# The quantum Harish-Chandra isomorphism for GLn

Joshua Jeishing Wen  Department of Mathematics, Northeastern University,
Boston, MA, USA<EMAIL_ADDRESS>
###### Abstract.

We prove that for generic parameters, the quantum radial parts map of
Varagnolo and Vasserot gives an isomorphism between the spherical double
affine Hecke algebra of $GL_{n}$ and a quantized multiplicative quiver
variety, as defined by Jordan.

###### Key words and phrases:

deformation quantization, double affine Hecke algebras, multiplicative quiver
varieties

##  1\. Introduction

This paper proves a quantum/multiplicative analogue of the Harish-Chandra
isomorphism, a result at the root of many fruitful directions of research. For
a complex reductive group $G$ with Lie algebra $\mathfrak{g}$, Cartan
subalgebra $\mathfrak{t}$, and Weyl group $W$, the classical isomorphism is
concerned with the ring of differential operators $D(\mathfrak{g})$. Harish-
Chandra’s radial parts map [HC64] is a homomorphism
$D(\mathfrak{g})^{G}\rightarrow D(\mathfrak{t})^{W}$ that is in some sense a
restriction to $\mathfrak{t}$ “performed with extra steps”. It was proven to
be surjective by Wallach [Wal93] and Levasseur–Stafford [LS95], and the latter
also determined its kernel [LS96]: the adjoint action induces a homomorphism
$\mu:\mathfrak{g}\rightarrow D(\mathfrak{g})$ and the kernel is the $G$
invariants of the left ideal $\mathfrak{I}:=D(\mathfrak{g})\mu(\mathfrak{g})$.
Altogether, we have a description of $D(\mathfrak{t})^{W}$ as a quantum
Hamiltonian reduction:

$\left[D(\mathfrak{g})\big{/}\mathfrak{I}\right]^{G}\cong D(\mathfrak{t})^{W}$

The radial parts map is of fundamental importance in the application of rings
of differential operators to geometric representation theory (see references
cited above).

In the case $G=GL_{n}$, this construction admits a 1-parameter deformation,
discovered by Etingof–Ginzburg [EG02]. The smash product
$D(\mathfrak{h})\rtimes W$ can be deformed via a parameter $c$ to the rational
Cherednik algebra $H_{n}(c)$ of the symmetric group $\Sigma_{n}$;
$D(\mathfrak{h})^{W}$ is then replaced with the spherical subalgebra
$SH_{n}(c)$. On the other side, we consider
$D(\mathfrak{gl}_{n}\times\mathbb{C}^{n})$. The adjoint and vector
representations give a map $\mu:\mathfrak{gl}_{n}\rightarrow
D(\mathfrak{gl}_{n}\times\mathbb{C}^{n})$, and the deformation parameter $c$
appears in the ideal via the trace character
$\mathrm{tr}:\mathfrak{gl}_{n}\rightarrow\mathbb{C}$:

$\displaystyle\mathfrak{I}_{c}$
$\displaystyle:=D(\mathfrak{gl}_{n}\times\mathbb{C}^{n})\big{(}(\mu-c\mathrm{tr})(\mathfrak{gl}_{n})\big{)}$
$\displaystyle\mathfrak{A}_{c}$
$\displaystyle=\left[D(\mathfrak{gl}_{n}\times\mathbb{C}^{n})\big{/}I_{c}\right]^{G}$

The deformed Harish-Chandra isomorphism was proved by Gan–Ginzburg [GG06]:

$\mathfrak{A}_{c}\cong SH_{n}(c)$ (1.1)

$\mathfrak{A}_{c}$ is a natural quantization of the Hilbert scheme $n$ points
in $\mathbb{C}^{2}$, constructed via Hamiltonian reduction as a Nakajima
quiver variety for the Jordan quiver:

(1.2)

Various constructions exist [BFG06, GS06, KR08] to microlocalize modules for
$\mathfrak{A}_{c}$ into coherent sheaves on the Hilbert scheme, and the
isomorphism (1.1) allows a rich interplay between such sheaves and
representations of $H_{n}(c)$; in particular, $H_{n}(c)$ itself microlocalizes
to Haiman’s Procesi bundle [Hai01].

The ladder of deformation affords us more rungs—the rational Cherednik algebra
is itself a degeneration of the double affine Hecke algebra (DAHA)
$\ddot{\mathcal{H}}_{n}(q,t)$, an algebra that has appeared across many fields
since its initial discovery and application by Cherednik in solving
conjectures from the theory of Macdonald polynomials [Che95]. It is natural to
ask if there is an analogue of the Harish-Chandra isomorphism for
$\ddot{\mathcal{H}}_{n}(q,t)$ and, more specifically, its spherical subalgebra
$\mathcal{S}\ddot{\mathcal{H}}_{n}(q,t)$. One can view the ring
$D(\mathfrak{gl}_{n})$ as
$\mathbb{C}[\mathfrak{gl}_{n}]\otimes\mathbb{C}[\mathfrak{gl}_{n}]$ with a
nontrivial commutation relation between the two tensorands. It is almost
immediately obvious that $D(\mathfrak{gl}_{n})$ would need to be replaced by
an algebra quantizing $\mathbb{C}[GL_{n}]\otimes\mathbb{C}[GL_{n}]$; this is
not the coordinate ring of a cotangent bundle and thus there is no longer a
quantization available to us “out of the box”. Moreover, to perform quantum
Hamiltonian reduction, the product of the quantization needs to be equivariant
with respect to whatever symmetry object replaces $GL_{n}$.

The correct quantization was defined by Varagnolo–Vasserot [VV10]: this is
their ring of quantum differential operators on $GL_{n}$, which we denote by
$\mathcal{D}$. Here, $GL_{n}$-equivariance is replaced with equivariance with
respect to the quantized universal enveloping algebra
$\mathcal{U}:=U_{q}(\mathfrak{gl}_{n})$. As a $\mathcal{U}$-module,
$\mathcal{D}$ is isomorphic to the tensor product of two copies of
$\mathcal{O}:=\mathcal{O}_{q}(GL_{n})$, an equivariant version of functions on
quantum $GL_{n}$. One can construct $\mathcal{O}$ from the braided monoidal
category of finite-dimensional $\mathcal{U}$-modules via a braided analogue of
Tannakian reconstruction discovered by Majid [Maj93], and it is also a
localization of what is called the reflection equation algebra [DM03].
$\mathcal{D}$ also appeared in prior work of Alekseev–Schomerus [AS96] on
quantizations of character varieties.

Varagnolo–Vasserot also address the other necessary ingredients, but we follow
the definitions of Jordan in his construction of quantized multiplicative
quiver varieties [Jor14]. Out of the same quiver data (1.2), this yields a
$\mathbb{C}(q,t)$-algebra $\mathcal{A}_{t}$ through a Hopf-algebraic version
of quantum Hamiltonian reduction. Our main result is the following:

######  Main Theorem.

The quantized multiplicative quiver variety $\mathcal{A}_{t}$ for the quiver
data (1.2) is isomorphic as a $\mathbb{C}(q,t)$-algebra to the spherical
$GL_{n}$-DAHA $\mathcal{S}\ddot{\mathcal{H}}_{n}(q,t)$.

Thus, the two algebras are isomorphic for generic values of the parameters $q$
and $t$. Prior to our result, the analogous isomorphism was proven in the
following cases:

  * •

when $q=1$ [Obl04];

  * •

when $q$ is a root of unity of sufficiently large order [VV10];

  * •

formally over the ring $\mathbb{C}[[\hbar]]$ where $q=e^{\hbar}$ [Jor14];

  * •

for any $q\in\mathbb{C}^{\times}$ and $n=2$ [BJ18].

