数理解析研究所講究録
1480 巻 2006 年 117-127

117

Elliptic Ruijsenaars operators and elliptic hypergeometricintegrals
名古屋大学・多元数理科学研究科 小森靖 (Yasushi Komori)Grad uate School of Mathema tics, Nagoya University*

ABSTRACT
We study a family of mutually commutative difference operators associated withroot systems and discuss their simultaneous eigenvector in a special case. For rootsystems with rank$n$

, we construct $3n$ commutative difference operators, which

are a generalization of elliptic Ruijsenaars operators. Especially, for theroot system, we construct an explicit simultaneous eigenvector of these operators$BC_{1}$

described in terms of elliptic hypergeometric integrals.

Introduction1

In [10] Ruijsenaars introduced the operators acting on the space of meromorphic

functions which are defined by$\mathrm{Y}_{n}=\sum_{I\subset\{1,..,l\}}.($$\prod_{j\in I,k\in I^{\mathrm{c}}}\frac{\sigma(x_{j}-x_{k}+\mu;\omega_{1},\omega_{2})}{\sigma(x_{j}-x_{k};\omega_{1},\omega_{2})})\prod_{j\in I}\tau_{j}(\omega_{3})$where$\omega_{1},$$\omega_{2}\in \mathbb{C}\backslash \{0\}$and the action of$\omega,$$\ldots,x_{n})$$\tau_{j}(\omega)$are arbitrary such thatis defined by,$\omega_{1}/\omega_{2}\not\in \mathrm{R}$

(1.1)

and

$\omega_{3},$$\mu\in \mathbb{C}\backslash \{0\}$$(\tau_{j}(\omega)f)(x_{1}, \ldots, x_{j}, \ldots, x_{n})=f(x_{1},$$\ldots,$$x_{j}-$. He showed that these operators are mutually commutative. The first

result of this article is a generalization of the elliptic Ruijsenaars operators. We

define$\mathrm{Y}_{n}^{(p)}=\sum_{I\subset\{1,..,l\}}.(,\prod_{J\in I}e^{\nu_{P}(x_{\mathrm{j}}-x_{k})}\frac{\sigma(x_{j}-x_{k}+\mu_{\mathrm{p}};\omega_{q},\omega_{f})}{\sigma(x_{j}-x_{k};\omega_{q},\omega_{f})})\prod_{j\in I}\tau_{j}(\omega_{p})$$|I|=n$,(1.2)$k\in I^{\mathrm{c}}$’Section 5and 6are ajointwork with 神戸大学・大学院自然科学研究科野滋正俊
Noumi) Department ofMatbematics, Grad uate School ofScience and Technology Kobe Univer$(\mathrm{M}\mathrm{a}8\mathrm{a}\mathrm{t}\mathrm{o}8\mathrm{h}\mathrm{i}$

味 ty

118

where

$p,$ $q,$ $r\in \mathbb{Z}/3\mathbb{Z}$$\omega_{p}/\omega_{q}\not\in \mathrm{R}$$\zeta$andare distinct, and for

if $p\neq q$ . Put$p\in \mathbb{Z}/3\mathbb{Z},$$\eta_{pq}=2\zeta(\omega_{p}/2;\omega_{p}, \omega_{q})$$a_{\tau}=\eta_{pq}\omega_{q}-\eta_{qp}\omega_{p}$, then $a_{f}=\pm 2\pi i$ .

$\nu_{\mathrm{p}}\omega_{q}+\mu_{\mathrm{p}}\eta_{q\mathrm{r}}=\nu_{q}\omega_{p}+\mu_{q}\eta_{pt}$for distinct$\omega_{p}\in \mathbb{C}\backslash \{0\}$are such thatwith Weierstrass’ zeta function
If$\nu_{\mathrm{p}},$$p,$ $q,$ $r\in \mathbb{Z}/3\mathbb{Z}$$\mu_{\mathrm{p}}$satisfy three equations, then all$\mathrm{Y}_{n}^{(\mathrm{p})}$are shown to

be mutually commutative. For instance, these equations are solved by$\nu_{1}=\frac{a_{1}(\nu_{3}\omega_{1}+\mu_{3}\eta_{12})-(a_{2}\mu_{2}+a_{3}\mu_{3})\eta_{32}}{a_{1}\omega_{3}}$$\nu_{2}=\frac{\nu_{3}\omega_{2}+\mu \mathrm{s}m_{1}-\mu_{2}\eta_{31}}{\omega_{3}}$$\mu_{1}=\frac{a_{2}\mu_{2}+a_{3}\mu_{3}}{a_{1}}$

,

,

(1.4)

,

are regard

(1.3)

(1.5)

as free parameters. Although the discussion above
is for the -type root system, the construction can be applied to arbitrary root

where$\nu_{3},$$\mu_{\mathit{2}},$$\mu_{3}$$e\mathrm{d}$$A$systems.

