Elliptic Quantum Groups
Hitoshi Konno
Department of Mathematics, Tokyo University of Marine Science and Technology,
Etchujima, Koto, Tokyo 135-8533, Japan
<EMAIL_ADDRESS>
###### Abstract

We expose the elliptic quantum groups in the Drinfeld realization associated
with both the affine Lie algebra ${g}$ and the toroidal algebra ${g}_{tor}$.
There the level-0 and level $\not=0$ representations appear in a unified way
so that one can define the vertex operators as intertwining operators of them.
The vertex operators are key for many applications such as a derivation of the
elliptic weight functions, integral solutions of the (elliptic) $q$-KZ
equation and a formulation of algebraic analysis of the elliptic solvable
lattice models. Identifying the elliptic weight functions with the elliptic
stable envelopes we make a correspondence between the level-0 representation
of the elliptic quantum group and the equivariant elliptic cohomology. We also
emphasize a characterization of the elliptic quantum groups as
$q$-deformations of the $W$-algebras.

Keywords : Elliptic quantum group, Affine Lie algebra, Toroidal algebra,
deformed $W$-algebra, $q$-KZ equation, Hypergeometric integral, Quantum
integrable system, Quiver variety, Elliptic cohomology, Stable envelope

Key points :

  * •

The elliptic quantum groups $U_{q,p}({g})$ and $U_{q,\kappa,p}({g}_{tor})$ are
formulated in terms of the Drinfeld generators in a uniform way111One may
unify them as “quiver elliptic quantum groups”, where $U_{q,p}({g})$ is a
finite Dynkin quiver type, whereas $U_{q,\kappa,p}({g}_{tor})$ is an affine
Dynkin quiver type.  and equipped with $H$-Hopf algebroid structure as a
coalgebra structure.

  * •

$U_{q,p}({g})$ is characterized both as a $p$-deformation of the quantum
affine algebra $U_{q}({g})$ and as a $q$-deformation of the $W$-algebra of the
coset type ${g}\oplus{g}\supset{g}_{\rm diag}$. In the same way,
$U_{q,\kappa,p}({g}_{tor})$ is characterized both as a $p$-deformation of the
quantum toroidal algebra $U_{q,\kappa}({g}_{tor})$ and as a
$(q,\kappa)$-deformation of affine quiver $W$-algebra.

  * •

The vertex operators defined as intertwiners of the tensor product of the
level-0 and $\not=0$ modules are key objects for many applications.

  * •

The Gelfand-Tsetlin basis for the level-0 representation of
$U_{q,p}(\widehat{{sl}}_{N})$ is constructed explicitly by using the elliptic
weight function as the change of basis matrix from the standard basis.

  * •

The elliptic weight function of type ${sl}_{N}$ is identified with the
elliptic stable envelope for the equivariant elliptic cohomology of the
cotangent bundle to the partial flag variety
$\mathrm{E}_{T}(T^{*}fl_{\lambda})$. This allows us to make a dictionary
between $U_{q,p}(\widehat{{sl}}_{N})$ and $\mathrm{E}_{T}(T^{*}fl_{\lambda})$.

##  1 Introduction

Quantum groups are algebraic systems associated with solutions of the Yang-
Baxter equation (YBE). They are in general associative algebras classified by
finite dimensional simple Lie algebras $\bar{{g}}$, affine Lie algebras ${g}$
or toroidal algebras ${g}_{tor}$, and equipped with certain coalgebra
structure. Typical examples are Yangians $Y(\bar{{g}})$, Yangian doubles
${\mathcal{D}}Y(\bar{{g}})$, quantum groups $U_{q}(\bar{{g}})$, affine quantum
groups $U_{q}({g})$ and quantum toroidal algebras $U_{q,\kappa}({g}_{tor})$.
See for example [11, 60, 26].

Similarly elliptic quantum groups (EQG) are quantum groups associated with
elliptic solutions of the YBE classified by affine Lie algebras ${g}$ or
toroidal algebras ${g}_{tor}$. In elliptic setting, it is known at least for
the cases classified by ${g}$ that there are two types of YBEs, the vertex
type and the face type (Fig.1 and Fig.2). Accordingly, there are two types of
elliptic quantum groups, the vertex type and the face type[29]. Both of them
are dynamical quantum groups, whose universal $R$ matrices[29] satisfy the
dynamical YBE (DYBE)[21]222 Obviously the face type YBE is dynamical[21]. The
vertex type is also dynamical, because the elliptic nome $p$ is a dynamical
parameter getting a shift by a central element $q^{-2c}$ and yields the new
elliptic nome $p^{*}=pq^{-2c}$[29]..

$\varepsilon_{3}$$\varepsilon_{3}^{\prime}$$\quad
u_{3}$$\varepsilon_{1}$$\varepsilon_{2}$$\varepsilon_{1}^{\prime}$$u_{1}$$\varepsilon_{2}^{\prime}$$u_{2}$$=$$\varepsilon_{3}$$\varepsilon_{3}^{\prime}$$\quad
u_{3}$$\varepsilon_{2}^{\prime}$$u_{2}$$\varepsilon_{1}^{\prime}$$u_{1}$$\varepsilon_{1}$$\varepsilon_{2}$

Figure 1: The vertex type Yang-Baxter equation

$a$$b$$c$$f$$d$$e$$\bullet$$u_{1}$$u_{2}$$\
u_{1}-u_{2}$$=$$a$$b$$c$$f$$d$$e$$\bullet$$u_{2}$$u_{1}$$\ u_{1}-u_{2}$

Figure 2: The face type Yang-Baxter equation

There are several formulations of the elliptic quantum groups. They can be
classified by their generators and coalgebra structures. See Table 1 :
${\mathcal{A}}_{q,p}(\widehat{{sl}}_{N})$ and
${{\mathcal{B}}_{q,\lambda}}({g})$ in terms of the Chevalley generators,
$U_{q,p}({g})$ in terms of the Drinfeld generators and
$E_{q,p}(\widehat{{gl}}_{N})$ in terms of the $L$ operators. Only
${\mathcal{A}}_{q,p}(\widehat{{sl}}_{N})$ is the vertex type. The others are
the face type. Coalgebra structures are the quasi-Hopf algebra structure for
${\mathcal{A}}_{q,p}(\widehat{{sl}}_{N})$, ${{\mathcal{B}}_{q,\lambda}}({g})$,
and the Hopf algebroid structure for $E_{q,p}(\widehat{{gl}}_{N})$,
$U_{q,p}({g})$, respectively. For expositions of these formulations with
historical remarks and arguments for their consistency, the reader can
consult[45].

| co-algebra structure | generators
---|---|---
$\begin{matrix}{\mathcal{A}}_{q,p}(\widehat{{sl}}_{N})\ \mbox{(vertex type)}\\\\[8.53581pt] {{\mathcal{B}}_{q,\lambda}}({{g}})\ \mbox{(face type)}\\\\[8.53581pt] \end{matrix}$ | quasi-Hopf algebra | Chevalley
$\begin{matrix}E_{q,p}({g})\ \mbox{(face type)}\\\\[8.53581pt] \end{matrix}$ | Hopf Algebroid |  $L$-operator
$\begin{matrix}U_{q,p}({{g}})\ \mbox{(face type)}\\\\[8.53581pt] \end{matrix}$ | Hopf Algebroid | Drinfeld
Table 1: Three formulations of the elliptic quantum groups

In this article, we expose the elliptic quantum group $U_{q,p}({g})$ and its
toroidal version $U_{q,\kappa,p}({g}_{tor})$. They are generated by the
Drinfeld generators, i.e. analogues of the loop generators of the affine Lie
algebra ${g}$[33]. After giving definitions of $U_{q,p}({g})$ and some typical
representations of $U_{q,p}(\widehat{{sl}}_{N})$, we introduce the vertex
operators of $U_{q,p}(\widehat{{sl}}_{N})$ and describe their applications :
derivations of the elliptic weight functions and integral solutions of the
elliptic $q$-KZ equation, construction of the level-0 action on the Gelfand-
Tsetlin basis, algebraic analysis of the elliptic solvable lattice models. We
then describe a geometric interpretation of the level-0 representation of
$U_{q,p}(\widehat{{sl}}_{N})$ in terms of the equivariant elliptic cohomology
of the cotangent bundle to the partial flag variety
$\mathrm{E}_{T}(T^{*}fl_{\lambda})$. We also emphasize a characterization of
$U_{q,p}({g})$ as a $q$-deformation of the $W$-algebra of the coset type. In
the final section we briefly describe a formulation of the elliptic quantum
toroidal algebras, $U_{q,\kappa,p}({g}_{tor})$, $U_{q,t,p}({{gl}}_{1,tor})$
and expose their connections to the Macdonald theory, affine quiver
$W$-algebra and application to instanton calculus of the super symmetric gauge
theories.

##  2 Elliptic Algebra $U_{q,p}(\mbox{\fourteeneufm g})$

Through this paper, let $p,q$ be generic complex numbers satisfying
$|p|,|q|<1$. Let $\bar{{g}}$ be a simple Lie algebra over ${\mathbb{C}}$ and
${g}=X^{(1)}_{l}$ the corresponding untwisted affine Lie algebra with the
generalized Cartan matrix $A=(a_{ij})_{i,j\in I}$, $I=\\{0\\}\cup\bar{I},\
\bar{I}=\\{1,\dots,l\\}$, $\mathrm{rank}A=l$. We denote by $B=(b_{ij})_{i,j\in
I},\,b_{ij}=d_{i}a_{ij}$ the symmetrization of $A$ and set $q_{i}=q^{d_{i}}$.
We also use the notations for $n\in\mathbb{Z}$,

$\displaystyle[n]_{q}=\frac{q^{n}-q^{-n}}{q-q^{-1}},\qquad[n]_{i}=\frac{q_{i}^{n}-q_{i}^{-n}}{q_{i}-q_{i}^{-1}}.$

We fix a realization $({h},\Pi,\Pi^{\vee})$ of $A$, i.e. ${h}$ is a
$l+2$-dimensional $\mathbb{C}$-vector space,
$\Pi=\\{\alpha_{0},\alpha_{1},\dots,\alpha_{l}\\}\subset{h}^{*}$ a set of
simple roots, and $\Pi^{\vee}=\\{h_{0},h_{1},\dots,h_{l}\\}\subset{h}$ a set
of simple coroots satisfying $\langle\alpha_{j},h_{i}\rangle=a_{ij}\ (i,j\in
I)$ for a canonical pairing
$\langle\,,\rangle:{h}^{*}\times{h}\to{\mathbb{C}}$[33]. Set also
${\mathcal{Q}}=\sum_{i\in I}{\mathbb{Z}}\alpha_{i}$. We take
$\\{h_{1},\dots,h_{l},c,d\\}$ as the basis of ${h}$ and
$\\{\bar{\Lambda}_{1},\cdots,\bar{\Lambda}_{l},\Lambda_{0},\delta\\}$ the dual
basis satisfying

$\displaystyle\langle\delta,d\rangle=1=\langle\Lambda_{0},c\rangle,\quad\langle\bar{\Lambda}_{i},h_{j}\rangle=\delta_{i,j},$

with the other pairings being 0.

Let $H_{P}$ be a ${\mathbb{C}}$-vector space spanned by
$P_{0},P_{1},\cdots,P_{l}$ and $H^{*}_{P}$ be its dual space spanned by
$Q_{0},Q_{1},\cdots,Q_{l}$ with a pairing $\langle Q_{i},P_{j}\rangle=a_{ij}$.
For $\alpha=\sum_{i\in I}c_{i}\alpha_{i}\in{h}^{*}$, we set
$P_{\alpha}=\sum_{i\in I}c_{i}P_{i}$, $Q_{\alpha}=\sum_{i\in I}c_{i}Q_{i}$. In
particular $P_{\alpha_{i}}=P_{i}$, $Q_{\alpha_{i}}=Q_{i}$. Define also an
analogue of the root lattice ${\mathcal{R}}_{Q}=\sum_{i\in
I}{\mathbb{Z}}Q_{i}$.

Let us consider $H:={h}\oplus H_{P}$ and $H^{*}:={h}^{*}\oplus H^{*}_{P}$ with
a pairing $\langle\ ,\ \rangle:H^{*}\times H\to{\mathbb{C}}$ by extending
those on ${h}^{*}\times{h}$ and $H^{*}_{P}\times H_{P}$ with
$\langle{h}^{*},H_{P}\rangle=0=\langle H^{*}_{P},{h}\rangle$. We denote by
${\mathbb{F}}={\mathcal{M}}_{H^{*}}$ the field of meromorphic functions on
$H^{*}$. We regard a meromorphic function $g(h,P)$ of $h\in{h},P\in H_{P}$ as
an element in ${\mathbb{F}}$ by
$g(h,P)(\mu)=g(\langle\mu,h\rangle,\langle\mu,P\rangle)$ for $\mu\in H^{*}$.

###  2.1 Definition

The elliptic algebra $U_{q,p}({g})$ is a topological algebra over
${\mathbb{F}}[[p]]$ generated by the Drinfeld generators

$\displaystyle\alpha_{i,m},\quad e_{i,n},\quad f_{i,n},\quad K_{i}^{\pm},\quad
q^{\pm c/2},\quad q^{d}\quad(i\in\bar{I},\quad
m\in\mathbb{Z}\backslash\\{0\\},\quad n\in\mathbb{Z}).$

In order to write down the defining relations, it is convenient to introduce
their generating functions $e_{i}(z),f_{i}(z)$ and $\phi^{\pm}_{i}(z)$ called
the elliptic currents.