Our strategy follows the well-established pattern from the rational case
[GG06, EGGO07]. Both sides of the isomorphism are invariant subalgebras, and
thus one does not have a presentation for either; from a general perspective,
it is an interesting problem to show two algebras are isomorphic without
generators and relations. In the rational case, the scheme of proof goes as
follows:

  1. (1)

embed $SH_{n}(c)$ into a ring of differential operators via a Dunkl
representation;

  2. (2)

map $\mathfrak{A}_{c}$ to that same ring via a deformed analogue of the
Harish-Chandra radial parts map;

  3. (3)

show that the radial parts map is injective and surjects onto the image of
$SH_{n}(c)$.

$\mathcal{S}\ddot{\mathcal{H}}_{n}(q,t)$ has an analogue of (1), the Dunkl-
Cherednik embedding into a ring of difference operators. Step (2) is not
straightforward, but Varagnolo–Vasserot [VV10] gave a brilliant definition for
a quantum radial parts map. Namely, the equivariance of $\mathcal{A}_{t}$
ensures that it acts on certain spaces of intertwiners, and the weighted
traces of these intertwiners have been identified with Macdonald polynomials
by Etingof–Kirillov [EK94]. We perform step (3) first for the case $t=q^{k}$,
wherein we use work of Jordan [Jor14] to perform a classical degeneration
$q\mapsto 1$. From here, we can bootstrap to general $t$.

### Future directions

We were quite delighted to encounter this appearance of Etingof–Kirillov’s
traces of intertwiners. Such functions have been proven to satisfy many
beautiful properties [EV00], but their role as vectors in a representation for
some quantum algebra has been largely unexplored. It seems like quantized
multiplicative quiver varieties should act as such algebras for at least a
large class of intertwiners.

For more geometric directions, $\mathcal{A}_{t}$ has also been constructed in
[BZBJ18a, BZBJ18b] via factorization homology as a quantized character variety
for the torus. Any way to assign to each
$\mathcal{S}\ddot{\mathcal{H}}_{n}(q,t)$-module a sheaf on the character
variety should be quite interesting.

### Acknowledgements

This work received support from NSF-RTG grant “Algebraic Geometry and
Representation Theory at Northeastern University” (DMS-1645877).

##  2\. Double affine Hecke algebras

In this section, we review the $GL_{n}$ DAHA and associated structures. Our
main reference is [Che05], although in order to make better contact with
Etingof-Kirillov theory, we follow the conventions from [Kir97].

###  2.1. Definition

Let $R:=\mathbb{C}[q^{\pm 1},t^{\pm 1}]$ and $K:=\mathbb{C}(q,t)$. The
$GL_{n}$ DAHA $\ddot{\mathcal{H}}_{n}(q,t)$ is the $K$-algebra with generators

$\\{T_{i},X_{j}^{\pm 1},\pi^{\pm 1}\,|\,i=1,\ldots,n-1\hbox{ and
}j=1,\ldots,n\\}$

and relations

$\displaystyle(T_{i}-t)(T_{i}+t^{-1})=0;$ $\displaystyle
X_{i}X_{j}=X_{j}X_{i};$ $\displaystyle T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1};$
$\displaystyle T_{i}T_{j}=T_{j}T_{i}\hbox{ for }j\not=i,i+1;$ $\displaystyle
T_{i}X_{j}=X_{j}T_{i}\hbox{ for }j\not=i,i+1;$ $\displaystyle
T_{i}X_{i}T_{i}=X_{i+1};$ $\displaystyle\pi T_{i}=T_{i+1}\pi;$
$\displaystyle\pi^{n}T_{i}=T_{i}\pi^{n};$ $\displaystyle\pi X_{i}=X_{i+1}\pi;$
$\displaystyle\pi X_{n}=q^{-2}X_{1}\pi$

We can also define this as an $R$-algebra, which we denote by
$\ddot{\mathcal{H}}^{R}_{n}(q,t)$.

####  2.1.1. Y-generators

The elements

$Y_{i}:=T_{i}\cdots T_{n-1}\pi^{-1}T_{1}^{-1}\cdots T_{i-1}^{-1}$ (2.1)

for $i=1,\ldots,n$ generate a polynomial subalgebra. They furnish an
alternative presentation of $\ddot{\mathcal{H}}_{n}(q,t)$, now with generators

$\left\\{T_{i},X_{j}^{\pm 1},Y_{j}^{\pm 1}\,|\,i=1,\ldots,n-1\hbox{ and
}j=1,\ldots,n\right\\}$

and relations

$\displaystyle(T_{i}-t)(T_{i}+t^{-1})=0;$ (2.2) $\displaystyle
T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1};$ $\displaystyle
T_{i}T_{j}=T_{j}T_{i}\hbox{ for }j\not=i,i+1;$ $\displaystyle
X_{i}X_{j}=X_{j}X_{i};$ $\displaystyle Y_{i}Y_{j}=Y_{j}Y_{i};$ $\displaystyle
T_{i}X_{i}T_{i}=X_{i+1}\hbox{ for }i\not=n;$ $\displaystyle
T_{i}^{-1}Y_{i}T_{i}^{-1}=Y_{i+1}\hbox{ for }i\not=n;$ $\displaystyle
T_{i}X_{j}=X_{j}T_{i}\hbox{ for }j\not=i,i+1;$ $\displaystyle
T_{i}Y_{j}=Y_{j}T_{i}\hbox{ for }j\not=i,i+1;$ $\displaystyle Y_{1}\cdots
Y_{n}X_{j}=q^{2}X_{j}Y_{1}\cdots Y_{n};$ $\displaystyle X_{1}\cdots
X_{n}Y_{j}=q^{-2}Y_{j}X_{1}\cdots X_{n};$ $\displaystyle
X_{1}Y_{2}=Y_{2}T^{2}_{1}X_{1}$

We note that this presentation is also valid for
$\ddot{\mathcal{H}}_{n}^{R}(q,t)$. From these relations, we can define a
bigrading on $\ddot{\mathcal{H}}_{n}^{R}(q,t)$ by setting:

$\displaystyle\deg(X_{j}^{\pm 1})$ $\displaystyle=(\pm 1,0)$
$\displaystyle\deg(Y_{j}^{\pm 1})$ $\displaystyle=(0,\pm 1)$
$\displaystyle\deg(T_{i})$ $\displaystyle=(0,0)$

####  2.1.2. Symmetrizer

The subalgebra $\mathcal{H}_{n}(t)$ of $\ddot{\mathcal{H}}_{n}(q,t)$ generated
by $T_{1},\ldots,T_{n-1}$ is isomorphic to the usual Hecke algebra for the
symmetric group $\Sigma_{n}$ with parameter $t$. As such, one can make sense
of elements $T_{w}\in\mathcal{H}_{n}(t)$ for any $w\in\Sigma_{n}$.
Specifically, first let $s_{i}\in\Sigma_{n}$ be the $i$th adjacent
transposition. For any reduced expression $w=s_{i_{1}}\cdots s_{i_{a}}$, the
element

$T_{w}:=T_{i_{1}}\cdots T_{i_{a}}$

is independent of the reduced expression. Letting $\ell(w)=a$ denote the
length, define the symmetrizer

$\tilde{\mathbf{s}}:=\sum_{w\in\Sigma_{n}}t^{\ell(w)}T_{w}$

It will be useful to note the following:

$\displaystyle T_{i}\tilde{\mathbf{s}}=\tilde{\mathbf{s}}T_{i}$
$\displaystyle=t\tilde{\mathbf{s}}\hbox{ for all $i$}$ (2.3)
$\displaystyle\sum_{w\in\Sigma_{n}}s^{\ell(w)}$ $\displaystyle=[n]_{s}!$
$\displaystyle\tilde{\mathbf{s}}^{2}$
$\displaystyle=[n]_{t^{2}}!\tilde{\mathbf{s}}$ (2.4)

where

$[\pm k]_{s}=\frac{1-s^{\pm k}}{1-s^{\pm
1}},\,[k]_{s}!=[k]_{s}[k-1]_{s}\cdots[1]_{s}$

for $k\in\mathbb{N}$. From (2.4),

$\mathbf{s}:=\frac{\tilde{\mathbf{s}}}{[n]_{t^{2}}!}$

is an idempotent.