The second result is a construction of a simultaneous eigenvector of the elliptic

Ruijsenaars operators of type

$BC_{1}$. So far, some class of eigenvectors was discussed

in [11], where certain transcendental equations should be solved. We first obtain an
explicit meromorphic eigenvector described in terms of the elliptic hypergeometricintegral.

Acknowledgement: The author expresses his sincere gratitude to Jun’ichi
Shiraishi, Yuji Yamada, Takashi Takebe and Michitomo Nishizawa who kindly

allowed him to speak about his research. Also thanks are due to Masatoshi Noumi,
Yasuhiko Yamada, Yasuhiro Ohta and Tetsu Masuda for fruitful discussions andhelpful comments.

2

Affine Root Systems
We summarize some facts about affine root systems and affine Weyl groups [1,2, 4, 5]. In this article, we will omit

$A_{2l}^{(2)}$-type root system because of simplicity,

though it is straightforward. The notation and symbols are a little different fromthose in the previous papers $[6, 7]$ in order to generalize the results.

119

Let

$\Delta$be the irreducible reduced finite root system of typevector space$V$with $\dim V=l$ and the inner productindices, II $=\{\alpha_{i}|i\in I\}\subset V$ the set of simple roots,

set of simple coroots,
and coweight lattices,

$Q$

and

$I=\{1, \ldots, l\}$$\langle\cdot, \cdot\rangle,$$\Pi^{\vee}=\{\alpha_{1}^{\vee}.

the root and coroot lattices, $P$ and$Q^{\vee}$$\{\Lambda_{\mathrm{i}}|i\in I\}$

fundamental coweights such that

and$\{\Lambda_{i}^{\vee}|i\in I\}$in a complex$X_{l}$

a set of|i\in I\}\subset V$$P^{\vee}$

the weightthe fundamental weights and$\langle\alpha_{i}, \Lambda_{j}^{\vee}\rangle=\langle\Lambda_{i}, \alpha_{j}^{\vee}\rangle=\delta_{ij}$. Then we have
(2.1)

,

$Q= \bigoplus_{i\in I}\mathbb{Z}\alpha_{i}\subset P=\bigoplus_{i\in I}\mathbb{Z}\Lambda_{i}\subset V$$Q^{\vee}=.

.

Let$\Delta_{+}$

Let

and

$\Delta_{-}$$\Delta_{\epsilon}\subset\Delta$roots. Let$r$$\langle\cdot, \cdot\rangle$

is normalized such thatbe the set of the shorter roots and$\langle\alpha:, \delta\rangle=\langle\delta, \delta\rangle=0$$\gamma_{\alpha}$

for a

if

$\gamma_{\alpha}=r$$\in\Delta$$\gamma_{\alpha}=1$

otherwise. Let$\hat{V}=V\oplus \mathfrak{G}$withis written as

and$\hat{\Delta}_{-}$$\alpha\in\hat{\Delta}$, let$\overline{v}$forthe natural projection on

$V$

.

.

(2.4)

,

,

$v\in V$

The Weyl group $W$ is generated by the fundamental

reflections $\{S::=s_{\alpha:}|i\in I\}$ on, where$v\in\hat{V}$be a reflection defined by$s_{\alpha}$$\alpha^{\vee}=2\alpha/\langle\alpha, \alpha\rangle$$\{S:|i\in I\}\wedge$

(2.3)

.

be the set of the positive affine roots and the negative affine roots$s_{\alpha}(v):=v-\langle\alpha, v\rangle\alpha^{\vee}$

where

the set of the longer

and its linear extension. Then the associated affine root system

respectively. We denote byFor$\Delta_{l}\subset\Delta$, such that in nontwisted cases all $\gamma_{a}=1$ ,$\alpha\in\Delta_{l},$$\hat{\Delta}=\{\alpha+n\gamma_{\alpha}\delta|\alpha\in\Delta, n\in \mathbb{Z}\}$$\hat{\Delta}_{+}$

for the longer roots a.

