数理解析研究所講究録第1977巻

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1 Ding-Iohara-Miki modular double Yosuke Saito Osaka City University Advanced Mathematical Institute Ding-Iohara-Miki Ruijsenaars Ding-Iohara-Miki Ding-Iohara-Miki modular double 1 Introduction Ding-Iohara-Miki modular Ding-Iohara-Miki modular double Ding-Iohara-Miki Ding-Iohara-Miki modular double 1.1 $Ding-Iohara-Mik\dot{\ovalbox{\tt\small REJECT}}$ Ding-Iohara-Miki 1997 Ding Iohara $U_{q}(\hat{sl_{2}})$ [DI]. $g(x^{-1})=g(x)^{-1}$ Ding-Iohara 2007 $W_{1+\infty}$ $q$- [Miki] Ding Iohara Ding-Iohara Ding-Iohara-Miki

2 34 Ding-Iohara-Miki Macdonald [Mac] 2009 Feigin-Hashizume-Hoshino-Shiraishi- Yanagida [FHHSY] $T_{q,x}$ q- : $T_{q,x}f(x):=f(qx)$. Macdonald $H_{N}(q, t)(n\in \mathbb{z}_{>0})$ $q$- $H_{N}(q, t):= \sum_{i=1}^{n}\prod_{j\neq i}\frac{tx_{i}-x_{j}}{x_{i}-x_{j}}t_{q,x_{i}}.$ Feigin Feigin-Odesskii Macdonald $q$- Ding-Iohara-Miki Macdonald Ding-Iohara-Miki $ p <1$ $p\in \mathbb{c}$ $(x;p)_{\infty}:= \prod_{n\geq 0}(1-xp^{n})(x\in \mathbb{c})$ $\Theta_{p}(x)$ $\Theta_{p}(x):=(p;p)_{\infty}(x;p)_{\infty}(px^{-1};p)_{\infty} (x\in \mathbb{c}^{\cross})$. Macdonald Ruijsenaars [R1] : $H_{N}(q, t, p):= \sum_{i=1}^{n}\prod_{j\neq i}\frac{\theta_{p}(tx_{i}/x_{j})}{\theta_{p}(x_{i}/x_{j})}t_{q,x_{t}}.$ Macdonald $\Theta_{p}(x)\vec{parrow 0}1-x$ $H_{N}(q, t, p)h_{n}(q, t)\vec{parrow 0}.$ Feigin Odesskii FFeigin-Odesskii [FO] Feigin Feigin-Odesskii Ruijsenaars $q$- Ruijsenaars 2009 Feigin Ruijsenaars 2 [Sal]. Ruijsenaars