$\displaystyle e_{i}(z)=\sum_{n\in{\mathbb{Z}}}e_{i,n}z^{-n},\qquad
f_{i}(z)=\sum_{n\in{\mathbb{Z}}}f_{i,n}z^{-n},$
$\displaystyle\phi^{+}_{i}(q^{-{c}/{2}}z)=K^{+}_{i}\exp\left(-(q_{i}-q_{i}^{-1})\sum_{n>0}\frac{\alpha_{i,-n}}{1-p^{n}}z^{n}\right)\exp\left((q_{i}-q_{i}^{-1})\sum_{n>0}\frac{p^{n}\alpha_{i,n}}{1-p^{n}}z^{-n}\right),$
$\displaystyle\phi^{-}_{i}(q^{{c}/{2}}z)=K^{-}_{i}\exp\left(-(q_{i}-q_{i}^{-1})\sum_{n>0}\frac{p^{n}\alpha_{i,-n}}{1-p^{n}}z^{n}\right)\exp\left((q_{i}-q_{i}^{-1})\sum_{n>0}\frac{\alpha_{i,n}}{1-p^{n}}z^{-n}\right).$

We also set $p^{*}=pq^{-2c}$. The defining relations are given as follows. For
$g(h,P)\in{\mathbb{F}}$,

$\displaystyle q^{\pm c/2}\ :\hbox{ central },$ $\displaystyle
g(h,{P})e_{j}(z)=e_{j}(z)g(h+\langle\alpha_{j},h\rangle,P-\langle
Q_{j},P\rangle),\quad
g(h,{P})f_{j}(z)=f_{j}(z)g(h-\langle\alpha_{j},h\rangle,P),$
$\displaystyle[g(h,P),\alpha_{i,m}]=0,\qquad[g(h,P),q^{d}]=0,\quad
g(h,{P})K^{\pm}_{j}=K^{\pm}_{j}g(h,P-\langle Q_{j},P\rangle),$
$\displaystyle[q^{d},\alpha_{j,m}]=q^{m}\alpha_{j,m},\quad[q^{d},x^{\pm}_{j}(z)]=x^{\pm}_{j}(q^{-1}z)$
$\displaystyle[K^{\pm}_{i},K^{\pm}_{j}]=[K^{\pm}_{i},K^{\mp}_{j}]=0=[K^{\pm}_{i},\alpha_{j,m}],$
$\displaystyle
K^{\pm}_{i}e_{j}(z)(K^{\pm}_{i})^{-1}=q^{\mp\langle\alpha_{j},h_{i}\rangle}e_{j}(z),\quad
K^{\pm}_{i}f_{j}(z)(K^{\pm}_{i})^{-1}=q^{\pm\langle\alpha_{j},h_{i}\rangle}f_{j}(z),$
$\displaystyle[\alpha_{i,m},\alpha_{j,n}]=\delta_{m+n,0}\frac{[a_{ij}m]_{i}}{m}\frac{q^{cm}-q^{-cm}}{q_{j}-q_{j}^{-1}}\frac{1-p^{m}}{1-p^{*m}}q^{-cm},$
$\displaystyle[\alpha_{i,m},e_{j}(z)]=\frac{[a_{ij}m]_{i}}{m}\frac{1-p^{m}}{1-p^{*m}}q^{-cm}z^{m}e_{j}(z),$
$\displaystyle[\alpha_{i,m},f_{j}(z)]=-\frac{[a_{ij}m]_{i}}{m}z^{m}f_{j}(z),$
$\displaystyle(z-q^{b_{ij}}w)g_{ij}(w/z;p^{*})e_{i}(z)e_{j}(w)=(q^{b_{ij}}z-w)g_{ij}(z/w;p^{*})e_{j}(w)e_{i}(z),{}$
$\displaystyle(z-q^{-b_{ij}}w)g_{ij}(w/z;p)^{-1}f_{i}(z)f_{j}(w)=(q^{-b_{ij}}z-w)g_{ij}(z/w;p)^{-1}f_{j}(w)f_{i}(z),{}$
$\displaystyle[e_{i}(z),f_{j}(w)]=\frac{\delta_{i,j}}{q_{i}-q_{i}^{-1}}\left(\delta\bigl{(}q^{c}{w}/{z}\bigr{)}\phi^{-}_{i}(q^{{c}/{2}}w)-\delta\bigl{(}q^{-c}{w}/{z}\bigr{)}\phi^{+}_{i}(q^{{c}/{2}}z)\right),$
\+ Serre relations.

Here we set

$\displaystyle
g_{ij}(z;s)=\exp\left(-\sum_{m>0}\frac{1}{m}\frac{q^{b_{ij}m}-q^{-b_{ij}m}}{1-s^{m}}(sz)^{m}\right)\
\in\ {\mathbb{C}}[[s]][[z]].$

These relations are treated as formal Laurent series in the argument of the
elliptic currents i.e. $z,w$ etc.. All the coefficients in $z,w$ etc. are well
defined in the $p$-adic topology.

It is easy to find that in the limit $p\to 0$ the above relations except for
those indicating the non-commutativity of ${\mathbb{F}}$ and
$e_{i}(z),f_{i}(z),K^{\pm}_{j}$ go to the defining relations of the quantum
affine algebra $U_{q}({g})$[15]. One also finds that the non-commutativity of
${\mathbb{F}}$ can be realized as the one of $P_{j}$ and $Q_{i}$ by setting
$[P_{i},Q_{j}]=a_{i,j}\ (i,j\in I)$. Hence by using the group algebra
${\mathbb{C}}[{\mathcal{R}}_{Q}]$ of ${\mathcal{R}}_{Q}$, i.e.
$e^{Q_{\alpha}},e^{Q_{\beta}},e^{Q_{\alpha}}e^{Q_{\beta}}=e^{Q_{\alpha}+Q_{\beta}},e^{0}=1\in{\mathbb{C}}[{\mathcal{R}}_{Q}]$,
one obtains the following isomorphism[17]. For generic $p,q$,

$\displaystyle U_{q,p}({g})/pU_{q,p}({g})\ \cong\
(U_{q}({g})\otimes{\mathbb{F}}[[p]])\sharp{\mathbb{C}}[{\mathcal{R}}_{Q}].$

Here the smash product $\sharp$ expresses the non-commutativity between
${\mathbb{F}}$ and ${\mathbb{C}}[{\mathcal{R}}_{Q}]$.

Remark. For representations, on which $q^{\pm c/2}$ take complex values e.g.
$q^{\pm k/2}$ (see Sec.4), we treat $p$ and $p^{*}=pq^{-2k}$ as generic
complex numbers satisfying $|p|<1$ and $|p^{*}|<1$. Then one has

$\displaystyle
g_{ij}(z;s)=\frac{(sq^{b_{ij}}z;s)_{\infty}}{(sq^{-b_{ij}}z;s)_{\infty}},\qquad(z;s)_{\infty}=\prod_{n=0}^{\infty}(1-zs^{n})$

for $|sq^{\pm b_{ij}}z|<1$, $s=p,p^{*}$. Hence in the sense of analytic
continuation, one can rewrite the relations of $e_{i}(z)$ and $e_{j}(w)$ and
of $f_{i}(z)$ and $f_{j}(w)$ as

$\displaystyle
z\theta_{p^{*}}(q^{b_{ij}}w/z)e_{i}(z)e_{j}(w)=-w\theta_{p^{*}}(q^{b_{ij}}z/w)e_{j}(w)e_{i}(z),$
(2.1) $\displaystyle
z\theta_{p}(q^{-b_{ij}}w/z)f_{i}(z)f_{j}(w)=-w\theta_{p}(q^{-b_{ij}}z/w)f_{j}(w)f_{i}(z),$
(2.2)

respectively. Here $\theta_{p}(z)$ denotes the odd theta function given by

$\displaystyle\theta_{p}(z)=(z;p)_{\infty}(p/z;p)_{\infty}(p;p)_{\infty}.$

The additive notations are also often used. Introduce $r,r^{*}=r-k\
\in{\mathbb{R}}_{>0}$ and set $p=q^{2r},p^{*}=pq^{-2k}=q^{2r^{*}}$,

$\displaystyle E_{i}(z):=e_{i}(z){z^{-\frac{P_{\alpha_{i}}-1}{r^{*}}}},\quad
F_{i}(z):=f_{i}(z){z^{\frac{(P+h)_{\alpha_{i}}-1}{r}}}.$

Then from (2.1), (2.2) and the non-commutativity of $h,P$ with
$e_{i}(z),f_{i}(z)$ one obtains

$\displaystyle
F_{i}(z_{1})F_{j}(z_{2})=\frac{[u_{1}-u_{2}-a_{ij}/2]}{[u_{1}-u_{2}+a_{ij}/2]}F_{j}(z_{2})F_{i}(z_{1}),$
$\displaystyle
E_{i}(z_{1})E_{j}(z_{2})=\frac{[u_{1}-u_{2}+a_{ij}/2]^{*}}{[u_{1}-u_{2}-a_{ij}/2]^{*}}E_{j}(z_{2})E_{i}(z_{1}).$

Here we set $z_{i}=q^{2u_{i}}\ (i=1,2)$. The symbols $[u]$ and $[u]^{*}$
denote Jacobi’s odd theta functions given by

$\displaystyle[u]=q^{\frac{u^{2}}{r}-u}\theta_{p}(q^{2u}),\quad[u]^{*}=q^{\frac{u^{2}}{r^{*}}-u}\theta_{p^{*}}(q^{2u}).$

They have the following quasi periodicity.

$\displaystyle[u+r]=-[u],\quad[u+r\tau]=-e^{-\pi i\tau}e^{-2\pi i{u}/{r}}[u],$
$\displaystyle[u+r^{*}]^{*}=-[u]^{*},\quad[u+r^{*}\tau^{*}]^{*}=-e^{-\pi
i\tau^{*}}e^{-2\pi i{u}/{r^{*}}}[u]^{*},$

where $p=e^{-2\pi i/\tau},\ p^{*}=e^{-2\pi i/\tau^{*}}$.

###  2.2 Coalgebra structure

In order to formulate a coalgebra structure over
${\mathbb{F}}={\mathcal{M}}_{H^{*}}$, which does not commute with the Drinfeld
generators, one needs to extend the Hopf algebra to the $H$-Hopf algebroid.
For the elliptic algebra ${\mathcal{U}}=U_{q,p}({g})$, it is roughly sketched
as follows. For expositions of the background concepts of $H$-Hopf algebroid,
the reader can consult[16, 35, 41].

Let $P+h=\sum_{i\in I}c_{i}(P_{i}+h_{i})\in H,\ c_{i}\in{\mathbb{C}}$. The
algebra ${\mathcal{U}}$ is a $H$-algebra by

$\displaystyle{\mathcal{U}}=\bigoplus_{\alpha,\beta\in{h}^{*}}{\mathcal{U}}_{\alpha,\beta}$
$\displaystyle({\mathcal{U}})_{\alpha\beta}=\left\\{x\in U\left|\
q^{P+h}xq^{-(P+h)}=q^{\langle\alpha,P+h\rangle}x,\quad q^{P}xq^{-P}=q^{\langle
Q_{\beta},P\rangle}x\quad\forall P+h,P\in H\right.\right\\}$

with the two moment maps $\mu_{l},\mu_{r}:{\mathbb{F}}\to{\mathcal{U}}_{0,0}$
defined by

$\displaystyle\mu_{l}(\widehat{f})=f(h,P+h,p)\in{\mathbb{F}}[[p]],\qquad\mu_{r}(\widehat{f})=f(h,P,p^{*})\in{\mathbb{F}}[[p^{*}]]$

for $\widehat{f}=f(h,P,p^{*})\in{\mathbb{F}}[[p^{*}]]$. Here $p^{*}=pq^{-2c}$
as before.

The tensor product ${\mathcal{U}}{\widetilde{\otimes}}\,{\mathcal{U}}$ is the
${h}^{*}$-bigraded vector space with

$\displaystyle({\mathcal{U}}{\widetilde{\otimes}}\,{\mathcal{U}})_{\alpha\beta}=\bigoplus_{\gamma\in{h}^{*}}({\mathcal{U}}_{\alpha\gamma}\otimes_{{\mathcal{M}}_{H^{*}}}{\mathcal{U}}_{\gamma\beta}),$

where $\otimes_{{\mathcal{M}}_{H^{*}}}$ denotes the ordinary tensor product
modulo the following relation.

$\displaystyle\mu_{r}(\widehat{f})a\,\widetilde{\otimes}\,b=a\,\widetilde{\otimes}\,\mu_{l}(\widehat{f})b\qquad
a,b\in{\mathcal{U}}.\ $

Let us regard $T_{\alpha}=e^{-Q_{\alpha}}\in{\mathbb{C}}[{\mathcal{R}}_{Q}]$
as a shift operator

$\displaystyle(T_{\alpha}\widehat{f})=f(h,P+\langle Q_{\alpha},P\rangle,p).$

Then ${\mathcal{D}}=\\{\widehat{f}e^{-Q_{\alpha}}\ |\
\widehat{f}\in{\mathbb{F}},e^{-Q_{\alpha}}\in{\mathbb{C}}[{\mathcal{R}}_{Q}]\\}$
becomes the $H$-algebra having the property

$\displaystyle{\mathcal{U}}\cong{\mathcal{U}}\,\widetilde{\otimes}\,{\mathcal{D}}\cong{\mathcal{D}}\,\widetilde{\otimes}\,{\mathcal{U}}$

by $a\cong a\,\widetilde{\otimes}\,T_{-\beta}\cong
T_{-\alpha}\,\widetilde{\otimes}\,a$ for all
$a\in{\mathcal{U}}_{\alpha\beta}$.

Let $\pi_{V},V=\oplus_{i}{\mathbb{C}}v_{i}$ be the vector representation of
$\bar{{g}}$ and consider the elliptic dynamical $R$-matrix
$R^{+}(z,\Pi)\in\mathrm{End}(V\otimes V)$ of type ${g}$[29, 39].

