Commutative Families of the Elliptic Macdonald Operator
In the paper [J. Math. Phys. 50 (2009), 095215, 42 pages], 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 trigono...
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| Цитувати: | Commutative Families of the Elliptic Macdonald Operator / Y. Saito // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-1468332019-02-12T01:25:29Z Commutative Families of the Elliptic Macdonald Operator Saito, Y. In the paper [J. Math. Phys. 50 (2009), 095215, 42 pages], 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. 2014 Article Commutative Families of the Elliptic Macdonald Operator / Y. Saito // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 8 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B37; 33D52 DOI:10.3842/SIGMA.2014.021 http://dspace.nbuv.gov.ua/handle/123456789/146833 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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In the paper [J. Math. Phys. 50 (2009), 095215, 42 pages], 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. |
| format |
Article |
| author |
Saito, Y. |
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Saito, Y. Commutative Families of the Elliptic Macdonald Operator Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Saito, Y. |
| author_sort |
Saito, Y. |
| title |
Commutative Families of the Elliptic Macdonald Operator |
| title_short |
Commutative Families of the Elliptic Macdonald Operator |
| title_full |
Commutative Families of the Elliptic Macdonald Operator |
| title_fullStr |
Commutative Families of the Elliptic Macdonald Operator |
| title_full_unstemmed |
Commutative Families of the Elliptic Macdonald Operator |
| title_sort |
commutative families of the elliptic macdonald operator |
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Інститут математики НАН України |
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2014 |
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http://dspace.nbuv.gov.ua/handle/123456789/146833 |
| citation_txt |
Commutative Families of the Elliptic Macdonald Operator / Y. Saito // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 8 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT saitoy commutativefamiliesoftheellipticmacdonaldoperator |
| first_indexed |
2025-07-11T00:43:35Z |
| last_indexed |
2025-07-11T00:43:35Z |
| _version_ |
1837309235754434560 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 021, 17 pages
Commutative Families
of the Elliptic Macdonald Operator?
Yosuke SAITO
Mathematical Institute of Tohoku University, Sendai, Japan
E-mail: yousukesaitou7@gmail.com
Received October 01, 2013, in final form February 25, 2014; Published online March 11, 2014
http://dx.doi.org/10.3842/SIGMA.2014.021
Abstract. In the paper [J. Math. Phys. 50 (2009), 095215, 42 pages], 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.
Key words: elliptic Ding–Iohara–Miki algebra; free field realization; elliptic Macdonald
operator
2010 Mathematics Subject Classification: 17B37; 33D52
Notations. In this paper, we use the following symbols.
Z : the set of integers, Z≥0 := {0, 1, 2, . . . }, Z>0 := {1, 2, . . . },
Q : the set of rational numbers, Q(q, t) : the field of rational functions of q, t over Q,
C : the set of complex numbers, C× := C \ {0},
C[[z, z−1]] : the set of formal power series of z, z−1 over C.
A partition is a sequence λ = (λ1, . . . , λN ) ∈ (Z≥0)
N , N ∈ Z>0 satisfying the condition
λi ≥ λi+1 for each i, 1 ≤ i ≤ N − 1. We denote the set of partitions by P. For a partition λ,
`(λ) := ]{i : λi 6= 0} denotes the length of λ and |λ| :=
`(λ)∑
i=1
λi denotes the size of λ.
Let q, p ∈ C be complex parameters satisfying |q| < 1, |p| < 1. We define the q-infinite
product as (x; q)∞ :=
∏
n≥0
(1− xqn) and the theta function as
Θp(x) := (p; p)∞(x; p)∞
(
px−1; p
)
∞.
We set the double infinite product as (x; q, p)∞ :=
∏
m,n≥0
(1 − xqmpn) and the elliptic gamma
function as
Γq,p(x) :=
(qpx−1; q, p)∞
(x; q, p)∞
.
?This paper is a contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa. The full
collection is available at http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html
yousukesaitou7@gmail.com
http://dx.doi.org/10.3842/SIGMA.2014.021
http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html
2 Y. Saito
1 Introduction
The elliptic Feigin–Odesskii algebra A(p) was originally introduced by Feigin and Odesskii [4]
as an algebra generated by a certain family of multivariate elliptic functions. The algebra A(p)
has a product called the star product ∗ (Definition 3.7), and in fact A(p) is unital, associative,
and commutative in terms of the star product ∗.
On the other hand, Miki [7] constructed a q-deformation of the W1+∞ algebra which has
a structure of Ding and Iohara’s quantum group [1]. Miki’s quantum group is also obtained
from the free field realization of the Macdonald operator [2]. We call the quantum group the
Ding–Iohara–Miki algebra U(q, t) (Definition 2.1). Then Feigin, Hashizume, Hoshino, Shiraishi,
and Yanagida showed the following [2]:
• There exists a trigonometric degeneration of the elliptic Feigin–Odesskii algebra A(p). We
call the algebra the trigonometric Feigin–Odesskii algebra A (Definition 2.6).
• By using the Ding–Iohara–Miki algebra U(q, t) and the trigonometric Feigin–Odesskii al-
gebra A, we can obtain commutative families of the Macdonald operator.
The trigonometric Feigin–Odesskii algebra A also has the star product ∗. In [2], the commu-
tativity of A with respect to the star product ∗ is shown by using some combinatorial tools
such as the Gordon filtrations. These tools are also available in the elliptic case. Therefore
we can say a combinatorial way to prove the commutativities of both trigonometric and elliptic
Feigin–Odesskii algebras has been found in [2].
The aim of this paper is to extend some results of Feigin, Hashizume, Hoshino, Shiarishi,
and Yanagida [2] to the elliptic case. That is, we construct commutative families of the elliptic
Macdonald operator by using the elliptic Ding–Iohara–Miki algebra U(q, t, p) and the elliptic
Feigin–Odesskii algebra A(p). We utilize our previous result [8], the free field realization of
the elliptic Macdonald operator and the elliptic Ding–Iohara–Miki algebra U(q, t, p), for this
purpose.
In the trigonometric case, due to the theory of the Macdonald symmetric functions, more
properties about the trigonometric Feigin–Odesskii algebra A and commutative families of the
Macdonald operator have been known. Details can be found in [2, 3].
Organization of this paper. In Section 2, we review the trigonometric case treated in [2].
In Section 3, first we recall related materials of the elliptic Ding–Iohara–Miki algebra and the
free field realization of the elliptic Macdonald operators. Then using the elliptic Ding–Iohara–
Miki algebra and the elliptic Feigin–Odesskii algebra, we derive commutative families of the
elliptic Macdonald operators.