3 $\tilde{f}$ $\tilde{f}$ 35 Ding-Iohara-Miki Ding-Iohara-Miki Ding-Iohara-Miki [Sal]. Feigin-Odesskii Ruijsenaars $q$- [Sa2] 1.2 modular double modular double 2 modular Faddeev [F] modular double $U_{q}(sl_{2}(\mathbb{R}))$ $U_{q,\overline{q}}(sl_{2}(\mathbb{R}))$ Virasoro $c_{\tau}=13-6( \tau+\frac{1}{\tau}) (\tau\in \mathbb{c})$ $c_{\tau}=c_{1/\tau}$ modular modular Faddeev $q=e^{2\pi i\tau}$ $U_{q}(sl_{2}(\mathbb{R}))$ $\overline{q}:=e^{2\pi i/\tau}$ $U_{\overline{q}}(sl_{2}(\mathbb{R}))$ $U_{q}(sl_{2}(\mathbb{R}))$ modular $U_{q,\overline{q}}(sl_{2}(\mathbb{R}))$ double modular double Ip $[Ip $, Nidaiev-Teschner [NT] 1.1 ( $U_{q}(sl_{2}(\mathbb{R}))$ modular double $U_{q,\overline{q}}(sl_{2}(\mathbb{R}))$ $q$ $\tau\in ). $:=e^{2\pi i\tau},$ $\overline{q}:=e^{2\pi i/\tau}$ $U_{q}(sl_{2}(\mathbb{R}))$ modular double \mathbb{r}\backslash \mathbb{q}$ $U_{q,\overline{q}}(sl_{2}(\mathbb{R})):=U_{q}(sl_{2}(\mathbb{R}))\otimes U_{\overline{q}}(sl_{2}(\mathbb{R}))$ $K^{\pm 1},$ $E,$ $F$, $\tilde{k}^{\pm 1},$ $\tilde{e},$ $\mathbb{c}$ $KK^{-1}=K^{-1}K=1,$ $KEK^{-1}=q^{2}E,$ $KFK^{-1}=q^{-2}F,$ $[E, F]= \frac{k-k^{-1}}{q-q-1},$ $\tilde{k}\tilde{k}^{-1}=\tilde{k}^{-1}\tilde{k}=1,$ $\tilde{k}\tilde{e}\tilde{k}^{-1}=\overline{q}^{2}\tilde{e},$ $\tilde{k}\tilde{f}\tilde{k}^{-1}=\overline{q}^{-2}\tilde{f},$ $[ \tilde{e}, \tilde{f}]=\frac{\tilde{k}-\tilde{k}^{-1}}{\overline{q}-\overline{q}^{-1}},$ $[X, Y]=0$ $(X=K^{\pm 1}, E, F, Y=\tilde{K}^{\pm 1},\tilde{E},\tilde{F})$. $U_{q,\overline{q}}(sl_{2}(\mathbb{R}))=U_{q}(sl_{2}(\mathbb{R})) \otimes U_{\overline{q}}(sl_{2}(\mathbb{R}))$ $K^{\pm 1},$ $E,$ $F$ $U_{q}(sl_{2}(\mathbb{R}))$ $\tilde{k}^{\pm 1},$ $\tilde{e},$ $ q = \overline{q} =1$ $U_{\overline{q}}(sl_{2}(\mathbb{R}))$ $\tau\in \mathbb{r}\backslash \mathbb{q}$

4 $S(\omega_{1}, \omega_{2};u)=\{\begin{array}{l}\frac{(e(u/\omega_{2});e(\omega_{1}/\omega_{2}))_{\infty}}{(e(-\omega_{2}/\omega_{1})e(u/\omega_{1});e(-\omega_{2}/\omega_{1}))_{\infty}} ({\rm Im}(\omega_{1}/\omega_{2})>0),\frac{(e(u/\omega_{1});e(\omega_{2}/\omega_{1}))_{\infty}}{(e(-\omega_{1}/\omega_{2})e(u/\omega_{2});e(-\omega_{1}/\omega_{2}))_{\infty}} ({\rm Im}(\omega_{2}/\omega_{1})>0).\end{array}$ 36 (modular ). $\taurightarrow\underline{1}$ $Xrightarrow\tilde{X}$ $(X=K, E, F)$. $U_{q}(sl_{2}(\mathbb{R}))$ $U_{\overline{q}}(sl_{2}(\mathbb{R}))$ modular double $U_{q,\overline{q}}(sl_{2}(\mathbb{R}))$ $\taurightarrow 1/\tau$ $U_{q,\overline{q}}(sl_{2}(\mathbb{R}))$ universal $R$ $(2 S(\omega_{1}, \omega_{2};u \omega_{1}, \omega_{2} {\rm Re}(\omega_{1})>0,$ ${\rm Re}(\omega_{2})>0$ 2 $S(\omega_{1}, \omega_{2};u)$ $S( \omega_{1}, \omega_{2};u):=\exp(\int_{\mathbb{r}+i0}\frac{e^{ku}}{(1-e^{\omega_{1}k})(1-e^{\omega_{2}k})}\frac{dk}{k})$ $(0<{\rm Re}(u)<{\rm Re}(\omega_{1}+\omega_{2}$ 2 $S(^{(}\omega_{1}, \omega_{2};u)$ $=S(\omega_{2}, \omega_{1};u)$ 1.3 $(2 S(\omega_{1}, \omega_{2};u)$ ). (1) $u\in \mathbb{c}$ $e(u):=e^{2\pi iu}$ ${\rm Im}(\omega_{1}/\omega_{2})\neq 0$ 2 $S(\omega_{1}, \omega_{2};u)$ (2) ( [K]) $x$ $ x $ $:= \min\{ x-n n\in \mathbb{z}\}$ $\omega_{1},$ $\omega_{2}\in \mathbb{r}_{>0}$ $\lim_{narrow\infty} n\omega_{1}/\omega_{2} ^{1/n}=1$ 2 $S(\omega_{1},\omega_{2};u)$ $S( \omega_{1}, \omega_{2};u)=\exp(-\sum_{n>0}\frac{e(nu/\omega_{2})}{1-e(n\omega_{1}/\omega_{2})}\frac{1}{n}-\sum_{n>0}\frac{e(nu/\omega_{1})}{1-e(n\omega_{2}/\omega_{1})}\frac{1}{n})$ $({\rm Im}(u)>0)$. 1.3 Ding-Iohara-Miki modular double Ding-Iohara-Miki modular double 2 Ding-Iohara-Miki 2 Ding-Iohara-Miki 2

