数理解析研究所講究録
第 1689 巻 2010 年 133-152

133

KERNEL FUNCTION AND QUANTUM ALGEBRAS
B. FEIGIN, A. HOSHINO, J. SHIBAHARA, J. SHIRAISHI AND S. YANAGIDAABSTRACT. We introduce an aiialogue $K_{t1}(x, z;q, t)$ of the Cauchy-type kernel function for the MMacdonald polynomials, being constructed in the tensor product of the ringof symmetric functions
and the commutative algebra
over the degenerate. We show that a certain restriction ofwith respect to the variable ls neatly described by the tableau sum formula of Macdonald polynomials. Next, we demonstrate that the level $rn$ representation of the Ding-Iohara quantum
algebranaturally produces the currents of the deformed. Then we remark that the
emerges in the highest-to-higliest correlation function of the deformed
algebra.
$\Lambda_{F}$$\mathbb{C}\mathbb{P}^{1}$$\mathcal{A}$$K_{n}(x,\tilde{k};q, t)$$z$$\mathcal{U}(q, t)$$V\backslash \prime_{q,p}(\epsilon 1_{r\iota})$$I\backslash ^{r}n(x, z;q, t)$$w_{J,p}^{\backslash }(\mathfrak{s}1_{\eta})$

1. KERNEL FUNCTION.

1.1. The algebra
We briefly recall the definition and the basic facts about the commutative
algebra introduced in [FHHSY]. Let
be two independent indeterminates and set $q_{3}:=1q_{1}q_{2}$ .
$\mathbb{N}:=\{0,1,2,\ldots\}$ and
We also use the symbols.
, we define two operators
For $n,$acting on the space of symmetric rational
functions in variables
by$\mathcal{A}$$\mathcal{A}$$q_{1},$$q_{2}$$\mathbb{N}_{+}:=\{1,2, \ldots\}$$\mathbb{F}:=\mathbb{Q}(q_{1}, q_{2}),$$\partial^{(0,k)},$$k\in \mathbb{N}+$$n$$x_{1},$:$\partial^{(0,k)}$$\partial^{(\infty,k)}$:$f$$\ldots,$$\mapsto$$\partial^{(\infty,k)}$$x_{n}$$\underline{n!}\lim f(x_{1}, \ldots, x_{n-k}, \xi x_{n-k+1}, \xi x_{n-k+2}, \ldots, \xi x_{n})$$(n-k).!\xiarrow 0$$f$$\mapsto$$\underline{n!}\lim f(x_{1}, \ldots, x_{n-k},\xi x_{n-k+1}, \xi x_{n-k+2}, \ldots, \xi x_{n})$$(n-k)!\xiarrow\infty$

whenever the limit exists. We also set
to be the identity operator.

$\partial^{(0k)}$)$c=0,$ $\partial^{(\infty,k)}c=0$ for$c\in \mathbb{F}$. Finally we define$\partial^{(0,0)}$and$\partial^{(\infty,0)}$, the vector space
Definition 1.1. For
is defined by the following conclitions(i), (ii), (iii) and (iv).
(i)
. For
is a rational function with coefficients in , andsymmetric with respect to the.
$0\leq k\leq n$ and(ii) For, the limits
andboth exist and coincide:$n\in \mathbb{N}$$n\in \mathbb{N}+,$$\mathcal{A}_{0};=\mathbb{F}$$A_{n}=\mathcal{A}_{n}(q_{1}, q_{2}, q_{3})$$f(x_{1}, \ldots, x_{n})\in \mathcal{A}_{n}$$\mathbb{F}$$x_{i}’ s$$n\in \mathbb{N},$$\partial^{(\infty,k)}f$$f\in \mathcal{A}_{n}$$\partial^{(\infty,k)}f=\partial^{(0_{2}k)}f$(degenerate$\mathbb{C}\mathbb{P}^{1}$$\partial^{(0,k)}f$

condition).

are located only on the diagonal $\{(x_{1}, \ldots, x_{n})|\exists(i,j), i\neq j, x_{i}=x_{j}\}$ , andthe orders of the poles are at most two.
(iii) The poles of(iv) For $n\geq 3,$$f\in \mathcal{A}_{n}$$!\in \mathcal{A}_{n}$satisfies the wheel conditions$f(x_{1}, q_{1}x_{1}, q_{1}q_{2}x_{1}, x_{4}, \ldots)=0$

Then we set the graded vector space,$f(x_{1}, q_{2}x_{1}, q_{1}q_{2}x_{1}, x_{4}, \ldots)=0$.

.

$\mathcal{A}=\mathcal{A}(q_{1}, q_{2}, q_{3}):=\oplus_{r\downarrow\geq 0}\mathcal{A}_{n}$Definition 1.2. For an m-variable symmetric rational function
and an n-variable symmetricrational function , we define an $(m+n)$ -variable symmetric rational function $f*g$ by
$f$

$g$

$(f*g)(x_{1}, \ldots, x_{m+n}):=$

Date: Feburary 12, 2010,

Sym$[\alpha’\beta m+1\leq\beta\leq m+n$ .

(11)

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B. FEIGIN, A. HOSHINO, J. SHIBAHARA, J. SHIRAISHI AND S. YANAGIDA

Hereis the rational function$\omega(x, y)$$\omega(x, y)=\omega(x.y;q_{1,12}r, q_{3}\backslash ):=\frac{(x-q_{1}y)(x-q_{2}y)(x-(13y)}{(x-\tau/)^{3}}$

,

(1.2)

and the symbol Syili means Sym $(f(x_{1}, \ldots, x_{n})):=(1/n!)\sum_{\sigma\in \mathfrak{S}_{n}}f(x_{\sigma(1)}, \ldots, x_{\sigma(n)})$.

is a unital
is closed with respect to , and tlre pair
Fact 1.3 ([FHHSY, Theorem 1.5]).
$assoc\cdot,iative$ commutative algebra. The Poincar6 series is $\sum_{n\geq 0}(\dim_{F}\mathcal{A}_{n})z^{n}=\prod_{m\geq 1}(1-z^{m})^{-1}$ .
$(\mathcal{A}, *)$$*$$\mathcal{A}$of symmetric functions. As for the notations and definitions concerning the1.2. The ringis a sequencebasicallyfollow the notation in [M]. A partition ofwepartitions,,.
We
defineof non-negative integers satisfying.
Webypartitionofa(transpose)
if I $=n$ . We denote the conjugate
and write$|\lambda|=|\mu|,$work with the dominanoe partial ordering defined as :$\mu\iota+\cdots+\mu$ ; for all $i\geq 1$ .
We recall some basic facts about the ring of symmetric functions. As was in [FHHSY], we set $q_{1}=$. Let be the ring of symmetric functions$q_{2}=t$ (lience $q_{3}=qt^{-1}$ ) andover the ba.se field , constructed in the category of graded ring with the projection operators$\Lambda_{F}$$\lambda=(\lambda_{1}, \lambda_{2}, \ldots)$$n\in \mathbb{N}$$|\lambda|:=\lambda_{1}+\lambda_{2}+\cdots,$ $\ell(\lambda):=\#\{i|\lambda_{i}\neq 0\}$$\lambda_{1}\geq\lambda_{2}\geq\cdots$$\lambda’$$\lambda$$\lambda\vdash n$$|\lambda$$\lambda\geq\mu\Leftrightarrow^{def}$$\mathbb{F}=\mathbb{Q}(q_{1}, q_{2})=\mathbb{Q}(q, t)$$q^{-1},$$\lambda_{1}+\cdots+\lambda_{i}\geq$$\Lambda_{F}$$\mathbb{F}$$\rho_{m_{2}n}:f(x_{1}, \ldots,x_{m})\mapsto f(x_{1}, \ldots, x_{n}, 0\ldots, 0)$.

, the monoinial
be the power sum function. For a partition
Let $p_{n}(x)$of .
permutations
distinct
all
the
,
runs
ovcr
where
by
symmetric function is defined$E(y):=\prod_{i}(1+x_{i}y)=$$e_{n}(x)$function
generating
is defined by the
The elementary symmetric function. Set $G(y):= \prod_{i}\{(tx_{i}y;q)_{\infty}/(x_{i}y;q)_{\infty}\}=\sum_{n\geq 0}g_{n}(x;q, t)y^{n}$ , wliere $(x;q)_{\infty}:= \prod_{i\geq 0}(1-$and. Siinilarly wc write
set. For a partition
.
of
form
bases
and
. It is known that
, whereRecall Macdonald’s scalar productwe haveanddenotes the number of parts in the partition . For any dual bases$\lambda=(\lambda_{1}, \lambda_{2}, \ldots)$$:= \sum_{i}x_{i}^{n}$$\lambda$$m_{\lambda}(x):= \sum_{\alpha}x^{\alpha}$$\alpha$$\sum_{n\geq 0}e_{n}(x)y^{n}$$\lambda=(\lambda_{1}, \lambda_{2}, \ldots)$$q^{i}x)$$p_{\lambda}$$\{p_{\lambda}\},$$:=g_{\lambda_{1}}g_{\lambda_{2}}\cdots$$g_{\lambda}$$e_{\lambda}$$:=p_{\lambda_{1}}p_{\lambda_{2}}\cdots$$\{m_{\lambda}\},$$\{e_{\lambda}\}$$:=e_{\lambda_{1}}e_{\lambda_{2}}\cdots$$\Lambda p$$\{g_{\lambda}\}$$\langle p_{\lambda,I\mu})\rangle_{q,t}:=\delta_{\lambda,\mu}\prod_{i\geq 1}i^{m_{i}}m_{i}!\prod_{j\geq 1}(1-q^{\lambda_{i}})’(1-t^{\lambda_{j}})$$\{v_{\lambda}\}$$\{u_{\lambda}\}$$\lambda$$i$$m_{i}$

.

.

(1.3)

$\Pi(x, y;q, t)$ $:= \prod_{i,j}\frac{(tx_{i}y_{j};q)_{\infty}}{(x_{i}\tau/j;q)_{\infty}}=\sum_{\lambda}u_{\lambda}(x)v_{\lambda}(y)$.
form dual bases, namely we haveand
It is known thatare uniquely characterized by (a) the triangular expansion
Macdonald polynomials.
if, and (b) the orthogonality
Se set$(m_{\lambda},$$\{g_{\lambda}\}$$\{m_{\lambda}\}$$g_{\mu}\rangle_{q,t}=\delta_{\lambda,\mu}$$P_{\lambda}(x;q, t)$$\langle P_{\lambda},$$m_{\lambda}+ \sum_{\mu<\lambda}a,m(a_{\lambda\mu}\in \mathbb{F})$$b_{\lambda}(q, t):=\langle P_{\lambda}(z;q, t),$Then

forms a dual basis to$\{Q_{\lambda}\}$$P_{\lambda}(z;q, t)\}_{q_{2}t}^{-1}$$\{P_{\lambda}\}$,$P_{\mu}\rangle_{q,t}=0$$P_{\lambda}=$$\lambda\neq\mu$$Q_{\lambda}(z;q, t):=b_{\lambda}(q, t)P_{\lambda}(z;q, t)$

.

(1.4)

.

. Both and are commutative rings having the saine
1.3. The isomorphism :$\sum_{n>0}(\dim_{F}\Lambda_{F}^{n})z^{n}=\sum_{n>0}(\dim_{F}\mathcal{A}_{n})z^{n}=\prod_{m>1}(1-z^{m})^{-1}$, wheredenotes the
Poincar\’e series
in [FHHSY] that there is a natural
it wa.
of degree .
ring of symmetricconstruction of the Macdonald
the
free
field
view
of
from the point ofandway to identify:
as follows.
isomorphism
operators. Based on the finding in [FHHSY] we give an$\iota$$\Lambda_{F}arrow A$$\Lambda_{F}$$\mathcal{A}$$\Lambda_{F}^{n}$$f\overline{\iota i}nctions$$\Lambda_{F}$$n$$\overline{M}orcover$$sshow\overline{n}$$\mathcal{A}$$\iota$For$\Lambda_{F}arrow \mathcal{A}$, let$p\in \mathbb{F}$$\epsilon_{n}(z_{1}, z_{2}, \ldots, z_{n};p):=\prod_{1\leq i<j\leq n}\frac{(z_{i}-pz_{j})(z_{i}-p^{-1}z_{j})}{(z_{i}-z_{j})^{2}}$and set$\epsilon\lambda(z;p)$$:=(\epsilon\lambda_{1}*\epsilon\lambda_{2}*\cdots*\epsilon\lambda_{l})(z;p)$for a multi-index

(1.5)

,

$\lambda=(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{l})$Fact 1.4 ([FHHSY, Propositions 2.20 &2.23]). For $i=1,2,3,$$\{\epsilon_{\lambda}(z;q_{i})\}_{\lambda\vdash n}$.

forms a basis of$\mathcal{A}_{n}$.

135
KERNEL FUNCTION AND QUANTUM ALGEBRASLet us write tlie expansions of$P_{\lambda}$

in the bases$P_{\lambda}(z;q, t)= \sum_{\mu\geq\lambda’}c_{\lambda_{J}}^{earrow P}(q, t)e_{l^{4}}(z;q, t)$A detailed study of the algebra
the following equality.

$\{e_{\mu}\}$,and$\{g_{\mu}\}$by$P_{\lambda}(x;q, t)= \sum_{\mu\geq\lambda}c_{\lambda\mu}^{garrow P}(q, t)g_{l^{4}}(x;q, t)$$\mathcal{A}$.

$f_{\lambda}^{(q^{-1})}(z;q, t):= \frac{t^{-|\lambda|}}{(1-t^{-1})^{|\lambda|}|\lambda|!}\sum_{\mu\geq\lambda’}c_{\lambda_{l^{l}}}^{rarrow P}(q, t)\epsilon_{\mu}(z;q)\frac{1l^{\iota|!}}{\prod_{i=1}^{\ell(\mu)}\mu_{i}!}$

Then we have,.