For example, the $\widehat{{sl}}_{N}$ type is given by

$\displaystyle R^{+}(z,\Pi)$ $\displaystyle=$
$\displaystyle{\rho}^{+}(z)\bar{R}(z,\Pi),$ (2.3)
$\displaystyle\bar{R}(z,\Pi)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}E_{j,j}\otimes E_{j,j}+\sum_{1\leq j_{1}<j_{2}\leq
N}\left(b(u,(P+h)_{j_{1},j_{2}})E_{j_{1},j_{1}}\otimes
E_{j_{2},j_{2}}+\bar{b}(u)E_{j_{2},j_{2}}\otimes E_{j_{1},j_{1}}\right.$
$\displaystyle\qquad\left.+c(u,(P+h)_{j_{1},j_{2}})E_{j_{1},j_{2}}\otimes
E_{j_{2},j_{1}}+\bar{c}(u,(P+h)_{j_{1},j_{2}})E_{j_{2},j_{1}}\otimes
E_{j_{1},j_{2}}\right),{}$

where $E_{i,j}v_{k}=\delta_{j,k}v_{i}$, $z=q^{2u}$,
$\Pi_{j,k}=q^{2(P+h)_{j,k}}$, $(P+h)_{j,k}=\sum_{i=j}^{k-1}(P_{i}+h_{i})$,

$\displaystyle{\rho}^{+}(z)=q^{-\frac{N-1}{N}}z^{\frac{N-1}{rN}}\frac{\Gamma(z;p,q^{2N})\Gamma(q^{2N}z;p,q^{2N})}{\Gamma(q^{2}z;p,q^{2N})\Gamma(q^{2N-2}z;p,q^{2N})},$
$\displaystyle
b(u,s)=\frac{[s+1][s-1][u]}{[s]^{2}[u+1]},\qquad\bar{b}(u)=\frac{[u]}{[u+1]},$
$\displaystyle
c(u,s)=\frac{[1][s+u]}{[s][u+1]},\qquad\bar{c}(u,s)=\frac{[1][s-u]}{[s][u+1]}.$

The symbol $\Gamma(z;p,q^{2N})$ denotes the elliptic Gamma function defined by

$\displaystyle\Gamma(z;p,q)=\frac{(pq/z;p,q)_{\infty}}{(z;p,q)_{\infty}},\qquad(z;p,q)_{\infty}=\prod_{m,n=0}^{\infty}(1-zp^{m}q^{n}),\quad|p|,|q|<1.$
(2.4)

Let
${L}^{+}(z)=\sum_{i,j}E_{i,j}L^{+}_{i,j}(z)\in\mathrm{End}(V)\,\widetilde{\otimes}\,{\mathcal{U}}[[z,z^{-1}]]$
be the the $L$-operator satisfying the dynamical $RLL$-relation[29]

$\displaystyle
R^{+(12)}(z_{1}/z_{2},\Pi){L}^{+(1)}(z_{1}){L}^{+(2)}(z_{2})={L}^{+(2)}(z_{2}){L}^{+(1)}(z_{1})R^{+*(12)}(z_{1}/z_{2},\Pi^{*}).$

Here $R^{+*}(z,\Pi^{*})$ denotes the same elliptic $R$-matrix
$R^{+}(z,\Pi^{*})$ with $\Pi^{*}_{i,j}=q^{2P_{i,j}}$ except for replacing the
elliptic nome $p$ by $p^{*}$. Define two $H$-algebra homomorphisms, the counit
$\varepsilon:{\mathcal{U}}\to{\mathcal{D}}$ and the comultiplication
$\Delta:{\mathcal{U}}\to{\mathcal{U}}\widetilde{\otimes}\;{\mathcal{U}}$, and
the antihomomorphism $S:{\mathcal{U}}\to{\mathcal{U}}$ by

$\displaystyle\varepsilon(L^{+}_{i,j}(z))=\delta_{i,j}{T}_{Q_{\epsilon_{i}}}\quad(n\in{\mathbb{Z}}),\qquad\varepsilon(e^{Q})=e^{Q},$
$\displaystyle\varepsilon(\mu_{l}({\widehat{f}}))=\varepsilon(\mu_{r}(\widehat{f}))=\widehat{f}T_{0},$
$\displaystyle\Delta(L^{+}_{i,j}(z))=\sum_{k}L^{+}_{i,k}(z)\widetilde{\otimes}L^{+}_{k,j}(z),$
$\displaystyle\Delta(e^{Q})=e^{Q}\,\widetilde{\otimes}\,e^{Q},$
$\displaystyle\Delta(\mu_{l}(\widehat{f}))=\mu_{l}(\widehat{f})\widetilde{\otimes}1,\quad\Delta(\mu_{r}(\widehat{f}))=1\widetilde{\otimes}\mu_{r}(\widehat{f}),$
$\displaystyle S(L^{+}_{ij}(z))=(L^{+}(z)^{-1})_{ij},$ $\displaystyle
S(e^{Q})=e^{-Q},\quad S(\mu_{r}(\hat{f}))=\mu_{l}(\hat{f}),\quad
S(\mu_{l}(\hat{f}))=\mu_{r}(\hat{f}).$

Here $\alpha_{i}=\epsilon_{i}-\epsilon_{i+1}\ (i\in\bar{I})$. Then the set
$(U_{q,p}({g}),H,{{\mathcal{M}}}_{H^{*}},\mu_{l},\mu_{r},\Delta,\varepsilon,S)$
becomes a $H$-Hopf algebroid[41, 44, 46, 45].

An explicit realization of $L^{+}(z)$ in terms of the elliptic currents of
${\mathcal{U}}$ was studied in[28, 36, 44, 46]. In general, the connection
between $L^{+}(z)$ and the elliptic currents is given as follows. Let
$\\{h^{1},\cdots,h^{l}\\}$ be the dual basis to $\\{h_{1},\cdots,h_{l}\\}$ of
$\bar{{h}}$ and $(\pi_{V},V)$ be the vector representation of $\bar{{g}}$.
Define[29]

$\displaystyle L^{-}(z)=({\rm
Ad}(q^{-2\theta_{V}(P)})\otimes\mathrm{id})(q^{2T_{V}}L^{+}(zpq^{-c}),$
$\displaystyle\theta_{V}(P)=-\frac{1}{2}\sum_{j}\left(\pi_{V}(h_{j})\pi_{V}(h^{j})+2P_{j}\pi_{V}(h^{j})\right),$
$\displaystyle T_{V}=\sum_{j}\pi_{V}(h_{j})\otimes h^{j}.$

One finds that $L^{\pm}(z)$ satisfy

$\displaystyle
R^{\pm(12)}(z_{1}/z_{2},\Pi){L}^{\pm(1)}(z_{1}){L}^{\pm(2)}(z_{2})={L}^{\pm(2)}(z_{2}){L}^{\pm(1)}(z_{1})R^{\pm*(12)}(z_{1}/z_{2},\Pi^{*}),$
(2.5) $\displaystyle
R^{+(12)}(q^{c}z_{1}/z_{2},\Pi){L}^{+(1)}(z_{1}){L}^{-(2)}(z_{2})={L}^{-(2)}(z_{2}){L}^{+(1)}(z_{1})R^{+*(12)}(q^{-c}z_{1}/z_{2},\Pi^{*}).$
(2.6)

Here $R^{\pm}(z,\Pi),R^{\pm*}(z,\Pi^{*})$ and $L^{+}(z)$ are related to
$R^{\pm}_{VV}(z,\lambda+h),R^{\pm}_{VV}(z,\lambda)$ and $L^{+}_{V}(z,\lambda)$
in [29] by

$\displaystyle R^{\pm}(z,\Pi)=R^{\pm}_{VV}(z,\lambda+h),\qquad
R^{\pm*}(z,\Pi^{*})=R^{\pm}_{VV}(z,\lambda),$ $\displaystyle
L^{+}(z)=L^{+}_{V}(z,\lambda)e^{-\sum_{i}E_{i,i}Q_{\epsilon_{i}}}$

with $\lambda=(r-k+h^{\vee})d+\sum_{i}(P_{\alpha^{\prime}_{i}}+1)\bar{h}^{i}$,
$\lambda+h=(r+h^{\vee})d+\sum_{i}(P_{\alpha^{\prime}_{i}}+h_{\alpha^{\prime}_{i}}+1)\bar{h}^{i}$,
$[P_{i,j},Q_{\epsilon_{k}}]=\delta_{i,k}-\delta_{j,k}$ and
$\alpha^{\prime}_{i}$ being the simple root of the Langlands dual Lie algebra
of $\bar{{g}}$[39]. Then consider the following Gauss decomposition of
$L^{\pm}(z)$.

$\displaystyle L^{\pm}(z)=F^{\pm}(z)K^{\pm}(z)E^{\pm}(z).$

Here $K^{\pm}(z)$ are diagonal matrices with entries $K^{\pm}_{j}(z)$, whereas
$F^{\pm}(z)$ (resp. $E^{\pm}(z)$) are upper (resp. lower) triangular matrices
with entries $F^{\pm}_{i,j}(z)$ (resp. $E^{\pm}_{j,i}(z)$) $(i<j)$ and all
diagonal entries 1. By combining the relations for these entries obtained from
(2.5)-(2.6), one finds the following identification with the elliptic currents
$E_{j}(z),F_{j}(z)$ and $\phi^{\pm}_{j}(z)$.

$\displaystyle
E_{j}(zq^{j-c/2})=\mu^{*}\left(E^{+}_{j+1,j}(zq^{c/2})-E^{-}_{j+1,j}(zq^{-c/2})\right),$
$\displaystyle
F_{j}(zq^{j-c/2})=\mu\left(F^{+}_{j,j+1}(zq^{-c/2})-F^{-}_{j,j+1}(zq^{c/2})\right),$
$\displaystyle\phi^{\pm}_{j}(zq^{-c/2}q^{j})=\kappa
K^{\pm}_{j}(z)K^{\pm}_{j+1}(z)^{-1}.$

Here $\mu,\mu^{*},\kappa$ are constants satisfying

$\displaystyle\mu\mu^{*}=-\kappa\frac{q}{q-q^{-1}}\frac{(p^{*};p^{*})_{\infty}}{\theta_{p^{*}}(q^{2})}.$

See for example [44, 45].

Note also that to formulate a $H$-Hopf algebroid structure one may also use
the Drinfeld comultiplication for the elliptic currents[47, 48] instead of the
standard comultiplication for $L^{+}(z)$ given in the above. The resultant
$H$-Hopf algebroid is completely different coalgebra structure from the above.
These two coalgebra structures have their own appropriate applications. In
particular, they define different vertex operators as intertwining operators
of $U_{q,p}({g})$-modules, even if the modules are the same. The standard one
is appropriate to discuss the problems exposed in the subsequent sections
Sec.5-10, whereas the Drinfeld comultiplication is appropriate to discuss
problems related to the deformed $W$-algebras. See Sec.3 and Sec.11. One
should also note that for the quantum toroidal algebras, whatever they are
trigonometric or elliptic, only the coalgebra structures associated with the
Drinfeld comultiplication are available at the moment, because of a lack of
expressions of the (elliptic) $R$-matrices and the $L$-operators.

##  3 $U_{q,p}(\mbox{\fourteeneufm g})$ as Deformed $W$-algebra

The elliptic algebra $U_{q,p}({g})$ has another characterization as a
$q$-deformation of the $W$-algebra ${\cal W}\bar{{g}}$ of the coset type
$({g})_{r-h^{\vee}-k}\oplus({g})_{k}\supset({g}_{\rm diag})_{r-h^{\vee}}$[27,
9, 12], or more precisely as an algebra of screening currents of the
deformation of ${\cal W}\bar{{g}}$. Here we assume ${g}$ is an untwisted
affine Lie algebra and realized as a central extension of the loop
algebra[33]. The generating functions of ${g}$ is called the currents. The
symbol $({g})_{k}$ means to consider ${g}$ at the level $k$ representation. A
key for understanding this characterization of $U_{q,p}({g})$ is the
(generalized) Feigin-Fuchs (FF) construction of ${\cal W}\bar{{g}}$ from
${g}$[19, 12]. There one starts from the level-$k$ currents of ${g}$ in the
Lepowsky-Wilson (LW) realization[53] and the corresponding energy-momentum
(EM) tensor of the Wess-Zumino-Witten (WZW) model, i.e. the generating
function of the Virasoro algebra. The LW realization of ${g}$ is a realization
of the currents in terms of the $\mathrm{rk}\;{{g}}$ number of free bosons and
the level-$k$ $Z$-algebra. The EM tensor is obtained by the Sugawara
construction in terms of the currents. Then the FF construction deforms the EM
tensor by introducing the so-called background charge term, which depends on a
parameter $r$, and at the same time deforms the free boson part of the
currents of ${g}$ by $r$ and the $Z$-algebra part by adding the dynamical
parameters. The deformed EM tensor becomes the generating function of the
Virasoro generators of the coset $W$-algebra ${\cal W}\bar{{g}}$, whereas the
deformed currents of ${g}$ becomes the so-called screening currents, i.e. the
primary field of conformal dimension one w.r.t. this new Virasoro generators.
All the generating functions of ${\cal W}\bar{{g}}$, the Virasoro as well as
the other generators of higher conformal dimensions, are characterized as the
intersection of the kernel of the screening operators i.e. certain contour
integral of the screening currents, on $U({g})$[18]. In this sense, we refer
the resultant $r$-deformation (+ modification by the dynamical parameters) of
the affine Lie algebra ${g}$ to the $W$-algebra ${\cal W}\bar{{g}}$. When one
consider a $q$-deformation of it, ${g}$ and its $r$-deformation ${\cal
W}\bar{{g}}$ turn out to give $U_{q}({g})$ and its $p$-deformation
$U_{q,p}({g})$ with $p=q^{2r}$, respectively.

For example, the currents of the level $k(\not=0)$ affine Lie algebra
$\widehat{{sl}}_{2}$ has the following LW realization.

$\displaystyle
e(z)=Z^{+}(z):\exp\left\\{\sqrt{2k}\sum_{m\not=0}\frac{a_{m}}{km}z^{-m}\right\\}:,\quad
f(z)=Z^{-}(z):\exp\left\\{-\sqrt{2k}\sum_{m\not=0}\frac{a_{m}}{km}z^{-m}\right\\}:,$

where $Z^{\pm}(z)$ are the generating functions of the level-$k$ $Z$-algebra,
and $a_{m}\ (m\in{\mathbb{Z}}\backslash\\{0\\})$ generate the Heisenberg
subalgebra of $\widehat{{sl}}_{2}$ satisfying

$\displaystyle[\sqrt{{2k}}a_{m},\sqrt{{2k}}a_{n}]=2mk\delta_{m+n,0}.$ (3.1)

The generating function
$\widetilde{\phi(z)}=\sum_{m\not=0}\frac{a_{m}}{m}z^{-m}$ is called free
boson. The $Z$-algebra is realized in terms of the
${\mathbb{Z}}_{k}$-parafermions $\Psi(z),\Psi^{\dagger}(z)$ and the zero mode
operators $\alpha$, $h$ satisfying $[h,\alpha]=2$ as follows.