5 37 Ruijsenaars $q$- $q$ $p$ Ruijsenaars [R1], Calogero-Moser Ruijsenaars $H_{N}(q, t,p)(n\in \mathbb{z}_{>0})$ $q$- $H_{N}(q, t, p):= \sum_{i=1j}^{n}\prod_{\neq i}\frac{\theta_{p}(tx_{i}/x_{j})}{\theta_{p}(x_{i}/x_{j})}t_{q,x_{i}}.$ $q,$ $p\in \mathbb{c}$ $ q <1,$ $ p <1$ $\Gamma_{q,p}(x)$ $\Gamma_{q,p}(x):=\frac{(qpx^{-1};q,p)_{\infty}}{(x;q,p)_{\infty}} (x\in \mathbb{c}^{\cross})$. Ruijsenaars $H_{N}(q, t,p)$ kernel function $\Pi_{MN}(q, t,p)(x, y)(m, N\in \mathbb{z}_{>0})$ $\Pi_{MN}(q, t,p)(x, y):=1\leq i\leq M\prod_{1\leq j\leq N}\frac{\Gamma_{q,p}(x_{i}y_{j})}{\Gamma_{q,p}(tx_{i}y_{j})}.$ kernel function Ruijsenaars [R2], [KNS] : $H_{N}(q, t,p)_{x}\pi_{nn}(q, t,p)(x, y)=h_{n}(q, t,p)_{y}\pi_{nn}(q, t,p)(x, y)$. $H_{N}(q, t,p)_{x}$ $x_{1}$,..., $x_{n}$ Ruijsenaars $x$ $y$ [Sa2] $\Gamma_{q,p}(x)=\Gamma_{p,q}(x)$ Ruijsenaars kernel function $\Pi_{MN}(q, t,p)(x, y)$ $q$ $p$ kernel function $\Pi_{MN}(q, t,p)(x, y)$ Ruijsenaars $H_{N}(q, t,p)$ $q$ $p$ $H_{N}(p, t, q):=h_{n}(q, t,p) _{qrightarrow p}= \sum_{i=1j}^{n}\prod_{\neq i}\frac{\theta_{q}(tx_{i}/x_{j})}{\theta_{q}(x_{i}/x_{j})}t_{p,x_{\iota}}$ kernel function Ruijsenaars $H_{N}(q, t,p)$ $q$ $p$ $qrightarrow p$ Ruijsenaars $H_{N}(p, t, q)$ : $[H_{N}(q, t, p), H_{N}(p, t, q)]=0.$