be the ratio of the square lengths of the longer roots and the shorterand in twisted cases,Let$\langle\alpha, \alpha\rangle=2$

be the set of positive roots and negative roots respectively.

roots. Fix parameters$\hat{\Delta}\subset\hat{V}$(2.2)\bigoplus_{*\in I}\mathbb{Z}\alpha_{i}^{\vee}\subset P^{\vee}=.\bigoplus_{*\in I}\mathbb{Z}\Lambda_{i}^{\vee}\subset V$

The inner product

the$I=\wedge I\cup\{0\}$

$V$

and

and the affine Weyl group$\alpha_{0}=\delta-\theta$$\wedge W$

is generated bywith the highest root in nontwisted$\theta$cases and the highest short root in twisted cases.
The defining relations are given by$(s_{1}s_{j})^{m_{1\mathrm{j}}}$$-=id$,

for$s_{i}^{2}=id$,$i\neq j\in I\wedge$

and the Coxeter relations:
(2.5)

120

where $m_{1j}=2$ if

and$\alpha_{i}$are disconnected in the Dynkin diagram and $m_{ij}=$$\alpha_{j}$$3,4,6$ if 1,2,3 lines respectively connect

phisms$\tau_{\mu}$of the vector space$\tau_{\mu}(\lambda):=\lambda-\langle\lambda, \mu\rangle\delta$

$V$

and

$\alpha_{i}$$\alpha_{j}$. For $\mu\in V$ , we define endomor-

by

.

(2.6)

. For an arbitrary lattice , we denote bythecorresponding group of translations of . Then one sees thatis the semidirectLet$M:=\mathbb{Z}(W\cdot\theta^{\vee})\subset V$

$L$

$L$

product$\overline{W}=W\ltimes T_{M}$affine Weyl group$\Omega$$\overline{W}$$\overline{M}$$\overline{W}$$P^{\vee}$,$P$$\overline{M}_{-}:=\oplus_{i\in I}\mathrm{z}_{\leq 0^{\lambda_{1}}}$The length$\ell(w)$$\{\lambda_{i}|i\in I\}$

nontwisted case,twisted case,,of$=s_{i_{1}}\ldots s_{i_{\ell}}\omega$$w\in\overline{W}$,. The extended$\overline{W}:=W\ltimes T_{\overline{M}}$which stabilizes the affine Weyl chamber

and its canonical basis$\overline{M}=\{$IV$\overline{M}:=\{\lambda\in V|\alpha\in\Delta, \langle\alpha, \lambda\rangle\in\gamma_{\alpha}\mathbb{Z}\}$is isomorphic to the semidirect productdescription of

We also use$\overline{W}$is defined by the semidirect product$\overline{W}$be the subgroup ofsees that. Let$T_{L}$$\overline{W}*\Omega$$C$. Here are the explicit:$\Lambda_{i}^{\vee}$,

nontwisted case,,twisted case.

The set$\Delta_{\hat{w}}$$ws_{1_{\ell}}(\alpha_{ip})\}$

,

(2.7)

.

is defined by the length of a reduced decomposition:$\ell$$i_{k}\in I\omega\in\Omega\wedge,$

.

(2.8)

It is equivalent to the number of the negative roots made positive by$\ell(\hat{w}):=|\Delta_{\dot{w}}|$. Then one$\lambda_{i}=\{$$\Lambda_{:}$. Let$\Delta_{\hat{w}}:=\hat{\Delta}_{+}\cap\hat{w}\hat{\Delta}_{-}$

is explicitly described as$\hat{w}$

:

.

(2.9)
$\alpha^{(\ell)}=$$\Delta_{\hat{w}}=\{\alpha^{(1)}=\alpha:_{1},$ $\alpha^{(2)}=s_{1_{1}}(\alpha:_{2}),$$\ldots,$. By definition, is independent of reduced expressions. One sees that
. A weightis said to be minuscule if.
$\Delta_{\hat{w}}$$\Omega=\{\omega\in\overline{W}|\ell(\omega)=0\}$$\lambda\in\overline{M}$$\Delta_{\tau_{-\lambda}}\subset\Delta_{+}$In the following, we use the constants$\rho_{x}:=\sum_{:\in I}x_{\alpha}.\Lambda_{i}=\frac{1}{2}\sum_{+\alpha\in\Delta}x_{\alpha}\alpha$$h_{x}^{\vee}:=\langle\rho_{x}, \theta\rangle+x_{\alpha 0}$

where$x_{\alpha}\in \mathbb{C}$,,which depends only on the length of roots,

We shall define the root algebras after Cherednik $[2, 7]$ .