In Appendix, we show a functional equation of the elliptic kernel function with the aid of the
free field realization of the elliptic Macdonald operators.
2 Review of the trigonometric case
In this section, we review the construction of the commutative families of the Macdonald operator
by Feigin, Hashizume, Hoshino, Shiraishi, and Yanagida [2].
2.1 Ding–Iohara–Miki algebra U(q, t)
The Ding–Iohara–Miki algebra is a quantum group obtained from the free field realization of
the Macdonald operator [2]. Let q, t ∈ C be parameters satisfying |q| < 1.
Commutative Families of the Elliptic Macdonald Operator 3
Definition 2.1 (Ding–Iohara–Miki algebra U(q, t)). Let us define the structure function g(x) ∈
C[[x]] as
g(x) :=
(1− qx)(1− t−1x)(1− q−1tx)
(1− q−1x)(1− tx)(1− qt−1x)
.
Let C be a central, invertible element and x±(z) :=
∑
n∈Z
x±n z
−n, ψ±(z) :=
∑
±n≥0
ψ±n z
−n be currents
satisfying the relations
[ψ±(z), ψ±(w)] = 0, ψ+(z)ψ−(w) =
g(Cz/w)
g(C−1z/w)
ψ−(w)ψ+(z),
ψ±(z)x+(w) = g
(
C±
1
2
z
w
)
x+(w)ψ±(z), ψ±(z)x−(w) = g
(
C∓
1
2
z
w
)−1
x−(w)ψ±(z),
x±(z)x±(w) = g
( z
w
)±1
x±(w)x±(z),
[x+(z), x−(w)] =
(1−q)(1−t−1)
1−qt−1
{
δ
(
C
w
z
)
ψ+
(
C1/2w
)
− δ
(
C−1
w
z
)
ψ−
(
C−1/2w
)}
.
Here we set the delta function by δ(x) :=
∑
n∈Z
xn. We define the Ding–Iohara–Miki algebra U(q, t)
to be an associative C-algebra generated by {x±n }n∈Z, {ψ±n }n∈Z, and C.
The free field realization of the Ding–Iohara–Miki algebra is known as follows In the follo-
wing, let q, t ∈ C and we assume |q| < 1. First we define the algebra B of bosons to be an
associative C-algebra generated by {an}n∈Z\{0} and the relation
[am, an] = m
1− q|m|
1− t|m|
δm+n,0, m, n ∈ Z \ {0}.
We set the normal ordering : • : as
: aman :=
{
aman, m < n,
anam, m ≥ n.
Let |0〉 be the vacuum vector which satisfies an|0〉 = 0, n > 0. For a partition λ, we set
a−λ := a−λ1 · · · a−λ`(λ) and define the boson Fock space F as the left B module
F := span{a−λ|0〉 : λ ∈ P}.
Let 〈0| be the dual vacuum vector which satisfies the condition 〈0|an = 0, n < 0, and define the
dual boson Fock space F∗ as the right B module
F∗ := span{〈0|aλ : λ ∈ P}, aλ := aλ1 · · · a`(λ).
We define nλ(a) := ]{i : λi = a} and zλ, zλ(q, t) as
zλ :=
∏
a≥1
anλ(a)nλ(a)!, zλ(q, t) := zλ
`(λ)∏
i=1
1− qλi
1− tλi
.
We define a bilinear form 〈•|•〉 : F∗ ×F → C by the following conditions:
(1) 〈0|0〉 = 1,
(2) 〈0|aλa−µ|0〉 = δλµzλ(q, t).
4 Y. Saito
Proposition 2.2 (free field realization of the Ding–Iohara–Miki algebra U(q, t)). Set γ :=
(qt−1)−1/2 and define operators η(z), ξ(z), ϕ±(z) : F → F ⊗C[[z, z−1]] as
η(z) :=: exp
−∑
n6=0
(1− tn)an
z−n
n
:, ξ(z) :=: exp
∑
n6=0
(1− tn)γ|n|an
z−n
n
:,
ϕ+(z) :=: η(γ1/2z)ξ(γ−1/2z) :, ϕ−(z) :=: η(γ−1/2z)ξ(γ1/2z) : .
Then the map
C 7→ γ, x+(z) 7→ η(z), x−(z) 7→ ξ(z), ψ±(z) 7→ ϕ±(z)
gives a representation of the Ding–Iohara–Miki algebra U(q, t).
Here we collect some notations of symmetric polynomials and symmetric functions [6]. Let
q, t ∈ C be parameters and assume |q| < 1. We denote the N -th symmetric group by SN and
define ΛN (q, t) := Q(q, t)[x1, . . . , xN ]SN as the space of N -variables symmetric polynomials over
Q(q, t). For α = (α1, . . . , αN ) ∈ (Z≥0)
N , we set xα := xα1
1 · · ·x
αN
N . For a partition λ, we define
the monomial symmetric polynomial mλ(x) as follows
mλ(x) :=
∑
α : α is a permutation of λ
xα.
As is well-known, {mλ(x)}λ∈P form a basis of ΛN (q, t). Let pn(x) :=
N∑
i=1
xni , n ∈ Z>0 be the
power sum, and for a partition λ we define pλ(x) := pλ1(x) · · · pλ`(λ)(x).
Let ρN+1
N : ΛN+1(q, t)→ ΛN (q, t) be the homomorphism defined by(
ρN+1
N f
)
(x1, . . . , xN ) := f(x1, . . . , xN , 0), f ∈ ΛN+1(q, t).
Define the ring of symmetric functions Λ(q, t) as the projective limit defined by
{
ρN+1
N
}
N≥1
Λ(q, t) := lim
←−
ΛN (q, t).
It is known that {pλ(x)}λ∈P form a basis of Λ(q, t). Then we define an inner product 〈 , 〉q,t as
follows
〈pλ(x), pµ(x)〉q,t = δλµzλ(q, t).
We define the order in P as follows: for λ, µ ∈ P,
λ ≥ µ ⇐⇒ |λ| = |µ| and for all i λ1 + · · ·+ λi ≥ µ1 + · · ·+ µi.
The existence of the Macdonald symmetric functions [6] is stated as follows: For each partition
λ, there exists a unique symmetric function Pλ(x) ∈ Λ(q, t) satisfying the following conditions:
(1) Pλ(x) =
∑
µ≤λ
uλµmµ(x), uλµ ∈ Q(q, t),
(2) λ 6= µ =⇒ 〈Pλ(x), Pµ(x)〉q,t = 0.