$:= \frac{(-1)^{|\lambda|}}{(1-q)^{|\lambda|}|\lambda|!}\sum_{\mu\geq\lambda}c_{\lambda\mu}^{garrow P}(q, t)\epsilon_{\mu}(z;t)\frac{|\mu|!}{\prod_{i=1}^{\ell(\mu)}xi!}$$f_{\lambda}^{(q^{-1})}(z;q, t)=f_{\lambda}^{(t)}(z;q, t)^{1}$

Definition 1.6. Let$F_{\lambda}(z;q, t)$$(1.\{;)$

with the help of tlre free field representation allowed us to establish$\mathcal{A}$Fact 1.5 ([FHHSY, \S 3 $E]$ ). Sct the next two elements in$f_{\lambda}^{(t)}(z;q, t)$

.

(1.8)

.

$:=f_{\lambda}^{(q^{-1})}(z;q, t)=f_{\lambda}^{(t)}(z;q, t)$

Definition 1.7. Define the isomorphism(1.7)$\iota$:.

by$\Lambda_{F}arrow \mathcal{A}$.

$\iota(e_{\lambda})=\frac{t^{-|\lambda|}}{(1-t^{-1})^{|\lambda|}}\frac{1}{\prod_{i=1}^{t(l^{l,})}\lambda_{i}!}\epsilon_{\lambda}(z;q)$

Proposition 1.8. (1) We have.

$\iota(g_{\lambda})=\frac{(-1)^{-|\lambda|}1}{(1-q)^{|\lambda|}\prod_{i=1}^{\ell(\mu)}\lambda_{i}!}\epsilon_{\lambda}(z;t)$(2) We have$\iota(P_{\lambda})=F_{\lambda}(z;q, t)$.

Proof. (1) By the Wronski relation given in [FHHSY, Proposition 3.11].
.
(2) By (1.6) and the definitions of and$\iota$$\square$$F_{\lambda}$, we recall the Gordon filtration
Remark 1.9. To explain tlie importance of the element, we (lefined a linear inap
andon . For$F_{\lambda}(z;q, t)$$\mathcal{A}$$\lambda=(\lambda_{\rfloor}, \ldots, \lambda_{l})\vdash n$$p\in \mathbb{F}$$(p)$$\varphi_{\lambda}$:$f(z_{1^{\mathcal{A}_{n}}},\ldots, z_{n})$$arrow^{\mapsto}$.

$\mathbb{F}(t/1,\ldots y_{l}.)f(y_{1},p\uparrow/1’,.$$\tau/2,pr/2,$$,p^{\lambda_{1}-1}y_{1}$$\ldots,p^{\lambda_{2}-1}y_{2}$$\uparrow/\iota,p\tau/\iota\cdots)p^{\lambda_{l}-1}\tau/l)$

,
,

(1.9)

,

for $i=1,2,3$ .
called the specialization mcip. The Gordon filtration is given by, is one diinensional and is spanned by.
Then by [FHHSY, Theorem 1.19],$\mathcal{A}_{n_{\backslash }\lambda}^{(q_{t})}$$:= \bigcap_{l^{\iota\not\leq\lambda}}ker\varphi_{\mu}^{(q_{i})}$$F_{\lambda}(z;q, t)$$\mathcal{A}_{7t}^{(q_{\mu}^{-1})}\cap \mathcal{A}_{n_{1}\mu}^{(t)}$1.4. The kernel function. Now we are ready to study the kernel function from the point of viewof the algebra$\mathcal{A}$.

Definition 1.10. Introduce$K_{n}(x, z;q, t)\in\Lambda_{\mathbb{F}}^{n}\otimes A_{n}$as$K_{n}(x, z;q, t):= \sum_{\lambda\vdash n}Q_{\lambda}(x)F_{\lambda}(z;q, t)$.

lNote that the first and second lines of Page 25 in $[FHHSY7$ contains typos and should be read as (1.7) and (1.8).

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B. FEIGIN, A. HOSHINO, J. SHIBAHARA, J. SHIRAISHI AND S. YANAGIDA

Remark 1.11. The name “kernel function’,liavc, in a suitablc completion of$coi\iota ies$from $\Pi(x, y)$ in (1.3). By Proposition 1.8 (2), we$\Lambda_{\mathbb{F}}\otimes \mathcal{A}$$\sum_{n\geq 0}K_{n}(x, z;q, t)=\sum_{\lambda}Q_{\lambda}(x)\iota(P_{\lambda}(y))$

,

where
runs over all the partitions of every non-negative integer.
component of the analogue of $\Pi(x, y)$ .
$\lambda$

Proposition 1.12. In$\Lambda_{F}\otimes \mathcal{A}$Thus$K_{n}$

is a homogeneous

we have(1.10)$K_{n}(x, z;q, t)= \frac{(-1)^{n}}{(1-q)^{n}n!}\sum_{\lambda\vdash n}m_{\lambda}(\alpha:)\epsilon_{\lambda}(z;t)\frac{|\lambda|!}{\prod^{p(\lambda)}i=1\lambda_{i}!}$

Proof. First we show
$m_{\lambda}(x)= \sum_{\mu\leq\lambda}c_{\mu\lambda}^{garrow P}(q, t)Q_{\mu}(x;q, t)$.

(1.11). Thenwith
Since $\{Q_{\mu}(x;q, t)\}$ is a basis of , we can expand, where weis calculated as
the pairing
are dual. On the other hand, by (1.6), we haveused the fact thatand
. Comparing both expressions, we obtain (1.11).
$m_{\lambda}(x)= \sum_{\nu}c_{J\text{ノ}\lambda}Q_{\nu}(x;q, t)$$\Lambda_{-}\neg\dot{\triangleleft}$$\langle m_{\lambda},$$P_{\mu}\rangle_{q,t}$$\{P_{\lambda}\}$$\langle m_{\lambda},$$\langle m_{\lambda},$$c_{\nu\lambda}\in \mathbb{F}$$P_{\mu} \}_{q,\ell}=\langle\sum_{l\text{ノ}}c_{\iota \text{ノ}\lambda}Q_{\nu}(z;q, t),$ $P_{\mu}\rangle_{q,t}=c_{\mu\lambda}$$\langle m_{\lambda},$$\{Q_{\lambda}\}$$P_{\mu}\}_{q,t}=$$\sum_{\nu\geq\mu}c_{\mu \text{ノ}}^{9^{arrow P},}(q, t)g_{\mu}\}_{q,t}=c_{\mu\lambda}^{garrow P}(q, t)$

Then we have

RHS of$(1.10)= \frac{(-1)^{n}}{(1-q)^{n}n!}\sum_{\lambda\vdash n}\sum_{\mu\leq\lambda}c_{\mu\lambda}^{garrow P}(q, t)Q_{\iota}(x;q, t)\epsilon_{\lambda}(z;t)\frac{|\lambda|!}{\prod_{i=1}^{\ell(\lambda)}\lambda_{i}!}$.

$= \frac{(-1)^{n}}{(1-q)^{n}n!}\sum_{l\vdash n}Q_{\mu}(x;q, t)\sum_{\lambda\geq\mu}c_{/\iota\lambda}^{garrow P}(q, t)\epsilon_{\lambda}(z;t)\frac{|\lambda|!}{\prod_{i=1}^{\ell(\lambda)}\lambda_{i}!}=\sum_{\prime\iota\vdash n}Q_{/\iota}(x;q, t)F_{\mu}(z;q, t)$$\square$Consider the case of finitely many variables and set $x=(x_{1}, x_{2}, \ldots, x_{m})$ . Also let $z=(z_{1}, z_{2}, \ldots, z_{n})$.
be the set of variables for the elements in$\mathcal{A}_{n}$Proposition 1.13. We have$K_{n}(x, z;q, t)= \frac{(-1)^{n}}{(1-q)^{n_{7t}}!}\sum_{i_{1}=1}^{m}\sum_{i_{2}=1}^{m}\cdots\sum_{i_{n}=1}^{m}x_{i_{1}}x_{i_{2}}\cdots x_{i_{\tau\iota}}\prod_{1\leq\alpha<\beta\leq n}\gamma_{i_{\alpha},i_{9}},(z_{\alpha’\beta]}z;c, t)$where the function$\gamma_{i_{2}j}(z, w;q, t)$,(1.12)

is given by$i=j$ ,$\frac{\frac{(z-tw)(z-t^{-1}w)}{(z-q^{-1}w)(z-tw)(z-w)^{2}(z-w)}(z-qt^{-1},w)}{3}$$i<j$ ,$\gamma_{i,j}(z, w;q, t):=\{$$\frac{(z-qw)(z-t^{-1}w)(z-q^{-1}tw)}{(z-w)^{3}}$(1.13)$i>j$ .

Proof. Note that we have$\gamma_{i.j}(z, w;q, t)=\{\begin{array}{ll}\epsilon_{2}(z, w;t) i=j,\omega(z, w;q^{-1}, t, qt^{-1}) i<j,\omega(z, w;q, t^{-1}, q^{-1}t)=\omega(w, z;q^{-1}, t, qt^{-1}) i>j,\end{array}$(1.14)

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KERNEL FUNCTION AND QLANTIJM ALGEBRAS

whidz is obtained from (1.2), (1.5) and (1.13). Thus we have$\sum_{i_{1}=1}^{m}\sum_{i_{2}=1}^{m}\cdots\sum_{i_{?l}=1}^{m}x_{i_{1}}x_{i_{3}}\cdots x_{i_{n}}\prod_{j},\gamma_{i_{\alpha},i_{3}}(z_{\alpha}, z_{\beta};q, t)\iota\leq\alpha<\leq n$$= \sum_{I_{1},\ldots,I_{l?l}}x_{1}^{c\iota 1}x_{2}^{(l2}\cdots x_{m}^{a_{m}}\prod_{k=1}^{?7t}\epsilon_{a_{k}}(zI_{k};t)\prod_{1\leq i<j\leq m}\prod_{\alpha\in I_{i},\beta\in I_{j}}\omega(z, z;q^{-1}, t, qt^{-1})$

where $I_{k}(k=1,2, \ldots, m)$ is a subset of 1, 2,
Using the multi-index notation $a=$

$n\}$

$\{$

$\ldots,$,such that $|I_{k}|=a_{k},$ $I_{1}\cup I_{2}\cup\cdots\cup I_{m}=\{1, \ldots, n\}$ .
, we have$(a_{1}, \ldots , a_{?n})\in \mathbb{N}^{m}$$= \sum_{a\in \mathbb{N}^{m},|a|=n}x^{(b}\frac{7\iota!}{\prod_{k=1}^{m}a_{k}!}\epsilon_{(\iota}(z;t)$

with

$|a|$

$:=a_{1}+\cdots+a_{m}$. Applying$\mathfrak{S}_{m}$on the running index$a$and averaging them, we have$= \frac{1}{n!}\sum_{\neg\sigma\in t_{m}^{\sim}}\sum_{a\in \mathbb{N}^{m})|a|=n}x^{\sigma(a)}\frac{n!}{\prod_{k=1}^{m}a_{k}!}\epsilon_{\sigma(a)}(z;t)$Dividing$\mathfrak{S}_{m}$by the stabilizer Stab $(a)$ of$= \frac{1}{n!}\sum_{a\in N^{m},|a|=n}$

#Stab

.

and using the commutativity of$a\in \mathbb{N}^{m}$$\mathcal{A}$, we have.

$(a) \frac{n!}{\prod_{k=1}^{m}a_{k}!}(\sum_{\overline{\sigma}\in 6_{m}/Stab(a)}x^{\overline{\sigma}(a)})\epsilon_{o}(z;t)$Then we obtain the result by taking a partition$\lambda$

as the running index.

$\square$1.5. Macdonald’s tableau sum formula. We recall the tableau sum forniula for the Macdonald

olynomials.
Let Tbdenotes the set of all tlie ways of drawing numbers 1, 2, . . . , $m$ into the Young diagramof shape without any conditions. Reading the numbers from left to right then top to bottom, namely
in the English reading manner, we get a bijection between Tb
and the set 1, 2, $\ldots)m\}^{n}$ .
Let RTb
denotes the subset of Tb
in which the numbers in each row are arranged
in non-decreasing manner. The element of RTb
is uniquely specified by the set of numbers
which denote the number of in the i-th row. We havefor $1\leq i\leq n$ . Next
we introduce a sequence. It is clear that we haveby setting. Note that
may not be a partition.
Let SSTbbe the set of semi-standard $tablea\iota ix$ . A semi-standard tableau $T$ is expressed asa sequence of partitions, where the skew diagrams$(k=1,2, \ldots, m)$ are horizontal strips. We havefor $i>j,$
for $1\leq i\leq n$ , and$p$$(\lambda;m)$$\lambda$$(\lambda;m)$$(\lambda;m)$$\{$$(\lambda;m)$$(\lambda;m)$$j$$\theta_{i_{2}j}$$\lambda_{i}=\sum_{k=1}^{n}\theta_{i,k}$$\lambda^{(j)}=(\lambda_{1}^{(j)}, \lambda_{2}^{(j)}, \ldots)$$(l=\lambda^{(0)}\subset\lambda^{(1)}\subset\cdots\subset\lambda^{(m)}=\lambda$$\lambda_{i}^{(j)}:=\sum_{k\cdot=1}^{j}\theta_{i,k}$$\lambda^{(j)}$$(\lambda;m)$$\emptyset=\lambda^{(0)}\subset\lambda^{(1)}\subset\cdots\subset\lambda^{(m)}=\lambda$$\theta_{i,j}=0$$\lambda^{(k)}\lambda^{(k-1)}$$\lambda_{i}=\sum_{k=i}^{n}\theta_{i,k}$$0 \leq\theta_{i,j}\leq\lambda_{i}-\lambda_{i+1}-\sum_{k=j+1}^{\ell(\lambda)}(\theta_{i_{2}k}-\theta_{i+1,k})$for $1\leq i<j\leq\ell(\lambda)$ .
It is known that the$b_{\lambda}(q, t)$in (1.4) has the factorized form.