$\displaystyle Z^{+}(z)=\Psi(z)e^{\alpha}z^{h/k},\qquad
Z^{-}(z)=\Psi^{\dagger}(z)e^{-\alpha}z^{-h/k}.$ (3.2)

The corresponding 2d conformal field theory (CFT) is the $\widehat{{sl}}_{2}$
WZW model generated by the EM tensor

$\displaystyle T(z)=\frac{1}{2}(\partial\widetilde{\phi(z)})^{2}+\mbox{(zero-
modes term)}+T_{PF}(z).$

Here $T_{PF}(z)$ denotes the EM tensor of the ${\mathbb{Z}}_{k}$-parafermions.
The central charge of the models is $c_{WZW}=\frac{3k}{k+2}$.

Then the FF construction is the following procedure. Deform $T(z)$ by adding
the background charge term to $T(z)$

$\displaystyle T(z)\ \mapsto\
T_{FF}(z)=T(z)+\sqrt{\frac{k}{2r(r-k)}}\partial^{2}\widetilde{\phi(z)}.$

This makes $T_{FF}(z)$ the generating function of the coset Virasoro algebra
$(\widehat{{sl}}_{2})_{r-2-k}\oplus(\widehat{{sl}}_{2})_{k}\supset(\widehat{{sl}}_{2})_{r-2}$
whose central charge is
$c_{Vir}=\frac{3k}{k+2}\left(1-\frac{2(k+2)}{r(r-k)}\right)$. At the same
time, deform the currents of $\widehat{{sl}}_{2}$ by scaling the coefficients
of the free boson and by adding the dynamical parameter $P$ and its conjugate
$Q$ satisfying $[P,Q]=2$ to the $Z$-algebra part :

$\displaystyle e(z)$ $\displaystyle\mapsto$ $\displaystyle
S^{+}(z)=Z^{+}(z)e^{-Q}z^{-\frac{P-1}{r-k}}:\exp\left\\{\sqrt{\frac{2kr}{r-k}}\sum_{m\not=0}\frac{a_{m}}{km}z^{-m}\right\\}:,$
(3.3) $\displaystyle f(z)$ $\displaystyle\mapsto$ $\displaystyle
S^{-}(z)=Z^{-}(z)z^{\frac{P+h-1}{r}}:\exp\left\\{-\sqrt{\frac{2k(r-k)}{r}}\sum_{m\not=0}\frac{a_{m}}{km}z^{-m}\right\\}:.$
(3.4)

It then turns out $S^{\pm}(z)$ become the screening currents of the Virasoro
algebra generated by $T_{FF}(z)$.

Now let us consider a $q$-deformation of these constructions. It is known that
the quantum affine algebra $U_{q}({g})$ has an analogue of the LW
realiozation. See for example [17]. In our case, the level-$k$ currents of
$U_{q}(\widehat{{sl}}_{2})$ are realized in terms of the Heisenberg generators
$\tilde{a}_{m}$ and the level-$k$ $q$-deformed $Z$ operators $Z^{\pm}_{q}(z)$.
The former satisfy

$\displaystyle[\tilde{a}_{m},\tilde{a}_{n}]=\frac{[2m]_{q}[km]_{q}}{m}q^{-k|m|}\delta_{m+n,0}.$
(3.5)

This is a $q$-deformation of (3.1), i.e. in the limit $q\to 1$ one recovers
(3.1). The operators $Z^{\pm}_{q}(z)$ are realized in the same was as (3.2) by
replacing $\Psi(z),\Psi^{\dagger}(z)$ with appropriate $q$-parafermions[38,
17]. The level-$k$ currents of $U_{q}(\widehat{{sl}}_{2})$ are hence realized
as

$\displaystyle
e_{q}(z)=Z_{q}^{+}(z):\exp\left\\{\sum_{m\not=0}\frac{\tilde{a}_{m}}{[km]_{q}}z^{-m}\right\\}:,\quad
f_{q}(z)=Z_{q}^{-}(z):\exp\left\\{-\sum_{m\not=0}\frac{\tilde{a}_{m}}{[km]_{q}}z^{-m}\right\\}:.$

Then a $q$-analogue of the FF construction (3.3)-(3.4) is obtained by noting
the following observations.

  * •

In (3.3) and (3.4), the $Z$-algebra part remains the same as in $e(z)$ and
$f(z)$ except for getting a modification by the dynamical parameters

  * •

The commutation relation of the Heisenberg generators $a_{m}$ with the scaled
coefficients can be read as from $S^{+}(z)$,

$\displaystyle[\sqrt{\frac{2kr}{r-k}}a_{m},\sqrt{\frac{2kr}{r-k}}a_{n}]=2mk\frac{r}{r-k}\delta_{m+n,0},$
(3.6)

and from $S^{-}(z)$,

$\displaystyle[\sqrt{\frac{2k(r-k)}{r}}a_{m},\sqrt{\frac{2k(r-k)}{r}}a_{n}]=2mk\frac{r-k}{r}\delta_{m+n,0}.$
(3.7)

The first point indicates that the $q$-$Z$-algebra operators $Z^{\pm}_{q}(z)$
in $e_{q}(z)$ and $f_{q}(z)$ should remain the same except for getting a
modification by the dynamical parameters $P$ and $Q$ in a $q$-deformed FF
process. The second point suggests to use the Heisenberg generators
$\alpha_{m}$ or $\alpha^{\prime}_{m}$ satisfying the following $p$-deformed
commutation relation from (3.5).

$\displaystyle[{\alpha}_{m},{\alpha}_{n}]=\frac{[2m]_{q}[km]_{q}}{m}\frac{1-p^{m}}{1-p^{*m}}q^{-km}\delta_{m+n,0},$
(3.8)
$\displaystyle[{\alpha}^{\prime}_{m},{\alpha}^{\prime}_{n}]=\frac{[2m]_{q}[km]_{q}}{m}\frac{1-p^{*m}}{1-p^{m}}q^{km}\delta_{m+n,0}.$
(3.9)

Here we set $p=q^{2r}$, $p^{*}=pq^{-2k}=q^{2(r-k)}$, and $\alpha_{m}$,
$\alpha^{\prime}_{m}$ are not independent

$\displaystyle{\alpha}^{\prime}_{m}=\frac{1-p^{*m}}{1-p^{m}}q^{km}{\alpha}_{m}.$

These are $q$-deformations of (3.6) and (3.7). Note that (3.8) is nothing but
the relations for the Heisenberg subalgebra of $U_{q,p}(\widehat{{sl}}_{2})$
at level $k$ given in Sec. 2.1. One hence obtains the following $q$-analogue
of the FF construction.

$\displaystyle e_{q}(z)$ $\displaystyle\mapsto$ $\displaystyle
S^{+}_{q}(z)=Z_{q}^{+}(z)e^{-Q}z^{-\frac{P-1}{r-k}}:\exp\left\\{\sum_{m\not=0}\frac{{\alpha}_{m}}{[km]_{q}}z^{-m}\right\\}:,$
$\displaystyle f_{q}(z)$ $\displaystyle\mapsto$ $\displaystyle
S^{-}_{q}(z)=Z_{q}^{-}(z)z^{\frac{P+h-1}{r}}:\exp\left\\{-\sum_{m\not=0}\frac{{\alpha}^{\prime}_{m}}{[km]_{q}}z^{-m}\right\\}:.$

Then the currents $S^{+}_{q}(z)$ and $S^{-}_{q}(z)$ turn out to give a
realization of the level-$k$ elliptic currents $e(z)$ and $f(z)$ of
$U_{q,p}(\widehat{{sl}}_{2})$, respectively[38, 17]. Hence regarding $e(z)$
and $f(z)$ as screening currents, one can define a deformation of the
$W$-algebra ${\cal W}{sl}_{2}$ of the coset type at level $k$ as the kernel of
the screening operators associated to either $e(z)$ or $f(z)$.

In general, in the FF process the $Z$-algebra structure of ${g}$ remains the
same as in the LW realization to realize the screening currents of ${\cal
W}\bar{{g}}$ of the coset type
$({g})_{r-h^{\vee}-k}\oplus({g})_{k}\supset({g}_{\rm diag})_{r-h^{\vee}}$
except for a modification by adding the dynamical parameters. Correspondingly,
the $q$-$Z$-algebra structure of $U_{q}({g})$ remains the same to realize the
elliptic currents $e_{i}(z),f_{i}(z)$ of $U_{q,p}({g})$ as a $p$-deformation
of $U_{q}({g})$ except for a modification by the dynamical parameters[38, 17].
Then by using $e_{i}(z)$ and $f_{i}(z)$ as screening currents, one defines a
deformation of ${\cal W}\bar{{g}}$ as the intersection of the kernel of the
screening operators associated with either $e_{i}(z)$’s or $f_{i}(z)$’s. We
denote the resultant deformed $W$-algebra by
${\mathcal{W}}_{p,p^{*}}\bar{{g}}$.

$\displaystyle U_{q}({g})$ $\displaystyle{\footnotesize q\mbox{-deform.}}\
\rotatebox[origin={c}]{45.0}{$\longrightarrow$}\hskip 28.45274pt$

$\longrightarrow$

$\displaystyle{\footnotesize{p=q^{2r}}\mbox{-deform.}+\mbox{dynamical
param.}}$ $\displaystyle{g}\hskip 65.44142pt$ $\displaystyle\quad
U_{q,p}({g})$ $\displaystyle{\footnotesize{r}\mbox{-deform.}+\mbox{dynamical
param.}}\ \rotatebox[origin={c}]{-45.0}{$\longrightarrow$}\hskip 28.45274pt$

$\longrightarrow$

$\displaystyle{\footnotesize q\mbox{-deform.}}$ (the Feigin-Fuchs
construction) $\displaystyle{\cal W}\bar{{g}}$ $\displaystyle\parallel$
$\displaystyle({g})_{r-h^{\vee}-k}\oplus({g})_{k}\supset({g})_{r-h^{\vee}}$

Figure 3 : Two chacterizations of $U_{q,p}({g})$

Note that here we have obtained two elliptic nomes $p$ and $p^{*}$ naturally
in a $q$-deformation of the FF construction. The parameters $p,p^{*}$
correspond to $q,t$ in Frenkel-Reshetikhin’s formulation of the deformed
$W$-algebra $W_{q,t}(\bar{{g}})$[24]333 Frenkel-Reshetikhin’s deformed
$W$-algebras are the Hamiltonian reduction type, which are isomorphic to the
coset type only for the simply laced $\bar{{g}}$. For example, for the case
$B_{l}$ the coset construction gives Fateev-Lukyanov’s $WB_{l}$ algebra[54],
which is different from $W(B_{l})$ obtained by the quantum Hamiltonian
reduction of $B^{(1)}_{l}$. See for example [9]. . It is remarkable that the
appearance of two elliptic nomes is also consistent to the quasi-Hopf
formulation of the face type EQG ${{\mathcal{B}}_{q,\lambda}}({g})$[29], where
$p$ is treated as a dynamical parameter and $p^{*}$ is nothing but a dynamical
shift of $p$ by $q^{-2c}$.

The generating function of ${\mathcal{W}}_{p,p^{*}}\bar{{g}}$ can also be
constructed explicitly in terms of the vertex operators of $U_{q,p}({g})$
w.r.t. the Drinfeld comultiplication. The same construction is valid also in
the elliptic quantum toroidal algebras, which give a realization of the
deformed affine quiver $W$-algebras. See Sec.11 for the case ${{gl}}_{1,tor}$.

Moreover in Sec.5, we discuss the vertex operators of
$U_{q,p}(\widehat{{sl}}_{N})$ derived as intertwining operators of
$U_{q,p}(\widehat{{sl}}_{N})$-modules w.r.t. the standard comultiplication.
Constructing correlation functions of them, one finds that they play the role
of the vertex operators of the corresponding deformed $W$-algebra
${\mathcal{W}}{}_{p,p^{*}}{{sl}_{N}}$. See Sec.7 and 8.

##  4 Representations

A representation of $H$-algebra is called a dynamical representation. We skip
all formal descriptions of it and expose only the level-0 and the level-1
dynamical representations of $U_{q,p}(\widehat{{sl}}_{N})$. Here one says that
a $U_{q,p}({g})$-module has level $k$, if $q^{c/2}$ acts as the scalar
$q^{k/2}$ on it. For detailed description of the dynamical representations,
the reader can consult[16, 35, 41].

###  4.1 The evaluation representation

Let $\displaystyle{{{\mathcal{V}}}=\oplus_{i=1}^{N}{\mathbb{F}}v_{i}\otimes
1}$ and set ${{\mathcal{V}}}_{z}={{\mathcal{V}}}[z,z^{-1}]$. The following
gives the level-0 dynamical action of $U_{q,p}(\widehat{{sl}}_{N})$ on
${{\mathcal{V}}}_{z}$.