For a Macdonald symmetric function Pλ(x), we define the N -variable symmetric polynomial
Pλ(x1, . . . , xN ) as Pλ(x1, . . . , xN ) := Pλ(x1, . . . , xN , 0, 0, . . . ), `(λ) ≤ N . We call it the N -
variables Macdonald polynomials. We set the q-shift operator by
Tq,xif(x1, . . . , xN ) := f(x1, . . . , qxi, . . . , xN )
Commutative Families of the Elliptic Macdonald Operator 5
and define the Macdonald operator HN (q, t) : ΛN (q, t)→ ΛN (q, t) as follows
HN (q, t) :=
N∑
i=1
∏
j 6=i
txi − xj
xi − xj
Tq,xi .
Then for each partition λ, `(λ) ≤ N , the Macdonald polynomial Pλ(x1, . . . , xN ) is an eigen-
function of the Macdonald operator [6]
HN (q, t)Pλ(x1, . . . , xN ) = εN (λ)Pλ(x1, . . . , xN ), εN (λ) :=
N∑
i=1
qλitN−i.
In the following [f(z)]1 stands for the constant term of f(z) in z.
Proposition 2.3 (free field realization of the Macdonald operator). Define the operator φ(z) :
F → F ⊗C[[z, z−1]] as follows
φ(z) := exp
(∑
n>0
1− tn
1− qn
a−n
zn
n
)
,
and set φN (x) :=
N∏
j=1
φ(xj).
(1) The operator η(z) reproduces the Macdonald operator HN (q, t) as follows
[η(z)]1φN (x)|0〉 = t−N{(t− 1)HN (q, t) + 1}φN (x)|0〉.
(2) The operator ξ(z) reproduces the Macdonald operator HN (q−1, t−1) as follows
[ξ(z)]1φN (x)|0〉 = tN
{(
t−1 − 1
)
HN
(
q−1, t−1
)
+ 1
}
φN (x)|0〉.
We also have the dual version of Proposition 2.3.
Proposition 2.4 (dual version of Proposition 2.3). Define the operator φ∗(z) : F∗ → F∗ ⊗
C[[z, z−1]] by
φ∗(z) := exp
(∑
n>0
1− tn
1− qn
an
zn
n
)
,
and set φ∗N (x) :=
N∏
j=1
φ∗(xj).
(1) The operator η(z) reproduces the Macdonald operator HN (q, t) as follows
〈0|φ∗N (x)[η(z)]1 = t−N{(t− 1)HN (q, t) + 1}〈0|φ∗N (x).
(2) The operator ξ(z) reproduces the Macdonald operator HN (q−1, t−1) as follows
〈0|φ∗N (x)[ξ(z)]1 = tN
{(
t−1 − 1
)
HN
(
q−1, t−1
)
+ 1
}
〈0|φ∗N (x).
Remark 2.5. Let us define the kernel function of the Macdonald operator ΠMN (q, t)(x, y),
M,N ∈ Z>0, by
ΠMN (q, t)(x, y) :=
∏
1≤i≤M
1≤j≤N
(txiyj ; q)∞
(xiyj ; q)∞
.
Then the kernel function ΠMN (q, t)(x, y) is reproduced from the operators φ∗M (x), φN (y) as
〈0|φ∗M (x)φN (y)|0〉 = ΠMN (q, t)(x, y).
6 Y. Saito
2.2 Trigonometric Feigin–Odesskii algebra A
In this subsection, we review basic facts of the trigonometric Feigin–Odesskii algebra [2]. In the
following let q, t ∈ C be parameters satisfying |q| < 1.
Definition 2.6 (trigonometric Feigin–Odesskii algebra A). Let εn(q;x), n ∈ Z>0 be a function
defined as
εn(q;x) :=
∏
1≤a<b≤n
(xa − qxb)(xa − q−1xb)
(xa − xb)2
.
We also define ω(x, y) as
ω(x, y) :=
(x− q−1y)(x− ty)(x− qt−1y)
(x− y)3
.
For an N -variable function f(x1, . . . , xN ), we define the action of the symmetric group SN of
order N on f(x1, . . . , xN ) by σ · (f(x1, . . . , xN )) := f(xσ(1), . . . , xσ(N)), σ ∈ SN . We define the
symmetrizer as
Sym[f(x1, . . . , xN )] :=
1
N !
∑
σ∈SN
σ · (f(x1, . . . , xN )).
For an m-variable function f(x1, . . . , xm) and an n-variable function g(x1, . . . , xn), we define
the star product ∗ as follows
(f ∗ g)(x1, . . . , xm+n) := Sym
f(x1, . . . , xm)g(xm+1, . . . , xm+n)
∏
1≤α≤m
m+1≤β≤m+n
ω(xα, xβ)
.
For a partition λ, we define ελ(q;x), x = (x1, . . . , x|λ|) as
ελ(q;x) := (ελ1(q; •) ∗ · · · ∗ ελ`(λ)(q; •))(x).
Set A0 := Q(q, t), An := span{ελ(q;x) : |λ| = n}, n ≥ 1. We define the trigonometric Feigin–
Odesskii algebra to be A :=
⊕
n≥0An whose algebra structure is given by the star product ∗.
Remark 2.7. The definition of the trigonometric Feigin–Odesskii algebra A above is a short-
handed one of [2]. For instance, there would be a question why the function εn(q;x) appears.
For more detail of the trigonometric Feigin–Odesskii algebra A, see [2].
Proposition 2.8 ([2]). The trigonometric Feigin–Odesskii algebra (A, ∗) is unital, associative,
and commutative.
2.3 Commutative families M, M′
Here we give an overview of the construction of the commutative families of the Macdonald
operator using the Ding–Iohara–Miki algebra and the trigonometric Feigin–Odesskii algebra.
Definition 2.9 (map O). Define a linear map O : A → End(F) as
O(f) :=
f(z1, . . . , zn)
∏
1≤i<j≤n
ω(zi, zj)
−1η(z1) · · · η(zn)
1
for f ∈ An, where [f(z1, . . . , zn)]1 denotes the constant term of f(z1, . . . , zn) in z1, . . . , zn, and
extend to A linearly.
Commutative Families of the Elliptic Macdonald Operator 7
From the relation
η(z)η(w) = g
( z
w
)
η(w)η(z),
we have the following
1
ω(z, w)
η(z)η(w) =
1
ω(w, z)
η(w)η(z).