$Q_{\lambda}(x;q, t)=b_{\lambda}(q, t)P_{\lambda}(x;q, t)$,$b_{\lambda}(q, t)= \prod_{s\in\lambda}\frac{1-q^{a(s)}t^{\ell(s)+1}}{1-q^{a(s)+1}t^{\ell(s)}}$,(1.15)

where for a box $s=(i,j)$ of $a(s):=\lambda_{i}-j$ is the arm-length and $\ell(s):=\lambda_{j}’-i$ is the leg-length.
The
has the tableau sum formula:$\lambda,$$P_{\lambda}(x;q, t)$$P_{\lambda}(x;q, t)= \sum_{T\in SSTb(\lambda;m)}x^{T}\psi_{T}(q, t)$.

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B. FEIGIN, A. HOSHINO, J. SHIBAHARA, J. SHIRAISHI AND S. YANAGIDA

Here the coefficient$’\psi(q, t)\in \mathbb{F}$is determined by$\psi_{T}(q, t):=\prod_{k=1}^{m}\psi_{\lambda^{(k)}’\lambda^{(k-1)}}(q, t)$,$\prod^{7l}$$\frac{f(q^{1_{\dot{i}}-l^{\iota_{j}}}t^{j-i})f(q^{\lambda_{j}-\lambda_{j+1}}t^{j-i})}{f(q^{\lambda-/\iota_{j}}t^{j-i})f(q^{\mu_{i}-\lambda_{j+1}}t^{j-i})}$$\psi_{\lambda/\mu}(q, t):=$(1.16).

$f( \uparrow\iota):=\frac{(tu;q)_{\infty}}{(qu;q)_{\infty}}$.

$1\leq i\leq j\leq l(l^{J})$The next proposition is obtained by simple combinatorics, and we omit the proof for lack of space.

and regard
SSTb
Proposition 1.14. Let $T\in$ RTb(1.16)
vanishes.
calculated fromabove. Then$(\lambda;m)$$(\lambda;m)\backslash$$T$as a sequence

explained as$\lambda^{(j)}$$\psi_{T}(q, t)$

1.6. Tableau sum formula and $K_{n}(x, z;q, t)$ . Now we investigate the relationship between thefunction $K_{n}(x, z;q, t)$ and the tableau formula of Macdonald polynomial. We fix a natural number
and consider the case $x=(x_{1}, \ldots, x_{m})$ .
In order to state the main result, we need to consider the composition of the specialization mapsby, we define the mapof andof (1.9). For a partition$77l$$\tilde{\varphi}_{\lambda}^{\{\zeta)}$$n$$\lambda=(\lambda_{1}, \ldots, \lambda_{l})$$\varphi_{\lambda}^{(p)}$$(\zeta)$$\triangleleft\zeta)$$\varphi_{\lambda}$$(q^{-1})_{;}$$\mathbb{F}(z_{1}, \ldots, z_{n})$$arrow$$f(z_{1}, \ldots, z_{n})$$\mapsto$$:=\varphi_{(l)}\circ\varphi_{\lambda}$$\zeta\in \mathbb{F}$$\mathbb{F}(y)$$f(y,$ $q^{-1}y,$ $\ldots,$ $q^{-(\lambda_{1}-1)}y$ ,$\zeta y,$$\ldots,$$C^{l-1}y,$ $q^{-1}\zeta^{l-1}y,$Here the map$\varphi_{(l)}^{(\zeta)}$denotes the substitution

Theorem 1.15. For partitions$l^{\iota,\lambda}$of$q^{-(\lambda_{l}-1)}\zeta^{l-1}y)$$\ldots,$$\varphi_{(l)}^{(\zeta)}g(y_{1}, \ldots, y_{l})=g(y, \zeta y, \ldots, \zeta^{l-1}y)$$n,\tilde{\varphi}_{\lambda}^{\langle\zeta)}(F_{\mu}F_{\lambda})$Our proof uses the tableau sum formula of(1.17),$q^{-(\lambda_{2}-1)}\zeta y$$q^{-1}\zeta y,$.

.

is regular at $\zeta=t$ and its value is$P_{\lambda}(x;q, t)$$\delta_{\lambda,\mu}$.

. Let us express the statement as.

$\lim_{\zetaarrow t}\tilde{\varphi}_{\lambda}^{(\zeta)}\frac{F_{\mu}(z;q,t)}{F_{\lambda}(z;q,t)}=\delta_{\lambda,\mu}$Then by using Proposition 1.12, it can be rewritten into the next equivalent form.
$\lim_{\zetaarrow t}\varphi_{\lambda}^{\zeta)}\triangleleft\frac{K_{n}(x,z;q,t,)}{F_{\lambda}(z;q,t)}=Q_{\lambda}(x;q, t)$

Regard $T=(i_{1}, i_{2}, \ldots, i_{n})\in\{1,2, \ldots, m\}^{n}$ as an element of Tb$\gamma_{T}(z):=\prod_{1\leq\alpha<\beta\leq n}\gamma_{i_{\alpha},i_{\beta}}(z_{\alpha\beta}z;q, t)$

We also use the same symbol for the cases $T\in$ RTb
1.12, (1.18) is equivalent to$(\lambda;7\gamma\iota)$.

(1.18)$(\lambda;m)$. For simplicity we set.

and $T\in$ SSTb$(\lambda;m)$. By Proposition.

$\frac{(-1)^{n}}{(1-q)^{n}n!}\sum_{T\in Tb(\lambda;m)}x^{T}\lim_{\zetaarrow t}\tilde{\varphi}_{\lambda}^{(\zeta)}\frac{\gamma_{T}(z)}{F_{\lambda}(z;q,t)}=Q_{\lambda}(x;q, t)$tliat all the terms with $T\in$ Tb
It is easy to see from the definition ofby. Thus we may replace
after the first specialization$T\in
RTb(\lambda;m)$
we
have
for
show
that
Hence it is enough to$\gamma_{i,j}$$\varphi_{\lambda}^{(q^{-1})}$

We prove this in two steps.

RTb$\sum_{T\in RTb(\lambda;m)}$$\sum_{T\in Tb(\lambda;m)}$$\frac{(-1)^{n}}{(1-q)^{n}n!}\lim_{\zetaarrow t}\tilde{\varphi}_{\lambda}^{(\zeta)}\frac{\gamma_{T}(z)}{F_{\lambda}(z;q,t)}=b_{\lambda}(q, t)\psi_{T}(q, t)$$(\lambda;m)\backslash$.

$(\lambda;m)$.

vanish139
KERNEL FUNCTION AND QUANTUM ALGEBRAS

Proposition 1.16. Let $D\in$ SSTb$(\lambda;m)$

givcn by$\theta_{i,i}=\lambda_{i}$and$\theta_{i,j}=0$$\frac{(-1)^{n}}{(1-q)^{n}n!}\lim_{\zetaarrow t}\varphi_{\lambda}\triangleleft()\frac{\gamma_{D}(z)}{F_{\lambda}(z;q,t)}=b_{\lambda}(q, t)$$\lim_{\zetaarrow t}\tilde{\varphi}_{\lambda}^{(\zeta)}\frac{\gamma_{T}(z)}{\gamma_{D}(z)}=\psi_{T}(q, t)$

for $i\neq j$ . Then we have
(1.19),.

(1.20)

Proof. The proof is postponed until 3.1.

$\square$

2. DING-IOHARA ALGEBRA AND KERNEL FUNCTION. We will also use $p:=q/t$ .
In this section all objects are defined on$\tilde{\mathbb{F}}:=\mathbb{Q}(q^{12}, t^{1\prime 2})$.

Recall that the Ding-Iohara algebra [DI] was
2.1. Review of the Ding-Iohara algebra
introduced as a generalization of the quantum affine algebra, wliich respects tlie structure of theof the DingDrinfeld coproduct. In [FHHSY, Appendix ], the authors introduced a version
Iohara algebra having two parameters and .
$\mathcal{U}(q, t)$$A$$\mathcal{U}(q, t)$

$t$

$q$

Definition 2.1. Set$g(z):= \frac{G^{+}(z)}{G^{-}(z)}$Then we define$\mathcal{U}(q, t)$,$G^{\pm}(z):=(1-q^{\pm 1}z)(1-t^{\mp 1}z)(1-q^{\mp 1}t^{\pm 1}z)$.

to be a umital associative algebra generated by the Drinfeld currents$x^{\pm}(z)= \sum_{n\in Z}x_{n}^{\pm}z^{-n}$and the central element$\gamma^{\pm 1/2}$,$\psi^{\pm}(z)=\sum_{\pm n\in \mathbb{N}}\psi_{n}^{\pm}z^{-n}$,, satisfying the defining relations$\psi^{\pm}(z)\psi^{\pm}(w)=’\psi^{\pm}(w)\psi^{\pm}(\nearrow_{\vee})$,$\psi^{+}(z)\psi^{-}(w)=\frac{g(\gamma^{+1_{t}}0’ z)}{g(\gamma^{-1}w’ z)}\psi^{-}(w)\psi^{+}(z)$$\psi^{+}(z)x^{\pm}(w)=g(\gamma^{\mp 1\prime 2}w\prime z)^{\mp 1}x^{\pm}(u;)\psi^{+}(z)$,,$\psi^{-}(z)x^{\pm}(w)=g(\gamma^{\mp 12}zw)^{\pm 1}x^{\pm}(’|/’)\psi^{-}(z)$$[x^{+}(z), x^{-}(w)]= \frac{(1-q)(1-1t)}{1-q’ t}(\delta(\gamma^{-1}zw)\psi^{+}(\gamma^{1’ 2}w)-\delta(\gamma zw)\psi^{-}(\gamma^{-1/2}w))$$G^{\mp}(z/w)x^{\pm}(z)x^{\pm}(w)=G^{\pm}(zw)x^{\pm}(w)x^{\pm}(z)$,.

Fact 2.2 ([FHHSY, Proposition A.2]). The algebra
Coproduct :$\mathcal{U}(q, t)$ha.s a Hopf algebra structure with$\Delta$$\Delta(\gamma^{\pm 1’ 2})=\gamma^{\pm 1/2}\otimes\gamma^{\pm 1\prime 2}$,,$\triangle(x^{+}(z))=x^{+}(z)\otimes 1+\psi^{-}(\gamma_{(1)}^{1\prime 2}z)\otimes x^{+}(\gamma_{(1)}z)$,$\triangle(\psi^{\pm}(z))=\psi^{\pm}(\gamma_{(2)}^{\pm 1/2}z)\otimes’\psi^{\pm}(\gamma_{(1)}^{\mp 1’ 2}z)$where$\gamma_{(1)}^{\pm 1’ 2}=\gamma^{\pm 1’ 2}\otimes 1$

and$\Delta(x^{-}(z))=x^{-}(\gamma_{(2)}z)\otimes\psi^{+}(\gamma_{(2)}^{12}z)+1\otimes x^{-}(z)$$\gamma_{(2)}^{\pm 1\prime 2}=1\otimes\gamma^{\pm 1’ 2}$.

Counit :$\epsilon$$\epsilon(\gamma^{\pm 1’ 2})=1$,$\epsilon(’\psi^{\pm}(z))=1$,$\epsilon(x^{\pm}(z))=0$.

Antipode :$a$$a(\gamma^{\pm 1’ 2})=\gamma^{\mp 1/2}$,$a(\psi^{\pm}(z))=\psi^{\pm}(z)^{-1}$$a(x^{+}(z))=-\psi^{-}(\gamma^{-1\prime 2}z)^{-1}x^{+}(\gamma^{-1}z)$,$a(x^{-}(z))=-x^{-}(\gamma^{-1}z)\psi^{+}(\gamma^{-1\prime 2}z)^{-1}$

,
.

,

,

140
B. FEIGIN, A. HOSHINO, J. SHIBAHARA, J. SHIRAISHI AND S. YANAGIDA. We a ropresentation of
2.2. Level one representation ofcentral element is realized by the constant $(t/q)^{k/2}=p^{-k’ 2}$ .
$\mathcal{U}(q, f)$

is of level$\mathcal{U}(q, t)$$sav\backslash$$k$, if the$\gamma$Fact 2.3 ([FHHSY, Proposition A.6]). Consider the Heisenberg Lie algebra
and the relations
generators$\mathfrak{h}$overwith the$\mathbb{F}$$a_{n}(n\in \mathbb{Z})$$[a_{m}, a_{n}]=m \frac{1-(J^{|m|}}{1-t^{|r\int\downarrow|}}\delta_{n+n,0}a_{0}$

.

(2.1)

for $n\geq 0$ (rosp. $n<0$ ). Consider the) be the subalgcbra generated by(rcsp.
acts by some fixed$a_{?1}(n>0)$
acts trivially and, whereone dimensional representation ofof . Let us also introduce
element of . Then one has the induced Fock representation
the following four vertex operators FHHSY, $(1.7),(3.23),(3.27),(3.28)]$ .
Let$\mathfrak{h}^{<0}$$\mathfrak{h}\geq 0$$a_{?1}$$\tilde{\mathcal{F}}$$\mathfrak{h}\geq 0$$a_{0}$$\tilde{\mathcal{F}}$$\mathcal{F}:=Ind_{\mathfrak{h}\geq 0}^{b}\tilde{\mathbb{F}}$$\mathfrak{h}$$[$,$\eta(z):=\exp(\sum_{n>0}\frac{1-t^{-n}}{7l}a_{-n}z^{n})\exp(-\sum_{n>0}\frac{1-t^{?\iota}}{n}a_{n}z^{-n})$$\xi(z):=\exp(-\sum_{n>0}\frac{1-t^{-n}}{n}p^{-n/2}a_{-n}z^{n})\exp(\sum_{?l>0}\frac{1-t^{r\iota}}{n}p^{-n’ 2}a_{n}z^{-n})$$\varphi^{+}(z):=\exp(-\sum_{n>0}\frac{1-t^{n}}{n}(1-p^{-n})p^{n/4}a_{n}z^{-n})$$\varphi^{-}(z):=\exp(\sum_{n>0}\frac{1-t^{-n}}{n}(1-p^{-n})p^{\uparrow\iota’\cdot 1}a_{-n}z^{n})$Then for a fixed$c\in\tilde{\mathbb{F}}^{\cross}$, we llave a level one representation$\rho_{c}(\gamma^{\pm 1’ 2})=p^{\mp 1\prime 4}$,$\rho_{c}(\psi^{\pm}(z))=\varphi^{\pm}(z)$

,

,

.