####  2.1.3. Triangular decomposition

For a vector $v=(v_{1},\ldots,v_{n})\in\mathbb{Z}^{n}$, denote the monomial

$Y^{v}:=Y_{1}^{v_{1}}\cdots Y_{n}^{v_{n}}$

The following is Theorem 2.3(ii) of [Che95]:

######  Theorem 2.1.

Any $H\in\ddot{\mathcal{H}}_{n}^{R}(q,t)$ can be uniquely written as

$H=\sum_{\begin{subarray}{c}w\in\Sigma_{n}\\\
v\in\mathbb{Z}^{n}\end{subarray}}Y^{v}f_{v,w}(X^{\pm 1}_{1},\ldots,X^{\pm
1}_{n})T_{w}$ (2.5)

for some Laurent polynomials $\\{f_{v,w}\\}$.

######  Corollary 2.1.

$\ddot{\mathcal{H}}_{n}^{R}(q,t)$ is a free $R$-module.

###  2.2. Spherical DAHA

The spherical subalgebra $\mathcal{S}\ddot{\mathcal{H}}_{n}(q,t)$ is defined
as

$\mathcal{S}\ddot{\mathcal{H}}_{n}(q,t):=\mathbf{s}\ddot{\mathcal{H}}_{n}(q,t)\mathbf{s}\subset\ddot{\mathcal{H}}_{n}(q,t)$

We denote the localizations $\tilde{R}:=R[(1-t^{2k})^{-1}]_{k>0}$ and
$\ddot{\mathcal{H}}^{\tilde{R}}_{n}(q,t):=\tilde{R}\otimes\ddot{\mathcal{H}}^{R}_{n}(q,t)$.
Note that $\mathbf{s}\in\ddot{\mathcal{H}}^{\tilde{R}}_{n}(q,t)$, and we
define

$\mathcal{S}\ddot{\mathcal{H}}_{n}^{R}(q,t):=\mathbf{s}\ddot{\mathcal{H}}_{n}^{\tilde{R}}(q,t)\mathbf{s}$

We then make sense of the specialization $t=q^{k}$ by setting

$\displaystyle\mathcal{S}\ddot{\mathcal{H}}_{n}^{R}(q,q^{k})$
$\displaystyle:=\mathcal{S}\ddot{\mathcal{H}}_{n}^{R}(q,t)\bigg{|}_{t=q^{k}}$
$\displaystyle\mathcal{S}\ddot{\mathcal{H}}(q,q^{k})$
$\displaystyle:=\mathbb{C}(q)\otimes\mathcal{S}\ddot{\mathcal{H}}_{n}^{R}(q,q^{k})$

####  2.2.1. Bigrading revisited

Multiplying both sides of (2.5) by $\mathbf{s}$ and absorbing the $T_{w}$ into
$\mathbf{s}$ using (2.3), we can see that
$\mathcal{S}\ddot{\mathcal{H}}_{n}^{R}(q,t)$ is spanned by elements of the
form

$\mathbf{s}\sum_{v\in\mathbb{Z}^{n}}Y^{v}f(X_{1}^{\pm 1},\ldots,X_{n}^{\pm
1})\mathbf{s}$

For such an element as above that is homogeneous with respect to the
bigrading, we can see from (2.2) that its bidegree can be discerned by
commuting with

$\displaystyle\mathbf{s}\mathbf{X}_{n}^{\pm 1}\mathbf{s}:=\mathbf{s}X_{1}^{\pm
1}\cdots X_{n}^{\pm 1}\mathbf{s}$ (2.6) $\displaystyle\hbox{ and
}\mathbf{s}\mathbf{Y}_{n}^{\pm 1}\mathbf{s}:=\mathbf{s}Y_{1}^{\pm 1}\cdots
Y_{n}^{\pm 1}\mathbf{s}$

######  Proposition 2.1.

$H\in\mathcal{S}\ddot{\mathcal{H}}_{n}^{R}(q,t)$ has bidegree $(a,b)$ if and
only if

$\displaystyle\mathbf{s}\mathbf{X}_{n}\mathbf{s}H\mathbf{s}\mathbf{X}_{n}^{-1}\mathbf{s}$
$\displaystyle=q^{-2a}H$
$\displaystyle\mathbf{s}\mathbf{Y}_{n}\mathbf{s}H\mathbf{s}\mathbf{Y}_{n}^{-1}\mathbf{s}$
$\displaystyle=q^{2b}H$

Let
$\ddot{\mathcal{H}}_{n}^{R}(q,t)_{\textrm{IV}}\subset\ddot{\mathcal{H}}_{n}^{R}(q,t)$
denote the subalgebra generated by

$\\{T_{i},X_{j},Y^{-1}_{j}\,|\,i=1,\ldots,n-1\hbox{ and }j=1,\ldots,n\\}$

i.e. we restrict to postiive powers of $X$-generator and negative powers of
$Y$-generators (the “fourth quadrant” in the bigrading). We then set

$\mathcal{S}\ddot{\mathcal{H}}_{n}^{R}(q,t)_{\textrm{IV}}:=\mathbf{s}\ddot{\mathcal{H}}_{n}^{R}(q,t)_{\textrm{IV}}\mathbf{s}\subset\mathcal{S}\ddot{\mathcal{H}}_{n}^{R}(q,t)$

The subalgebras $\mathcal{S}\ddot{\mathcal{H}}_{n}(q,t)_{\mathrm{IV}}$,
$\mathcal{S}\ddot{\mathcal{H}}_{n}^{R}(q,q^{k})_{\mathrm{IV}}$, and
$\mathcal{S}\ddot{\mathcal{H}}_{n}(q,q^{k})_{\mathrm{IV}}$ are defined
similarly. Let $\mathcal{S}^{R}_{\mathrm{IV}}[a,b]$ denote the homogeneous
piece of $\mathcal{S}\ddot{\mathcal{H}}_{n}^{R}(q,t)$ of bidegree $(a,b)$ and
likewise for

$\displaystyle\mathcal{S}_{t,\mathrm{IV}}[a,b]$
$\displaystyle\subset\mathcal{S}\ddot{\mathcal{H}}_{n}(q,t)_{\mathrm{IV}}$
$\displaystyle\mathcal{S}_{q^{k},\mathrm{IV}}[a,b]$
$\displaystyle\subset\mathcal{S}\ddot{\mathcal{H}}_{n}(q,q^{k})_{\mathrm{IV}}$

Finally, let

$\mathbb{C}[\mathbf{x}_{n},\mathbf{y}_{n}]:=\mathbb{C}[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}]$

The symmetric group $\Sigma_{n}$ acts on
$\mathbb{C}[\mathbf{x}_{n},\mathbf{y}_{n}]$ by permuting subscripts.
$\mathbb{C}[\mathbf{x}_{n},\mathbf{y}_{n}]$ is also bigraded where
$\deg(x_{i})=(1,0)$ and $\deg(y_{i})=(0,1)$. Denote by
$\mathbb{C}[\mathbf{x}_{n},\mathbf{y}_{n}]^{\Sigma_{n}}_{a,b}$ the subspace of
homogeneous elements of bidegree $(a,b)$.