6 38 $qrightarrow p$ Ruijsenaars modular double $U_{q}(sl_{2}(\mathbb{R}))$ $U_{q,\overline{q}}(sl_{2}(\mathbb{R}))$ modular double $=$ Ruijsenaars Ding-Iohara-Miki $=$ : $\Gamma_{el1}(\omega_{1}, \omega_{2};u):=\frac{(e(\omega_{1}+\omega_{2}-u);e(\omega_{1}),e(\omega_{2}))_{\infty}}{(e(u);e(\omega_{1}),e(\omega_{2}))_{\infty}}.$ 1.4 $(} \Gamma_{el1}(\omega_{1}, \omega_{2};u)$ modular [FV] [Naru]). $\omega_{1},$ $\omega_{2}\in \mathbb{c}$ ${\rm Im}(\omega_{1})>0,$ ${\rm Im}(\omega_{2})>0$, ${\rm Im}(\omega_{1}/\omega_{2})>0$ $\Gamma_{el1}(\omega_{1}, \omega_{2};u)$ $\Gamma_{el1}(\omega_{1}, \omega_{2};u)=e^{-\pi iq(\omega_{1},\omega_{2};u)}\gamma_{el1}(\frac{\omega_{1}}{\omega_{2}}, -\frac{1}{\omega_{2}};\frac{u}{\omega_{2}})\gamma_{el1}(-\frac{\omega_{2}}{\omega_{1}}, -\frac{1}{\omega_{1}};\frac{u-\omega_{2}}{\omega_{1}})^{-1}$ $u\in \mathbb{c}$ $Q(\omega_{1}, \omega_{2};u)$ $Q(\omega_{1}, \omega_{2};u)$ $= \frac{u^{3}}{3\omega_{1}\omega_{2}}-\frac{1}{2}(\frac{1}{\omega_{1}}+\frac{1}{\omega_{2}}-\frac{1}{\omega_{1}\omega_{2}})u^{2}+\frac{1}{6}[\frac{\omega_{1}}{\omega_{2}}+\frac{\omega_{2}}{\omega_{1}}+3-3(\frac{1}{\omega_{1}}+\frac{1}{\omega_{2}})+\frac{1}{\omega_{1}\omega_{2}}]u$ $- \frac{1}{12}(\omega_{1}+\omega_{2}-\frac{\omega_{1}}{\omega_{2}}-\frac{\omega_{2}}{\omega_{1}}-3+\frac{1}{\omega_{1}}+\frac{1}{\omega_{2}})$. 1.5 $(} \Gamma_{e11}(\omega_{1}, \omega_{2};u)$ ). $\omega_{1},$ ${\rm Im}(\omega_{1})>0,$ ${\rm Im}(\omega_{2})>0$ $\omega_{2}\in \mathbb{c}$ $\Gamma_{el1}(\omega_{1}, \omega_{2};u)$ 2 $S(\omega_{1}, \omega_{2};u)$ $\lim_{rarrow 0}e^{\frac{\pi}{12r\omega\omega 2}i(2u-\omega_{1}-\omega_{2})_{\Gamma_{el1}(r\omega_{1},r\omega_{2};ru)}}=e^{-\frac{\pi}{2}B_{2,2}(\omega_{1},\omega_{2};u)}S(\omega_{1}, \omega_{2};u)^{-1}.$ $B_{2,2}(\omega_{1}, \omega_{2};u)$ 2 Bernoulli $B_{2,2}( \omega_{1}, \omega_{2};u):=\frac{u^{2}}{\omega_{1}\omega_{2}}-(\frac{1}{\omega_{1}}+\frac{1}{\omega_{2}})u+\frac{1}{6}(\frac{\omega_{1}}{\omega_{2}}+\frac{\omega_{2}}{\omega_{1}})+\frac{1}{2}.$ modular double 2 universl $R$ modular