(2.10)(2.11)

121

Definition 2.1. Rootand {$\hat{\Delta}\}$$\tau_{\lambda}|$A$\in\overline{M}$is generated by independent va riables$algeb^{l}ra\mathcal{R}$} with the following defining relations:for$\tau_{\lambda}R_{\alpha}=R_{\tau_{\lambda}\alpha}\tau_{\lambda}$$\tau_{\lambda}\tau_{\lambda’}=\tau_{\lambda+\lambda’}$$\{R_{\alpha}|\alpha\in$(2.12)$w\in\overline{W}$,(2.13).

(2.14)

1. There exists a unique set

Theorem 2.2 (Cherednik).

$\{R_{w}|w\in\overline{W}\}\subset \mathcal{R}$satisfying the relations:,$R_{vw}=R_{v}vR_{w}$where$\omega\in\Omega_{f}v,$,$(i\in I)\wedge$$R_{s:}=R_{\alpha:}$$R_{\omega}=1$,(2.15)

and $\ell(vw)=\ell(v)+\ell(w)$ , and$w\in\overline{W}$$v(R_{a_{1}}\ldots R_{\alpha:})=$$R_{v\alpha_{1}}\ldots R_{v\alpha_{1}}$2. We have the$R$-matrix for$w\in\overline{W}$and its arbitrary reduced decompositionas$w=s_{i_{1}}\ldots s_{i_{\ell}}\omega$$R_{w}=R_{\alpha^{(1)}}\ldots R_{a^{(\ell\rangle}}$$\alpha^{(1)}=\alpha_{i_{1}}$,,$\alpha^{(2)}=s_{i_{1}}(\alpha_{i_{2}})$

Theorem 2.3. The subalgebra.. . ,,$\alpha^{(\ell)}=ws_{ip}(\alpha_{1_{\ell}})\in\Delta_{w}$generated by {$S\subset \mathcal{R}$forms a commutative algebra and is generated by$\mathrm{Y}^{\lambda}:=R_{\tau_{\lambda}}\tau_{\lambda}|$$\{\mathrm{Y}^{-\lambda:}|i\in I\}$

(2.16).
A$\in\overline{M}_{-}$}.

Representation and Difference Operators

3
Let
and

$\gamma_{\alpha}^{(1)}=\gamma_{\alpha}$$\overline{M}^{(2)},\overline{M}^{(3)}$and$\gamma_{\alpha}^{(2)},$$\gamma_{\alpha}^{(3)}$be taken similarly as

be taken similarly. Herebe the set of meromorphic functions on

sufficient to specify the action of$s_{1}$$\gamma_{\alpha}^{(:)}$

$V$

and

$\gamma_{\alpha}^{(1)}$$\gamma_{\alpha}^{(j)}$. Accordingly let

may differ if $i\neq j$ . Let. To define the action of

for $i\in I$ and$\tau_{\lambda}$$\overline{M}^{(1)}=\overline{M}$for A$\in\overline{M}^{(1)}$$\overline{W}$

. For

on

$\lambda 4$$\mathcal{M}$, it is$f\in \mathcal{M}$, wedefine$s_{i}(f)(v)=f(s:(v))$ ,$\tau_{\lambda}(f)(v)=\tau_{\lambda}^{(1)}(f)(v)=f(v-\omega_{1}\lambda)$

.