$\displaystyle\pi_{z}(q^{c/2})=1,\quad\pi_{z}(d)=-z\frac{d}{dz},$
$\displaystyle\pi_{z}(\alpha_{j,m})=\frac{[m]_{q}}{m}(q^{j-N+1}z)^{m}(q^{-m}E_{j,j}-q^{m}E_{j+1,j+1}),$
$\displaystyle\pi_{z}(e_{j}(w))=\frac{(pq^{2};p)_{\infty}}{(p;p)_{\infty}}E_{j,j+1}\delta\left(q^{j-N+1}{z}/{w}\right)e^{-Q_{\alpha_{j}}},$
$\displaystyle\pi_{z}(f_{j}(w))=\frac{(pq^{-2};p)_{\infty}}{(p;p)_{\infty}}E_{j+1,j}\delta\left(q^{j-N+1}{z}/{w}\right),$
$\displaystyle\pi_{z}(\phi_{j}^{+}(w))=q^{-\pi(h_{j})}e^{-Q_{\alpha_{j}}}\frac{\theta_{p}(q^{-j+N-1+2\pi(h_{j})}\frac{w}{z})}{\theta_{p}(q^{-j+N-1}{w}/{z})},$
$\displaystyle\pi_{z}(\phi_{j}^{-}(w))=q^{\pi(h_{j})}e^{-Q_{\alpha_{j}}}\frac{\theta_{p}(q^{j-N+1-2\pi(h_{j})}\frac{z}{w})}{\theta_{p}(q^{j-N+1}{z}/{w})},\qquad
j=1,\cdots,N-1.$

Here $\pi(h_{j})=E_{j,j}-E_{j+1,j+1}$, and the element
$e^{Q}\in{\mathbb{C}}[{\mathcal{R}}_{Q}]$ acts on $f(h,P)v\otimes 1$ by
$e^{Q}\cdot(f(h,P)v\otimes 1)=f(h,P-\langle P,Q\rangle)v\otimes e^{Q}$. This
is called the evaluation representation associated with the vector
representation with the evaluation parameter $z$. In particluar, one has

$\displaystyle\pi_{z}({L^{+}_{i,j}(w)})_{k,l}=R^{+}(w/z,\Pi^{*})_{ik}^{jl}.$

###  4.2 The level-1 highest weight representation

Let $\Lambda_{0}$ and $\Lambda_{a}=\Lambda_{0}+\bar{\Lambda}_{a}\
(a=1,\cdots,N-1)$ be the fundamental weights of $\widehat{{sl}}_{N}$. For
generic $\nu\in{h}^{*}$, set

$\displaystyle{{\mathcal{V}}}(\Lambda_{a}+\nu,\nu)$ $\displaystyle=$
$\displaystyle{\mathbb{F}}\otimes_{\mathbb{C}}({\mathcal{F}}_{\alpha}\otimes
e^{{\Lambda}_{a}}{\mathbb{C}}[{{\mathcal{Q}}}])\otimes
e^{Q_{{\nu}}}{\mathbb{C}}[{{\mathcal{R}}}_{Q}],$

where ${\mathcal{F}}_{\alpha}={\mathbb{C}}[\alpha_{j,-m}\ (j=1,\cdots,N-1,\
m\in{\mathbb{N}}_{>0})]$ denotes the Fock module of the Heisenberg algebra. We
here consider the group algebra
${\mathbb{C}}[{{\mathcal{Q}}}]\otimes{\mathbb{C}}[{\mathcal{R}}_{Q}]$ with the
following central extension.

$\displaystyle
e^{\alpha_{j}}e^{\alpha_{k}}=(-1)^{(\alpha_{j},\alpha_{k})}q^{\frac{1}{r}(\delta_{j,k+1}-\delta_{k,j-1})}e^{\alpha_{k}}e^{\alpha_{j}},$
$\displaystyle
e^{Q_{\alpha_{j}}}e^{Q_{\alpha_{k}}}=q^{\left(\frac{1}{r}-\frac{1}{r^{*}}\right)(\delta_{j,k+1}-\delta_{j,k-1})}e^{Q_{\alpha_{k}}}e^{Q_{\alpha_{j}}},$
$\displaystyle
e^{Q_{\alpha_{j}}}e^{\alpha_{k}}=q^{\frac{1}{r}(\delta_{j,k+1}-\delta_{k,j-1})}e^{\alpha_{k}}e^{Q_{\alpha_{j}}}.$

Then the space ${{\mathcal{V}}}(\Lambda_{a}+\nu,\nu)$ is the level-1
irreducible highest weight $U_{q,p}(\widehat{{sl}}_{N})$-module with the
highest weight $(\Lambda_{a}+\nu,\nu)$ w.r.t. the set $(P+h,P)$ by the action

$\displaystyle E_{j}(z)=\;:\exp\left\\{-\sum_{n\neq
0}\frac{1}{[n]_{q}}\alpha_{j,n}z^{-n}\right\\}:e^{{\alpha}_{j}}e^{-Q_{\alpha_{j}}}z^{h_{\alpha_{j}}+1}(q^{N-j}z)^{-\frac{P_{\alpha_{j}}-1}{r^{*}}},$
$\displaystyle F_{j}(z)=\;:\exp\left\\{\sum_{n\neq
0}\frac{1}{[n]_{q}}\alpha^{\prime}_{j,n}z^{-n}\right\\}:e^{-{\alpha}_{j}}z^{-h_{\alpha_{j}}+1}(q^{N-j}z)^{\frac{(P+h)_{\alpha_{j}}-1}{r}},$
$\displaystyle\phi^{+}_{j}(q^{-{1}/{2}}z)=q^{-h_{j}}e^{-Q_{\alpha_{j}}}:\exp\left\\{(q-q^{-1})\sum_{m\not=0}\frac{\alpha_{j,m}}{1-p^{m}}p^{m}z^{-m}\right\\}:,$
$\displaystyle\phi^{-}_{j}(q^{{1}/{2}}z)=q^{h_{j}}e^{-Q_{\alpha_{j}}}:\exp\left\\{(q-q^{-1})\sum_{m\not=0}\frac{\alpha_{j,m}}{1-p^{m}}z^{-m}\right\\}:\quad(1\leq
j\leq N-1).$

The highest weight vector is given by $1\otimes e^{\Lambda_{a}}\otimes
e^{Q_{{\nu}}}$.

Note that one has the following natural decomposition.

$\displaystyle{{\mathcal{V}}}(\Lambda_{a}+\nu,\nu)=\bigoplus_{\xi,\eta\in{\mathcal{Q}}}{\mathcal{F}}_{a,\nu}(\xi,\eta),\qquad{\mathcal{F}}_{a,\nu}(\xi,\eta)={\mathbb{F}}\otimes_{\mathbb{C}}({\mathcal{F}}_{\alpha}\otimes
e^{{\Lambda}_{a}+{\xi}})\otimes e^{Q_{{\nu}+\eta}}.$

It turns out that ${\mathcal{F}}_{a,\nu}(\xi,\eta)$ is isomorphic to the Verma
module of the $W$-algebra ${\mathcal{W}}{sl}_{N}$ with the central charge
$(N-1)\left(1-\frac{N(N+1)}{r(r-1)}\right)$ and the highest weight
$\frac{1}{r(r-1)}\bigl{|}r(\nu+\eta+\rho)-(r-1)(\Lambda_{a}+\nu+\xi+\eta+\rho)\bigr{|}^{2}$[17].

##  5 Vertex Operators

The vertex operators defined for the infinite dimensional quantum groups such
as $U_{q}({g})$, $U_{q,p}({g})$ and $U_{q,\kappa,p}({g}_{tor})$ have proven to
be key objects in many applications[23, 30, 37, 10, 47]. Here we expose the
vertex operators of $U_{q,p}(\widehat{{sl}}_{N})$[28, 36], which are used in a
derivation of the elliptic weight functions (Sec.6), integral solutions of the
elliptic $q$-KZ equation and the vertex function (Sec.7) and also in the
algebraic analysis of the elliptic lattice models (Sec.8).

Let ${{\mathcal{V}}}_{z}$ be as in Sec.4.1 and
${{\mathcal{V}}}(\Lambda_{a}+\nu,\nu)$ denote the irreducible level-$1$
highest weight $U_{q,p}(\widehat{{sl}}_{N})$-module with the highest weight
$(\Lambda_{a}+\nu,\nu)$ in Sec.4.2. Let $\Lambda_{a+N}=\Lambda_{a}$. The (type
I) vertex operator $\Phi(z)$ is the intertwinner of the
$U_{q,p}(\widehat{{sl}}_{N})$-modules

$\displaystyle{\Phi}(z)$ $\displaystyle:$
$\displaystyle{{\mathcal{V}}}(\Lambda_{a}+\nu,\nu)\to{{\mathcal{V}}}_{z}\,\widetilde{\otimes}\,{{\mathcal{V}}}(\Lambda_{a-1}+\nu,\nu)$

satisfying the intertwining relations

$\displaystyle\Delta(x){\Phi}(z)={\Phi}(z)x\qquad\forall x\in
U_{q,p}(\widehat{{sl}}_{N}).$ (5.1)

The components of the vertex operator are defined by

${\Phi}(zq^{-1})u=\sum_{\mu=1}^{N}v_{\mu}\,\widetilde{\otimes}\,\Phi_{\mu}\left(z\right)u,\quad\qquad\forall
u\in{{\mathcal{V}}}(\lambda,\nu).$

One can solve the linear equation (5.1) by using the representations in the
last section uniquely up to an overall constant factor. The result is
summarized as follows.

$\displaystyle\Phi_{\mu}(z)$ $\displaystyle=$ $\displaystyle
a_{\mu,N}\oint_{{\mathbb{T}}^{N-\mu}}\prod_{m=\mu}^{N-1}\frac{dt_{m}}{2\pi
it_{m}}\Phi_{N}(z)F_{N-1}(t_{N-1})F_{N-2}(t_{N-2})\cdots
F_{\mu}(t_{\mu})\varphi_{\mu}(z,t_{\mu},\cdots,t_{N-1};\Pi),{}$
$\displaystyle\Phi_{N}(z)$ $\displaystyle=$
$\displaystyle:\exp\left\\{\sum_{m\neq
0}(q^{m}-q^{-m}){{\mathcal{E}}}_{m}^{{}^{\prime}N}z^{-m}\right\\}:e^{-{\bar{\epsilon}}_{N}}(-z)^{-h_{\bar{\epsilon}_{N}}}z^{\frac{1}{r}(P+h)_{\bar{\epsilon}_{N}}},$
(5.2)

where we set $\bar{\epsilon}_{i}=\epsilon_{i}-\sum_{j=1}^{N}\epsilon_{j}/N$,
$\Pi=\\{\Pi_{\mu,m}\ (m=\mu+1,\cdots,N)\\}$,

$\displaystyle{\mathbb{T}}^{N-\mu}=\\{t\in{\mathbb{C}}^{N-\mu}\ |\
|t_{\mu}|=\cdots=|t_{N-1}|=1\\},$

and

$\displaystyle{\varphi_{\mu}(z,t_{\mu},\cdots,t_{N-1};\Pi)=\prod_{m=\mu}^{N-1}\frac{[v_{m+1}-v_{m}+(P+h)_{\mu,m+1}-\frac{1}{2}][1]}{[v_{m+1}-v_{m}+\frac{1}{2}][(P+h)_{\mu,m+1}]}}$

with $z=q^{2u},t_{m}=q^{2v_{m}}$, $v_{N}=u$. We assume $|p|<|z|<1$. The symbol
${\mathcal{E}}_{m}^{{}^{\prime}j}$ denotes a linear combination of
$\alpha^{\prime}_{j,m}$ obtained by solving

$\displaystyle\alpha^{\prime}_{j,m}=[m]_{q}^{2}(q-q^{-1})({\mathcal{E}}_{m}^{{}^{\prime}j}-q^{-m}{\mathcal{E}}_{m}^{{}^{\prime}j+1}),\qquad\sum_{j=1}^{N}q^{(j-1)m}{\mathcal{E}}^{{}^{\prime}j}_{m}=0.$

The most important property of $\Phi_{\mu}(z)$ is the following commutation
relation.

$\displaystyle\Phi_{\mu_{2}}(z_{2})\Phi_{\mu_{1}}(z_{1})$ $\displaystyle=$
$\displaystyle\sum_{\mu_{1}^{\prime},\mu_{2}^{\prime}=1}^{N}R(z_{1}/z_{2},\Pi)_{\mu_{1}\mu_{2}}^{\mu_{1}^{\prime}\mu_{2}^{\prime}}\
\Phi_{\mu_{1}^{\prime}}(z_{1})\Phi_{\mu_{2}^{\prime}}(z_{2}),$ (5.3)

where

$\displaystyle
R(z,\Pi)=\mu(z){\overline{R}}(z,\Pi),\qquad\mu(z)=z^{-\frac{r-1}{r}\frac{N-1}{N}}\frac{\Gamma(pz;p,q^{2N})\Gamma(q^{2N}z;p,q^{2N})}{\Gamma(q^{2}z;p,q^{2N})\Gamma(pq^{2N-2}z;p,q^{2N})}.$

For example, this yields the transition property of the elliptic weight
function (Sec.6) as well as the $R$-matrix coefficients in the elliptic $q$-KZ
connection (Sec.7).

##  6 Elliptic Weight Functions

The weight functions are objects in the theory of ($q$-) hypergeometric
integrals. They play the role of a basis of the (twisted) de Rham
cohomology[4, 56, 59, 70]. The elliptic version of the weight function was
first introduced for the ${sl}_{2}$ type in [70]444Strictly speaking, the
elliptic weight functions in [70] were used as a pole subtraction matrix, in
the terminology of [2], in the $q$-hypergeometric integral solution of the
$q$-KZ equation. They specifies the cycles of the integral. On the other hand,
there the role of a basis of the co-cycle was played by the trigonometric
weight functions. . We here expose the elliptic weight functions of the
${sl}_{N}$ type, which can be derived by considering a composition of the
vertex operators given in the last section. In the next section, we show that
they play the role in elliptic hypergeometric integral solutions of the
elliptic $q$-KZ equation. Moreover, the elliptic weight functions are
identified with the elliptic stable envelopes[2] associated with Nakajima
quiver variety $X$. This makes clear a connection between representation
theory of EQG and geometry of the elliptic cohomology (Sec.10).