This relation shows that the operator-valued function∏
1≤i<j≤N
ω(xi, xj)
−1η(x1) · · · η(xN )
is symmetric in x1, . . . , xN . From this fact, we have the following proposition.
Proposition 2.10. The map O and the star product ∗ are compatible: for f, g ∈ A, we have
O(f ∗ g) = O(f)O(g).
The trigonometric Feigin–Odesskii algebra A is commutative with respect to the star prod-
uct ∗, 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 ⊂ EndC(V ), we use the
symbol M |W := {T |W : T ∈M}.
Corollary 2.11 (commutative family M).
(1) Set M := O(A). The space M consists of operators commuting: [O(f),O(g)] = 0,
f, g ∈ A.
(2) The space M|CφN (x)|0〉 is a set of commuting q-difference operators containing the Mac-
donald operator HN (q, t).
Proof. (1) This statement follows from the commutativity of A in terms of the star product ∗
and Proposition 2.10.
(2) Due to the free field realization of the Macdonald operator HN (q, t), the operator
O(εr(q; z)), r ∈ Z>0, acts on φN (x)|0〉, N ∈ Z>0, as an r-th order q-difference operator. By the
fact that M = O(A) is generated by {O(εr(q; z))}r∈Z>0 , and the relation
O(ε1(q; z))φN (x)|0〉 = [η(z)]1φN (x)|0〉 = t−N{(t− 1)HN (q, t) + 1}φN (x)|0〉
and (1), the statement holds. �
The Macdonald operator HN (q−1, t−1) is reproduced from the operator ξ(z). By this fact,
we can construct another commutative family of the Macdonald operator.
Definition 2.12 (map O′). Define a function ω′(x, y) as
ω′(x, y) :=
(x− qy)(x− t−1y)(x− q−1ty)
(x− y)3
.
Define a linear map O′ : A → End(F) as
O′(f) :=
f(z1, . . . , zn)
∏
1≤i<j≤n
ω′(zi, zj)
−1ξ(z1) · · · ξ(zn)
1
for f ∈ An, and extend to A linearly.
8 Y. Saito
From the relation ω(x, y) = ω′(y, x) we have
Lemma 2.13. Define another star product ∗′ as follows
(f ∗′ g)(x1, . . . , xm+n) := Sym
f(x1, . . . , xm)g(xm+1, . . . , xm+n)
∏
1≤α≤m
m+1≤β≤m+n
ω′(xα, xβ)
.
Then in the trigonometric Feigin–Odesskii algebra A, we have ∗′ = ∗.
We can check the map O′ and the star product ∗′ are compatible in the similar way of the
proof of Proposition 2.10. Furthermore since ∗′ = ∗, we have
Corollary 2.14 (commutative family M′).
(1) Set M′ := O′(A). Then the space M′ consists of commuting operators.
(2) The space M′|CφN (x)|0〉 is a set of commuting q-difference operators containing the Mac-
donald operator HN (q−1, t−1).
From the relation [[η(z)]1, [ξ(w)]1] = 0, we have the following proposition.
Proposition 2.15. The commutative families M, M′ satisfy [M,M′] = 0.
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 Proposition 2.15, commutative families M|CφN (x)|0〉, M′|CφN (x)|0〉 also commute:
[M|CφN (x)|0〉,M′|CφN (x)|0〉] = 0.
3 Elliptic case
In this section, we construct commutative families of the elliptic Macdonald operators by using
the elliptic Ding–Iohara–Miki algebra and the elliptic Feigin–Odesskii algebra. Let q, t, p ∈ C
with |q| < 1, |p| < 1.
3.1 Elliptic Ding–Iohara–Miki algebra U(q, t, p)
The elliptic Ding–Iohara–Miki algebra is an elliptic analog of the Ding–Iohara–Miki algebra
introduced in [8]. First we recall the definition of the elliptic Ding–Iohara–Miki algebra and its
free field realization.
Definition 3.1 (elliptic Ding–Iohara–Miki algebra U(q, t, p)). Define the structure function
gp(x) ∈ C[[x, x−1]] as
gp(x) :=
Θp(qx)Θp(t
−1x)Θp(q
−1tx)
Θp(q−1x)Θp(tx)Θp(qt−1x)
.
Let x±(p; z) :=
∑
n∈Z
x±n (p)z−n, ψ±(p; z) :=
∑
n∈Z
ψ±n (p)z−n be currents and C be a central, inver-
tible element satisfying the following relations
[ψ±(p; z), ψ±(p;w)] = 0, ψ+(p; z)ψ−(p;w) =
gp(Cz/w)
gp(C−1z/w)
ψ−(p;w)ψ+(p; z),
Commutative Families of the Elliptic Macdonald Operator 9
ψ±(p; z)x+(p;w) = gp
(
C±
1
2
z
w
)
x+(p;w)ψ±(p; z),
ψ±(p; z)x−(p;w) = gp
(
C∓
1
2
z
w
)−1
x−(p;w)ψ±(p; z),
x±(p; z)x±(p;w) = gp
( z
w
)±1
x±(p;w)x±(p; z),
[x+(p; z), x−(p;w)] =
Θp(q)Θp(t
−1)
(p; p)3∞Θp(qt−1)
×
{
δ
(
C
w
z
)
ψ+(p;C1/2w)− δ
(
C−1
w
z
)
ψ−(p;C−1/2w)
}
.
We define the elliptic Ding–Iohara–Miki algebra U(q, t, p) to be the associative C-algebra gene-
rated by {x±n (p)}n∈Z, {ψ±n (p)}n∈Z and C.
Let Ba,b be the associative C-algebra generated by {an}n∈Z\{0}, {bn}n∈Z\{0} and the following
relations
[am, an] = m(1− p|m|)1− q|m|
1− t|m|
δm+n,0, [bm, bn] = m
1− p|m|
(qt−1p)|m|
1− q|m|
1− t|m|
δm+n,0,
[am, bn] = 0, m, n ∈ Z \ {0}.
We define the normal ordering : • : as usual
: aman :=
{
aman, m < n,
anam, m ≥ n,
: bmbn :=
{
bmbn, m < n,
bnbm, m ≥ n.
Let |0〉 be the vacuum vector which satisfies the condition an|0〉 = bn|0〉 = 0, n > 0, and set the
boson Fock space F as the left Ba,b module
F = span{a−λb−µ|0〉 : λ, µ ∈ P}.