(3.1)

122

Fix

$\xi^{(1)},$$\zeta^{(1)}\in V$

and let$\mu_{\alpha}^{(1)},$$\nu_{\alpha}^{(1)}$be constants depending only on the length ofroots. Then one can check that (3.1) and$(R_{\alpha}f)(v):=H_{\alpha}(\mu_{\alpha}^{(1)}, \nu_{\alpha}^{(1)})f(v)-H_{\alpha}(\langle\xi^{(1)}, \alpha^{\vee}\rangle, \langle\zeta^{(1)}, \alpha^{\vee}\rangle)f(s_{\alpha}v)$

satisfy the defining relations of the root algebra [8], where$H_{\alpha}(\eta, \kappa)$

,

(3.2)

is the mero-morphic function defined by$H_{\alpha}(\eta, \kappa)(v):=e^{\kappa\alpha(v)_{\frac{\sigma(\mu_{a}^{(1)};\gamma_{\alpha}^{(2)}\omega_{2},\gamma_{\alpha}^{(3)}\omega_{3})}{\sigma(\eta;\gamma_{a}^{(2)}\omega_{2},\gamma_{\alpha}^{(3)}\omega_{3})}\frac{\sigma(\alpha(v)+\eta;\gamma_{\alpha}^{(2)}\omega_{2},\gamma_{\alpha}^{(3)}\omega_{3})}{\sigma(\alpha(v);\gamma_{\alpha}^{(2)}\omega_{2},\gamma_{\alpha}^{(3)}\omega_{3})}}}$

and a root

and$\alpha$acts on$\alpha=\alpha’+n\delta,$

as an affine linear functional$\alpha’\in\Delta$

Theorem 3.1. Let$-\rho_{\mu^{(1)}},$$V$$\zeta^{(1)}=-\rho_{\nu^{(1)}}$

Theorem 3.2. Let$\alpha(v)=\langle\alpha, v\rangle+n\omega_{1}$(3.3)

for $v\in V$. Then we have the following theorems $[6, 7]$$\mathcal{V}:=\mathcal{M}^{W}$. Then, the $W$ -invariant subspace of$\mathrm{Y}_{\lambda}^{(1)}:=\mathrm{Y}^{\lambda}\in \mathrm{E}\mathrm{n}\mathrm{d}_{\mathbb{C}}\mathcal{V}$$(-\lambda)\in\overline{M}^{(1)}$$\mathcal{M}$and let$\xi^{(1)}=$.

be minuscule. Then we have$\mathrm{Y}^{\lambda}|_{\mathcal{V}}=\frac{1}{|W_{\lambda}|}\sum_{w\in W}w(\alpha\in$$\prod_{+,(\lambda,\alpha\rangle=-\gamma_{\alpha}\mathrm{A}}H_{\alpha}(\mu_{\alpha}^{(1)}, \nu_{\alpha}^{(1)})\tau_{\lambda}^{(1)})|_{\mathcal{V}}$

where

,

,

(3.4)

is the stabilizer of A in $W$ .

$W_{\lambda}$

Commutativity in Case of Minuscule Weights

4Let$\mathrm{Y}_{\lambda}^{(2)}$be the operator obtained by changing the role of the indices 1 and 2

in the construction ofsees that$\mathrm{Y}_{\lambda}^{(j)}$for A$\mathrm{Y}_{\lambda}^{(1)}$$\in\overline{M}^{(j)}$, and$\mathrm{Y}_{\nu}^{(k)}$be obtained in the same manner. Then one

are commutative [7]. In the following, we assume that$\mu_{\alpha}^{(j)}\eta_{kl}+\nu_{\alpha}^{(j)}\omega_{k}=\mu_{\alpha}^{(k)}\eta_{j1}+\nu_{\alpha}^{(k)}\omega_{j}$and$\mathrm{Y}_{\lambda}^{(3)}$where $j\neq k\neq l\neq j$ . We demonstrate thatare also commutative if-A$\mathrm{a}\mathrm{n}\mathrm{d}-\nu$Lemma 4.1 (Proposition 7.7 in [7]). Let$\mathrm{Y}_{\lambda}^{(j)}$are minuscule.
$x_{a}$

be constants dependent only on

the length of roots. Then we have$- \sum_{\alpha\in\Delta_{\tau_{\lambda}}}x_{\alpha}\overline{\alpha}=h_{x}^{\vee}\lambda$

.

(4.1)

123

Theorem 4.2. Let$j,$ $k\in \mathbb{Z}/3\mathbb{Z}$Then

are commutative.