7 39 modular double Ding-Iohara-Miki 2 Ding lohara-miki modular double modular $+$ 2 $r_{modular}$ $+$ Ding-Iohara-Miki Ding-Iohara-Miki modular $+$ Ding-Iohara-Miki modular double 2.1 Ding-lohara-Miki $\mathcal{u}(\omega, \sigma, \tau)$ Ding-Iohara-Miki $\mathcal{u}(\omega, \sigma, \tau)$ ${\rm Im}(\tau)>0$ $\theta_{\tau}(u)$ $\theta_{\tau}(u):=(e(u);e(\tau))_{\infty}(e(\tau-u);e(\tau))_{\infty} (u\in \mathbb{c})$. 2.1 $ \mathcal{u}(\omega, \sigma, \tau)$ $( Ding- Iohara-$Miki [Sal]). $\sigma\in \mathbb{c}$ $\omega,$ ${\rm Im}(\omega)>0,$ ${\rm Im}(-\sigma)>0$ $e(\omega)^{a}e(\sigma)^{b}\neq 1(\forall(a, b)\in \mathbb{z}^{2}\backslash (0,0))$ $\mathbb{k}:=\mathbb{q}(e(\omega/4), e(\sigma/4))$ $f^{\pm}(\tau;u)$, $g_{\tau}(u)$ $f^{+}(\tau;u):=\theta_{\tau}(\omega+u)\theta_{\tau}(-\sigma+u)\theta_{\tau}(-\omega+\sigma+u)$, $f^{-}(\tau;u):=\theta_{\tau}(-\omega+u)\theta_{\tau}(\sigma+u)\theta_{\tau}(\omega-\sigma+u)$, $g_{\tau}(u):=f^{+}(\tau;u)/f^{-}(\tau;u)$. $x^{\pm}(\tau;u)$, $\psi^{\pm}(\tau;u)$ $x^{\pm}( \tau;u)=\sum_{d\geq 0}\sum_{n\in \mathbb{z}}x_{d}^{\pm}[n]e(-nu)p^{d},\psi^{\pm}(\tau;u)=\sum_{d\geq 0}\sum_{n\in \mathbb{z}}\psi_{d}^{\pm}[n]e(-nu)p^{d}.$ $p=e(\tau)$ Ding-Iohara-Miki $\mathcal{u}(\omega, \sigma, \tau)$ $\{x_{d}^{\pm}[n]\}_{n\in \mathbb{z}}^{d\geq 0},$ $\{\psi_{d}^{\pm}[n]\}_{n\in \mathbb{z}}^{d\geq 0}$ $\mathbb{k}[$ $c$ $p]$ ]

8 40 $[\psi^{\pm}(\tau;u), \psi^{\pm}(\tau;v)]=0,$ $\psi^{+}(\tau;u)\psi^{-}(\tau;v)=\frac{g_{\tau}(c-u+v)}{g_{\tau}(-c-u+v)}\psi^{-}(\tau;v)\psi^{+}(\tau;u)$, $\psi^{+}(\tau;u)x^{\pm}(\tau;v)=g_{\tau}(\mp\frac{c}{2}-u+v)^{\mp 1}x^{\pm}(\tau;v)\psi^{+}(\tau;u)$, $\psi^{-}(\tau;u)x^{\pm}(\tau;v)=g_{\tau}(\mp\frac{c}{2}-u+v)^{\pm 1}x^{\pm}(\tau;v)\psi^{-}(\tau;u)$, $-e(u-v)^{3}f^{\pm}(\tau;-u+v)x^{\pm}(\tau;u)x^{\pm}(\tau;v)=f^{\pm}(\tau;u-v)x^{\pm}(\tau;v)x^{\pm}(\tau;u)$, $[x^{+}(\tau;u), x^{-}(\tau;v$ $=c( \omega, \sigma, \tau)\{\delta(c-u+v)\psi^{+}(\tau;\frac{c}{2}+v)-\delta(-c-u+v)\psi^{-}(\tau;-\frac{c}{2}+v)\}.$ $\delta(u)$ $:= \sum_{n\in \mathbb{z}}e(nu)$ $c(\omega, \sigma, \tau)\in \mathbb{k}[$ $p]$ ] $c( \omega, \sigma, \tau):=\frac{\theta_{\tau}(\omega)\theta_{\tau}(-\sigma)}{(p;p)_{\infty}^{2}\theta_{\tau}(\omega-\sigma)}.$ Ding-Iohara-Miki $\mathcal{u}(\omega, \sigma, \tau)$ $\mathcal{u}(\omega, \sigma, \tau)$ Ding-Iohara-Miki $\tauarrow i\infty(parrow 0)$ Ding-Iohara-Miki $\mathcal{u}(\omega, \sigma)$ 2.2 $Ding-\ovalbox{\tt\small REJECT} ohara-m\dot{\ovalbox{\tt\small REJECT}}k\dot{\ovalbox{\tt\small REJECT}}$ Ding-Iohara-Miki $\mathcal{u}(\omega, \sigma, \tau)$ 2.2 $( Ding- Iohara-$Miki $ \mathcal{u}(\omega, \sigma, \tau)$ ). $x$ $T_{\omega,x}$ $x$ $\omega$ : $T_{\omega,x}f(x):=f(x+\omega)$. $\pi_{0}$ : $\mathcal{u}(\omega, \sigma, \tau)arrow End_{\mathbb{K}[[p]]}(\mathbb{K}[[p]][e(\pm x)])$ $\mathcal{u}(\omega, Ding-Iohara-Miki \sigma, \tau)$ $\pi_{0}[c]:=0,$ $\pi_{0}[x^{+}(\tau;u)]:=\frac{\theta_{\tau}(-\sigma)}{(p;p)_{\infty}^{2}}\delta(-\omega+\sigma+x-u)t_{\omega,x}^{-1},$ $\pi_{0}[x^{-}(\tau;u)]:=\frac{\theta_{\tau}(\sigma)}{(p;p)_{\infty}^{2}}\delta(\sigma+x-u)t_{\omega,x},$