Let 〈0| be the dual vacuum vector which satisfies the condition 〈0|an = 〈0|bn = 0, n < 0, and
〈0|a0 = 0. We define the dual boson Fock space as the right Ba,b module
F∗ := span{〈0|aλbµ : λ, µ ∈ P}.
For a partition λ, set nλ(a) = ]{i : λi = a}, zλ =
∏
a≥1
anλ(a)nλ(a)! and define zλ(q, t, p),
zλ(q, t, p) by
zλ(q, t, p) := zλ
`(λ)∏
i=1
(1− pλi)1− qλi
1− tλi
, zλ(q, t, p) := zλ
`(λ)∏
i=1
1− pλi
(qt−1p)λi
1− qλi
1− tλi
.
We define a bilinear form 〈•|•〉 : F∗ ×F → C by the following conditions:
(1) 〈0|0〉 = 1,
(2) 〈0|aλ1bλ2a−µ1b−µ2 |0〉 = δλ1µ1δλ2µ2zλ1(q, t, p)zλ2(q, t, p).
Theorem 3.2 (free field realization of the elliptic Ding–Iohara–Miki algebra U(q, t, p)). Define
operators η(p; z), ξ(p; z), ϕ±(p; z) : F → F ⊗C[[z, z−1]] as follows (γ := (qt−1)−1/2)
η(p; z) :=: exp
−∑
n6=0
1− t−n
1− p|n|
p|n|bn
zn
n
exp
−∑
n6=0
1− tn
1− p|n|
an
z−n
n
:,
10 Y. Saito
ξ(p; z) :=: exp
∑
n 6=0
1− t−n
1− p|n|
γ−|n|p|n|bn
zn
n
exp
∑
n6=0
1− tn
1− p|n|
γ|n|an
z−n
n
:,
ϕ+(p; z) :=: η(p; γ1/2z)ξ(p; γ−1/2z) :, ϕ−(p; z) :=: η(p; γ−1/2z)ξ(p; γ1/2z) : .
Then the map
C 7→ γ, x+(p; z) 7→ η(p; z), x−(p; z) 7→ ξ(p; z), ψ±(p; z) 7→ ϕ±(p; z)
gives a representation of the elliptic Ding–Iohara–Miki algebra U(q, t, p).
The elliptic Macdonald operator HN (q, t, p), N ∈ Z>0, is defined as follows
HN (q, t, p) :=
N∑
i=1
∏
j 6=i
Θp(txi/xj)
Θp(xi/xj)
Tq,xi .
By the operators η(p; z), ξ(p; z) in Theorem 3.2, we can reproduce the elliptic Macdonald ope-
rator as follows [8].
Theorem 3.3 (free field realization of the elliptic Macdonald operator). Let us define an ope-
rator φ(p; z) : F → F ⊗C[[z, z−1]] as follows
φ(p; z) := exp
(∑
n>0
(1− tn)(qt−1p)n
(1− qn)(1− pn)
b−n
z−n
n
)
exp
(∑
n>0
1− tn
(1− qn)(1− pn)
a−n
zn
n
)
,
and put φN (p;x) :=
N∏
j=1
φ(p;xj).
(1) The elliptic Macdonald operator HN (q, t, p) is reproduced by the operator η(p; z) as follows
[η(p; z)− t−N (η(p; z))−(η(p; p−1z))+]1φN (p;x)|0〉 =
t−N+1Θp(t
−1)
(p; p)3∞
HN (q, t, p)φN (p;x)|0〉.
Here we use the notation (η(p; z))± as
(η(p; z))± := exp
(
−
∑
±n>0
1− t−n
1− p|n|
p|n|bn
zn
n
)
exp
(
−
∑
±n>0
1− tn
1− p|n|
an
z−n
n
)
.
(2) The elliptic Macdonald operator HN (q−1, t−1, p) is reproduced by the operator ξ(p; z) as
follows
[ξ(p; z)− tN (ξ(p; z))−(ξ(p; p−1z))+]1φN (p;x)|0〉 =
tN−1Θp(t)
(p; p)3∞
HN
(
q−1, t−1, p
)
φN (p;x)|0〉.
Here we use the notation (ξ(p; z))± as
(ξ(p; z))± := exp
(∑
±n>0
1− t−n
1− p|n|
γ−|n|p|n|bn
zn
n
)
exp
(∑
±n>0
1− tn
1− p|n|
γ|n|an
z−n
n
)
.
To state the next theorem, we introduce zero mode generators a0, Q satisfying
[a0, Q] = 1, [an, a0] = [bn, a0] = 0, [an, Q] = [bn, Q] = 0, n ∈ Z \ {0}.
We also set the condition a0|0〉 = 0. For a complex number α ∈ C, we define |α〉 := eαQ|0〉.
Then we can check a0|α〉 = α|α〉. For α ∈ C, we set Fα := span{a−λb−µ|α〉 : λ, µ ∈ P}.
Commutative Families of the Elliptic Macdonald Operator 11
Theorem 3.4. Set η̃(p; z) := (η(p; z))−(η(p; p−1z))+, ξ̃(p; z) := (ξ(p; z))−(ξ(p; p−1z))+ and
define
E(p; z) := η(p; z)− η̃(p; z)t−a0 , F (p; z) := ξ(p; z)− ξ̃(p; z)ta0 .
Then the elliptic Macdonald operators HN (q, t, p), HN (q−1, t−1, p) are reproduced by the opera-
tors E(p; z), F (p; z) as follows
[E(p; z)]1φN (p;x)|N〉 =
t−N+1Θp(t
−1)
(p; p)3∞
HN (q, t, p)φN (p;x)|N〉,
[F (p; z)]1φN (p;x)|N〉 =
tN−1Θp(t)
(p; p)3∞
HN
(
q−1, t−1, p
)
φN (p;x)|N〉.
Dual versions of Theorems 3.3 and 3.4 are also available. For α ∈ C, set 〈α| := 〈0|e−αQ.
Then we have 〈α|a0 = α〈α|. For α ∈ C, we set F∗α := span{〈α|aλbµ : λ, µ ∈ P}.
Theorem 3.5 (dual versions of Theorems 3.3 and 3.4). Let us define the operator φ∗(p; z) :
F∗ → F∗ ⊗C[[z, z−1]] as follows
φ∗(p; z) := exp
(∑
n>0
(1− tn)(qt−1p)n
(1− qn)(1− pn)
bn
z−n
n
)
exp
(∑
n>0
1− tn
(1− qn)(1− pn)
an
zn
n
)
,
and set φ∗N (p;x) :=
N∏
j=1
φ∗(p;xj).