$\mathrm{Y}_{\lambda}^{(j)}$and$\mathrm{Y}_{\nu}^{(k)}$and$-\lambda\in\overline{M}^{(j)},$ $-\nu\in\overline{M}^{(k)}$be minuscule andProof. If $j=k$ , then the statement follows from Theorem 6.3 in [7]. So it issufficient to show the case $j\neq k$ and the commutativity of each term. Let$F_{j,\lambda}:=$$\prod_{+,(\lambda,\alpha\rangle=-\gamma_{a}\alpha\in\Delta}e^{\nu_{a}^{0)}(\alpha,v)_{\frac{\sigma(\langle\alpha,v\rangle+\mu_{\alpha}^{(j)};\gamma_{\alpha}^{(k)}\omega_{k},\gamma_{\alpha}^{(l)}\omega_{l})}{\sigma(\langle\alpha,v\rangle;\gamma_{\alpha}^{(k)}\omega_{k},\gamma_{\alpha}^{(l)}\omega_{l})}}}.$

where $j\neq l\neq k$ . Since for$u,$$w\in W$

(4.2)

,

,
(4.3)

$\tau_{u\nu}^{(k)}(e^{\nu_{\alpha}^{(j\rangle}(w\alpha,v\rangle}\frac{\sigma(\langle w\alpha,v\rangle+\mu_{\alpha}^{(j)};\gamma_{\alpha}^{(k)}\omega_{k},\gamma_{\alpha}^{(l)}\omega_{l})}{\sigma(\langle w\alpha,v\rangle;\gamma_{a}^{(k)}\omega_{k},\gamma_{\alpha}^{(l)}\omega_{l})})$$=e^{\nu_{\alpha}^{(j)}(w\alpha,v-\omega_{k}\mathrm{u}\nu\rangle_{\frac{\sigma(\langle w\alpha,v-\omega_{k}u\nu\rangle+\mu_{\alpha}^{\mathrm{C})};\gamma_{\alpha}^{(k)}\omega_{k},\gamma_{\alpha}^{(l)}\omega_{l})}{\sigma(\langle w\alpha,v-\omega_{k}u\nu\rangle;\gamma_{\alpha}^{(k)}\omega_{k},\gamma_{\alpha}^{(l)}\omega_{l})}}}$$=e^{-(\mu_{\alpha}^{\langle \mathrm{j})}\eta kl+\nu_{\alpha}^{(j)}\omega_{k})\langle w\alpha,u\nu\rangle}e^{\nu_{\alpha}^{(j)}\langle w\alpha,v\rangle_{\frac{\sigma(\langle w\alpha,v\rangle+\mu_{\alpha}^{(j)};\gamma_{\alpha}^{(k)}\omega_{k},\gamma_{\alpha}^{(l)}\omega_{l})}{\sigma(\langle w\alpha,v\rangle;\gamma_{a}^{(k)}\omega_{k},\gamma_{\alpha}^{(l)}\omega_{l})}}}$, (4.4)

we have$\tau_{\mathrm{u}\nu}^{(k)}(F_{j,w\lambda})=e^{A\langle w\lambda,u\nu\rangle}(F_{j,w\lambda})$

(4.5)

,

where by Lemma 4. 1

.

$-$

(4.6)

$\sum_{+,\langle\lambda,\alpha\rangle=-\gamma_{\alpha}\alpha\in\Delta}(\mu_{\alpha}^{(j)}\eta_{kl}+\nu_{\alpha}^{(j)}\omega_{k})\alpha=A\lambda$

Thus$F_{k,u\nu}\tau_{u\nu}^{(k)}F_{j,w\lambda}\tau_{w\lambda}^{(j)}=e^{A\langle w\lambda,u\nu\rangle}F_{k,u\nu}F_{j,w\lambda}\tau_{\mathrm{u}\nu}^{(k)}\tau_{w\lambda}^{(j)}$$=F_{j,w\lambda}\tau_{w\lambda}^{(j)(k\rangle}F_{k,u\nu^{\mathcal{T}_{u\nu}}}$For minuscule weights$-\lambda$

.

(4.7)

, the periodicity of the coefficients is easily obtainedfor general
are calculated. However
A is complicated and the proof of the commutativity requires a further investigation

since the explicit forms of the operatorsand is omitted here.

$\mathrm{Y}^{\lambda}$$\mathrm{Y}^{\lambda}$

124

5

$BC_{1}$-type Operators and Eigenvector

Generally, it is very difficult to construct explicit eigenvectors of the elliptic
Ruijsenaars operators. However we can construct a simultaneous eigenvector intheroot system. In this root system, there are three mutually commutative$BC_{1}$

operators.