######  Proposition 2.2.

We have

$\dim_{K}\mathcal{S}_{t,\mathrm{IV}}[a,-b]=\dim_{\mathbb{C}(q)}\mathcal{S}_{q^{k},\mathrm{IV}}[a,-b]=\dim_{\mathbb{C}}\mathbb{C}[\mathbf{x}_{n},\mathbf{y}_{n}]^{\Sigma_{n}}_{a,b}$

###### Proof.

$\mathcal{S}^{R}_{\mathrm{IV}}[a,b]$ is a direct summand of the free
$R$-module $\ddot{\mathcal{H}}_{n}^{R}(q,t)_{\mathrm{IV}}$ (Corollary 2.1).
Therefore, it is also free and the dimensions of
$\mathcal{S}_{t,\mathrm{IV}}[a,b]$ and $\mathcal{S}_{q^{k},\mathrm{IV}}[a,b]$
are both equal to its rank. To compute this rank, we can set $q=t=1$, in which
case

$\displaystyle\ddot{\mathcal{H}}_{n}^{R}(q,t)_{\mathrm{IV}}\big{|}_{q=t=1}$
$\displaystyle\cong\mathbb{C}[\mathbf{x}_{n},\mathbf{y}_{n}]\rtimes\mathbb{C}[\Sigma_{n}]$
$\displaystyle\mathcal{S}\ddot{\mathcal{H}}_{n}^{R}(q,t)_{\mathrm{IV}}\big{|}_{q=t=1}$
$\displaystyle\cong\mathbb{C}[\mathbf{x}_{n},\mathbf{y}_{n}]^{\Sigma_{n}}$

where $x_{i}$ and $y_{i}$ are the images of $X_{i}$ and $Y_{i}^{-1}$,
respectively. ∎

####  2.2.2. Generators

We use ideas from the proofs of Proposition 2.5 of [SV11] and Lemma 5.2 of
[FFJ+11] to produce nice generating sets. For $r=1,\ldots,n$, Let $e_{r}$
denote the $r$th elementary symmetric polynomial in $n$ variables and set

$\displaystyle e_{r}(\mathbf{X}_{n}^{\pm 1})$ $\displaystyle:=e_{r}(X_{1}^{\pm
1},\ldots,X_{n}^{\pm 1})$ (2.7) $\displaystyle e_{r}(\mathbf{Y}_{n}^{\pm 1})$
$\displaystyle:=e_{r}(Y_{1}^{\pm 1},\ldots,Y_{n}^{\pm 1})$

######  Lemma 2.1.

We have the following:

  1. (1)

$\mathcal{S}\ddot{\mathcal{H}}_{n}(q,t)_{\mathrm{IV}}$ and
$\mathcal{S}\ddot{\mathcal{H}}_{n}(q,q^{k})_{\mathrm{IV}}$ are generated by

$\left\\{\mathbf{s}e_{r}(\mathbf{X}_{n})\mathbf{s},\mathbf{s}e_{r}(\mathbf{Y}_{n}^{-1})\mathbf{s}\,\middle|\,r=1,\ldots,n\right\\}$
(2.8)

  2. (2)

$\mathcal{S}\ddot{\mathcal{H}}_{n}(q,t)$ and
$\mathcal{S}\ddot{\mathcal{H}}_{n}(q,q^{k})$ are generated by the set (2.8)
along with $\mathbf{s}\mathbf{X}_{n}^{-1}\mathbf{s}$ and
$\mathbf{s}\mathbf{Y}_{n}\mathbf{s}$.

###### Proof.

Part (1) easily implies (2), so we will focus on the former. For
$(a,b)\in\mathbb{Z}_{\geq 0}^{2}$, let

$P_{a,-b}=\mathbf{s}\left(\sum_{i=1}^{n}X_{i}^{a}Y_{i}^{-b}\right)\mathbf{s}$

First note that by a classical theorem of Weyl [Wey39], the invariant
polynomial ring $\mathbb{C}[\mathbf{x}_{n},\mathbf{y}_{n}]^{\Sigma_{n}}$ is
generated by power sums of the form

$P_{a,-b}\bigg{|}_{q=t=1}=\sum_{i=1}^{n}x_{i}^{a}y_{i}^{b}$

Thus, the $P_{a,b}$ generate
$\mathcal{S}\ddot{\mathcal{H}}_{n}^{R}(q,t)_{\mathrm{IV}}$ modulo the ideal
$(q-1,t-1)$. By applying Nakayama’s Lemma to each (finite rank) bigraded
piece, we get that they generate $\mathcal{S}\ddot{\mathcal{H}}_{n}(q,t)$ and
$\mathcal{S}\ddot{\mathcal{H}}_{n}(q,q^{k})$.

Next, we follow the proof of Lemma 5.2 [FFJ+11] and specialize
$\mathcal{S}\ddot{\mathcal{H}}_{n}^{R}(q,t)$ at $t=1$. Note that
$P_{1,0}=\mathbf{s}e_{1}(\mathbf{X}_{n})\mathbf{s}$ and
$P_{0,-1}=\mathbf{s}e_{1}(\mathbf{Y}_{n}^{-1})\mathbf{s}$. By (2.1), $Y_{i}$
becomes the $q^{2}$-shift operator on $X_{i}$ at $t=1$. We thus have

$\displaystyle\mathrm{ad}\left(P_{1,0}\right)^{a}P_{0,-1}$
$\displaystyle=(1-q^{-2})^{a}P_{a,-1}$
$\displaystyle\mathrm{ad}\left(P_{0,-1}\right)^{b}P_{a,-1}$
$\displaystyle=(q^{-2a}-1)^{b}P_{a,-1-b}$

Thus, after localizing to $\mathbb{C}(q)[t^{\pm 1}]$, any $P_{a,-b}$ with
$a,b>0$ can be written in terms of $\mathbf{s}e_{1}(\mathbf{X}_{n})\mathbf{s}$
and $\mathbf{s}e_{1}(\mathbf{Y}_{n}^{-1})\mathbf{s}$ modulo the ideal $(t-1)$.
We include the other elementary symmetric polynomials to cover the cases $a$
or $b=0$. The result follows by again apply Nakayama’s Lemma to each bigraded
piece. ∎

###  2.3. Polynomial representation

While Lemma 2.1 gives us generators for
$\mathcal{S}\ddot{\mathcal{H}}_{n}(q,t)$ and
$\mathcal{S}\ddot{\mathcal{H}}_{n}(q,q^{k})$, we do not have relations; in
compensation, we have instead a faithful representation.

####  2.3.1. Demazure-Lusztig operators

Let $K[\mathbf{x}_{n}^{\pm}]:=K[x_{1}^{\pm 1},\ldots x_{n}^{\pm 1}]$ be the
ring of Laurent polynomials in $n$ variables. We similarly use the
abbreviation $R[\mathbf{x}_{n}^{\pm}]$. The symmetric group $\Sigma_{n}$ acts
on both rings by permuting variables, and as before, we let
$s_{i}\in\Sigma_{n}$ denote the usual adjacent transposition.

######  Theorem 2.2 ([Che95] Theorem 2.3).

The following defines a faithful represention $\rho$ of
$\ddot{\mathcal{H}}_{n}^{R}(q,t)$ on $R[\mathbf{x}_{n}^{\pm}]$: for $f\in
K[\mathbf{x}_{n}^{\pm}]$,

$\displaystyle\rho(T_{i})$
$\displaystyle=ts_{i}+(t-t^{-1})\frac{s_{i}-1}{x_{i}/x_{i+1}-1}$
$\displaystyle\rho(X_{i})f$ $\displaystyle=x_{i}f$
$\displaystyle\rho(\pi)f(x_{1},x_{2},\ldots,x_{n})$
$\displaystyle=f(x_{2},x_{3},\ldots,x_{n},q^{-2}x_{1})$

This representation remains faithful on any specialization of $(q,t)$ so long
as $q$ is not sent to a root of unity.