$\rho_{c}(\cdot)$

of

,

on

$\mathcal{U}(q, t)$$\rho_{c}(x^{+}(z))=c\eta(z)$Remark 2.4. We can rephrase this fact as follows. Let us define$\psi^{+}(z)=’\psi_{0}^{+}\exp(+\sum_{n>0}b_{l}\gamma^{n’ 2}z^{-n})$,$b_{n}$,

by setting$\mathcal{F}$$\rho_{c}(x^{-}(z))=c^{-1}\xi(z)$’s by the expansion of$\psi^{-}(z)=\psi_{0}^{-}\exp(-\sum_{n>0}b_{-n}\gamma^{n/2}z^{n})$.

$\psi^{\pm}$

.

:
(2.2)

Then we have$[b_{m}, b_{n}]= \frac{1}{m}(1-q^{-m})(1-t^{m})(1-p^{m})(\gamma^{?n}-\gamma^{-m})\gamma^{-|n\iota|}\delta_{m+n,0}$

and the coproduct for$b_{n}$

(2.3)

reads
$\triangle(b_{n})=b_{n}\otimes\gamma^{-|n|}+1\otimes b_{n}$Then the representation,

is given )$1$$\rho_{r}$$y\gamma^{\pm 12}\mapsto p^{\mp 1\prime 4}$$b_{n} \mapsto-\frac{1-t^{n}}{|n|}(p^{|n|/2}-p^{-|n|’ 2})a_{n}$,$\psi_{0}^{\pm}\mapsto 1$

,

.

(2.4)

an(1
$x^{+}(z)\mapsto c\eta(z)$Definition 2.5. Consider the m-fold tensor representationinductively by
Define,$x^{-}(z)\mapsto c^{-1}\xi(z)$$\rho_{y_{1}}\otimes\cdots\otimes\rho_{y_{m}}$on$\mathcal{F}^{\otimes r)l}$for.

$m\in \mathbb{Z}_{\geq 2}$.

$\triangle^{(m)}$$\Delta^{(2)}$Since we have$:=\triangle$,$\Delta^{(rn)}$

$:=$

$($

id

$\otimes\cdots\otimes$id$\otimes\triangle)\circ\triangle^{(m-1)}$.

, the level is $m$ . We also define$\rho_{y_{1}}\otimes\cdots\otimes\rho_{y_{m}}\triangle^{(m)}(\gamma)=\gamma_{(1)}\cdots\gamma_{(m)}=p^{-m/2}$$\rho_{y}^{(m)}:=\rho_{y_{1}}\otimes\cdots\otimes\rho_{\tau/m}\circ\Delta^{(n\iota)}:\mathcal{U}(q, t)arrow \mathcal{F}^{\otimes m}$

.

(2.5)

141
KERNEL FUNCTION AND QUANTUM ALGEBRAS

Lemma 2.6. We have$\rho_{y}^{(rn)}(x^{+}(z))=\sum_{i=1}^{7n}y_{i}\tilde{\Lambda}_{i}(z)$where the$\tilde{\Lambda}_{i}(z)$$\tilde{\Lambda}_{i}(z),\tilde{\Lambda}_{i}^{*}(z)$,$\rho_{t/}^{(m)}(x^{-}(z))=\sum_{i=1}^{m}y_{i}^{-1}\tilde{\Lambda}_{i}^{*}(z)$,are defined to be,$:=\varphi^{-}(p^{-1’ 4}z)\otimes\varphi^{-}(p^{-3/4}z)\otimes\cdots\otimes\varphi^{-}(p^{-(2i-3)/4}z)\otimes\eta(p^{-(i-1)\prime 2}z)\otimes 1\otimes\cdots\otimes 1$

(2.7)

,

$\tilde{\Lambda}_{i}^{*}(z):=1\otimes\cdots\otimes 1\otimes\xi(p^{-(m-i)\prime 2}z)\otimes\varphi^{+}(p^{-(271l-2i-1)’ 4}z)\otimes\cdots\otimes\varphi^{+}(p^{-14}z)$where$\eta(p^{-(i-1)\prime 2}z)$(2.6)

and $\xi(p^{-(m-i)’ 2}z)$ sit in the i-th tensor component.

Proof. By the definition (2.5), Fact 2.3 and Remark 2.4.
2.3. New currents $t(z)$ and$t^{*}(z)$$\square$.

Definition 2.7. We define$t(z)$ $:=\alpha(z)x^{+}(z)\beta(z)$,$t^{*}(z)$$:=\alpha(p^{-1}z)^{-1}x^{-}(p^{-1}\gamma^{-1}z)\beta(\gamma^{-2}p^{-1}z)^{-1}$

.

(2.8)

Here we used auxiliary vertex operators$\alpha(z):=\exp(-\sum_{n=1}^{\infty}\frac{1}{\gamma^{n}-\gamma^{-n}}b_{-n}z^{n})$Here the part 1$(\gamma^{n}-\gamma^{-n})$,$\beta(z):=\exp(\sum_{n=1}^{\infty}\frac{1}{\gamma^{n}-\gamma^{-n}}b_{n}z^{-n})$

is considered to be the formal power sum

Remark 2.8. The definition of$t^{*}(z)$$\sum_{j=0}^{\infty}\gamma^{-(2i+1)n}$

.

(2.9)

.

can be read as$t^{*}(\gamma pz)=\alpha(\gamma z)^{-1}x^{-}(z)\beta(\gamma^{-1}z)^{-1}$.

This form is convenient in the actual calculations.

and$[t(z), \alpha(w)]=[t(z), \beta(w)]=[t^{*}(z), \alpha(u)]=[t^{*}(z), \beta(w)]=0$ ,Proposition 2.9. (1) The elements $t(z)$ and$t^{*}(z)$commutes with$[t(z), \psi^{\pm}(w)]=[t^{*}(z), \psi^{\pm}(w)]=0$$\alpha(w),$$\beta(w)$$\psi^{\pm}(w)$:.

(2) Set(2.10)$A(z)$ $:= \exp(\sum_{n=1}^{\infty}\frac{1}{n}\frac{(1-q^{n})(1-t^{-n})(1-p^{-n}\gamma^{-2n})}{1-\gamma^{-2n}}z^{n})$ ,where the part 1 $(1-\gamma^{-2n})$ is considered to be the formal power sum$\sum_{i=0}^{\infty}\gamma^{-2in}$. Then we have

$A( \frac{w}{z})t(z)t(w)-A(\frac{z}{w})t(w)t(z)=\frac{(1-q)(1-t^{-1})}{1-p}[\delta(p^{-1}\frac{w}{z})t^{(2)}(z)-\delta(p\frac{w}{z})t^{(2)}(w)]$,$A( \frac{w}{z})t^{*}(z)t^{*}(w)-A(\frac{z}{w})t^{*}(w)t^{*}(z)=\frac{(1-q^{-1})(1-t)}{1-p-1}[\delta(p\frac{w}{z})t^{*(2)}(z)-\delta(p^{-1}\frac{w}{z})t^{*(2)}(u)]$(2.11),(2.12)

where $\delta(z):=\sum_{n\in N}z^{n}+z^{-1}\sum_{n\in N}z^{-n}$ is the formal delta function, and$t^{(2)}(z):=\alpha(pz)\alpha(z)x^{+}(pz)x^{+}(z)\beta(pz)\beta(z)$,$t^{*(2)}(z):=\alpha(\gamma pz)^{-1}\alpha(\gamma z)^{-1}x^{-}(pz)x^{-}(z)\beta(\gamma^{-1}pz)^{-1}\beta(\gamma^{-1}z)^{-1}$.

(3) As in (2), set$B(z)$ $:= \exp(\sum_{n=1}^{\infty}\frac{1}{n}\frac{(1-q^{n})(1-t^{-n})(p^{-2n}\gamma^{-271}-p^{-n}\gamma^{-2n})}{1-\gamma^{-2n}}z^{n})$.

(2.13)

142
B. FEIGIN, A. HOSHINO, J. SHIBAHARA, J. SHIRAISHI$A\backslash \tau D$

S. YANAGIDA

Then$B( \frac{w}{z})t(z)t^{*}(w)-B(\gamma^{2}p^{2}\frac{z}{w})t^{*}(w)t(z)=\frac{(1-q)(1-t^{-1})}{1-p}(\delta(p^{-1}\frac{w}{z})\psi_{0}^{+}-\delta(\gamma^{-2}p^{-1}\frac{w}{z})\psi_{0}^{-}))$

Proof. See \S 3.2.

.

(2.14)$\square$In the next subsection we sliow tliat the currents $t(z),$.
deformed
algebra in the Fock representation of$\mathcal{W}$$t^{*}(z)$are connected to the realization of$\mathcal{U}(q, t)$

2.4. Deformed algebra. We basically follow the description ofin [FF, \S 4]. As
for the coimection between the singular vectors of theand the Macdonald polynomials,
see [SKAO, AKOS].
$\mathcal{W}_{q,p}(\epsilon t_{m})$$\mathcal{W}_{q,p}(\mathfrak{s}1_{m})$$\mathcal{W}_{q,p}(\epsilon 1_{m})$Definition 2.10. Set.

$f_{k_{2}l}(z):= \exp(\sum_{n=1}^{\infty}\frac{(1-q^{n})(1-t^{-n})(p^{(k-1)n}-p^{(l-1)n})}{1-p^{ln}}z^{n})$Remark 2.11. Our functions $A(z)$ and $B(z)$ give special cases of this function under$\rho_{y}^{(m)}(A(z))=f_{1,m}(z)$,$\rho_{y}^{(m)}(B(z))=f_{m-1,m}(z)$$\rho_{y}^{(m)}$, that is,.

Definition 2.12. Set$T(z)=T_{1}(z)$ $:=\rho_{y}^{(m)}(t(z))$ ,$T^{*}(z)=T_{1}^{*}(z)$ $:=p_{y}^{(m)}(t^{*}(z))$.

(2.15)Let us also define$\Lambda_{i}(z):=\rho_{y}^{(m)}(\alpha(z))\tilde{\Lambda}_{i}(z)\rho_{y}^{(m)}(\beta(z))$,$\Lambda_{i}^{*}(z):=\rho_{y}^{(m)}(\alpha(p^{-1}z)^{-1})\tilde{\Lambda}_{i}^{*}0^{y^{(m-2)’ 2}}z)\rho_{y}^{(m)}(\beta(\gamma^{-2}p^{-1}z)^{-1})$.

(2.16)

Then by Definition 2.7 aiid Lemma 2.6 we have$T_{1}(z)= \sum_{i=1}^{m}\iota_{i}/\Lambda_{i}(z)$For $i=2,$$\ldots,$$m$,$T_{1}^{*}(z)= \sum_{i=1}^{m}y_{i}^{-1}\Lambda_{i}^{*}(z)$.

(2.17), we further define$T_{i}(z):= \sum_{1\leq j_{1}<\cdots<j_{i}\leq m}y_{j_{1}}y_{j_{2}}\cdots y_{j_{i}}:\Lambda_{j_{1}}(z)\Lambda_{j_{2}}(zp)\cdots\Lambda_{j}.(zp^{i-1}):$$T_{i}^{*}(z)$$:= \sum_{1\leq j_{1}<\cdots<j,\leq m}\uparrow J_{j_{1}j_{2}}^{-1-1}r/\cdots\{J_{j_{i}}^{-1}$

Proposition 2.13. (1) The operator product of:,$\Lambda_{j_{1}}^{*}(z)\Lambda_{j_{2}}^{*}(zp^{-1})\cdots\Lambda_{j_{2}}^{*}(zp^{-i+1})$$\Lambda_{i}(z)$and(2.18):.

(2.19)

is given by$\Lambda_{j}(u’)$(2.20)$f_{1,\eta\iota}( \frac{w}{z})\Lambda_{i}(z)\Lambda_{j}(w)=:\Lambda_{i}(z)\Lambda_{j}(w):\cross\{\begin{array}{ll}1 i=j,\gamma+(z, w;q,p) i<j,\gamma_{-}(z, w;q,p) i>j.\end{array}$Here we used the symbol$\gamma+(z, w;q, t)$ $:= \frac{(z-q^{-1}w)(z-qt^{-1}w)}{(z-w)(z-t^{-1}w)}$,$\gamma_{-}(z, w;q, t)$ $:= \frac{(z-qw)(z-q^{-1}tw)}{(z-w)(z-tw)}$(2) We have$:\Lambda_{1}(z)\Lambda_{2}Q)z)\cdots\Lambda_{m}(p^{m-1}z):=1$

Therefore $T_{m}(z)=y_{1}y_{2}\cdots?Jm$ .

.

.

(2.21)

143
KERNEL $FU\backslash \tau CTION$ AND QUANTUM ALGEBRAS(3) The$\Lambda_{i}(z)$and$\Lambda_{j}^{*}(z)$are connected by the following equation.
:.

$\Lambda_{k}^{*}(z)=:\prod_{1i=}^{k-1}\Lambda_{i}(p^{k-1}z)\prod_{l=k+1}^{rtb}\Lambda_{l}(p^{l-2}x)$(2.22)

Thus we also Iiave$T_{1}^{*}(z)=y_{1}^{-1}?\cdots\uparrow T_{m-1}(z)$(4) The operator product of$\Lambda_{i}^{*}(z)$and$\Lambda_{j}^{*}(w)$.

(2.23)

is given}$)y$(2.24)$f_{1,m}( \frac{w}{z})\Lambda_{i}^{*}(\nearrow_{\vee})\Lambda_{j}^{*}(w)=:\Lambda_{\dot{x}}^{*}(z)\Lambda_{j}^{*}(w):\cross\{\begin{array}{ll}1 i=j,\gamma_{-}(z, w;q,p) i<j,\gamma+(z, w;q,p) i>j.\end{array}$(5) We have$:\Lambda_{1}^{*}(z)\Lambda_{2}^{*}(p^{-1}z)\cdots\Lambda_{m}^{*}(p^{-m+1}z):=1$

Therefore$T_{m}^{*}(z)=y_{12J_{m}^{-1}}^{-1-1}?/\cdots\tau$.