Let $\Phi_{\mu}(z)$ be the vertex operators of $U_{q,p}(\widehat{{sl}}_{N})$
in the last section and consider their composition

$\displaystyle\phi_{\mu_{1}\cdots\mu_{n}}(z_{1},\cdots,z_{n}):=\Phi_{\mu_{1}}(z_{1})\cdots\Phi_{\mu_{n}}(z_{n})\
:\ {\mathcal{F}}_{a,\nu}(\xi,\eta)\ \to\
{\mathcal{F}}_{a^{\prime},\nu}(\xi,\eta).$

Here $a^{\prime}\equiv a-n$ mod $N$. Let $[1,n]=\\{1,\cdots,n\\}$ and define
the index sets $I_{l}:=\\{i\in[1,n]\ |\ \mu_{i}=l\\}$ $(l=1,\cdots,N)$. Set
also $\lambda_{l}:=|I_{l}|$, $\lambda:=(\lambda_{1},\cdots,\lambda_{N})$. Then
$I=(I_{1},\cdots,I_{N})$ is a partition of $[1,n]$, i.e.

$\displaystyle I_{1}\cup\cdots\cup I_{N}=[1,n],\quad I_{k}\cap
I_{l}=\emptyset\quad\mbox{$(k\not=l)$}.$

Thus obtained partition $I$ is often denoted by $I_{\mu_{1},\cdots\mu_{n}}$.
For $\lambda=(\lambda_{1},\cdots,\lambda_{N})\in{\mathbb{N}}^{N}$, let
${\mathcal{I}}_{\lambda}$ be the set of all partitions
$I=(I_{1},\cdots,I_{N})$ satisfying $|I_{l}|=\lambda_{l}\ (l=1,\cdots,N)$. Set
also $\lambda^{(l)}:=\lambda_{1}+\cdots+\lambda_{l}$,
$I^{(l)}:=I_{1}\cup\cdots\cup I_{l}$ and let
$I^{(l)}=:\\{i^{(l)}_{1}<\cdots<i^{(l)}_{\lambda^{(l)}}\\}$. Note that
${\mathcal{I}}_{\lambda}$ specifies the coordinate flags of the partial flag
variety $fl_{\lambda}$ : $0=V_{0}\subset V_{1}\subset\cdots\subset
V_{N-1}\subset V_{N}={\mathbb{C}}^{n}$ with $\dim V_{l}=\lambda^{(l)}$.

Now let us substitute the realization of $\Phi_{\mu}(z)$ (5.2) to
$\phi_{\mu_{1}\cdots\mu_{n}}(z_{1},\cdots,z_{n})$. In each $\Phi_{\mu}(z)$, we
assign the integration variable $t^{(l)}_{a}$ to the elliptic current
$F_{l}(\ast)$ $(l=1,\cdots,N-1)$ as its argument if $\Phi_{\mu}(z)$ is the
$i^{(l)}_{a}$-th vertex operator i.e. $z=z_{i^{(l)}_{a}}$. After rearranging
the order of the elliptic currents $F_{l}$’s and $\Phi_{N}$’s and taking their
normal ordering as specified in $\widetilde{\Phi}(t,z)$ in the below, one
obtains the following expression for $I=I_{\mu_{1}\cdots\mu_{n}}$,

$\displaystyle\phi_{\mu_{1}\cdots\mu_{n}}(z_{1},\cdots,z_{n})=\oint_{{\mathbb{T}}^{M}}\prod_{l=1}^{N-1}\prod_{a=1}^{\lambda^{(l)}}\frac{dt^{(l)}_{a}}{2\pi
it^{(l)}_{a}}\
\widetilde{\Phi}(t,z)\widetilde{W}_{I}(t,z,\Pi),\qquad|p|<|z_{1}|,\cdots,|z_{n}|<1,{}$
(6.1)
$\displaystyle\widetilde{\Phi}(t,z)=:\Phi_{N}(z_{1})\cdots\Phi_{N}(z_{n})::F_{N-1}(t^{(N-1)}_{1})\cdots
F_{N-1}(t^{(N-1)}_{\lambda^{(N-1)}}):\cdots:F_{1}(t_{1}^{(1)})\cdots
F_{1}(t^{(1)}_{\lambda^{(1)}}):{}$
$\displaystyle\qquad\qquad\times\prod_{1\leq k<l\leq
n}<\Phi_{N}(z_{k})\Phi_{N}(z_{l})>^{Sym}\prod_{l=1}^{N-1}\prod_{1\leq
a<b\leq\lambda^{(l)}}<F_{l}(t^{(l)}_{a})F_{l}(t^{(l)}_{b})>^{Sym},$
$\displaystyle\widetilde{W}_{I}(t,z,\Pi)={\rm Sym}_{t^{(1)}}\cdots{\rm
Sym}_{t^{(N-1)}}\widetilde{U}_{I}(t,z,\Pi),$
$\displaystyle\widetilde{U}_{I}(t,z,\Pi)$ $\displaystyle=$
$\displaystyle\prod_{l=1}^{N-1}\prod_{a=1}^{\lambda^{(l)}}\left(\frac{[v^{(l+1)}_{b}-v^{(l)}_{a}+(P+h)_{\mu_{s},l+1}-C_{\mu_{s},l+1}(s)][1]}{[v^{(l+1)}_{b}-v^{(l)}_{a}+1]\left[(P+h)_{\mu_{s},l+1}-C_{\mu_{s},l+1}(s)\right]}\right|_{i^{(N)}_{s}=i^{(l+1)}_{b}=i^{(l)}_{a}}$
$\displaystyle\qquad\qquad\qquad\times\prod_{b=1\atop
i^{(l+1)}_{b}>i^{(l)}_{a}}^{\lambda^{(l+1)}}\frac{[v^{(l+1)}_{b}-v^{(l)}_{a}]}{[v^{(l+1)}_{b}-v^{(l)}_{a}+1]}\prod_{b=a+1}^{\lambda^{(l)}}\frac{[v^{(l)}_{a}-v^{(l)}_{b}-1]}{[v^{(l)}_{a}-v^{(l)}_{b}]}\Biggr{)},$

where we set $t^{(l)}_{a}=q^{2v^{(l)}_{a}}$, $t^{(N)}_{s}=z_{s}=q^{2u_{s}}$,
$M=\sum_{l=1}^{N-1}\lambda^{(l)}$ and
$C_{\mu_{s},l+1}(s)=\sum_{j=s+1}^{n}\langle\bar{\epsilon}_{\mu_{j}},h_{\mu_{s},l+1}\rangle$.
The symbol $<\ \ >^{Sym}$ denotes the symmetric part of the operator product
expansion coefficient $<\ \ >$ defined by for ${\mathcal{O}}=\Phi_{N},F_{l}$,

$\displaystyle{\mathcal{O}}(w_{a}){\mathcal{O}}(w_{b})=<{\mathcal{O}}(w_{a}){\mathcal{O}}(w_{b})>:{\mathcal{O}}(w_{a}){\mathcal{O}}(w_{b}):.$

Hence $\widetilde{\Phi}(t,z)$ is an operator valued symmetric function in
$z=(z_{1},\cdots,z_{n})$ as well as in
$t^{(l)}=(t^{(l)}_{1},\cdots,t^{(l)}_{\lambda^{(l)}})$ for each $l$. The
function $\widetilde{W}_{I}(t,z,\Pi)$ is the elliptic weight function of type
${sl}_{N}$. It is obtained by collecting all non-symmetric part of $<\ \ >$’s
and factors arisen from the exchange among $\Phi_{N}$’s and $F_{l}$’s and by
making symmetrizations ${\rm Sym}_{t^{(l)}}$ of all entries in $t^{(l)}$.
Hence $\widetilde{W}_{I}(t,z,\Pi)$ is a symmetric function in $v^{(l)}_{a}\
(a=1,\cdots,\lambda^{(l)})$ for each $l$.

The weight function $\widetilde{W}_{I}(t,z,\Pi)$ has several nice properties
such as

  * •

The triangular property: For $I,J\in{\mathcal{I}}_{\lambda}$,

    * (1)

$\widetilde{W}_{J}(z_{I},z,\Pi)=0$ unless $I\leqslant J$.

    * (2)

${\displaystyle\widetilde{W}_{I}(z_{I},z,\Pi)=\prod_{1\leq k<l\leq
N}\prod_{a\in I_{k}}\prod_{b\in I_{l}\atop
a<b}\frac{[u_{b}-u_{a}]}{[u_{b}-u_{a}+1]}}$

Here $\leqslant$ denotes the partial ordering defined by

$\displaystyle I\leqslant J\Leftrightarrow i^{(l)}_{a}\leq
j^{(l)}_{a}\qquad\forall l,a.$

for $I^{(l)}=\\{i^{(l)}_{1}<\cdots<i^{(l)}_{\lambda^{(l)}}\\}$ and
$J^{(l)}=\\{j^{(l)}_{1}<\cdots<j^{(l)}_{\lambda^{(l)}}\\}$ $(l=1,\cdots,N)$.
We also denote by $t=z_{I}$ the specialization $t^{(l)}_{a}=z_{i^{(l)}_{a}}$
$(l=1,\cdots,N-1,a=1,\cdots,\lambda^{(l)})$[66].

  * •

The transition property:

$\displaystyle\widetilde{W}_{I_{\cdots\
\mu_{i+1}\mu_{i}\cdots}}(t,\cdots,z_{i+1},z_{i},\cdots,\Pi){}$
$\displaystyle=\sum_{\mu_{i}^{\prime},\mu_{i+1}^{\prime}}{\overline{R}}(z_{i}/z_{i+1},\Pi
q^{-2\sum_{j=i}^{n}\langle\bar{\epsilon}_{\mu_{j}},h\rangle})_{\mu_{i}\mu_{i+1}}^{\mu_{i}^{\prime}\mu_{i+1}^{\prime}}\
\widetilde{W}_{I_{\cdots\
\mu^{\prime}_{i}\mu^{\prime}_{i+1}\cdots}}(t,\cdots,z_{i},z_{i+1},\cdots,\Pi)$

as well as the orthogonality, quasi-periodicity, the shuffle product formula
compatible to the wheel conditions etc.[42]. These properties are used to
identify $\widetilde{W}_{I}(t,z,\Pi)$ with the elliptic stable envelope for
the cotangent bundle to the partial flag variety (Sec.10).

##  7 Integral Solution of the Elliptic $q$-KZ equation

Let us consider the $n$-point function
$\phi_{\mu_{1}\cdots\mu_{n}}(z_{1},\cdots,z_{n})$ with the zero weight
condition $\sum_{i=1}^{n}\bar{\epsilon}_{\mu_{i}}=0$, which is equivalent to
consider $\lambda=(m^{N})$ with $mN=n$. In this case, one can take a trace
over the space ${\mathcal{F}}_{a,\nu}(\xi,\eta)$. It then turns out that the
trace gives a solution of the elliptic $q$-KZ equation due to the cyclic
property of the trace and the commutation relation of the vertex operators
(5.3)[22, 42]. From (6.1), one gets the following elliptic hypergeometric
integral.

$\displaystyle{\rm
tr}_{{\mathcal{F}}_{a,\nu}}\left(Q^{-d}\Phi_{\mu_{1}}(z_{1})\cdots\Phi_{\mu_{N}}(z_{N})\right)=\oint_{{\mathbb{T}}^{M}}\prod_{l=1}^{N-1}\prod_{a=1}^{\lambda^{(l)}}\frac{dt^{(l)}_{a}}{2\pi
it^{(l)}_{a}}\ e(t,\Pi){\Phi}(t,z)\widetilde{W}_{I}(t,z,\Pi),$ $\displaystyle
e(t,\Pi)=\exp{\left\\{\frac{\sum_{l=1}^{N-1}\log(\Pi_{l}/\Pi_{l+1})\sum_{a}\log(t^{(l)}_{a})}{\log
p}\right\\}},$
$\displaystyle{\Phi}(t,z)=\prod_{l=1}^{N-1}\left[\prod_{a=1}^{\lambda^{(l)}}\prod_{b=1}^{\lambda^{(l+1)}}\frac{\Gamma(t^{(l)}_{a}/t^{(l+1)}_{b};p,Q)}{\Gamma(p^{*}t^{(l)}_{a}/t^{(l+1)}_{b};p,Q)}\prod_{1\leq
a<b\leq\lambda^{(l)}}\frac{\Gamma(p^{*}t^{(l)}_{a}/t^{(l)}_{b},p^{*}t^{(l)}_{b}/t^{(l)}_{a};p,Q)}{\Gamma(t^{(l)}_{a}/t^{(l)}_{b},t^{(l)}_{b}/t^{(l)}_{a};p,Q)}\right].$

Note that only the weight function $\widetilde{W}_{I}(t,z,\Pi)$ carries the
tensor indices through $I$. We expect that
$\widetilde{W}_{I}(t,z,\Pi),{I\in{\mathcal{I}}_{(m^{N})}}$ form a basis of the
twisted de Rham cohomology associated with $T_{Q,z_{i}}\Phi(t,z)/\Phi(t,z)$,
$i=1,\cdots,n$, where $T_{Q,z_{i}}$ is the shift operator
$T_{Q,z_{i}}f(\cdots,z_{i},\cdots)=f(\cdots,Qz_{i},\cdots)$. Furthermore, to
specify the cycle of the integral one may insert to the integral an additional
elliptic weight function $\widetilde{W}_{J}(t,z,\Pi)$,
$J\in{\mathcal{I}}_{(m^{N})}$, whose elliptic nome is $Q$ instead of $p$. Thus
obtained elliptic hypergeometric integrals labeled by
$I,J\in{\mathcal{I}}_{(m^{N})}$ are conjectured to give fundamental solutions
of the elliptic $q$-KZ equation. See [70] for the trigonometric ${sl}_{2}$
case.

In the trigonometric limit $Q\to 0$, the trace goes to the following vacuum
expectation value.