We will abuse notation and use $\rho$ to also denote its versions obtained via
base-change from $R$. From the formula for $\rho(T_{i})$, we can see that

$f\in\mathbb{C}_{q,t}[\mathbf{x}_{n}^{\pm}]^{\Sigma_{n}}\hbox{ if and only if
}\rho(T_{i})f=tf$

for all $i$. It follows from (2.3) that the restriction of $\rho$ to
$\mathcal{S}\ddot{\mathcal{H}}_{n}(q,t)$ and
$\mathcal{S}\ddot{\mathcal{H}}_{n}^{R}(q,t)$ preserves the subring
$\Lambda_{n}^{\pm}(q,t):=K[\mathbf{x}_{n}^{\pm}]^{\Sigma_{n}}$. We similarly
define $\Lambda_{n}^{\pm}(q)$ in the obvious way.

The following is well known:

######  Proposition 2.3.

The elements $\mathbf{s}e_{r}(\mathbf{X}_{n}^{\pm 1})\mathbf{s}$ and
$\mathbf{s}e_{r}(\mathbf{Y}_{n}^{\pm 1})\mathbf{s}$ act on
$f\in\Lambda^{\pm}_{n}(q,t)$ by:

$\displaystyle\rho\left(\mathbf{s}e_{r}(\mathbf{X}_{n}^{\pm
1})\mathbf{s}\right)f$ $\displaystyle=e_{r}(x_{1}^{\pm 1},\ldots,x_{n}^{\pm
1})f$ $\displaystyle\rho\left(\mathbf{s}e_{r}(\mathbf{Y}_{n}^{\pm
1})\mathbf{s}\right)f$
$\displaystyle=\sum_{I\subset\\{1,\ldots,n\\}}\left(\prod_{\begin{subarray}{c}i\in
I\\\ j\not\in
I\end{subarray}}\frac{t^{2}\left(\dfrac{x_{i}}{x_{j}}\right)^{\pm
1}-1}{\left(\dfrac{x_{i}}{x_{j}}\right)^{\pm 1}-1}\right)\prod_{i\in
I}T_{q^{2},x_{i}}^{\pm 1}f$

where
$T_{q^{2},x_{i}}f(x_{1},\ldots,x_{i},\ldots,x_{n})=f(x_{1},\ldots,q^{2}x_{i},\ldots,x_{n})$.

####  2.3.2. Macdonald polynomials

We will need a nice basis for the representation of
$\mathcal{S}\ddot{\mathcal{H}}_{n}(q,t)$ on $\Lambda^{\pm}_{n}(q,t)$. A
natural starting point is the basis of monomial symmetric functions: for
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in\mathbb{Z}^{n}$,

$m_{\lambda}:=\sum_{w\in\Sigma_{n}/\mathrm{Stab}(\lambda)}x^{w\lambda}$

where $\Sigma_{n}$ acts on $\mathbb{Z}^{n}$ by permutations and for
$\mu=(\mu_{1},\ldots,\mu_{n})\in\mathbb{Z}^{n}$,

$x^{\mu}=x_{1}^{\mu_{1}}x_{2}^{\mu_{2}}\cdots x_{n}^{\mu_{n}}$

We say that $\lambda\in\mathbb{Z}^{n}$ is dominant if
$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}$; it is easy to see that
the $m_{\lambda}$ for dominant $\lambda$ provide a basis of
$\Lambda_{n}^{\pm}(q,t)$.

######  Theorem 2.3.

We have the following:

  1. (1)

For each dominant $\lambda$, there exists a unique
$P_{\lambda}(q,t)\in\Lambda_{n}^{\pm}(q,t)$ satisfying

     * •

for $f\in\Lambda_{n}^{\pm}(q,t)$,

$\rho\left(\mathbf{s}f(Y_{1},\ldots,Y_{n})\mathbf{s}\right)P_{\lambda}(q,t)=f\left(q^{2\lambda_{1}}t^{n-1},q^{2\lambda_{2}}t^{n-3},\ldots,q^{2\lambda_{n}}t^{1-n}\right)P_{\lambda}(q,t)$

     * •

the coefficient of $m_{\lambda}$ is $1$.

Moreover, $\left\\{P_{\lambda}(q,t)\right\\}$ form a basis of
$\Lambda_{n}^{\pm}(q,t)$.

  2. (2)

For any integer $k>0$, $P_{\lambda}(q,t)$ can be specialized to $t=q^{k}$.

###### Proof.

Part (1) is classical. For (2), one can consider the tableaux sum formula for
$P_{\lambda}(q,t)$ (cf. [Mac15] VI.6). ∎

$P_{\lambda}(q,t)$ is the Macdonald polynomial associated to $\lambda$. By
Theorem 2.3(2), when we can specialize the representation of
$\mathcal{S}\ddot{\mathcal{H}}_{n}^{R}(q,t)$ on $\Lambda_{n}^{\pm}(q,t)$ to
$t=q^{k}$ by taking the $\mathbb{C}(q)[t^{\pm 1}]$-sublattice spanned by
$\left\\{P_{\lambda}(q,t)\right\\}$.

######  Remark 2.1.

When $\lambda$ is a partition (i.e., $\lambda_{n}\geq 0$), our
$P_{\lambda}(q,t)$ is actually the $P_{\lambda}(q^{2},t^{2})$ found in
[Mac15]. As stated in the start of this section, we make this choice to better
align with the Etingof-Kirillov approach to Macdonald polynomials.

##  3\. Quantum groups

###  3.1. Quantum enveloping algebra

In this subsection, we review the algebra
$\mathcal{U}:=U_{q}(\mathfrak{gl}_{n})$ and some of its basic structures.

####  3.1.1. Roots and weights

Let $\epsilon_{i}\in\mathbb{R}^{n}$ be the $i$th coordinate vector. The root
system $R$ is the set $\left\\{\epsilon_{i}-\epsilon_{j}\right\\}$, and the
set of positive roots $R^{+}$ is the subset where $i<j$. Within $R^{+}$, we
call the $n-1$ elements

$\alpha_{i}=\epsilon_{i}-\epsilon_{i+1}\hbox{ for }i=1,\ldots,n-1$

the simple roots. The root lattice $Q$ is the lattice spanned by $R$, and we
denote by $Q^{+}$ the semigroup spanned by $R^{+}$. We equip $\mathbb{R}^{n}$
with the usual symmetric pairing $\langle-,-\rangle$ for which
$\left\\{\epsilon_{i}\right\\}$ is an orthonormal basis.

The set of weights is the subset

$\left\\{\omega\in\mathbb{R}^{n}\,|\,\langle\omega,\alpha\rangle\in\mathbb{Z}\hbox{
for all }\alpha\in R\right\\}$

For $i=1,\ldots,n$, we let

$\omega_{i}:=\epsilon_{1}+\epsilon_{2}+\cdots+\epsilon_{i}$

denote the $i$th fundamental weight. The lattice spanned by
$\left\\{\omega_{i}\right\\}$ is called the weight lattice $P$, which is equal
to the lattice spanned by $\left\\{\epsilon_{i}\right\\}$. For $\lambda,\mu\in
P$, we order $\lambda\geq\mu$ if $\lambda-\mu\in Q^{+}$. A weight is dominant
if it pairs nonnegatively with all roots. Let $P^{+}\subset P$ denote the
subset of dominant elements. This is not the set of dominant weights but
rather the subset of dominant weights whose coefficient for $\omega_{n}$ is an
integer. Finally,

$\rho:=\left(\frac{n-1}{2},\frac{n-3}{2},\ldots,\frac{1-n}{2}\right)$

is called the Weyl vector. It satisfies $\langle\rho,\alpha_{i}\rangle=1$ for
all $i$ and $\langle\rho,\omega_{n}\rangle=0$. Note that $2\rho\in P$.