9 $\pi_{0}[\psi^{+}(\tau;u)]:=\frac{\theta_{\tau}(x-u)\theta_{\tau}(-\omega+2\sigma+x-u)}{\theta_{\tau}(\sigma+x-u)\theta_{\tau}(-\omega+\sigma+x-u)},$ 41 $\pi_{0}[\psi^{-}(\tau;u)]:=\frac{\theta_{\tau}(-x+u)\theta_{\tau}(\omega-2\sigma-x+u)}{\theta_{\tau}(-\sigma-x+u)\theta_{\tau}(\omega-\sigma-x+u)}.$ $\mathcal{u}(\omega, \sigma, \tau)$ Ding-Iohara-Miki $\tauarrow i\infty(parrow 0)$ $\mathcal{u}(\omega, \sigma)$ Ding-Iohara-Miki 2.3 $\theta_{\mathcal{t}}(u)=(e(u);e(\tau))_{\infty}(e(\tau-u);e(\tau))_{\infty}$ modular $\tauarrow-1/\tau$ : $\theta_{-1/\tau}(u/\tau)=\exp[\pi i\{\frac{u^{2}}{\tau}+(\frac{1}{\tau}-1)u+\frac{1}{6}(\tau+\frac{1}{\tau})-\frac{1}{2}\}]\theta_{\tau}(u)$. 2.2 Ding-Iohara-Miki () Ding-Iohara-Miki modular 2.3 $( Ding- Iohara-$Miki modular ). $ \mathcal{u}(\omega, \sigma, \tau)$ Ding-Iohara-Miki modular $\mathcal{u}(\omega, \sigma, \tau)$ $M_{\tau}[\pi_{0}[x^{+}(\tau;u$ $= \exp[-\pi i\{\frac{\sigma^{2}}{\tau}-(\frac{1}{\tau}-1)\sigma\}]\frac{\theta_{-1/\tau}(-\sigma/\tau)}{(e(-1/\tau);e(-1/\tau))_{\infty}^{2}}\delta_{\tau}(-\omega+\sigma+x-u)t_{\omega,x}^{-1},$ $M_{\tau}[\pi_{0}[x^{-}(\tau;u$ $= \exp[-\pi i\{\frac{\sigma^{2}}{\tau}+(\frac{1}{\tau}-1)\sigma\}]\frac{\theta_{-1/\tau}(\sigma/\tau)}{(e(-1/\tau);e(-1/\tau))_{\infty}^{2}}\delta_{\tau}(\sigma+x-u^{k})t_{\omega,x},$ $M_{\tau}[\pi_{0}[\psi^{+}(\tau;u$ $= \exp[2\pi i\frac{\sigma(\omega-\sigma)}{\tau}]\frac{\theta_{-1/\tau}((x-u)/\tau)\theta_{-1/\tau}((-\omega+2\sigma+x-u)/\tau)}{\theta_{-1/\tau}((\sigma+x-u)/\tau)\theta_{-1/\tau}((-\omega+\sigma+x-u)/\tau)},$ $M_{\tau}[\pi_{0}[\psi^{-}(\tau;u$ $= \exp[2\pi i\frac{\sigma(\omega-\sigma)}{\tau}]\frac{\theta_{-1/\tau}((-x+u)/\tau)\theta_{-1/\tau}((\omega-2\sigma-x+u)/\tau)}{\theta_{-1/\tau}((-\sigma-x+u)/\tau)\theta_{-1/\tau}((\omega-\sigma-x+u)/\tau)}.$ $\delta_{\tau}(u):=\sum_{n\in \mathbb{z}}e(nu/\tau)$ modular $M_{\tau}$