.

Proof. See \S 3.3.

$\square$Proposition 2.14. We have$f_{1,m}( \frac{w}{z})T_{1}(z)T_{i}(w)-f_{1,m}(p^{1-i}\frac{z}{w})T_{i}(w)T_{1}(z)=\frac{(1-q)(1-t^{-1})}{1-p}[\delta(p^{-1}\frac{w}{z})T_{i.+1}(z)-\delta(j)^{i}\frac{w}{z})T_{i+1}(w)]$,(2.25)$f_{1,m}( \frac{w}{z})T_{m-1}(z)T_{m-1}(w)-f_{1,m}(\frac{z}{w})T_{m-1}(w)T_{m-1}(z)$$= \frac{(1-q^{-1})(1-t)}{1-p-1}[\delta(p\frac{w}{z})T_{2}^{*}(z)-\delta(p^{-1}\frac{u:}{\sim\gamma})T_{2}^{*}(w)]$(2.26).

Proof. (2.25) follows from (2.17), (2.18) and (2.20). See [FF, Theorem 2] for detail2.
(2.26) is also shown by the same method using (2.23), (2.19) and and (2.24).

2.5. Deformed$\mathcal{W}$$\square$algebra and kernel function. Our final consequence of this paper relates thewith the finite kernel function.

vacuum expectation values of the deformed algebra$\mathcal{W}_{q,p}$Theorem 2.15. Let
be tlre vacuum of , that is,
and $a_{n}|0\rangle=0$ for $n>0$ . Let
to be the dual vacuum. We denote the tensor
bysame symbol . We use thesimilar abbreviation for the tensored dual vacuum. Then, denoting $y=(y_{1,\ldots,/m}\tau)$ , we have$|0\rangle$$\mathcal{F}$$a_{0}|0\rangle=|0\rangle$$|0\}^{\otimes m}\in \mathcal{F}^{\Theta m}$$\langle 0|$$t1_{1}e$$|0\rangle$$\frac{(-1)^{n}}{(1-q)^{n}n!}\prod_{i<j}f_{1,m}(Z_{i}’ 7_{j})\langle 0|T_{1}(z_{1})T_{1}(z_{2})\cdots T_{1}(z_{n})|0\rangle=K_{n}(y, z;q,p)$.

Proof. This follows from (2.17), the operator product (2.20) and the definition (1.12).
$2_{It}$seems that [FF] contains some typo. In (6.2) of that paper, the term$f_{m,N}(\begin{array}{l}\simeq-w\end{array})$

should be $f_{m,N}(p^{1-m}w\simeq)$ .

$\square$144
B. FEIGIN, A. HOSHINO, J. SHIBAHARA, J. SHIRAISHI AND S. YANAGIDA

3. PROOFS OF THE PHOPOSITIONS

3.1. Proof of Proposition 1.16. Usiiig the$\frac{\omega(z,u’)}{\epsilon_{2}^{(t)}(z,w)}=\gamma+(z, w;q, t)$,$\gamma\pm$

defined in (2.21), we have$\frac{\omega(w,z)}{\epsilon_{2}^{(t)}(w,z)}=\gamma-(z, w;q, t)$

For later purpose, we prepare the following formulae. Let$\theta$,$\frac{\omega(w,z)}{\omega(z,w)}=\frac{\gamma_{-}(z,w;q,t)}{\gamma+(z,w;q,t)}$and$\rho$.

be natural numbers. Tlien(3.2)$\prod_{\downarrow\leq i<j\leq\theta}\gamma+(q^{-i}z, q^{-j}u’;q, t)=(\frac{1-tz’ w}{1-q’\prime})^{\theta}\frac{(qz\prime w)_{\theta}}{(tz/w)_{\theta}}$$\prod$

(3.1)

(3.3)

$\gamma_{-}(q^{-i}z, q^{-j}w;q, t)=(\frac{1-z’ u\prime}{1-qt^{-1}z’ w})^{\theta}\frac{(qt^{-1}z’ w)_{\theta}}{(z’ w)_{\theta}}$$1\leq i<j\leq\theta$(3.4)$\prod_{l=1}^{\theta}\prod_{k=1}^{\rho}\gamma+(q^{-l}z, q^{-k}w;q, t)=\frac{(q^{-\rho}u\rangle/z)_{\theta}(qt^{-1}w’ z)_{\theta}}{(w’ z)_{\theta}(qtw/z)_{\theta}}$

(3.5)

,

$= \frac{(q^{\rho-\theta+1}zw)_{\theta}}{(q-\theta+1_{7}u\prime)_{\theta}}\frac{(q^{-\theta}t\nearrow/w)_{\theta}}{(q^{J-\theta}t.z’ w)_{\theta}}$(3.6)$\prod_{l=1}^{\theta}\prod_{k=1}^{\rho}\gamma-(q^{-l}z, q^{-k}w;q, t)=\frac{(qw’ z)_{\theta}(q^{-\rho}tw’ z)_{\theta}}{(q-\rho+1w’ z)_{\theta}(tw/z)_{\theta}}$$= \frac{(q^{-\theta}z\prime w)_{\theta}}{(q^{\rho-\theta}z’ w)_{\theta}}\frac{(q^{\rho-\theta+1}t^{-1}z\prime cv)_{\theta}}{(q^{-\theta+1}t^{-1}z’ w)_{\theta}}$

.

(3.7)

Here we used $( \tau\iota)_{n}:=(u;q)_{n}=\prod_{i=1}^{n}(1-uq^{i-1})$ . These equations are checked by simple calculations.

3.1.1. Proof of (1.19). By (1.7) we have$\frac{(1-q)^{n}n!}{(-1)^{n}}F_{\lambda}(z;q, t)=(\frac{1-q}{1-t})^{|\lambda|}\sum_{\mu\geq\lambda’}c_{\dot{\lambda}_{l^{l}}}^{earrow P}(q, t)\epsilon_{\mu}(z;q)\frac{|\mu|!}{\prod_{i=1}^{\ell(\mu)}\ell\iota_{i}!}$.

Recalling the argument of [FHHSY, Proposition 2.19], we find that under the specialization
vanish. The specialization result is 3
insurvives and tbe other terms
only the term$\varphi_{\lambda}^{(q^{-1})}$$\epsilon_{\lambda’}$$F_{\lambda}(z;q, t)$$\epsilon_{\mu}$$\varphi_{\lambda}^{(q^{-1})_{\epsilon}}\lambda’(y)=\frac{\prod_{h=1}^{\ell(\lambda)}\lambda_{h}’!}{n!}\prod_{i=1}^{\ell(\lambda)}\epsilon_{\lambda’}.(y_{1}, \ldots, y_{\lambda_{i}’};q)\prod_{1\leq j<k\leq\ell(\lambda’)}\prod_{\alpha=1}^{\lambda_{j}’}\prod_{\beta=1}^{\lambda_{k}’}\omega(q^{-j+1}y_{\alpha}, q^{-k+1}y_{\theta})$$= \frac{\prod_{h=1}^{\ell(\lambda)}\lambda_{h}’!}{n!}\prod_{i=1}^{\ell(\lambda)}\epsilon_{\lambda_{i}’}(y_{1}, \ldots, y_{\lambda_{i}’};q)\cross\prod_{\alpha=1}^{\ell(\lambda)}\prod_{1\leq i<j\leq\lambda_{\alpha}}\omega(q^{-i+1}y_{\alpha}, q^{-j+1}?/\alpha)$$\cross\prod_{1\leq\alpha<\beta\leq t(\lambda)}[/\beta$$\cross\prod_{i=1j}^{\lambda\rho}\prod_{=\lambda_{\beta}+1}^{\lambda_{\alpha}}\omega(q^{-i+1}\tau/\beta, q^{-j+1}y_{\alpha})]$We also note that$c_{\lambda\lambda}^{earrow P}(q, t)=1$.

.

expression is given at the last equation in the proof of [FHHSY, Proposition 2.19], although it contains atypo. The range $u1\leq j<k\leq l$ ” of the third product should be $1\leq j<k\leq P(\lambda’)$”$3_{This}$

145
KERNEL $FU\backslash vCTION$ AND QUANTUM ALGEBRAS

Recalling (1.14), we can also calculate the first specialization$\varphi_{\lambda}^{(q^{-1})}$of the numerator in (1.19) as$\varphi_{\lambda}^{(q^{-1})}\gamma_{D}(z)=[\prod_{k\cdot=1}^{\ell(\lambda)}\prod_{1\leq i<j\leq\lambda_{k}}\epsilon_{2}(q^{-i}, q^{-j};t)][\prod_{o=1,}^{\ell(\lambda)}\prod_{9=\alpha}^{\ell(\lambda)}\prod_{i=1j}^{\lambda_{\alpha}}\prod_{=1}^{\cap}\omega(q^{-i}y_{\alpha}, q^{-j}y_{\beta})]\lambda_{l)}$$=[ \prod_{k=1}^{\ell(\lambda)}\prod_{1\leq i<j\leq\lambda_{k}}\epsilon_{2}(q^{-i}, q^{-j};t)]\prod_{1\leq\alpha<\beta\leq\ell(\lambda)}[(\prod_{1\leq i<j\leq\lambda_{\partial}}\omega(q^{-i}y_{\alpha}, q^{-j}y_{\beta}))(\prod_{1\leq j<i\leq\lambda_{\theta}}\omega(q^{-i_{\uparrow J\alpha}}, q^{-j}y_{\beta}))$$( \prod_{j=1}^{\lambda_{\beta}}\prod_{i=\lambda_{\beta}+1}^{\lambda_{a}}\omega(q^{-i}?J\alpha’ q^{-j}?J_{l^{9}}))(\prod_{i=1}^{\lambda_{\alpha}}\omega(q^{-i}y_{\alpha}, q^{-i}y_{\beta}))]$.

Tlius we have$\frac{(-1)^{n}}{(1-q)^{n}n!}\tilde{\varphi}_{\lambda}^{\langle\zeta)}\frac{\gamma_{D}(z)}{F_{\lambda}(7)q,t)}=(\frac{1-t}{1-q})^{|\lambda|}\prod_{\alpha=1}^{\ell(\lambda)}\prod_{1\leq i<j\leq\lambda_{\alpha}}\frac{\epsilon_{2}^{(t)}(t^{-i+1_{\gamma_{\alpha}}}1/,q^{-j+1_{\{J\beta}})}{\omega(q\uparrow q)}\cross$$\prod_{1\leq\alpha<\beta\leq l(\lambda)}[(\prod_{1\leq i<j\leq\lambda_{\beta}}\frac{\omega(q^{-i+1}y_{\mathfrak{a}},q^{-j+1}y_{\beta})}{\omega(q/,qy_{\alpha})})(\prod_{i=1j}^{\lambda_{\beta}}\prod_{=\lambda_{\beta}+1}^{\lambda_{a}}\frac{\omega(q^{-j+1}y_{\mathfrak{c}\iota},q^{-i+\iota_{?J\beta}})}{\omega(q^{-i+1}\uparrow/\beta,q^{-j+1_{t/\alpha}})})(\frac{\omega(yc)}{\epsilon_{2}^{(q)}(y_{\beta},y_{\alpha})})^{\lambda_{l}\prime f}]$Then recalling (3.1) and using (3.2) and (3.3), one.

$I_{1}as$$\prod_{1\leq i<j\leq\lambda_{\alpha}}\frac{\epsilon_{2}^{(t)}(q^{-i+1}y_{\alpha},q^{-j\sim\vdash 1}y/3)}{\omega(q-i+1y_{\alpha},q^{-j+1}y_{\beta})}=(\frac{1-q}{1-t})^{\lambda_{\alpha}}\frac{(t)_{\lambda_{\alpha}}}{(q)_{\lambda_{a}}}$,$\prod_{1\leq i<j\leq\lambda\rho}\frac{\omega(q^{-i+1_{l/\alpha}},q^{-j+1}y_{\beta})}{\omega(q-i+1_{l}J\beta,q-j+1_{l/\alpha})}[\frac{\omega(y_{\alpha},y_{\beta})}{\epsilon_{2}^{(q)}(?//3,y_{\alpha})}]^{\lambda_{\beta}}=\frac{(t_{J\beta}?\prime\tau_{\alpha}/)_{\lambda_{\alpha}}(qt^{-1}y_{\beta}’ y_{\alpha})_{\lambda_{\alpha}}}{(qy_{\beta}/y_{\alpha})_{\lambda_{a}}(’?)_{\lambda_{\alpha}}}$,$\prod_{i=1j}^{\lambda\rho}\prod_{=\lambda_{\beta}+1}^{\lambda_{a}}\frac{\omega(q^{-j+1}y_{\alpha},q^{-i+1}y_{\beta})}{\omega(q-i+\perp_{J\beta)}?q-j+1y_{(\}})}$.

$= \frac{(qy_{\beta/\uparrow/\alpha})_{\lambda_{\beta}}}{(ty’\tau)_{\lambda_{\beta}}}\frac{(?t_{\alpha}}{(qt^{-1_{?J\beta/J\alpha}}\uparrow)_{\lambda_{\beta}}(q^{\lambda_{\alpha}-\lambda_{\beta_{8/\beta//\alpha}}}r)_{\lambda_{\theta}}}\frac{(’\iota)_{\lambda_{\beta}}}{(q^{\lambda_{a}-\lambda_{\beta}+1_{\eta/\beta/\uparrow/\alpha}})_{\lambda_{\beta}}}$Combining these factors, we obtain$\frac{(-1)^{n}}{(1-q)^{n}n!}\lim_{\zetaarrow t}\tilde{\varphi}_{\lambda}^{\langle\zeta)}\frac{\gamma_{D}(z;t)}{F_{\lambda}(z;q,t)}=\prod_{\alpha=1}^{\ell(\lambda)}\frac{(t)_{\lambda_{\alpha}}}{(q)_{\lambda_{a}}}\prod_{1\leq\alpha<\beta\leq\ell(\lambda)}\frac{(q^{\lambda_{\alpha}-\lambda_{\beta}}ty_{\beta}’ y_{\alpha})_{\lambda_{\beta}}(q^{\lambda_{\alpha}-\lambda\rho+1}t_{J\beta}^{-1_{l}}/y_{\alpha})_{\lambda_{\beta}}}{(q^{\lambda_{\alpha}-\lambda_{\beta}}y_{\beta?/\alpha})_{\lambda_{\theta}}(q^{\lambda_{\alpha}-\lambda\rho+1}y_{\beta}’ y_{\alpha})_{\lambda_{\beta}}}$.