####  3.1.2. Definition

$\mathcal{U}$ is the $\mathbb{C}(q)$-algebra with generators

$\left\\{E_{i},F_{i},q^{h}\,|\,i=1,\ldots,n-1\hbox{ and }h\in P\right\\}$

and relations

$\displaystyle q^{\vec{0}}=1,\,q^{h_{1}}q^{h_{2}}=q^{h_{1}+h_{2}},$
$\displaystyle
q^{h}E_{i}q^{-h}=q^{\langle\alpha_{i},h\rangle}E_{i},\,q^{h}F_{i}q^{-h}=q^{-\langle\alpha_{i},h\rangle}F_{i},$
$\displaystyle
E_{i}F_{j}-F_{j}E_{i}=\delta_{ij}\dfrac{q^{\alpha_{i}}-q^{-\alpha_{i}}}{q-q^{-1}},$
$\displaystyle
E_{i}^{2}E_{i+1}-(q+q^{-1})E_{i}E_{i+1}E_{i}+E_{i}E_{i+1}^{2}=0,$
$\displaystyle
F_{i}^{2}F_{i+1}-(q+q^{-1})F_{i}F_{i+1}F_{i}+F_{i}F_{i+1}^{2}=0,$

where $\delta_{ij}$ is the Kronecker delta. We can endow it with a Hopf
algebra structure where the coproduct $\Delta$, counit $\epsilon$, and
antipode $S$ are given by

$\displaystyle\Delta(E_{i})=E_{i}\otimes q^{\alpha_{i}}+1\otimes E_{i},\hskip
14.22636pt\Delta(F_{i})=F_{i}\otimes 1+q^{-\alpha_{i}}\otimes F_{i},\hskip
14.22636pt\Delta(q^{h})=q^{h}\otimes q^{h};$
$\displaystyle\epsilon(E_{i})=\epsilon(F_{i})=0,\hskip
14.22636pt\epsilon(q^{h})=1;$ $\displaystyle
S(E_{i})=-E_{i}q^{-\alpha_{i}},\hskip
14.22636ptS(F_{i})=-q^{\alpha_{i}}F_{i},\hskip 14.22636ptS(q^{h})=q^{-h};$

For coproducts, we will use Sweedler notation:

$\Delta(x)=x_{(1)}\otimes x_{(2)}$

With this Hopf algebra structure, $\mathcal{U}$ acts on itself via the adjoint
action: for $u,v\in\mathcal{U}$,

$\mathrm{ad}_{v}u:=v_{(1)}uS(v_{(2)})$

Finally, we will denote by $\mathcal{U}_{c}$ the same algebra equipped with
the co-opposite coalgebra structure.

###  3.2. R-matrix

The universal R-matrix $\mathcal{R}$ is an invertible element in a suitable
completion of $\mathcal{U}^{\otimes 2}$. We will vaguely write

$\mathcal{R}=\sum_{s}\tensor[_{s}]{r}{}\otimes\tensor[]{r}{{}_{s}}$

and even employ an Einstein-like notation:
$\mathcal{R}=\tensor[_{s}]{r}{}\otimes\tensor[]{r}{{}_{s}}$. Let
$\mathcal{R}_{xy}$ denote the tensor where $\tensor[_{s}]{r}{}$ is inserted in
the $x$th tensorand and $\tensor[]{r}{{}_{s}}$ is inserted in the $y$th
tensorand, and let $b:\mathcal{U}^{\otimes 2}\rightarrow\mathcal{U}^{\otimes
2}$ denote the tensor flip. The following equations distinguish $\mathcal{R}$:

$\displaystyle(\Delta\otimes
1)\mathcal{R}=\mathcal{R}_{13}\mathcal{R}_{23},\,(1\otimes\Delta)\mathcal{R}=\mathcal{R}_{13}\mathcal{R}_{12},$
(3.1) $\displaystyle b\Delta(h)=\mathcal{R}\Delta(h)\mathcal{R}^{-1}$ (3.2)

Some consequences of (3.1) are:

$\displaystyle(\epsilon\otimes 1)\mathcal{R}=(1\otimes\epsilon)\mathcal{R}=1$
(3.3) $\displaystyle(S\otimes 1)\mathcal{R}=(1\otimes
S)\mathcal{R}=\mathcal{R}^{-1}$ (3.4)
$\displaystyle\mathcal{R}_{12}\mathcal{R}_{13}\mathcal{R}_{23}=\mathcal{R}_{23}\mathcal{R}_{13}\mathcal{R}_{12}$
(3.5)

Equation (3.5) is called the quantum Yang-Baxter equation. Finally, we recall
a factorization of $\mathcal{R}$. Let $\mathcal{U}^{+}$ be the subspace of
$\mathcal{U}$ spanned by products of $\left\\{E_{i}\right\\}$ and
$\mathcal{U}^{-}$ be the subspace spanned by products of
$\left\\{F_{i}\right\\}$. We then have

$\mathcal{R}=q^{-\sum\epsilon_{i}\otimes\epsilon_{i}}(1\otimes
1+\mathcal{R}^{*})$ (3.6)

where $\mathcal{R}^{*}$ is in a suitable completion of
$\mathcal{U}^{+}\otimes\mathcal{U}^{-}$.

$\mathcal{R}$ is an infinite sum, but its action on tensors of finite-
dimensional representations is well-defined. For two representations $V$ and
$W$ of $\mathcal{U}$, let $b_{V,W}:V\otimes W\rightarrow W\otimes V$ be the
tensor flip. As a result of (3.2), we have that the composition

$\beta_{V,W}:=b_{V,W}\mathcal{R}\big{|}_{V,W}:V\otimes W\rightarrow W\otimes
V$

is an isomorphism of $\mathcal{U}$-modules. We note that

$\beta_{V,W}^{-1}=b_{W,V}\mathcal{R}^{-1}_{21}\big{|}_{W,V}$

####  3.2.1. Vector representation

Let $\mathbb{V}:=V_{\omega_{1}}\cong\mathbb{C}^{n}$ be the vector
representation of $\mathcal{U}$. We denote by $E_{j}^{i}$ the matrix unit
sending

$E_{j}^{i}e_{j}=e_{i}$

The specialized R-matrix
$R^{\prime}:=\mathcal{R}\big{|}_{\mathbb{V},\mathbb{V}}$ has the form

$R^{\prime}=q\sum_{i}E_{i}^{i}\otimes E_{i}^{i}+\sum_{i\not=j}E_{i}^{i}\otimes
E_{j}^{j}+(q-q^{-1})\sum_{i<j}E_{i}^{j}\otimes E_{j}^{i}$

In accordance with [Jor14] (cf. Remark 3.2 below), we will more often work
with the transposed version:

$R=q\sum_{i}E_{i}^{i}\otimes E_{i}^{i}+\sum_{i\not=j}E_{i}^{i}\otimes
E_{j}^{j}+(q-q^{-1})\sum_{i>j}E_{i}^{j}\otimes E_{j}^{i}$ (3.7)

######  Remark 3.1.

Because our R-matrix $R^{\prime}$, which is dictated by the coproduct
$\Delta$, is different from that of [Jor14], some of our formulas will differ
from those of loc. cit., e.g. (4.8).