$\displaystyle\langle
0|\Phi_{\mu_{1}}(z_{1})\cdots\Phi_{\mu_{N}}(z_{N}){|0\rangle}=\oint_{{\mathbb{T}}^{M}}\prod_{l=1}^{N-1}\prod_{a=1}^{\lambda^{(l)}}\frac{dt^{(l)}_{a}}{2\pi
it^{(l)}_{a}}\ e(t,\Pi){\Phi}^{trig.}(t,z)\widetilde{W}_{I}(t,z,\Pi),$
$\displaystyle{\Phi}^{trig.}(t,z)=\prod_{l=1}^{N-1}\left[\prod_{a=1}^{\lambda^{(l)}}\prod_{b=1}^{\lambda^{(l+1)}}\frac{(p^{*}t^{(l)}_{a}/t^{(l+1)}_{b};p)_{\infty}}{(t^{(l)}_{a}/t^{(l+1)}_{b};p)_{\infty}}\prod_{1\leq
a<b\leq\lambda^{(l)}}\frac{(t^{(l)}_{a}/t^{(l)}_{b},t^{(l)}_{b}/t^{(l)}_{a};p)_{\infty}}{(p^{*}t^{(l)}_{a}/t^{(l)}_{b},p^{*}t^{(l)}_{b}/t^{(l)}_{a};p)_{\infty}}\right].$

This gives the integral formula for the vertex function for the equivariant
$\mathrm{K}$-theory $\mathrm{K}_{T}(T^{*}fl_{\lambda})$ with
$T=({\mathbb{C}}^{\times})^{n}\times{\mathbb{C}}^{\times}$, which is the
generating function of counting the quasi-maps from ${\mathbb{P}}^{1}$ to
$T^{*}fl_{\lambda}$[49]. Here the elliptic weight function $\widetilde{W}_{I}$
plays the role of the pole subtraction matrix[2]. In particular, for the full
flag variety case i.e. $\lambda^{(l)}=l\ (l=1,\cdots,N-1)$ with $n=N$, it
gives the integral formula for the Macdonald symmetric function[59], which is
also known as the 3d vortex partition function of the $\mathrm{T}[U(N)]$
theory, if one drops the weight function $\widetilde{W}_{I}$ from the
integrand. Indeed the first and the second blocks in ${\Phi}(t,z)$ are
elliptic analogues of the Cauchy type reproducing kernel and the Macdonald
weight function, respectively, by reading $(p,p^{*})$ as $(q,t)$ in the
Macdonald theory. Hence the vertex operators of $U_{q,p}(\widehat{{sl}}_{N})$
play the role of the vertex operators of the deformed $W$-algebra
${\mathcal{W}}_{p,p^{*}}{sl}_{N}$[3, 25].

##  8 Algebraic Analysis of the Elliptic Lattice Models

The algebraic analysis [30] is a scheme of mathematical formulation of the
solvable 2d lattice models defined by the $R$-matrices. It formulates the
model on the infinite lattice directly by using both finite and infinite
dimensional representations of the corresponding affine quantum groups
$U_{q}({g})$ or elliptic quantum groups $U_{q,p}({g})$. In particular, the
infinite dimensional representations are identified with the spaces of states
of the model specified by the ground state boundary condition, where the
configurations at enough far sites from the center of the lattice is given by
the ground state configurations. In addition, the two types intertwining
operators, the type I and II vertex operators, of the quantum groups realize
the local operators of the model such as spin operators acting on the infinite
dimensional representations (type I) and the creation operators of the
physical excitations (type II), respectively. Then the traces of compositions
of the vertex operators give correlation functions as well as form factors of
the models. Hence these are characterized as solutions of the (elliptic)
$q$-KZ equations[23, 22, 30] as discussed in the last section for the type I
case. In this sense, the algebraic analysis is an off-critical extension of
CFT [8].

There are series of elliptic solvable lattice models defined by the elliptic
solutions of the face type YBE, i.e. the elliptic dynamical $R$-matrices,
associated with affine Lie algebras[5, 32, 13, 50, 51]. The critical behavior
of such models is described by the $W$-algebra ${\mathcal{W}}\bar{{g}}$ of the
coset type $({g})_{r-h^{\vee}-k}\oplus({g})_{k}\supset({g}_{\rm
diag.})_{r-h^{\vee}}$[14, 13]. Here $r$ is taken as a real positive parameter
and called the restriction height. Accordingly the elliptic quantum group
$U_{q,p}({g})$ plays a central role in the algebraic analysis of these models
as a deformation of the $W$-algebra ${\cal W}_{p,p^{*}}\bar{{g}}$. See Sec.3.
The spaces of states with the ground state boundary condition are identified
with the irreducible ${\cal W}_{p,p^{*}}\bar{{g}}$-modules, whose direct sum
are isomorphic to the $U_{q,p}({g})$-modules (Sec.4.2). The type I and II
vertex operators of $U_{q,p}({g})$ provide correlation functions and form
factors of the model. See [22, 55, 37].

In the case ${g}=\widehat{{sl}}_{N}$, one also has the vertex type elliptic
lattice models[6, 7]. These are related to the face models by the so-called
vertex-face correspondence[6, 31]. The algebraic analysis of the vertex models
is carried out based on the one of the face models by applying the vertex-face
correspondence[52, 37, 10].

##  9 Level-0 Action on the Gelfand-Tsetlin Basis

Let $(\pi_{z},{{\mathcal{V}}}_{z})$ be the evaluation representation of
$U_{q,p}(\widehat{{sl}}_{N})$ in Sec.4.1. The action of the $L$-operator
$L^{+}(1/w)$ on
${{\mathcal{V}}}_{w}\,\widetilde{\otimes}\,{{\mathcal{V}}}_{z}$ is extended to
the tensor space
${{\mathcal{V}}}_{w}\,\widetilde{\otimes}\,{{\mathcal{V}}}_{z_{1}}\,\widetilde{\otimes}\,\cdots\,\widetilde{\otimes}\,{{\mathcal{V}}}_{z_{n}}$
by the comultiplication

$\displaystyle(\pi_{z_{1}}\otimes\cdots\otimes\pi_{z_{n}}){\Delta^{\prime}}^{(n-1)}(L^{\pm}(1/w))$
$\displaystyle=\bar{R}^{(0n)}(z_{n}/w,\Pi^{*}q^{2\sum_{j=1}^{n-1}h^{(j)}}){{\overline{R}}}^{(0n-1)}(z_{n-1}/w,\Pi^{*}q^{2\sum_{j=1}^{n-2}h^{(j)}})\cdots\bar{R}^{(01)}(z_{1}/w,\Pi^{*}).$

Here we take the opposite comultiplication $\Delta^{\prime}$ following [43].
This is the action on the standard basis

$\displaystyle
v_{I}:=v_{\mu_{1}}\,\widetilde{\otimes}\,\cdots\,\widetilde{\otimes}\,v_{\mu_{n}},\qquad
I=I_{\mu_{1}\cdots\mu_{n}}\in{\mathcal{I}}_{\lambda},\
\lambda\in{\mathbb{N}}^{N},\ |\lambda|=n.$

Let us consider the following change of basis[43].

$\displaystyle\xi_{I}=\sum_{J\in{\mathcal{I}}_{\lambda}}\widetilde{W}_{J}(z^{-1}_{I},z^{-1},\Pi
q^{2\sum_{j=1}^{n}<\bar{\epsilon}_{\mu_{j}},h>})v_{J}.$ (9.1)

The transition matrix is given by the specialized elliptic weight function so
that it is lower triangular (Sec.6). It turns out that the new vectors
$\xi_{I}$
$(I\in{\mathcal{I}}_{\lambda},\lambda\in{\mathbb{N}}^{N},|\lambda|=n)$ form
the Gelfand-Tsetlin (GT) basis of
${{\mathcal{V}}}_{z_{1}}\,\widetilde{\otimes}\,\cdots\,\widetilde{\otimes}\,{{\mathcal{V}}}_{z_{n}}$,
on which the commuting subalgebra generated by $\alpha_{j,m}$
$(j=1,\cdots,N-1,\ m\in{\mathbb{Z}}\backslash\\{0\\})$ is diagonalized
simultaneously. In fact the level-0 action of the elliptic currents of
$U_{q,p}(\widehat{{sl}}_{N})$ on $\xi_{I}$ is given by

$\displaystyle\phi_{j}^{\pm}(w)\xi_{I}=\left.\prod_{a\in
I_{j}}\frac{[u_{a}-v+1]}{[u_{a}-v]}\right|_{\pm}\left.\prod_{b\in{I_{j+1}}}\frac{[u_{b}-v-1]}{[u_{b}-v]}\right|_{\pm}\
e^{-Q_{\alpha_{j}}}\xi_{I}$ $\displaystyle E_{j}(w)\xi_{I}=a^{*}\sum_{i\in
I_{j+1}}\delta(z_{i}/w)\prod_{k\in
I_{j+1}\atop\not=i}\frac{[u_{i}-u_{k}+1]}{[u_{i}-u_{k}]}\
e^{-Q_{\alpha_{j}}}\xi_{I^{i^{\prime}}}$ $\displaystyle
F_{j}(w)\xi_{I}=a\sum_{i\in I_{j}}\delta(z_{i}/w)\prod_{k\in
I_{j}\atop\not=i}\frac{[u_{k}-u_{i}+1]}{[u_{k}-u_{i}]}\
\xi_{I^{{}^{\prime}i}}$

where ${aa^{*}=-\frac{1}{q-q^{-1}}\frac{[0]^{\prime}}{[1]}}$, and
$I=(I_{1},\cdots,I_{N})$,
$I^{i^{\prime}}\in{\mathcal{I}}_{(\lambda_{1},\cdots,\lambda_{j}+1,\lambda_{j+1}-1,\cdots,\lambda_{N})}$,
$I^{{}^{\prime}i}\in{\mathcal{I}}_{(\lambda_{1},\cdots,\lambda_{j}-1,\lambda_{j+1}+1,\cdots,\lambda_{N})}$
with

$\displaystyle(I^{i^{\prime}})_{j}=I_{j}\cup\\{i\\},\quad(I^{i^{\prime}})_{j+1}=I_{j+1}-\\{i\\},\quad(I^{i^{\prime}})_{k}=I_{k}\
\ (k\not=j,j+1),$
$\displaystyle(I^{{}^{\prime}i})_{j}=I_{j}-\\{i\\},\quad(I^{{}^{\prime}i})_{j+1}=I_{j+1}\cup\\{i\\},\quad(I^{{}^{\prime}i})_{k}=I_{k}\
\ (k\not=j,j+1).$

Note that this action is given in terms of the partitions in completely a
combinatorial way. In the next section, this is identified with a geometric
action of $U_{q,p}(\widehat{{sl}}_{N})$ on the equivariant elliptic cohomology
${\displaystyle\bigoplus_{\lambda\in{\mathbb{N}}^{N},|\lambda|=n}\mathrm{E}_{T}(T^{*}fl_{\lambda})}$
under the identification of the elliptic weight function with the elliptic
stable envelope.

##  10 Geometric Interpritation

The notion of stable envelopes was initiated in [57, 65]. They form a good
basis of equivariant cohomology and $\mathrm{K}$-theory of Nakajima quiver
varieties[61, 62] and has nice applications to enumerative geometry, geometric
representation theory and quantum integrable systems. In particular, the
transition matrices between stable envelopes defined for different chambers
give the $R$-matrices satisfying the YBE so that they provide a new geometric
formulation of quantum groups as well as quantum integrable systems associated
with the quiver varieties. The elliptic version of stable envelopes was
defined in [2] for the equivariant elliptic cohomology $\mathrm{E}_{T}(X)$ of
the Nakajima quiver varieties $X$. Remarkably they provide the elliptic
dynamical $R$-matrix as their transition matrix, which coincides for example
with (2.2) for the case $X=T^{*}fl_{\lambda}$. We here expose an
identification of the elliptic stable envelopes for
$\mathrm{E}_{T}(T^{*}fl_{\lambda})$ with the elliptic weight functions for
$U_{q,p}(\widehat{{sl}}_{N})$ described in Sec.6. This makes more transparent
the correspondence between geometry of quiver variety and representation of
EQG. For the definition of the elliptic cohomology and the elliptic stable
envelopes, the reader can consult[2].

The elliptic stable envelope $\mathrm{Stab}_{\mathfrak{C}}(F_{I})$ for
$\mathrm{E}_{T}(T^{*}fl_{\lambda})$ at the torus fixed point $F_{I}$ labeled
by the partition $I\in{\mathcal{I}}_{\lambda}$ can be constructed explicitly
by the abelianization procedure[2, 43]. On the other hand, let us consider the
elliptic weight function modified as

$\displaystyle{\mathcal{W}}_{I}(t,z,\Pi)=\frac{H_{\lambda}(t,z)\widetilde{W}_{I}(t,z,\Pi)}{E_{\lambda}(t)},$
$\displaystyle
H_{\lambda}(t,z)=\prod_{l=1}^{N-1}\prod_{a=1}^{\lambda^{(l)}}\prod_{b=1}^{\lambda^{(l+1)}}\left[v^{(l+1)}_{b}-v^{(l)}_{a}+1\right],\quad
E_{\lambda}(t)=\prod_{l=1}^{N-1}\prod_{a=1}^{\lambda^{(l)}}\prod_{b=1}^{\lambda^{(l)}}[v^{(l)}_{b}-v^{(l)}_{a}+1].$

This yields the expresion

$\displaystyle{\mathcal{W}}_{I}(t,z,\Pi)$ $\displaystyle=$ $\displaystyle{\rm
Sym}_{t^{(1)}}\cdots{\rm Sym}_{t^{(N-1)}}\ {U}_{I}(t,z,\Pi),$ $\displaystyle
U_{I}(t,z,\Pi)$ $\displaystyle=$
$\displaystyle\prod_{l=1}^{N-1}\frac{\prod_{a=1}^{\lambda^{(l)}}u^{(l)}_{I}(t^{(l)}_{a},t^{(l+1)},\Pi_{\mu_{\mbox{\tiny${i^{(l)}_{a}}$}},l+1}q^{-2C_{\mu_{\mbox{\tiny${i^{(l)}_{a}}$}},l+1}(i^{(l)}_{a})})}{\prod_{1\leq
a<b\leq\lambda^{(l)}}{[v^{(l)}_{a}-v^{(l)}_{b}]}{[v^{(l)}_{b}-v^{(l)}_{a}-1]}},$
$\displaystyle u^{(l)}_{I}(t^{(l)}_{a},t^{(l+1)},\Pi_{j,k})$ $\displaystyle=$
$\displaystyle\left.\frac{\left[v^{(l+1)}_{b}-v^{(l)}_{a}+(P+h)_{j,k}\right]}{[(P+h)_{j,k}]}\right|_{i^{(l+1)}_{b}=i^{(l)}_{a}}{}$
$\displaystyle\qquad\times\prod_{b=1\atop
i^{(l+1)}_{b}>i^{(l)}_{a}}^{\lambda^{(l+1)}}{\left[v^{(l+1)}_{b}-v^{(l)}_{a}\right]}\prod_{b=1\atop
i^{(l+1)}_{b}<i^{(l)}_{a}}^{\lambda^{(l+1)}}{\left[v^{(l+1)}_{b}-v^{(l)}_{a}+{1}\right]}.$