$= \prod_{1\leq\alpha<\beta\leq\ell(\lambda)}\frac{(q^{\lambda_{Q}-\lambda_{\theta}}t^{\beta-\alpha+1})_{\lambda_{3}-\lambda_{\beta+1}}}{(q^{\lambda_{\alpha}-\lambda_{\beta}+1}t^{\beta-a})_{\lambda_{\theta}-\lambda_{3+1}}}$

But one can easily find that the last expression equals to

3.1.2. Proof of (1.20), For a tableau $T\in$ RTbby the direct calculation we have$(\lambda;m)$$b_{\lambda}(q, t)$, define$\theta_{\alpha,k}$using the forin (1.15).

and$\tilde{\varphi}_{\lambda}^{(\zeta)}\frac{\gamma\tau(z)}{\gamma_{D}(z)}=\prod_{1\leq\alpha<\beta\leq\ell(\lambda)}\prod_{i=1j}^{\lambda_{\alpha}}\prod_{=1}^{\lambda\rho}\gamma_{-\vdash}(q^{-i}\zeta^{\alpha}, q^{-j}\zeta^{\beta})^{-1}$$\lambda_{\alpha}^{(k)_{\dot{\zeta}}}$)$s$

explained in \S 1.5. Then

(3.8)

146
B. FEIGIN, A. HOSHINO, J. SHIBAHARA, J. SHIRAISHI$n\iota p(\lambda)$$A\backslash vD$

S. YANAGIDA$\ell(\lambda)\theta_{a,k}^{\lambda_{8}^{(k-1)}}$(3.9)$\cross\prod_{k=1}\prod_{a=1}\prod_{\{9=\alpha+1}\prod_{i=1}\prod_{j=1}\gamma_{-}(q^{-i-\lambda_{a}^{(k-1)}}\zeta^{\alpha}, q^{-j}\zeta^{\beta})$$\cross\prod\prod^{m}\prod^{p(\lambda)}\prod^{\ell(\lambda)}\theta_{\alpha,k}$$\prod^{\lambda_{\beta}}$(3.10)$\gamma+(q^{-i-\lambda_{a}^{(k-1)}}\zeta^{a}, q^{-r}\zeta^{\beta})$$k=1\alpha=1\beta=\alpha i=1_{j=1+\lambda_{\beta}^{(k)}}$By the formula (3.4) we find that$\zetaarrow t1in1(3.8)=\prod_{1\leq\alpha<\beta\leq\ell(\lambda)}\frac{(t^{\beta-\alpha})_{\lambda_{\alpha}}}{(q^{-\lambda_{\beta}}t^{\beta-\alpha})_{\lambda_{a}}}\frac{(q^{-\lambda_{\beta}+1}t^{\beta-\alpha-1})_{\lambda_{\alpha}}}{(qt^{\beta-\alpha-1})_{\lambda_{\alpha}}}$(3.11),

Note that the regularity of (3.8) at $\zeta=t$ is included in this equation. Similarly by the fornmla (3.6),(3.9) is regular at $\zeta=t$ and its value is$( arrow thm(3.9)=\prod_{k=1}^{m}\prod_{\alpha=1}^{C(\lambda)}\prod_{\beta=\alpha+1}^{\ell(\lambda)}\frac{(q^{\lambda_{a}^{(k-1)}+1}t^{\beta-\alpha})_{\theta_{\alpha,k}}(q^{\lambda_{\alpha}^{(k-1)}-\lambda_{\beta}^{(k-1)}}t^{\beta-\alpha+1})_{\theta_{\alpha.k}}}{(q^{\lambda_{\alpha}^{(k-1)}-\lambda_{\beta}^{(k-1)}}t^{\beta-\alpha+1})_{\theta_{\alpha.k}}(q^{\lambda_{a}^{\langle k-1)}-\lambda_{\beta}^{(k-1)}+1}t^{\beta-\alpha})_{\theta_{\alpha,k}}}$.

(3.12)The rest term (3.10) is calculated by the formula (3.4) and (3.5):$\lim_{\zetaarrow t}(3.10)=\prod_{k=1}^{m}\prod_{\alpha=1}^{\ell(\lambda)}[\frac{(t,)_{\theta_{\alpha,k}}}{(q)_{\theta_{\alpha.k}}}\frac{(q^{\lambda_{a}-\lambda_{a}^{(k)}+1})_{\theta_{\alpha,k}}}{(q^{\lambda_{\alpha}-\lambda_{\alpha}^{(k)}})_{\theta_{a,k}}}$(3.13)$\prod_{\beta=\alpha+1}^{\ell(\lambda)}\frac{(q^{\lambda_{\alpha}^{(k\cdot-1)}-\lambda_{\beta}^{(k-1)}+1}t^{\beta-\alpha-1},)_{\theta_{\alpha.k}}}{(q^{\lambda_{a}^{(k-1)}-\lambda_{\beta}^{(k)}}t^{\beta-\alpha})_{\theta_{ak}}}\frac{(q^{\lambda_{\alpha}^{(k-1)}-\lambda_{\beta}}t^{\beta-\alpha})_{\theta_{a,k}}}{(t}]$.

Note that some parts of (3.12) and (3.13) are combined into the next form.

$[ \prod_{k=1}^{m}\prod_{\alpha=1}^{\ell(\lambda)}\prod_{\beta=\alpha+1}^{\ell(\lambda)}\frac{(q^{\lambda_{\alpha}^{(k-1)}+1}t^{\beta-\alpha})_{\theta_{\alpha,k}}}{(q^{\lambda_{\alpha}^{(k-1)}-\lambda_{\beta}^{(k-1)}}t^{\beta-\alpha+1})_{\theta_{\alpha,k}}}]$$\cross[\prod_{k\cdot=1}^{m}\prod_{\alpha=1}^{\ell(\lambda)}\frac{(q^{\lambda_{\alpha}-\lambda_{\alpha}^{(k)}+1})_{\theta_{\alpha,k}}}{(q^{\lambda_{\alpha}-\lambda_{a}^{(k)}})_{\theta_{\alpha.k}}}\prod_{\beta=\alpha+1}^{\ell(\lambda)}\frac{(q^{\lambda_{\alpha}^{(k-1)}-\lambda_{\beta}}t^{\beta-\alpha})_{\theta_{\alpha,k}}}{(q^{\lambda_{a}^{(k\cdot-1)}-\lambda_{\beta}+1}t^{\beta-\alpha-1})_{\theta_{\alpha.k}}}]$$=[ \prod_{\alpha=1}^{t(\lambda)}\frac{(q)_{\lambda_{a}}}{(t)_{\lambda_{a}}}][\prod_{1\leq\alpha<\beta\leq\ell(\lambda)}\frac{(q^{-\lambda\rho}t^{\beta-\alpha})_{\lambda_{\alpha}}(qt^{\beta-\alpha})_{\lambda_{\alpha}}}{(q^{-\lambda_{\beta}+1}t^{\beta-a-1})_{\lambda_{a}}(t^{\beta-\alpha+1})_{\lambda_{\alpha}}}]$$=[ \prod_{1\leq\alpha<\beta\leq\ell(\lambda)}\frac{(q^{-\lambda_{3}}t^{\beta-\alpha})_{\lambda_{\alpha}}}{(q^{-\lambda_{\beta}+1}t^{\beta-\alpha-1})_{\lambda_{\alpha}}}][\prod_{1\leq\alpha<\beta\leq\ell(\lambda)+1}\frac{(qt^{\beta-\alpha+1})_{\lambda_{a}}}{(t^{\beta-\alpha})_{\lambda_{\alpha}}}]=(3.13)^{-1}\cross\prod_{\alpha=1}^{p(\lambda)}\frac{(qt^{p(\lambda)-\alpha})_{\lambda_{\alpha}}}{(t^{\ell(\lambda)-\alpha+1})_{\lambda_{a}}}$Tlierefore we have$\lim_{\zetaarrow t}\varphi_{\lambda}\triangleleft()\frac{\gamma_{T}(z)}{\gamma_{D}(z)}=[\prod_{\alpha=1}^{\ell(\lambda)}\frac{(qt^{\ell(\lambda)-\alpha})_{\lambda_{\alpha}}}{(t^{\ell(\lambda)-\alpha+1})_{\lambda_{a}}}\prod_{k=1}^{m}\frac{(t)_{\theta_{\alpha.k}}}{(q)_{\theta_{\alpha_{\iota}k}}}]\cross$$\prod_{k=1}^{m}\prod_{1\leq\alpha<\beta\leq\ell(\lambda)}\frac{(q^{\lambda_{a}^{(k-1)}-\lambda_{\beta}^{(k\sim 1)}}t^{\beta-\alpha+1})_{\theta_{a.k}}(q^{\lambda_{\alpha}^{(k-1)}-\lambda_{\beta}^{(k-1)}+1}t^{9-\alpha+1}l)_{\theta_{\alpha,k}}}{(q^{\lambda_{\alpha}^{(k-1)}-\lambda_{\rho+\iota_{t^{\beta-\alpha}}}^{(k-1)}})_{\theta_{a.k}}(q^{\lambda_{a}^{(k-1)}-\lambda_{\beta}^{(k)}}t^{\beta-\alpha},)_{\theta_{\alpha,k}}}$.

147
KERNEL FUNCTION AND QUANTUM ALGEBRAS$= \prod_{k=1}^{m}\prod_{1\leq\alpha\leq\beta\leq\ell(\lambda)}\frac{(q^{\lambda_{Q}^{(k-1)}-\lambda_{\theta}^{(k-1)}}\cdot t^{\beta-\alpha+1})_{\theta_{\alpha,A}}}{(q^{\lambda_{a}^{(k-1)}-\lambda_{\beta}^{(k-1)}+1}t^{\beta-\alpha})_{\theta_{\alpha,k}}}\cross\prod_{k=1}^{m}\prod_{1\leq\alpha\leq\beta\leq\ell(\lambda)}\frac{(q^{\lambda_{a}^{(k-1)}-\lambda_{3+1}^{(k)}+1}t^{\beta-\alpha})_{\theta_{\alpha,k}}}{(q^{\lambda_{\alpha}^{(A-1)}-\lambda_{3+1^{+1}}^{(k)}}t^{\theta-\alpha+1})_{\theta_{\alpha,h}}}$Note that the function $f(u)$(3.14) can be rewritten into
(3.14)
Finally, if $T\in$ SSTb
one can see that$:=(t\{\iota)_{\infty}/(qu)_{\infty}$

satisfies. (3.14)$f(u)f(q^{-\theta}u)=(q^{-\theta+1}u)_{\infty}(q^{-\theta}t\tau\iota)_{\infty}$. Then.

$= \prod^{m}$$\prod_{k=11\leq\alpha\leq\beta\leq l(\lambda)}\frac{f(q^{\lambda_{\mathfrak{a}}^{(k-1)}-\lambda_{3}^{(k-1)}}t^{\beta-\alpha})f(r]^{\lambda_{a}^{(k)}-\lambda_{\beta+t^{\beta-\alpha})}^{(k)_{1}}}}{f(q^{\lambda_{\alpha}^{(h)}-\lambda_{\theta}^{(k-1)}}t^{\beta-\alpha})f(q^{\lambda_{\alpha}^{(k-1)}-\lambda_{\beta+1}^{(k)}}t^{\beta-\alpha})}$$(\lambda;m)$, we have$k\geq\ell(\lambda^{(k)})$, Therefore if$\beta\geq k$

then$(3.1_{0}^{\ulcorner})$$\lambda_{\beta+1}^{(k)}=\lambda_{\beta}^{(k-1)}=0$. Thus$(3.15)= \prod^{m}$,$k=1_{1\leq\alpha\leq\beta\leq\ell(\lambda)} \prod_{(k-1)}\frac{f(q^{\lambda_{a}^{(k-1)}-\lambda_{\theta}^{(k-1)}}t^{\beta-\alpha})}{f(q^{\lambda_{\alpha}^{(k)}-\lambda_{\beta}^{(k\cdot-1)}}t^{\beta-\alpha})}\frac{f(\prime}{f(q^{\lambda_{\alpha}^{(k\sim 1)}-\lambda_{\beta+1}^{(k)}}t^{\beta-\alpha})}=\prod_{k=1}^{m}\psi_{\lambda\prime\lambda^{(k-1)}}(k)(q, t)$which is. On the other hand if $T\in$ RTb
Using Proposition 1.14, we have the desired equality.
$\psi_{T}(q, t)$$(\lambda;m)\backslash$SSTb

3.2. Proof of Proposition 2.9. First we rewrite the relation of
2.1 into the next adjoint form.
$e,xp$ $( \sum_{n>0}$$(\lambda;m)$, one can see that (3.15) $=0$ .

$\psi^{\pm}(z)$and $x^{\pm}(w)$ given in Definition

ad $b_{n} \gamma^{n\prime 2}z^{-n})x^{\pm}(w)=\exp(\mp\sum_{n>0}\frac{1}{n}(1-q^{n})(1-t^{-n})(1-p^{-n})\gamma^{\mp n’ 2}(\frac{w}{z})^{n})x^{\pm}(w)$ ,.

$\exp(-\sum_{n>0}ad_{b-n}\gamma^{n/2}z^{n})x^{\pm}(w)=\exp(\pm\sum_{n>0}\frac{1}{n}(1-q^{n})(1-t^{-n})(1-p^{-n})\gamma^{\mp n\prime 2}(\frac{u;}{z})^{n})x^{\pm}(w)$Here we used the exponential form (2.2) of$\psi^{\pm}$. Then we see that$\alpha(z)x^{\pm}(w)\alpha(z)^{-1}=\exp(-\sum_{n>0}ad_{b-n}\frac{z^{n}}{\gamma^{n}-\gamma^{-n}})x^{\pm}(w)$$= \exp(\pm\sum_{n>0}\frac{1}{n}\frac{(1-q^{n})(1-t^{-n})(1-p^{-n})}{\gamma^{n}-\gamma^{-n}}\gamma^{-n’ 2\mp n/2}(\frac{z}{w})^{n})x^{\pm}(w)$,(3.16).