10 ( modular ). $\omega,$ $\sigma,$ $x$ $r$ $(r\in \mathbb{r})$, $rarrow 0$ 2.3 Ding-Iohara-Miki modular $M_{\tau}[ \pi_{0}[x^{+}(\tau;u arrow\exp(\pi i\frac{\sigma}{\tau})[1-e(-\sigma/\tau)]\delta_{\tau}(-\omega+\sigma+x-u)t_{\omega,x}^{-1}$, (2.1) $M_{\tau}[ \pi_{0}[x^{-}(\tau;u arrow\exp(-\pi i\frac{\sigma}{\tau})[1-e(\sigma/\tau)]\delta_{\tau}(\sigma+x-u)t_{\omega,x}$, (2.2) $M_{\tau}[ \pi_{0}[\psi^{+}(\tau;u arrow\frac{[1-e((x-u)/\tau)][1-e((-\omega+2\sigma+x-u)/\tau)]}{[1-e((\sigma+x-u)/\tau)][1-e((-\omega+\sigma+x-u)/\tau)]},$ (2.3) $M_{\tau}[ \pi_{0}[\psi^{-}(\tau;u arrow\frac{[1-e((-x+u)/\tau)][1-e((\omega-2\sigma-x+u)/\tau)]}{[1-e((-\sigma-x+u)/\tau)][1-e((\omega-\sigma-x+u)/\tau)]}$. (2.4) 2.4 Ding-Iohara-Miki $\omega$ Miki $\mathcal{u}(\tau, \sigma,\omega)$ ( $\omega$ ) $\mathcal{u}(\omega/\tau, \sigma/\tau)$ Ding-Iohara- $(\omega$ ) modular : $M_{\omega}[ \pi_{0}[x^{+}(\omega;u arrow\exp(\pi i\frac{\sigma}{\omega})[1-e(-\sigma/\omega)]\delta_{\omega}(-\tau+\sigma+x-u)t_{\tau,x}^{-1}$, (2.5) $M_{\omega}[ \pi_{0}[x^{-}(\omega;u arrow\exp(-\pi i\frac{\sigma}{\omega})[1-e(\sigma/\omega)]\delta_{\omega}(\sigma+x-u)t_{\tau,x}$, (2.6) $[ \pi_{0}[\psi^{+}(\omega;u arrow\frac{[1-e((x-u)/\omega)][1-e((-\tau+2\sigma+x-u)/\omega)]}{[1-e((\sigma+x-u)/\omega)][1-e((-\tau+\sigma+x-u)/\omega)]},$ (2.7) $M_{\omega}[ \pi_{0}[\psi^{-}(\omega;u arrow\frac{[1-e((-x+u)/\omega)][1-e((\tau-2\sigma-x+u)/\omega)]}{[1-e((-\sigma-x+u)/\omega)][1-e((\tau-\sigma-x+u)/\omega)]}$. (2.8) Ding-Iohara-Miki $\mathcal{u}(\tau/\omega, \sigma/\omega)$ (2.1) $\sim(2.4)$ $\mathcal{u}(\omega/\tau, \sigma/\tau)$ (2.5) $\sim(2.8)$ $\mathcal{u}(\tau/\omega, \sigma/\omega)$ 2 2