First we give the explicit forms of these operators. Let
such that$s^{\infty}\omega_{1},\omega_{2}>0$

and$2\mu 0+\mu_{1}+\cdots+\mu_{6}+\omega_{1}+\omega_{\mathit{2}}=0$$\omega_{1},\omega_{2},\mu_{0},$$\ldots,$$\mu_{6}\in \mathbb{C}$. Then the commutative

operators are given by$\theta(x-\mu_{0}+\omega_{1};\omega_{2})\prod^{6}\theta(x-\mu_{j};\omega_{2})$

,

(5.1)

,

(5.2)

$\mathrm{Y}^{(1)}=e^{2\pi 1x}\frac{j=0}{\theta(2x;\omega_{2})\theta(2x+\omega_{1};\omega_{\mathit{2}})}(\tau(\omega_{1})-1)$$+(Xrightarrow-x)$$\theta(x-\mu_{0}+\omega_{2};\omega_{1})\prod^{6}\theta(x-\mu_{j};\omega_{1})$$\mathrm{Y}^{(2)}=e^{2\pi ix}\frac{j=0}{\theta(2x;\omega_{1})\theta(2x+\omega_{2)}\omega_{1})}(\tau(\omega_{2})-1)$$+(Xrightarrow-X)$$\mathrm{Y}^{(3)}=\tau(1)+\tau(-1)-2$

where$\theta$,(5.3)

denotes the Jacobi odd theta function. For these commutative difference

operators we can construct an explicit simultaneous eigenvector:

Theorem 5.1. Let$\prod^{7}\Gamma(pqc_{0}/c_{i};p, q)$$\phi(c)=\frac{i=1}{\mathrm{r}(pqc_{0};p,q)\prod_{1\leq i<j\leq 7}\Gamma(pqc_{0}/c_{\dot{*}}c_{j^{1p,q)}}}$(5.4)$\Gamma(pq/(pqc_{0})^{1/\mathit{2}}z^{\pm 1}; p, q)\prod\Gamma((pqc_{0})^{1/2}/c_{j}z^{\pm 1}; p, q)7$$\cross\int_{C}\frac{j=1}{\Gamma(z^{\mathit{2}},z^{-2};p,q)}\frac{dz}{z}$

,

125

with
$c_{0}=e^{2:(\mu 0-\mu_{1}-\omega_{1}-\omega_{2})}\pi$$c_{3}=e^{2\pi\dot{*}(\mu 0+\mu_{4})}$,$c_{6}=e^{2\pi i(-x-\mu_{1})}$$p=e^{2\pi\dot{u}d_{1}}$,,$c_{1}=e^{2\pi i(\mu 0+\mu_{2})}$$c_{4}=e^{2\pi i(\mu 0+\mu_{5})}$,$c_{7}=e^{2\pi\dot{*}(x-\mu_{1})}$$q=e^{2\pi 1\omega_{2}}$,,$c_{\mathit{2}}=e^{2\pi i(\mu 0+\mu s)}$$c_{6}=e^{2\pi i(\mu 0+\mu_{6})}$

,

,

,

,

and $C$ a closed circle taken appropriately, and $\Gamma(z;p, q)$ elliptic Gamma function

defined by$\Gamma(z;p, q)=\frac{(pqz^{-1};p,q)_{\infty}}{(z;p,q)_{\infty}}$Then$\phi(c)$.

is a simultaneous eigenvector of$E^{(1)}=e^{-2\pi i\mu_{0}} \prod_{6}^{6}\theta(\mu_{0}+\mu|;\omega_{2})i=1$$E^{(\mathit{2})}=e^{-2\pi\dot{*}\mu 0} \prod_{i=1}\theta(\mu_{0}+\mu_{i};\omega_{1})$’$\mathrm{Y}^{(j)}$with the eigenvalues$E^{(3)}=0$,.

sketch of proof. One can show the following contiguous relations directly or by useof the result in [9]:$\frac{\langle \mathrm{q}\rangle\prod_{k\neq\dot{*}i}\langle pqc_{0}/c_{j}c_{k}\rangle}{\langle pqc_{0}/c_{j},p^{2}q^{2}c_{0}/c_{j}\rangle}\phi^{+}(c_{j}^{-})-\frac{\langle c_{j}\rangle\prod_{k\neq,;}\langle pqc_{0}/c_{i}c_{k}\rangle}{\langle pqc_{0}/cp^{2}q^{2}c_{0}/c_{i}\rangle}::;\emptyset^{+}(c^{-}\dot{.})$