(1) The elliptic Macdonald operators HN (q, t, p), HN (q−1, t−1, p) are reproduced by the ope-
rators η(p; z), ξ(p; z) as follows
〈0|φ∗N (p;x)
[
η(p; z)− t−N (η(p; z))−(η(p; p−1z))+
]
1
=
t−N+1Θp(t
−1)
(p; p)3∞
HN (q, t, p)〈0|φ∗N (p;x),
〈0|φ∗N (p;x)
[
ξ(p; z)− tN (ξ(p; z))−(ξ(p; p−1z))+
]
1
=
tN−1Θp(t)
(p; p)3∞
HN
(
q−1, t−1, p
)
〈0|φ∗N (p;x).
(2) The operators E(p; z), F (p; z) reproduce the elliptic Macdonald operators HN (q, t, p),
HN (q−1, t−1, p) as follows
〈N |φ∗N (p;x)[E(p; z)]1 =
t−N+1Θp(t
−1)
(p; p)3∞
HN (q, t, p)〈N |φ∗N (p;x),
〈N |φ∗N (p;x)[F (p; z)]1 =
tN−1Θp(t)
(p; p)3∞
HN
(
q−1, t−1, p
)
〈N |φ∗N (p;x).
Remark 3.6. Let ΠMN (q, t, p)(x, y), M,N ∈ Z>0, be the kernel function of the elliptic Mac-
donald operator defined as
ΠMN (q, t, p)(x, y) :=
∏
1≤i≤M
1≤j≤N
Γq,p(xiyj)
Γq,p(txiyj)
.
Then the kernel function ΠMN (q, t, p)(x, y) is reproduced from the operators φ∗M (p;x), φN (p; y)
as
〈0|φ∗M (p;x)φN (p; y)|0〉 = ΠMN (q, t, p)(x, y).
12 Y. Saito
3.2 Elliptic Feigin–Odesskii algebra A(p)
The elliptic Feigin−Odesskii algebra is defined quite similar as in the trigonometric case except
for the emergence of elliptic functions [2]. Let q, t, p ∈ C be complex parameters satifying |q| < 1,
|p| < 1.
Definition 3.7 (elliptic Feigin–Odesskii algebra A(p)). Define an n-variable function εn(q, p;x),
n ∈ Z>0, as follows
εn(q, p;x) :=
∏
1≤a<b≤n
Θp(qxa/xb)Θp(q
−1xa/xb)
Θp(xa/xb)2
.
Define a function ωp(x, y) as
ωp(x, y) :=
Θp(q
−1y/x)Θp(ty/x)Θp(qt
−1y/x)
Θp(y/x)3
.
Define the star product ∗ as
(f ∗ g)(x1, . . . , xm+n) := Sym
f(x1, . . . , xm)g(xm+1, . . . , xm+n)
∏
1≤α≤m
m+1≤β≤m+n
ωp(xα, xβ)
.
For a partition λ, we set ελ(q, p;x), x = (x1, . . . , x|λ|) as
ελ(q, p;x) := (ελ1(q, p; •) ∗ · · · ∗ ελ`(λ)(q, p; •))(x).
Set A0(p): = C, An(p): = span{ελ(q, p;x) : |λ|=n}, n≥1. We define the elliptic Feigin–Odesskii
algebra as A(p) :=
⊕
n≥0An(p) whose algebra structure is given by the star product ∗.
As in the trigonometric case (Proposition 2.8), we have
Proposition 3.8. The elliptic Feigin–Odesskii algebra (A(p), ∗) is an unital, associative, com-
mutative algebra.
3.3 Commutative families M(p), M′(p)
For the operators E(p; z), F (p; z) in Theorem 3.4, we have [8]
Proposition 3.9. The following relations hold
E(p; z)E(p;w) = gp
( z
w
)
E(p;w)E(p; z), (3.1)
F (p; z)F (p;w) = gp
( z
w
)−1
F (p;w)F (p; z), (3.2)
[E(p; z), F (p;w)] =
Θp(q)Θp(t
−1)
(p; p)3∞Θp(qt−1)
δ
(
γ
w
z
){
ϕ+
(
p; γ1/2w
)
− ϕ+
(
p; γ1/2p−1w
)}
. (3.3)
From the relation (3.3) we have [[E(p; z)]1, [F (p;w)]1] = 0. This corresponds to the commu-
tativity of the elliptic Macdonald operators [HN (q, t, p), HN (q−1, t−1, p)] = 0.
Define a function ω′p(x, y) as
ω′p(x, y) :=
Θp(qy/x)Θp(t
−1y/x)Θp(q
−1ty/x)
Θp(y/x)3
.
Commutative Families of the Elliptic Macdonald Operator 13
Due to the relations (3.1) and (3.2), operator-valued functions∏
1≤i<j≤N
ωp(xi, xj)
−1E(p;x1) · · ·E(p;xN ),
∏
1≤i<j≤N
ω′p(xi, xj)
−1F (p;x1) · · ·F (p;xN )
are symmetric in x1, . . . , xN .
Definition 3.10 (map Op). We define a linear map Op : A(p)→ End(Fα), α ∈ C as follows
Op(f) :=
f(z1, . . . , zn)
∏
1≤i<j≤n
ωp(zi, zj)
−1E(p; z1) · · ·E(p; zn)
1
for f∈An(p), where [f(z1, . . . , zn)]1 denotes the constant term of f(z1, . . . , zn) in z1, . . . , zn, and
extend linearly to A(p).
In the similar way of the trigonometric case, we can check the following
Proposition 3.11. The map Op and the star product ∗ are compatible: for f, g ∈ A(p), we
have Op(f ∗ g) = Op(f)Op(g).
Theorem 3.12 (commutative family M(p)).
(1) Set M(p) := Op(A(p)). The space M(p) is commutative.
(2) The spaceM(p)|CφN (p;x)|N〉 is a set of commuting elliptic q-difference operators containing
the elliptic Macdonald operator HN (q, t, p).
A commutative family containing the elliptic Macdonald operator HN (q−1, t−1, p) is also
constructed as follows
Definition 3.13 (map O′p). We define a linear map O′p : A(p)→ End(Fα), α ∈ C as follows.
O′p(f) :=
f(z1, . . . , zn)
∏
1≤i<j≤n
ω′p(zi, zj)
−1F (p; z1) · · ·F (p; zn)
1
for f ∈ An(p), and extend linearly to A(p).