(3.17)$\beta(z)x^{\pm}(w)\beta(z)^{-1}=c^{3}xp(\sum_{n>0}ad_{b_{f1}}\frac{z^{-n}}{\gamma^{n}-\gamma^{-n}})x^{\pm}(u’)$$= \exp(\mp\sum_{n>0}\frac{1}{n}\frac{(1-q^{n})(1-t^{-n})(1-p^{-n})}{\gamma^{n}-\gamma^{-n}}\gamma^{-n’ 2\mp n’ 2}(\frac{w}{7})^{n})x^{\pm}(w)$We also prepare the operator product of
and(2.9) and the commutation relations (2.3) of:$\alpha(w)$$\beta(z)$, which is eaisily obtained from the definitions$b_{n}’ s$$\beta(z)\alpha(w)=\alpha(w)\beta(z)\exp(\sum_{n>0}\frac{1}{n}\frac{(1-q^{n})(1-t^{-n})(1-p^{-n})}{\gamma^{n}-\gamma^{-n}}\gamma^{-n}(\frac{w}{z})^{n})$.

3.2.1. Proof of (1). Using (3.18) and (3.16), we see that$\alpha(z)t(w)\alpha(z)^{-1}=\alpha(z)\alpha(w)x^{+}(w)\beta(w)\alpha(z)^{-1}$$= \alpha(w)\alpha(z)x^{+}(w)\alpha(z)^{-1}\beta(w)\cross cxp(-\sum_{n>0}\frac{1}{\tau\iota}\frac{(1-q^{n})(1-t^{-n})(1-p^{-n})}{\gamma^{n}-\gamma^{-n}}\gamma^{-n}(\frac{z}{w})^{n})$$= \alpha(w)x^{+}(w)\beta(w)\cross\exp(-\sum_{n>0}\frac{1}{n}\frac{(1-q^{n})(1-t^{-n})(1-p^{-n})}{\gamma^{n}-\gamma^{-n}}\gamma^{-n}(\frac{z}{w})^{n}$(3.18)

148
B. FEIGIN, A. HOSHINO, J. SHIBAHARA, J. SHIRAISHI AND S. YANAGIDA$+ \sum_{n>0}\frac{1}{n}\frac{(1-q^{n})(1-t^{-n})(1-p^{-n})}{\gamma^{\eta}-\gamma^{-n}}\gamma^{-n}(\frac{z}{w})^{n})=t(w)$.

Tlius we have $[t(z),$ $\alpha(w)]=0$ . The other relations $[t(z),$ $\beta(w)]=0,$ $[t^{*}(z),$ $\alpha(w)]=[t^{*}(z),$ $\beta(w)]=0$ ,$[t(z), \psi^{\pm}(w)]=[t^{*}(z), \psi^{\pm}(w)]=0$ also follow from equations (3.16)-(3.18) and we omit the detail.

3.2.2. Proof of (2). Using tlie commutativity $[t(z)$ , cu $(u’)]=0$ given in (1), we have$A(wz)t(z)t(\tau v)=A(w’ z)\alpha(z)x^{+}(z)\beta(z)\alpha(w)x^{+}(w)\beta(\iota v)=\alpha(z)\alpha(uf)x^{+}(z)x^{+}(w)\beta(z)\beta(tA)$.

$\cross\exp(\sum_{n>0}\frac{1}{n}\frac{(1-q^{n})(1-t^{-n})(1-p^{-r\iota}\gamma^{-2n})}{1-\gamma^{-2n}}(\frac{w}{7,})^{n}-\sum_{n>0}\frac{1}{7l}\frac{(1-q^{n})(1-t^{-n})(1-p^{-n})}{\gamma^{n}-\gamma^{-n}}\gamma^{-n}(\frac{w}{z})^{n})$Here the first summation in the exponential comes from the $A(w/z)$ , and the second from transposition of $\beta(w)x^{+}(z)$ using (3.17). Thus we haveA(w/z)t(z)t(w) $= \alpha(z)\alpha(w)x^{+}(z)x^{+}(w)\beta(z)\beta(w)\cross\exp(\sum_{n>0}\frac{1}{n}(1-q^{n})(1-t^{-n})(\frac{\uparrow 0}{z})^{n})$$= \frac{(1-q)(1-t^{-1}\frac{w}{)\approx})}{(1-\frac{\frac z\omega w}{z})(1-I)\frac{w}{z}}\alpha(z)\alpha(w)x^{+}(z)x^{+}(w)\beta(z)\beta(u’)$.

Then$A(w/z)t(z)t(w)-A(z/w)t(w)t(z)=\alpha(z)\alpha(w)$(3.19)$\cross[\frac{(1-q)(1-t^{-1},\frac{w}{)z})}{(1-\frac{\frac zww}{z})(1-p\frac{w}{z}}x^{+}(z)x^{+}(u’)-\frac{(1-q)(1-t^{-1}\frac{z}{w)})}{(1-\frac{\frac wzz}{w})(1-p\frac{z}{w}}x^{+}(w)x(z)]$$\cross\beta(z)\beta(w)$.

Now recall the relation of $x^{+}(z)$ and $x^{+}(w)$ given in Definition 2.1:$-( \frac{z}{w})^{3}G^{+}(\frac{z}{w})x^{+}(z)x^{+}(w)=G^{+}(\frac{z}{w})x^{+}(?v)x^{+}(z)$.

(3.20)

Using this equation, the line (3.19) is rewritten into$(3.19)=[ \frac{1}{(1-\frac{w}{z})(1-p\frac{w}{z})(1-p-1\frac{w}{z})}+\frac{(w\simeq^{3}\sim}{(1-\frac{z}{w})(1-p\frac{z)}{w})(1-p-1\frac{z}{w})}]G^{+}(\frac{w}{z})x^{+}(cv)x^{+}(z)$.

$=[ \frac{\delta(\frac{w}{z)})}{(1-p^{-1}(1-p)}+\frac{\delta(p\frac{w}{(1z})}{(1-p-1)-p^{-2})}+\frac{\delta(p^{-1}\frac{w}{z})}{(1-p)(1-p^{2})}]G^{+}(\frac{w}{z})x^{+}(w)x^{+}(z)$Now from (3.20) and $G^{+}(1)\neq 0$ , we see that $\delta(wz)G^{+}(wz)x^{+}(10)x^{+}(z)=0$ . We also find from(3.20) and $G^{+}(p^{-1})=0$ tlrat $\delta(p\frac{w}{z})G^{+}(\frac{w}{z})x^{+}(\iota v)x^{+}(z)=\delta(p\frac{w}{z})G^{+}(p^{-1})x^{+}(pw)x^{+}(\tau v)$ . Similarly
from (3.20) and $G^{+}(p)\neq 0$ we have $\delta(p^{-1}\frac{w}{z})G^{+}(\frac{w}{\sim\gamma})x^{+}(w)x^{+}(z)=\delta(p^{-1}\frac{w}{z})G^{+}Q))x^{+}(pz)x^{+}(z)$ . Thus
after a short calculation we have$(3.19)= \frac{(1-t^{-1})(1-q)}{1-p}[\delta(p^{-1}\frac{w}{\approx})x^{+}(pz)x^{+}(z)-\delta(p\frac{w}{z})x^{+}(pw)x^{+}(w)]$.

Then we have the desired consequence (2.11).
The equation (2.12) can be similarly shown, so that we omit the detail.

3.2.3. Proof of (3). We apply the same method as in (2). Recalling Reinark (2.8), we calculate. From the definition (2.13) of $B(z)$ , the commu$[t(z),\alpha(w)]=0$
tativity
given in (1) and the formula (3.18), we have$B(\gamma pwz)t(z)t^{*}(\gamma pw)-B(\gamma^{-1}p^{-1}zw)t^{*}(\gamma pw)t(z)$$B( \gamma pw\prime z)t(z)t^{*}(\gamma\mu\iota’)=B(\gamma p\frac{w}{\sim})\alpha(z)x^{+}(z)\beta(z)\alpha(\gamma w)^{-1}x^{-}(w)\beta(\gamma^{-1}\iota v)^{-1}$$=\alpha(z)\alpha(\gamma w)^{-1}x^{+}(z)x^{-}(w)\beta(z)\beta(\gamma^{-1}w)^{-1}$

149
KERNEL FUNCTION AND QUANTUM ALGEBRAS$\cross\exp(\sum_{n>0}\frac{1}{n}\frac{(1-q^{n})(1-t^{-n})(1-p^{-n})}{\gamma^{n}-\gamma^{-n}}(_{\sim,d}^{\underline{w}})^{n}$$+ \sum_{n>0}\frac{1}{n}\frac{(1-q^{n})(1-t^{-n})(\gamma^{-2n}p^{-2n}-\gamma^{-2n}p^{-n})}{1-\gamma^{-2n}}\gamma^{n}p^{-n}(\frac{w}{:7\vee})^{n})$$=\alpha(z)\alpha(\gamma w)^{-1}x^{+}(z)x^{-}(w)\beta(z)\beta(\gamma^{-1}u’)^{-1}$.

A similar calculation shows tliat $B( \gamma p\frac{z}{w})t^{*}(\gamma pw)t(z)=\alpha(z)\alpha(\gamma w)^{-1}x^{-}(w)x^{+}(z)\beta(z)\beta(\gamma^{-1}w)^{-1}$ .
Thus we have$B( \gamma p\frac{w}{z})t(z)t^{*}(\gamma pw)-B(\gamma p\frac{z}{w})t^{*}(\gamma pw)t(z)$$=\alpha(z)\alpha(\gamma w)^{-1}[x^{+}(z)x^{-}(w)-x^{-}(w)x^{+}(z)]\beta(z)\beta(\gamma^{-1}w)^{-1}$

Using the expression of $[x^{+}(7), x^{-}(w)]$ given in Definition 2.1, the expansion (2.2) and the defnition(2.9), one may immediately find that
$B( \gamma p\frac{w}{z})t(z)t^{*}(\gamma pw)-B(\gamma p_{w}’\simeq’)t^{*}(\gamma prv)t(z)=\frac{(1-q)(1-t^{-1})}{1-p}(\delta(\gamma^{-1}\frac{z}{w})\psi_{0}^{+}-\delta(\gamma\frac{z}{w})\uparrow/J_{0}^{-})$

Replacing$w$

in the above equation with$\gamma^{-1}p^{-1}w$

3.3. Proof of Proposition 2.13. Let us define.

, we have the desired equation (2.14).
$a_{n,(i)}$$:=1\otimes\cdots\otimes 1\otimes a_{n}\otimes 1\otimes\cdots 1$, where$a_{n}$

sitsin the i-th tensor component. Then from (2.4) and (2.5) one finds that $\rho_{\tau}^{(m)}(f_{J_{n})}=-\sum_{i=1}^{m}a_{n,(i)}(1-$. Thus we have$t^{n})(1-p^{-|n|})p^{(m-?+1)|n|\prime 2}|n|$$t)_{7/}(m)( \alpha(z))=\prod_{i=1}^{m}\alpha_{(i)}^{m}(z)$$\rho_{\tau}^{(m)}/(\beta(z))=\prod_{i=1}^{m}\beta_{(i)}^{m}(z)$

,

,

,

$\alpha_{(i)}^{n\iota}(z)$$:= \exp(\sum_{n>0}\frac{1}{n}\frac{p^{(m-i+1)n’ 2}(1-t^{-n})(1-p^{-n})}{p-mn’ 2-p^{mn’ 2}}a_{-n,(i)}z^{n})$$\beta_{(i)}^{m}(z)$$:= \exp(-\sum_{n>0}\frac{1}{n}\frac{p^{(7\prime\iota-i+1)n/2}(1-t^{n})(1-p^{-n})}{p^{-mn/2}-p^{n\iota n’ 2}}a_{n,(i)}z^{-n})$3.3.1. Proof of (1). We calculate each tensor component ofIf $k>i$ , then the k-th tensor component coines fromnormal ordering, the following coefficient arises.

$\Lambda_{i}(z)\Lambda_{j}(w)$(3.21).

(3.22). First assume $i=j$ .
. Under the$\alpha_{(k)}^{m}(z)\beta_{(k)}^{m}(z)\alpha_{(k)}^{m}(w)\beta_{(k)}^{m}(z)$$\exp(-\sum_{n>0}\frac{1}{n}(1-q^{n})(1-t^{-n})(\frac{1-p^{-n}}{1-p^{mn}})^{2}p^{(2m-k+1)n}(\frac{w}{z})^{n})$.

(3.23)For $k=i$ , the i-th tensor compoiient comes from
Under the normal ordering, the following coefficient arises.

$\alpha_{(k)}^{m}(z)\eta(p^{-(i-1)/2}z)\beta_{(k)}^{m}(x)\alpha_{(k)}^{m}(w)\eta(p^{-(i-1)\prime 2}w)\beta_{(k)}^{m}(w)$$\exp(-\sum_{n>0}\frac{1}{n}(1-q^{n})(1-t^{-n})(\frac{w}{7_{\vee}})^{n}$(3.24)$\{(\frac{1-p^{-n}}{1-p^{mn}})^{2}p^{(2m-i+1)n}+\frac{1-p^{-n}}{1-p^{mn}}p^{mn}+\frac{1-p^{-n}}{1-p^{mn}}p^{(m-i+1)n}+1\})$

If $k<i$ , then the k-th tensor component is. The normal ordering coefficient is.