(5.5)

,

$= \frac{\langle c_{i}/c_{j}\rangle\prod_{k\neq i,j}\{pqc_{0}/c_{k}\rangle}{\langle pqc_{0},p^{2}q^{2}c_{0}\rangle}\emptyset$$\phi-\phi(c_{j}^{-}, c_{i}^{+})$$= \frac{\langle pq\mathrm{q}/c_{j}\rangle\langle pqc_{0}/c_{i}c_{j}\rangle\langle pqc_{0}\rangle\langle p^{2}q^{2}c_{0}\rangle}{\langle pqc_{0}/c_{j}\rangle\langle p^{2}q^{2}c_{0}/c_{j}\rangle\langle c_{0}/\mathrm{q}\rangle\langle pqc_{0}/c;\rangle}\prod_{k\neq i_{\dot{O}}}\frac{\langle c_{k}\rangle}{\langle pqc_{0}/c_{k}\rangle}\phi^{+}(c_{j}^{-})$

where$\langle\cdot\rangle$

is the multiplicative form of$\phi(c_{j}^{\pm})=\phi(c_{0}, c_{1}, \ldots, q^{\pm 1}c_{j}, \ldots, c_{7})$an eigenvector of$\mathrm{Y}^{(1)}$$\phi^{+}=\phi(q^{2}c_{0}, qc_{1}, \ldots, qc_{7})$. Combining these relations, we see thatand thus is an eigenvector ofand , and that by the definition of$q$$\theta(\cdot;p),$

(5.6)

,

$c_{6}$

and$c_{7},$$\phi(c)$$\mathrm{Y}^{(2)}$and$\phi(c)$due to the symmetry ofis an eigenvector of$\mathrm{Y}^{(3)}$

.

is
$p$

126

Special Cases

6

In this section, we clarify the relation between

metric series$10^{E_{9}}$$\phi(c)$and the elliptic hypergeo-

introduced by Frenkel and Turaev [3].

Theorem 6.1. If $c_{1}=p^{-M}q^{-N}$ , then$\phi(c)=\tilde{\phi}(c_{0}, \ldots, q^{-N}, \ldots,c_{7};p)\cross\tilde{\phi}(c_{0}, \ldots,p^{-M}, \ldots, c_{7};q)$

,

(6.1)

where
,

$\tilde{\phi}(c_{0}, \ldots, c_{7};r)=\sum_{k=0}^{\infty}\frac{\langle(pq)^{2k}c_{0};r\rangle}{\langle c_{0};r\rangle}\frac{\langle\langle c_{0};r\rangle\rangle_{k}}{\langle\langle pq;r\rangle\rangle_{k}}\prod_{i=1}^{7}\frac{\langle\langle \mathrm{q};r\rangle\rangle_{k}}{\langle\langle pq\mathrm{c}_{0}/\mathrm{q};r\rangle\rangle_{k}}$and$\langle\langle u;r\rangle\rangle_{k}=\langle u;r\rangle\cdots\langle u(pq)^{k-1};r\rangle$

(6.2)

.

sketch of proof. We see that both the right hand side of (6.1) and the elliptic

hypergeometric integral (5.4) satisfy the contiguous relation (5.6), with the same
initial condition.

Now we state the relation between$10E_{9}(b_{0;}b_{1},$where one of$b_{i}$$\phi(c)$and$10^{E_{9}}$which is defined by$b_{7;r)=\sum_{k=0}^{\infty}\frac{\langle q^{2k}b_{0};r\rangle}{\langle b_{0;r}\rangle}\frac{\langle b_{0};r\rangle_{k}}{\langle r;r\rangle_{k}}\prod_{i=1}^{7}\frac{\langle b_{i};r\rangle_{k}}{\langle rb_{0}/b_{i;r}\rangle_{k}}}$$\ldots,$

should be$r^{-N}$and$\langle u;r\rangle_{k}=\langle u;r\rangle\cdots\langle ur^{k-1};r\rangle$,. The following isshown by a direct calculation.
Theorem 6.2.
$\tilde{\phi}(c_{0}, \ldots, c_{7};p)=_{10}E_{9}(pc_{0};c_{1}, \ldots,pc_{k}, \ldots, c_{7};p)$which is independent of the choice of

$k$

,

.

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