As in the trigonometric case we have
Lemma 3.14. Define another star product ∗′ as
(f ∗′ g)(x1, . . . , xm+n) := Sym
f(x1, . . . , xm)g(xm+1, . . . , xm+n)
∏
1≤α≤m
m+1≤β≤m+n
ω′p(xα, xβ)
.
In the elliptic Feigin–Odesskii algebra A(p), we have ∗′ = ∗.
Theorem 3.15 (commutative family M′(p)).
(1) Set M′(p) := O′p(A(p)). The space M′(p) is commutative.
(2) The spaceM′(p)|CφN (p;x)|N〉 is a set of commuting elliptic q-difference operators containing
the elliptic Macdonald operator HN (q−1, t−1, p).
Similar to Proposition 2.15, we can show that the commutative families M(p), M′(p) com-
mute with each other.
14 Y. Saito
Theorem 3.16. Commutative families M(p), M′(p) commute: [M(p),M′(p)]=0.
Theorem 3.16 is an elliptic analog of Proposition 2.15. But we can’t prove Theorem 3.16 in
the similar way of the proof of Proposition 2.15, because we don’t have an elliptic analog of the
Macdonald symmetric functions. Hence we will show Theorem 3.16 in a direct way. We need
the following lemma
Lemma 3.17. Assume that an r-variable function A(x1, . . . , xr) and an s-variable function
B(x1, . . . , xs) have a period p, i.e.
Tp,xiA(x1, . . . , xr) = A(x1, . . . , xr), 1 ≤ i ≤ r,
Tp,xiB(x1, . . . , xs) = B(x1, . . . , xs), 1 ≤ i ≤ s.
Then we have[
[A(z1, . . . , zr)E(p; z1) · · ·E(p; zr)]1, [B(w1, . . . , ws)F (p;w1) · · ·F (p;ws)]1
]
= 0.
Proof. Recall the general formula of commutator
[A1 · · ·Ar, B1 · · ·Bs] =
r∑
i=1
s∑
j=1
A1 · · ·Ai−1B1 · · ·Bj−1[Ai, Bj ]Bj+1 · · ·BsAi+1 · · ·Ar. (3.4)
Set c(q, t, p) := Θp(q)Θp(t
−1)/(p; p)3∞Θp(qt
−1) and let ∆pf(z) := f(pz) − f(z) stands for the
p-difference of f(z). By the identities (3.3) and (3.4), we have the following
[A(z1, . . . , zr)E(p; z1) · · ·E(p; zr), B(w1, . . . , ws)F (p;w1) · · ·F (p;ws)]
=
r∑
i=1
s∑
j=1
A(z1, . . . , zr)B(w1, . . . , ws)E(p; z1) · · ·E(p; zi−1)F (p;w1) · · ·F (p;wj−1)
× [E(p; zi), F (p;wj)]F (p;wj+1) · · ·F (p;ws)E(p; zi+1) · · ·E(p; zr)
= c(q, t, p)
r∑
i=1
s∑
j=1
E(p; z1) · · ·E(p; zi−1)F (p;w1) · · ·F (p;wj−1)
×A(z1, . . . ,
i-th︷︸︸︷
γwj , . . . , zr)B(w1, . . . ,
j-th︷︸︸︷
wj , . . . , ws)δ
(
γ
wj
zi
)
∆pϕ
+(p; γ1/2p−1wj)
× F (p;wj+1) · · ·F (p;ws)E(p; zi+1) · · ·E(p; zr). (3.5)
By picking up the constant term of zi, wj dependent part of (3.5), we have
[
A(z1, . . . ,
i-th︷︸︸︷
γwj , . . . , zr)B(w1, . . . ,
j-th︷︸︸︷
wj , . . . , ws)δ
(
γ
wj
zi
)
∆pϕ
+(p; γ1/2p−1wj)
]
zi,wj ,1
=
[
A(z1, . . . ,
i-th︷︸︸︷
γwj , . . . , zr)B(w1, . . . ,
j-th︷︸︸︷
wj , . . . , ws)∆pϕ
+(p; γ1/2p−1wj)
]
wj ,1
=
[
A(z1, . . . ,
i-th︷︸︸︷
γwj , . . . , zr)B(w1, . . . ,
j-th︷︸︸︷
wj , . . . , ws)ϕ
+(p; γ1/2wj)
]
wj ,1
−
[
A(z1, . . . ,
i-th︷︸︸︷
γwj , . . . , zr)B(w1, . . . ,
j-th︷︸︸︷
wj , . . . , ws)ϕ
+(p; γ1/2p−1wj)
]
wj ,1
.
Commutative Families of the Elliptic Macdonald Operator 15
We recall [f(z)]1=[f(az)]1, a∈C, and the assumption that both A(z1, . . . , zr) and B(w1, . . . , ws)
have the period p. Therefore we have
[
A(z1, . . . ,
i-th︷︸︸︷
γwj , . . . , zr)B(w1, . . . ,
j-th︷︸︸︷
wj , . . . , ws)δ
(
γ
wj
zi
)
∆pϕ
+(p; γ1/2p−1wj)
]
zi,wj ,1
= 0
for any i, j, proving the lemma. �
Proof of Theorem 3.16. It is enough to show
[Op(εr(q, p; z)),O′p(εs(q, p;w))] = 0, r, s ∈ Z>0.
By the definition of Op, O′p, the operators Op(εr(q, p; z)), O′p(εs(q, p;w)) are the constant terms
of the following operators
εr(q, p; z)
∏
1≤i<j≤r
ωp(zi, zj)
−1E(p; z1) · · ·E(p; zr), (3.6)
εs(q, p;w)
∏
1≤i<j≤s
ω′p(wi, wj)
−1F (p;w1) · · ·F (p;ws). (3.7)
Then their functional parts take the following forms
(Functional part of (3.6))
= εr(q, p; z)
∏
1≤i<j≤r
ωp(zi, zj)
−1 =
∏
1≤i<j≤r
Θp(zi/zj)Θp(q
−1zi/zj)
Θp(t−1zi/zj)Θp(q−1tzi/zj)
, (3.8)
(Functional part of (3.7))
= εs(q, p;w)
∏
1≤i<j≤s
ω′p(wi, wj)
−1 =
∏
1≤i<j≤s
Θp(wi/wj)Θp(qwi/wj)
Θp(twi/wj)Θp(qt−1wi/wj)
. (3.9)
We can check (3.8), (3.9) have a period p. By Lemma 3.17, we have Theorem 3.16. �
By Theorem 3.16, commutative families M(p)|CφN (p;x)|N〉, M′(p)|CφN (p;x)|N〉 also commute:
[M(p)|CφN (p;x)|N〉,M′(p)|CφN (p;x)|N〉] = 0.