####  3.2.2. Symmetric and exterior powers

One can check that $R$ satisfies the Hecke condition:

$(R-q)(R+q^{-1})=0$ (3.8)

In the tensor power $\mathbb{V}^{\otimes m}$, let $\mathbb{V}^{i}$ denote the
$i$th tensorand. The Yang-Baxter equation (3.5) and the Hecke condition (3.8)
imply that the map

$T_{i}\mapsto\beta_{\mathbb{V}^{i},\mathbb{V}^{i+1}}$

yields a representation of the $\Sigma_{m}$ Hecke algebra $\mathcal{H}_{m}(q)$
on $\mathbb{V}^{\otimes m}$. Similar to 2.1.2, we can $q$-symmetrize and
$q$-antisymmetrize by applying the operators

$\sum_{w\in\Sigma_{m}}q^{\ell(w)}T_{w}\hbox{ and
}\sum_{w\in\Sigma_{m}}(-q)^{-\ell(w)}T_{w}$

respectively to $\mathbb{V}^{\otimes m}$. The results are denoted by
$S_{q}^{m}\mathbb{V}$ and $\wedge_{q}^{m}\mathbb{V}$. Since
$\beta_{\mathbb{V},\mathbb{V}}$ is an isomorphism of $\mathcal{U}$-modules,
these symmetric and exterior powers are also $\mathcal{U}$-modules.

We call $\mathbb{1}_{1}:=\wedge^{n}_{q}\mathbb{V}^{n}$ the determinant
representation. It is one-dimensional, and thus $\mathcal{U}$ acts via a
character that we denote by $\chi$:

$\begin{gathered}\chi(E_{i})=\chi(F_{i})=0\\\ \chi(q^{h})=q^{\langle
h,\omega_{n}\rangle}\end{gathered}$ (3.9)

The action on the tensor powers
$\mathbb{1}_{k}:=(\wedge^{n}_{q}\mathbb{V})^{\otimes k}$ is then via
$\chi^{k}$. We similarly define
$\mathbb{1}_{-1}:=\wedge_{q}^{n}\mathbb{V}^{*}$ and $\mathbb{1}_{-k}$.

####  3.2.3. Drinfeld element

Define the Drinfeld element as

$u:=m(S\otimes 1)\mathcal{R}_{21}=S(\tensor[]{r}{{}_{s}})\tensor[_{s}]{r}{}$

where $m$ is the multiplication map. This is an infinite sum defined in a
suitable completion of $\mathcal{U}$. The following is proved in [Dd89a]:

######  Proposition 3.1.

The Drinfeld element $u$ satisfies:

$\displaystyle u^{-1}$ $\displaystyle=m(S^{-1}\otimes
1)\mathcal{R}_{21}^{-1}=\tensor[]{r}{{}_{s}}\tensor[_{s}]{r}{}$ (3.10)
$\displaystyle S^{2}(x)$ $\displaystyle=uxu^{-1}$ (3.11)
$\displaystyle\Delta(u)$
$\displaystyle=(\mathcal{R}_{12}\mathcal{R})^{-1}(u\otimes u)$ (3.12)

###  3.3. Representations

We now turn our attention to the category $\mathcal{C}$ of finite-dimensional
$\mathcal{U}$-modules. $\mathcal{C}$ is semisimple with simple objects indexed
by $P^{+}$; for $\lambda\in P^{+}$, we denote the corresponding irreducible
representation by $V_{\lambda}$. Since $\mathcal{U}$ is a Hopf algebra,
$\mathcal{C}$ is a monoidal category. The constructions from the previous
subsection involving the R-matrix endow $\mathcal{C}$ with the structure of a
ribbon category. As worked out in [RT90], such categories come with a
graphical calculus for working with morphisms. Here, we will review this
calculus as presented in Section 2.3 of [BK01].

####  3.3.1. Arrows

We will depict a morphism $f:V\rightarrow W$ of $\mathcal{U}$-modules as an
upward oriented arrow decorated with a coupon marked by $f$. When $V=W$ and
$f$ is the identity, we will omit the coupon:

Tensor product of objects and morphisms will be denoted by horizontal
juxtaposition:

####  3.3.2. Duality

For $V\in\mathcal{C}$, we endow $V^{*}$ with the structure of a
$\mathcal{U}$-module via

$x\cdot f(v)=f(S(x)v)$

for $x\in\mathcal{U}$, $f\in V^{*}$, and $v\in V$. We set $(V\otimes
W)^{*}=W^{*}\otimes V^{*}$ as they are isomorphic $\mathcal{U}$-modules under
the natural tensor flip. In our graphical calculus, we will denote $V^{*}$
using $V$ but use downward pointing arrows. Note that $V$ and $V^{**}$ are
isomorphic under the nontrivial isomorphism $S^{2}$. It is easy to see that

$S^{2}(x)=\mathrm{ad}_{q^{2\rho}}(x)$

Since $\Delta(q^{2\rho})=q^{2\rho}\otimes q^{2\rho}$, we can use
$q^{2\rho}:V\rightarrow V^{**}$ to identify the two modules in a manner that
respects tensor products. We will do so and write $V$ instead of $V^{**}$,
which will always imply a twist by $q^{-2\rho}$.

####  3.3.3. Evaluation and coevaluation

Let $\left\\{v_{i}\right\\}$ be a basis of $V$ with corresponding dual basis
$\left\\{v^{i}\right\\}$ and let $\mathbb{1}\in\mathcal{C}$ be the trivial
representation. The canonical maps

$\displaystyle c$ $\displaystyle\mapsto cv_{i}\otimes v^{i}\in V\otimes V^{*}$
$\displaystyle V^{*}\otimes V\ni v^{i}\otimes v_{j}$
$\displaystyle\mapsto\delta_{ij}$

are homomorphisms $\mathrm{coev}_{V}:\mathbb{1}\rightarrow V\otimes V^{*}$ and
$\mathrm{ev}_{V}:V^{*}\otimes V\rightarrow\mathbb{1}$, respectively. If $V$ is
irreducible, they are the unique such homomorphisms. Graphically, we will omit
depicting $\mathbb{1}$; $\mathrm{ev}_{V}$ and $\mathrm{coev}_{V}$ appear as
caps and cups oriented towards the left:

The ordering of tensor factors matter in $\mathrm{ev}_{V}$ and
$\mathrm{coev}_{V}$. To define maps with the opposite ordering, we will use
$q^{2\rho}$ to identify $V$ and $V^{**}$.

$\displaystyle c$ $\displaystyle\mapsto cv^{i}\otimes q^{-2\rho}v_{i}\in
V^{*}\otimes V$ $\displaystyle V\otimes V^{*}\ni q^{-2\rho}v_{i}\otimes v^{j}$
$\displaystyle\mapsto\delta_{ij}$

We denote these maps by $\mathrm{qcoev}_{V}$ and $\mathrm{qev}_{V}$,
respectively. Graphically, they will be depicted as cups and caps with
orientations opposite from before:

####  3.3.4. Adjunction

The caps and cups allow us to define (right) adjoints: for $f:V\rightarrow W$,
$f^{*}:W^{*}\rightarrow V^{*}$ is given by

$f^{*}:=(\mathrm{ev}_{W}\otimes 1_{V^{*}})(1_{W^{*}}\otimes f\otimes
1_{V^{*}})(1_{W^{*}}\otimes\mathrm{coev}_{V})$

Note that the adjoint of the identity map of $V$ is the identity map of
$V^{*}$.

####  3.3.5. Braiding

We will depict $\beta_{V,W}$ and $\beta_{V,W}^{-1}$ as braid crossings:

The quantum Yang-Baxter equation (3.5) implies that $\beta_{V,W}$ endows
$\mathcal{C}$ with the structure of a braided monoidal category.

