the equation (3.3).