In the trigonometric case, relations between the commutative familiesM,M′ and the higher-
order Macdonald operators are studied in [2]. In contrast, the elliptic case relations between the
commutative families M(p), M′(p) and the higher-order elliptic Macdonald operator [5] still
remain unclear.
A Appendix
A.1 Trigonometric kernel function and its functional equation
The following theorem is shown by Komori, Noumi, and Shiraishi [5].
Theorem A.1 ([5]). Define the Macdonald operator HN (q, t), N ∈ Z>0, as
HN (q, t) :=
N∑
i=1
∏
j 6=i
txi − xj
xi − xj
Tq,xi , Tq,xf(x) := f(qx),
16 Y. Saito
and its kernel function ΠMN (q, t)(x, y), M,N ∈ Z>0, as
ΠMN (q, t)(x, y) :=
∏
1≤i≤M
1≤j≤N
(txiyj ; q)∞
(xiyj ; q)∞
.
Then we have the following functional equation
{
HM (q, t)x−tM−NHN (q, t)y
}
ΠMN (q, t)(x, y) =
1−tM−N
1−t
ΠMN (q, t)(x, y). (A.1)
Here we denote the Macdonald operator which acts on functions of x1, . . . , xM by HM (q, t)x.
In the following, we will show an elliptic analog of Theorem A.1 by the free field realization
of the elliptic Macdonald operator.
A.2 Elliptic kernel function and its functional equation
By the free field realization, we can show the following theorem.
Theorem A.2 (functional equation of the elliptic kernel function). Define the elliptic kernel
function ΠMN (q, t, p)(x, y) by
ΠMN (q, t, p)(x, y) :=
∏
1≤i≤M
1≤j≤N
Γq,p(xiyj)
Γq,p(txiyj)
.
We also define CMN (p;x, y) as
CMN (p;x, y) :=
〈0|φ∗M (p;x)[(η(p; z))−(η(p; p−1z))+]1φN (p; y)|0〉
ΠMN (q, t, p)(x, y)
=
M∏
i=1
Θp(t
−1xiz)
Θp(xiz)
N∏
j=1
Θp(z/yj)
Θp(t−1z/yj)
1
.
For the elliptic Macdonald operator and the elliptic kernel function ΠMN (q, t, p)(x, y), we have
the following functional equation
{HM (q, t, p)x − tM−NHN (q, t, p)y}ΠMN (q, t, p)(x, y)
=
(1− tM−N )(p; p)3∞
Θp(t)
CMN (p;x, y)ΠMN (q, t, p)(x, y). (A.2)
Proof. The proof is straightforward. Using Theorems 3.3 and 3.5, we calculate the matrix
element 〈0|φ∗M (p;x)[η(p; z)]1φN (p; y)|0〉 in two different ways as follows
〈0|φ∗M (p;x)[η(p; z)]1φN (p; y)|0〉 =
t−M+1Θp(t
−1)
(p; p)3∞
HM (q, t, p)xΠMN (q, t, p)(x, y)
+ t−MCMN (p;x, y)ΠMN (q, t, p)(x, y)
=
t−N+1Θp(t
−1)
(p; p)3∞
HN (q, t, p)yΠMN (q, t, p)(x, y) + t−NCMN (p;x, y)ΠMN (q, t, p)(x, y).
Therefore we obtain Theorem A.2. �
Commutative Families of the Elliptic Macdonald Operator 17
Remark A.3. We can check the following:
CMN (p;x, y) =
M∏
i=1
Θp(t
−1xiz)
Θp(xiz)
N∏
j=1
Θp(z/yj)
Θp(t−1z/yj)
1
−−−→
p→0
M∏
i=1
1− t−1xiz
1− xiz
N∏
j=1
1− z/yj
1− t−1z/yj
1
= 1.
Hence, by taking the limit p→ 0, equation (A.2) reduces to equation (A.1).
Acknowledgements
The author would like to thank Koji Hasegawa and Gen Kuroki for helpful discussions and
comments. The author also would like to thank referees for their valuable comments on im-
provements of the present paper.
References
[1] Ding J., Iohara K., Generalization of Drinfeld quantum affine algebras, Lett. Math. Phys. 41 (1997), 181–
193.
[2] Feigin B., Hashizume K., Hoshino A., Shiraishi J., Yanagida S., A commutative algebra on degenerate CP1
and Macdonald polynomials, J. Math. Phys. 50 (2009), 095215, 42 pages, arXiv:0904.2291.
[3] Feigin B., Hoshino A., Shibahara J., Shiraishi J., Yanagida S., Kernel function and quantum algebras,
arXiv:1002.2485.
[4] Feigin B., Odesskii A., A family of elliptic algebras, Int. Math. Res. Not. 1997 (1997), no. 11, 531–539.
[5] Komori Y., Noumi M., Shiraishi J., Kernel functions for difference operators of Ruijsenaars type and their
applications, SIGMA 5 (2009), 054, 40 pages, arXiv:0812.0279.
[6] Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs,
Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
[7] Miki K., A (q, γ) analog of the W1+∞ algebra, J. Math. Phys. 48 (2007), 123520, 35 pages.
[8] Saito Y., Elliptic Ding–Iohara algebra and the free field realization of the elliptic Macdonald operator,
arXiv:1301.4912.
http://dx.doi.org/10.1023/A:1007341410987
http://dx.doi.org/10.1063/1.3192773
http://arxiv.org/abs/0904.2291
http://arxiv.org/abs/1002.2485
http://dx.doi.org/10.1155/S1073792897000354
http://dx.doi.org/10.3842/SIGMA.2009.054
http://arxiv.org/abs/0812.0279
http://dx.doi.org/10.1063/1.2823979
http://arxiv.org/abs/1301.4912
1 Introduction
2 Review of the trigonometric case
2.1 Ding–Iohara–Miki algebra U(q,t)
2.2 Trigonometric Feigin–Odesskii algebra A
2.3 Commutative families M, M
3 Elliptic case
3.1 Elliptic Ding–Iohara–Miki algebra U(q,t,p)
3.2 Elliptic Feigin–Odesskii algebra A(p)
3.3 Commutative families M(p), M(p)
A Appendix
A.1 Trigonometric kernel function and its functional equation
A.2 Elliptic kernel function and its functional equation
References
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