$\alpha_{(k)}^{m}(z)\varphi^{-}(p^{-(2k-1)\prime 4}z)\beta_{(k)}^{m}(z)\alpha_{(k)}^{m}(w)\varphi^{-}(p^{-(2k-1)’ 4}w)$$\beta_{(k)}^{m}(w)$$\exp(-\sum_{n>0}\frac{1}{n}(1-q^{n})(1-t^{-n})(\frac{w}{z})^{n}\{(\frac{1-p^{-n}}{1-p^{mn}})^{2}p^{(2m-k+1)n}+\frac{(1-p^{-n})^{2}}{1-p^{mn}}p^{(m-k+1)n}\})$.

(3.25)

By simple calculations, the product of (3.23), (3.24) and (3.25) is shown to be $f_{1,m}(wz)^{-1}$ . Thus
the statement holds.

.

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B. FEIGIN, A. HOSHINO, J. SHIBAHARA, J. SHIRAISHI AND S. YANAGIDA

Next we consider the case $i<j$ . If $k<i$ , then the normal order coefficient is the same as (3.25).
For $k=i$ , thc normal order coefficient is$\exp(-\sum_{n>0}\frac{1}{n}(1-q^{n})(1-t^{-n})(\frac{w}{z})^{n}$(3.26)$\{(\frac{1-p^{-n}}{1-p^{2nn}})^{2}p^{(2m-i+1)n}+\frac{1-p^{-n}}{1-p^{mn}}p^{nm}+\frac{(1-p^{-n})^{2}}{1-p^{mn}}p^{(m-i+1)n}+1-p^{-n}\})$.

If $i<k<j$ , then the normal order coefficient is.

$\exp(-\sum_{n>0}\frac{1}{n}(1-q^{n})(1-t^{-n})(\frac{w}{z})^{n}\{(\frac{1-p^{-n}}{1-p^{mn}})^{2}p^{(2m-k+1)n}+\frac{(1-p^{-n})^{2}}{1-I)^{l1\iota n}}p^{(m-k+1)n}\})$(3.27)

If $k=j$ , then the normal order coefficient is$\exp(-\sum_{n>0}\frac{1}{n}(1-q^{n})(1-t^{-n})(\frac{u;}{z})^{n}\{(\frac{1-p^{-n}}{1-p^{mn}})^{2}p^{(2m-j+1)n}+\frac{1-p^{-n}}{1-p^{mn}}p^{(m-j+1)n}\})$.

(3.28)

If $k>j$ , then the normal order coefficient is (3.23). The product of (3.25), (3.26), (3.27), (3.28),(3.23) is equal to $f_{1,m}(wz)^{-1}\gamma+(z, w;q,p)$ . Thus we obtain the result.
The case $i>j$ is similar, so we omit the detail.
3.3.2. Proof of (2). The desired equation is equivalent to.

$\rho_{1/}^{(m)}(\alpha(z)\cdots\alpha(p^{m-1}z)):\prod_{k=1}^{?n}\tilde{\Lambda}_{k}(p^{k-1}z):\rho_{I}^{(m)}/(\beta(z)\cdots\beta(p^{n\iota-1}z))=1$

We will show this equation by comparing each tensor component.
is equal toBy (3.21), the k-th tensor component of$\rho_{y}^{(n\iota)}(\alpha(z)\cdots\alpha(p^{\tau n-2}z))$$\exp(-\sum_{n>0}\frac{1}{n}(1-t^{-n})p^{(2m-k-1)n’ 2}a_{-n}7^{n})$.

(3.29)

Similarly, the k-th tensor component of $\rho_{y}^{(m)}(\beta(z)\cdots\beta(p^{m-1}z))$ is equal to$\exp(\sum_{n.>0}\frac{1}{n}(1-t^{-n})p^{(-k+1)n/2}a_{n}z^{-n})$

The k-th tensor component of:$\prod_{k=1}^{m}\tilde{\Lambda}_{k}(p^{k-1}z)$.

(3.30): is$:\eta(p^{-(k-1)’ 2}p^{k-1}z)\varphi^{-}(p^{-(2k-1)\prime 4}p^{k}z)\varphi^{-}(p^{-(2k-1)’ 4}p^{k+1}z)\cdots\varphi^{-}(p^{-(2k-1)’ 4}p^{m-1}z)$$= \exp(\sum_{n>0}\frac{1-t^{-n}}{n}p^{n(2m-k-1)’ 2}a_{-n}z^{n})\exp(-\sum_{n>0}\frac{1-t^{n}}{n}p^{-n(k-1)’ 2}a_{n}z^{-n})$:(3.31)

It is easy to see that (3.29),(3.30) and (3.31) cancel. Thus we have the consequnce.

3.3.3. Proof of (3). The desired equation is equivalent to$\rho_{\tau}^{(m)}(\alpha(p^{-1}z)\cdots\alpha(p^{m-2}z)):\prod_{k=1}^{i-1}\tilde{\Lambda}_{k}(p^{k-1}z)\prod_{l=i+1}^{nl}\tilde{\Lambda}_{l}(p^{l-2}z):\rho_{?J}^{(m)}(\beta(z)\cdots\beta \mathfrak{a}J^{m-1}Z))$$=\tilde{\Lambda}_{i}^{*}(p^{(m-2)\prime 2}z)$.

We will show this equation by $compai\cdot ing$ each tensor component.

(3.32)

151
KERNEL $FU\backslash 7CTION$ AND QUANTUM ALGEBRASAs in (3.29), the k-th tensor component of$p_{l}^{(?7t)}/(\alpha(p^{-1}z)\cdots\alpha(p^{m-2}z))$$\exp(\frac{1}{n}(1-t^{n})p^{(m-k-3)n\prime 2}a_{-n}z^{n})$is equal to.

(3.33)The k-th tensor component of $\rho_{y}^{(m)}(\beta(z)\cdots\beta(p^{m-1}z))$ is given by (3.30).
The k-th tensor component of:: depends on . If $k=i$ , thenby Lemma 2.6 and somc simple calculations, the component turns out to be$\prod_{k=1}^{i-1}\tilde{\Lambda}_{k}(p^{k-1}z)\prod_{l=i+1}^{m}\tilde{\Lambda}_{l}(p^{l-2}z)$$k$$\varphi^{-}(p^{-(2i-1)\prime 4}p^{i-1}z)\varphi^{-}(p^{-(2i-1)’ 4}p^{i}z)\cdots\varphi^{-}(I)^{-(2i-1)’ 4_{p}m-2_{Z)}}$$= \exp(-\sum_{n>0}\frac{1-t^{-n}}{n}p^{(i-3)’ 2}(1-p^{n(rn-i)})a_{-n}z^{n})$.

(3.34)

Similarly, if $k<i$ , then by Leinma 2.6 the component is$:\eta(p^{-(k-1)’ 2}p^{k-1}z)\varphi^{-}(p^{-(2k-1)/4}p^{k}z)\varphi^{-}(p^{-(2k-1)/4}p^{k+1}z)\cdots\varphi^{-}(p^{-(2k-1)’ 4}p^{m-2_{7_{\vee}}})$$= \exp(\sum_{n>0}\frac{1-t^{-n}}{n}p^{n(2m-k-3)\prime 2}a_{-n}z^{n})\exp(-\sum_{n>0}\frac{1-t^{n}}{n}p^{-n(k-1)’ 2}a_{n}z^{-n})$:(3.35):

If $k>i$ , then by Lemma 2.6 the component is$:\eta(p^{-(k-1)/2}p^{k-2}z)\varphi^{-}(p^{-(2k-1)/4}p^{k-1}z)\varphi^{-}(p^{-(2k-1)/4}p^{k}z)\cdots\varphi^{-}(p^{-(2k-1)/4}p^{m-2}z)$$= \exp(\sum_{n>0}\frac{1-t^{-n}}{n}p^{n(2m-k-3)’ 2}a_{-n}z^{n})\exp(-\sum_{n>0}\frac{1-t^{n}}{n}p^{-n(k-3)’ 2}a_{n}z^{-n})$;:(3.36)

Then the i-th tensor component of (3.32) is the product of (3.33),(3.30) and (3.34). After a short
calculation, one finds that it is $\xi(p^{(i-2)/2}z)$ , which is the i-th component of.
If $k<i$ , then the k-th tensor component of (3.32) is the product of (3.33),(3.30) and (3.35). It is
1, that is, the k-th component of.
Finally, for $k>i$ , the k-th tensor component of (3.32) is the product of (3.33),(3.30) and (3.36).
It tums out to be $\varphi^{-}(p^{(2j-5)’ 4}z)$ , which is the k-th component of.
$\tilde{\Lambda}_{i}^{*}(p^{(m-2)\prime 2}z)$$\tilde{\Lambda}_{i}^{*}(p^{(m-2)’ 2}z)$$\tilde{\Lambda}_{i}^{*}(p^{(m-2)’ 2}z)$3.3.4. Proof of (4). From the known identities (2.20) and (2.22), it is not difficult to calculate
in terms of.
First we consider the case $i=j$ . From the operator product (2.20), we have$[ \prod_{k,l=1}^{m-1}f_{1.m}(F^{-k+l}?\iota;/z)]\Lambda_{i}^{*}(z)\Lambda_{j}^{*}(w)$$\Lambda_{k}’ s$$[ \prod_{k,l=1}^{m-1}f_{1,m}Q)^{-k+\iota_{\frac{w}{\sim\gamma})]\Lambda_{i}^{*}(Z)\Lambda_{i}^{*}(w)}}$$=[ \prod_{k=1}^{m-2}\prod_{l=k+1}^{m-1}\gamma_{\dashv}-(p^{-k+l}\frac{w}{\approx})][\prod_{k=2}^{m-1}\prod_{l=1}^{k-1}\gamma_{-}(p^{-k+l}\frac{w}{z})]:\Lambda_{i}^{*}(z)\Lambda_{i}^{*}(w)$:$= \exp(\sum_{n>0}\frac{1}{n}(1-q^{n})(1-t^{-n})\frac{1-p^{-n(m-2)}\prime}{1-p-n}\frac{1-p^{n(m-1)}}{1-p^{n}}(\frac{w}{z})^{n}):\Lambda_{i}^{*}(z)\Lambda_{i}^{*}(w)$Here we used the abbreviation$\gamma\pm(\frac{w}{z})$$:=\gamma\pm(z, w;q,p)$:.

. Then we have$[ \prod_{k,l=1}^{m-1}f_{1,m}(p^{-k+l}\frac{w}{\approx})]\cross\exp(-\sum_{n>0}\frac{1}{n}(1-q^{n})(1-t^{-n})\frac{1-p^{-n(m-2)}}{1-p-n}\frac{1-p^{n(m-1)}}{1-p^{n}}(\frac{w}{z})^{n})$$= \exp(-\sum_{r\iota>0}\frac{1}{n}\frac{(1-q^{\prime l})(1-t^{-n})(1-p^{(m-1)n}}{1-p^{mn}}(\frac{u}{z})^{n})=f_{1,\tau n}(\frac{w}{z})$.

152
B. FEIGIN, A. HOSHINO, J. SHIBAHARA, J. SHIRAISHI AND S. YANAGIDA: is proved.
: A7
Thus the desired equation
Next, note that thc calculation of tlie case $i\neq./’$ reduces to that of $k=i$ . If $i<$ ), then$f_{1,m}( \frac{w}{\approx})\Lambda_{i}^{*}(z)\Lambda_{i}^{*}(w)=$$(z)\Lambda_{i}^{*}(u’)$$f_{1,m}( \frac{w}{z})\Lambda_{i}^{*}(z)\Lambda_{j}^{*}(w)=:\Lambda_{i}^{*}(z)\Lambda_{j}^{*}(\iota v):\frac{\gamma_{-}(\frac{w}{z})^{J^{-?}}}{\gamma+(p_{\sim}^{\underline{u}},1)^{j-i-1}}=:\Lambda_{i}^{*}(z)\Lambda_{j}^{*}(w):\gamma_{-}(\frac{lL^{1}}{\approx}I\cdot$

At the last line we used the formula $\gamma_{-}(z)/\gamma+(pz)=1$ . For the final case $i>j$ , we have.

$f_{1,m}( \frac{w}{z})\Lambda_{i}^{*}(z)\Lambda_{j}^{*}(w)=:\Lambda_{i}^{*}(z)\Lambda_{j}^{*}(w):\frac{\gamma+(\frac{1}{}i-j}{\gamma_{-}(p^{-1}\frac{z\iota^{}w)}{z})^{i-j-1}}=:\Lambda_{i}^{*}(z)\Lambda_{j}^{*}(w):\gamma+(\frac{w}{z})$Thus all the cases are proved.

3.3.5. Proof of$(v\ulcorner)$. Tliis is similary shown as (2) and (3), so we omit the detail.

Acknowledgements. S.Y. is supported by JSPS Fellowships for Young Scientists (No.21-2241).
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B. Feigin, E. Frenkel,-algebras and elliptic$\nu v_{N}$$\iota_{(}l$$Qi\iota antt\downarrow m\mathcal{W}$$al_{9^{(}}:f_{l7}$653-678.

[FHHSY] B. Feigin, K. Hashizume, A. Hosliino, J. Shiraishi, S. Yanagida, A commutative algebm on degenemte CPlan (f Macdonald polynomials, J. Math. Phys. 50 (2009), 095215.
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I.G. Macdonald, Symmetric functions and Hall polynornials, 2nd ed. Oxford MMathematical Monographs,
Oxford University Press, (1995).
[SKAO] J. Shiraishi, H. Kubo, H. Awata, S. Odake, A quantum deformation of the Virasoro algebra and the Macdonald symmetre c functions, Lett. Math. Phys. 38 (1996), no. 1, 33-51.

BF: $LAND\Lambda U$ INSTITUTE FOR TIIEORETICAL PHYSICS, RUSSIA, CIIERNOGOLOVKA, 142432, PROSP. AKADEMIKA$1A$ ,
A 1 L., 20 AND
HIGHER SCHOOL OF $Eco\backslash OMlC’ S$ , RUSSIA, bfoscow, 101000,
INDEPENDENT UNIVERiSITY OF Mosclow, RUSSIA, $bIoscow$ , 119002, BOL’SHOI VLAS’EVSKI PER., 11
E-mail address: bf iginQgmail. om$SE\},IENOVA,$$\backslash I\backslash \prime ASNITSKAY$

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