Comparing these expressions, one finds $\mathrm{Stab}_{\mathfrak{C}}(F_{I})$
and ${\mathcal{W}}_{I}(z,t,\Pi)$ are identical as

$\displaystyle\mathrm{Stab}_{\mathfrak{C}}(F_{I})={\mathcal{W}}_{\sigma_{0}(I)}({\tilde{t}},\sigma_{0}(z^{-1}),\Pi^{-1}).$
(10.1)

Here for the longest element $\sigma_{0}$ in $\mathfrak{S}_{n}$ and
$I=I_{\mu_{1}\cdots\mu_{n}}$, we define
${\sigma}_{0}(I)=I_{\mu_{{\sigma}_{0}(1)}\cdots\mu_{{\sigma}_{0}(n)}}$ and set
${\sigma}_{0}(I)=(\tilde{I}_{1},\dots,\tilde{I}_{N})$,
$\tilde{I}^{(l)}=\tilde{I}_{1}\cup\cdots\cup\tilde{I}_{l}$ and
$\tilde{I}^{(l)}=\\{\tilde{i}^{(l)}_{1}<\cdots<\tilde{i}^{(l)}_{\lambda^{(l)}}\\}$.
Then ${\tilde{t}}$ denotes the set of all $t^{(l)}_{a}$ corresponding to
$\tilde{i}^{(l)}_{a}$. The chamber $\mathfrak{C}$ is taken as
$|z_{1}|<\cdots<|z_{n}|$. In addition, the restriction $t=z_{J}$ defined in
Sec.6 can be identified with the restriction of the stable class
$\mathrm{Stab}_{\mathfrak{C}}(F_{I})$ to the fixed point $F_{J}$. One then
finds

$\displaystyle{\rm
Stab}_{\mathfrak{C}}(F_{I})|_{F_{J}}={\mathcal{W}}_{\sigma_{0}(I)}(z^{-1}_{J},\sigma_{0}(z^{-1}),\Pi^{-1}).$
(10.2)

Such coincidence as well as the integral formula for the vertex function
obtained as correlation function of the vertex operators in Sec.7 lead us to
the following dictionary from the level-0 representation of
$U_{q,p}(\widehat{{sl}}_{N})$ to the equivariant elliptic cohomology
$\mathrm{E}_{T}(T^{*}fl_{\lambda})$.

$\displaystyle\mbox{evaluation parameters}\ z_{1},\cdots,z_{n}\ $
$\displaystyle\leftrightarrow$ $\displaystyle\ \mbox{equivariant parameters}\
\in({\mathbb{C}}^{\times})^{n}$ $\displaystyle\mbox{dynamical parameters}\
\Pi_{j}\ $ $\displaystyle\leftrightarrow$ $\displaystyle\ \mbox{K\"{a}hler
parameters}\in\mathrm{Pic}_{T}(X)\otimes_{\mathbb{Z}}E\qquad$ integration
variables $\displaystyle\leftrightarrow$ Chern roots of the tautological
vector bundle ${\mathcal{V}}_{l}$ $\displaystyle\ t^{(l)}_{a}\
(a=1,\cdots,\lambda^{(l)})\quad$ of $\mathrm{rk}\;\lambda^{(l)}$on the $l$-th
vertex of the quiver diagram $\displaystyle\mbox{the weight}\
(\mu_{1},\cdots,\mu_{n})\ $ $\displaystyle\leftrightarrow$ $\displaystyle\
\mbox{the fixed point}\ F_{I_{\mu_{1},\cdots,\mu_{n}}}\ $

Furthermore, denoting the fixed point classes by $[I]$ one has the
localization formula for the stable envelope.

$\displaystyle{\rm
Stab}_{\mathfrak{C}}(F_{K})=\sum_{I\in{\mathcal{I}}_{\lambda}}\frac{{\rm
Stab}_{\mathfrak{C}}(F_{K})|_{F_{I}}}{N(z_{I})}\ [I],$

where $N(z_{I})$ is a certain normalization. Under the identification (10.2),
one can compare this with the change of basis relation from the GT basis
$\xi_{I}$ to the standard basis $v_{K}$, i.e. the reversed relation of (9.1)
obtained by appying the orthogonality property of the elliptic weight
functions. Then one finds that these two are identical under the
correspondence[43]

$\displaystyle\mbox{standard bases}\ v_{I}\ $ $\displaystyle\leftrightarrow$
$\displaystyle\ \mbox{stable classes }\ \mathrm{Stab}_{\mathfrak{C}}(F_{I})$
$\displaystyle\mbox{the Gelfand-Tsetlin bases}\ \xi_{I}\ $
$\displaystyle\leftrightarrow$ $\displaystyle\ \mbox{fixed point classes}\
[I].$

This allows us to interpret the level-0 action of
$U_{q,p}(\widehat{{sl}}_{N})$ on the GT basis of
${\mathcal{V}}_{z_{1}}\otimes\cdots\otimes{\mathcal{V}}_{z_{n}}$ in Sec.9 as
an action on the fixed point classes in
$\bigoplus_{\lambda\in{\mathbb{N}}^{N},|\lambda|=n}\mathrm{E}_{T}(T^{*}fl_{\lambda})$.

##  11 Elliptic Quantum Toroidal Algebras

A toroidal algebra ${g}_{tor}$ associated with a complex simple Lie algebra
$\bar{{g}}$ is a canonical two-dimensional central extension of the double
loop Lie algebra
${\mathbb{C}}^{\times}\times{\mathbb{C}}^{\times}\to\bar{{g}}$. Its quantum
version, i.e. quantum toroidal algebra, was introduced in [26]. We denote it
by $U_{q,\kappa}({g}_{tor})$. It is formulated as an extension of the quantum
affine algebra $U_{q}({{g}})$ in the Drinfeld realization[15] by introducing
the new generators, i.e. the 0-th generators w.r.t. the index set $I$, and
replacing the finite type Cartan matrix $(a_{i,j})\ i,j\in\bar{I}$ appearing
in the defining relation of $U_{q}({{g}})$ with the affine type generalized
Cartan matrix $(a_{i,j})\ i,j\in I$. This procedure is called the quantum
affinization. In particular, for the case $\bar{{g}}={{gl}}_{N}$ one can
introduce an extra parameter $\kappa$ due to a cyclic property of the affine
Dynkin diagram of type $A^{(1)}_{N-1}$.

In the same way, the elliptic quantum toroidal algebra
$U_{q,\kappa,p}({g}_{tor})$ is formulated as the quantum affinization of the
elliptic algebra $U_{q,p}({g})$ by adding the 0-th generators
$\alpha_{0,m},e_{0,n},f_{0,n},K^{\pm}_{0}\
(m\in{\mathbb{Z}}\backslash\\{0\\},n\in{\mathbb{Z}})$[48]. As a result, the
element $\prod_{i\in I}(K^{+}_{i})^{a_{i}^{\vee}}$ becomes a central element.
Here $a^{\vee}_{i}\ (i\in I)$ denote the colabel of the affine Dynkin
diagram[33]. In addition, ${U}_{q,\kappa,p}({g}_{tor})$ possesses two
subalgebras, both of which are isomorphic to the elliptic algebra
$U_{q,p}({g})$: the one generated by
$\alpha_{i,l},k_{i}^{\pm},e_{i,n},f_{i,n},q^{\pm{c}/{2}},q^{d}$
$(i\in\bar{I},l\in{\mathbb{Z}}\backslash\\{0\\},n\in{\mathbb{Z}})$ and the
other by $e_{i,0},f_{i,0},K^{\pm}_{i},q^{d}$ $(i\in I)$. These are elliptic
analogues of those in $U_{q,\kappa}({g}_{tor})$[26].

On the other hand, the ${{gl}}_{1}$ version of the quantum toroidal algebra
$U_{q,t}({{gl}}_{1,tor})$ was introduced in [58] as a $q,t$-deformation of the
$W_{1+\infty}$ algebra. Remarkably, $U_{q,t}({{gl}}_{1,tor})$ is isomorphic to
the elliptic Hall algebra[68]. It is also remarkable that representations of
$U_{q,t}({{gl}}_{1,tor})$ have a deep connection to the Macdonald theory[58,
20, 69, 25]. In addition, representations of $U_{q,t}({{gl}}_{1,tor})$ provide
a relevant scheme of calculation of the instanton partition functions[63] as
well as a study of the Alday-Gaiotto-Tachikawa (AGT) correspondence[1] for the
5d and 6d lifts of the 4d ${{\mathcal{N}}=2}$ SUSY gauge theories, which are
known as linear quiver gauge theories.

The elliptic quantum toroidal algebra $U_{q,t,p}({{gl}}_{1,tor})$ was
formulated in [47]. It has several nice representations such as the
level-(0,0) representation realized in terms of the elliptic Ruijsenaars
difference operators[67] as well as an elliptic analogue of the $q$-Fock
representation of $U_{q,t}({{gl}}_{1,tor})$ which is expected to give a
geometric representation on the equivariant elliptic cohomology of the Hilbert
scheme of points on ${\mathbb{C}}^{2}$,
$\bigoplus_{n}\mathrm{E}_{T}(\mathrm{Hilb}_{n}({\mathbb{C}}^{2}))$. These
suggest a deep connection of $U_{q,t,p}({{gl}}_{1,tor})$ to an expected
elliptic analogue of the theory of Macdonald symmetric function. It is also
remarkable that $U_{q,t,p}({{gl}}_{1,tor})$ realizes a deformation of the
Jordan quiver $W$-algebra ${\mathcal{W}}_{p,p^{*}}(\Gamma(\widehat{A}_{0}))$
introduced in [34] in the same way as the elliptic algebra $U_{q,p}({g})$
realizes the deformed $W$-algebra ${\cal W}_{p,p^{*}}\bar{{g}}$ discussed in
Sec.3. Namely, the level-$(1,N)$ elliptic currents of
$U_{q,t,p}({{gl}}_{1,tor})$ gives a realization of the screening currents and
a composition of the vertex operators w.r.t. the Drinfeld comultiplication
gives a realization of the generating functions of
${\mathcal{W}}_{p,p^{*}}(\Gamma(\widehat{A}_{0}))$. Moreover thus obtained
realization turns out to provide a relevant scheme to the instanton calculus
of the 5d and 6d lifts of the 4d ${\mathcal{N}}=2^{*}$ SUSY gauge theories,
i.e. the ${\mathcal{N}}=2$ SUSY gauge theories coupled with the adjoint
matter[63], known as the Jordan quiver gauge theories555It is also called the
ADHM quiver gauge theories.. There one of the essential feature is that
$U_{q,t,p}({{gl}}_{1,tor})$ at the level (1,$N$) possesses the four parameters
$q,t,p,p^{*}$ satisfying one constraint $p/p^{*}=t/q$, which play the role of
the $SU(4)$ $\Omega$-deformation parameters[64].

##  12 Summary

We have exposed the EQGs $U_{q,p}({g}),U_{q,\kappa,p}({g}_{tor})$ and
$U_{q,t,p}({{gl}}_{1,tor})$ in the Drinfeld realization. The main features of
them are summarized as follows.

  * •

They allow to treat both the level-0 and level $\not=0$ representations in a
unified way. In particular the level $\not=0$ representation yields the two
elliptic nomes $p,p^{*}$, which are identified with $q,t$ in the Macdonald
theory and the deformed $W$-algebras. In the same way
$U_{q,t,p}({{gl}}_{1,to})$ possesses the $SU(4)$ $\Omega$-deformation
parameters $q,t,p,p^{*}$ in the level $(1,N)$ representation.

  * •

The coalgebra structure of them allows to formulate the vertex operators as
intertwining operators of modules, where the level-0 and level $\not=0$
representations are mixed as tensor product.

  * •

The vertex operators are important objects having many applications to
formulations of mathematical and physical quantities such as elliptic weight
functions, integral solutions of the elliptic $q$-KZ equation, vertex
functions, correlation functions and form factors in algebraic analysis of
elliptic solvable lattice models etc..

  * •

They realize a deformation of the W-algebras including the affine quiver
types. In particular, the vertex operators of EQGs provide those of the
deformed $W$-algebras.

We also have shown a correspondence between the level-0 representation of
$U_{q,p}(\widehat{{sl}}_{N})$ and the equivariant elliptic cohomology
$\mathrm{E}_{T}(T^{*}fl_{\lambda})$ by deriving the elliptic weight functions,
the integral solutions of the (elliptic) $q$-KZ equation as well as the
level-0 action on the Gelfand-Tsetlin basis. In this process, the vertex
operators of $U_{q,p}(\widehat{{sl}}_{N})$ play an essential role. However, in
the geometry side neither the non-zero level representations nor the vertex
operators intertwining them have yet been understood. It should be outstanding
to formulate the vertex operators of EQGs geometrically.

In addition, knowing that the formulations of elliptic quantum groups and
equivariant elliptic cohomologies possess the equivariant and Kähler
parameters in a unified way, one of their promising application seems to be
the 3d-mirror symmetry for a pair of supersymmetric gauge theories with
${\mathcal{N}}$ = 4 supersymmetry[2]. This provides an important example of
the deeper mathematical problem of symplectic duality for conical symplectic
resolutions. In such gauge theories, Nakajima quiver varieties are known to
give moduli spaces of vacua called the Higgs branch. The 3d-mirror symmetry
states that the Higgs branch of the dual theory conjecturally coincides with
the Coulomb branch of the original one in such a way that the roles of the
equivariant and the Kähler parameters of the dual theories are exchanged. To
show the duality at the level of physical quantities such as vertex functions
would be outstanding.

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