11 $\bullet$ $\bullet$ 43 (1) Ding-Iohara-Miki $\mathcal{u}(\omega, \sigma, \tau)$ modular Ding-Iohara-Miki $\mathcal{u}(\omega/\tau, \sigma/\tau)$ (2) (1) $\omega$ Ding-Iohara- Miki $\mathcal{u}(\tau/\omega, \sigma/\omega)$ (3) (1), (2) 2 $\omegarightarrow\tau$ Miki modular double $\mathcal{u}(\omega/\tau, Ding-Iohara- $\omegarightarrow\tau$ \sigma/\tau)\otimes \mathcal{u}(\tau/\omega, \sigma/\omega)$ modular double modular Ding-Iohara-Miki $\mathcal{u}(\omega, \sigma, \tau)$ $\mathcal{u}(\omega, \sigma, \tau)$ modular $\mathcal{u}(\omega, \sigma, \tau)$ $\mathcal{u}(\omega, \sigma, \tau)$ modular $\omega,$ $\sigma,$ Ding-Iohara-Miki modular double RIMS Conference 2015 [DI] J. Ding, K. Iohara. Generalization of Drinfeld quantum affine algebras. Lett. Math. Phys. 41 (1997), no. 2, [F] L. Faddeev. Modular double of quantum group. (1999) $arxiv:math/ $

12 44 [FHHSY] B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi, S. Yanagida. A commutative algebra on degenerate $\mathbb{c}\mathbb{p}^{1}$ (2009) $arxiv: $ and Macdonald polynomials. J. Math. Phys. 50 [FO] B. Feigin, A. Odesskii. A family of elliptic algebras. (1997) Internat. Math. Res. Notices. no.ll. [FV] G. Felder, A. Varchenko. The Elliptic Gamma Function and. Advances in Mathematics (2000): $SL(3, \mathbb{z})\ltimes \mathbb{z}^{3}$ [Ip] Ivan C. H. Ip. Positive representations of split real quantum groups : the universal $R$ operator. (2012) $arxiv: $ [K] (2013) ISBN [KNS] Y. Komori, M. Noumi, J. Shiraishi. Kernel functions for difference operators of Ruijsenaars type and their applications. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications. Volume 5 (2009) $arxiv: $ [Mac] I. G. Macdonald. Symmetric functions and Hall polynomials. 2nd edition, Oxford University Press (1995). [Miki] K. Miki. $A(q, \gamma)$ analog of the $W_{1+\infty}$ algebra. J. Math. Phys. 48, ; $doi: / $ (2007). [Naru] A. Narukawa. The modular properties and the integral representations of the multiple elliptic gamma functions. (2003) $arxiv:math/ $ [NT] I. Nidaiev, J. Teschner. On the relation between the modular double $of\mathcal{u}_{q}(\mathfrak{s}\mathfrak{l}(2, and the quantum Teichm\"uller theory. (2013) $arxiv: $ \mathbb{r}))$ [R1] S. N. M. Ruijsenaars. Complete integrability of relativistic Calogero-Moser systems and elliptic function identities. Commun. Math. Physic. Volume 110. (1987). [R2] S. N. M. Ruijsenaars. Zero-eigenvalue eigenfunctions for differences of elliptic relativistic Calogero-Moser Hamiltonians. Theoretical and mathematical physics (2006): [Sal] Yosuke Saito. Elliptic Ding-Iohara algebra and the free field realization of the elliptic Macdonald operator. Publ. Res. Inst. Math. Sci. 50 (2014), doi: 10.$4171/PRIMS/139,$ $arxiv: $ [Sa2] Yosuke Saito. Commutative families of the elliptic Macdonald operator. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 10 (2014): 021. arxiv:

コホモロジー的AGT対応とK群類似

コホモロジー的AGT対応とK群類似 AGT K ( ) Encounter with Mathematics October 29, 2016 AGT L. F. Alday, D. Gaiotto, Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010), arxiv:0906.3219.

数理解析研究所講究録 第1977巻

数理解析研究所講究録第1977巻