####  3.3.6. Ribbon structure

By (3.11), $u$ gives an isomorphism $V\rightarrow V^{**}$ and thus
$q^{-2\rho}u$ is an automorphism of $V$, i.e. it is central. We define the
ribbon element $\nu$ by

$\nu:=\left(q^{-2\rho}u\right)^{-1}=u^{-1}q^{2\rho}$

The following was computed in [Dd89a]:

######  Proposition 3.2.

On $V_{\lambda}$, $u$ acts as
$q^{-\langle\lambda,\lambda+2\rho\rangle}q^{2\rho}$. Thus, the ribbon element
$\nu$ acts by the scalar $q^{\langle\lambda,\lambda+2\rho\rangle}$.

Using (3.10), we can see that $\nu$ can be drawn in two ways:

It is not quite true that morphisms in $\mathcal{C}$ only depend on the
isotopy type of its diagram in $\mathbb{R}^{3}$ under the graphical calculus.
Rather, we should view each strand as a ribbon with a front side and back
side, and we require the front side to always face the reader at the start and
end of the ribbon, . The back side may appear in the middle if the loop given
by $\nu$ appears, in which case the ribbon in twisted twice. Our graphical
calculus assigns to each morphism in $\mathcal{C}$ a $\mathcal{C}$-colored
ribbon tangle.

######  Theorem 3.1 ([RT90]).

A morphism in $\mathcal{C}$ only depends on the isotopy type of its associated
tangle.

We refer the reader to the original source as well as [BK01] for details. In
practice, we will instead work with strands but keep track of loops
representing $\nu$.

###  3.4. Reflection equation algebra

Here, we introduce a quantization of the Hopf algebra of functions on
$GL_{n}$. We would like this Hopf algebra to be a
$\mathcal{U}=U_{q}(\mathfrak{gl}_{n})$-module, and, critically, we want its
structure maps to be $\mathcal{U}$-homomorphisms. This requires a braided
variant of Tannakian reconstruction, defined by Majid [Maj93]. The resulting
algebra is a localization of what is known as the reflection equation algebra.

####  3.4.1. Majid reconstruction

Recall that $\mathcal{C}$ is the category of finite-dimensional
$\mathcal{U}$-modules. We define the $\mathcal{U}$-module $\mathcal{O}$ as a
space of matrix elements:

$\mathcal{O}:=\left(\bigoplus_{V\in\mathcal{C}}V^{*}\otimes
V\right)\bigg{/}\left\langle f^{*}(w^{*})\otimes v-w^{*}\otimes
f(v)\,\big{|}\,v\in V,w^{*}\in
W^{*},f\in\mathrm{Hom}_{GL_{n}}(V,W)\right\rangle$ (3.13)

For the categorically-minded, $\mathcal{O}$ is the coend of the identity
functor of $\mathcal{C}$. By considering for $f$ in (3.13) the projections
onto irreducible representations, we obtain an analogue of the Peter-Weyl
Theorem:

$\mathcal{O}=\bigoplus_{\lambda\in P^{+}}V_{\lambda}^{*}\otimes V_{\lambda}$
(3.14)

As we will see below, we can use operations on representations to define a
Hopf algebra structure on $\mathcal{O}$ that recovers at $q=1$ the classical
Hopf algebra structure on the ring of functions on $GL_{n}$.

  * •

Coalgebra structure: The coalgebra structure is identical to that of the
classical case. For $v^{*}\otimes v\in V^{*}\otimes V$, we define the
coproduct $\nabla(v^{*}\otimes v)$ as

$\nabla(v^{*}\otimes v)=v^{*}\otimes\mathrm{coev}_{V}(1)\otimes v$

The evaluation map $\mathrm{ev}_{V}$ on $V^{*}\otimes V$ yields the counit.

  * •

Algebra structure: In the classical case, the product structure entails
permuting tensorands. To make such an operation a $\mathcal{U}$-morphism, we
utilize the braiding. For $v^{*}\otimes v\in V^{*}\otimes V$ and $w^{*}\otimes
w\in W^{*}\otimes W$,

$m(v^{*}\otimes v\otimes w^{*}\otimes
w)=\tensor[]{r}{{}_{t}}\tensor[]{r}{{}_{s}}w^{*}\otimes\tensor[_{t}]{r}{}v^{*}\otimes\tensor[_{s}]{r}{}v\otimes
w$ (3.15)

The inclusion
$\mathbb{1}\rightarrow\mathbb{1}^{*}\otimes\mathbb{1}\in\mathcal{O}$ provides
the unit.

  * •

Antipode: The antipode $\iota$ is given by

$\iota(v^{*}\otimes v)=\nu\tensor[]{r}{{}_{s}}v\otimes\tensor[_{s}]{r}{}v^{*}$
(3.16)

We note that the relations for the coend (3.13) are necessary to show that
$\iota$ is an antipode. For example, in computing
$m\circ(1\otimes\iota)\circ\nabla$, we use the coend relation in the first
equality below:

####  3.4.2. Generating matrix

Since the finite-dimensional representions of $\mathcal{U}$ can be built out
of tensor functors applied to the vector representation $\mathbb{V}$, it is
perhaps unsurprising that $\mathcal{O}$ has a presentation written in terms of
matrix elements of $\mathbb{V}$. The generators of this presentation are a set
of symbols $\left\\{m_{j}^{i}\,|\,i,j=1,\ldots n\right\\}$. We arrange them
into a matrix $M=(M_{j}^{i})$ where

$M^{i}_{j}=E_{j}^{i}\otimes m^{i}_{j}$

######  Definition 3.1.

The reflection equation algebra $\mathfrak{R}$ is generated by
$\left\\{m^{i}_{j}\right\\}$ and has relations

$R_{21}M_{13}R_{12}M_{23}=M_{23}R_{21}M_{13}R_{12}$ (3.17)

######  Theorem 3.2 ([DM03]).

The map $m_{j}^{i}\mapsto e^{i}\otimes e_{j}$ gives an embedding from
$\mathfrak{R}$ to $\mathcal{O}$. It is an isomorphism after inverting a
central element of $\mathfrak{R}$ called the quantum determinant
$\det_{q}(M)$, which is mapped to $\mathrm{qcoev}_{\mathbb{1}_{1}}(1)$.

Equation(3.17) is known as the reflection equation. For an explicit formula
for the quantum determinant, see [JW20].

######  Remark 3.2.

Notice that the entries of $M$ are assigned to the transposed matrix element,
i.e the row index $i$ is sent to the dual basis vector $e^{i}$. This is
consistently done in [Jor14] (cf. formula (12) of loc. cit.), which we will
follow. For comparison, note that the reflection equation is written
differently in Proposition 10.16 of [KS97]. We will encounter this transpose
later when discussing classical degeneration in 5.1.

We will abuse notation and conflate $\mathfrak{R}$ with its image in
$\mathcal{O}$. Let $M^{-1}:=(1\otimes\iota)(M)$. By the definition of the
antipode, we have:

$M^{-1}M=MM^{-1}=I$

######  Corollary 3.1.

$\mathcal{O}$ is generated by $\det_{q}(M)$ and the entries of $M^{-1}$.

###### Proof.

First, note that

$\textstyle\iota(\det_{q}(M))=\iota^{-1}(\det_{q}(M))=\det_{q}(M)^{-1}$

Thus, by writing $\det_{q}(M)$ in terms of the entries of $M$ and applying
$\iota$, we obtain an expression for $\det_{q}(M)^{-1}$ in terms of the
entries of $M^{-1}$. Similarly, we can write the entries of
$(1\otimes\iota^{-1})(M)$ in terms of the entries of $M$ and
$\det_{q}(M)^{-1}$. Applying $\iota$, we obtain an expression for the entries
of $M$ in terms of those of $M^{-1}$ and $\det_{q}(M)$. ∎

####  3.4.3. Killing form

Matrix elements give functionals on $\mathcal{U}$ in the natural way: for
$v^{*}\otimes v\in V^{*}\otimes V$ and $x\in\mathcal{U}$:

$(v^{*}\otimes v)(x)=v^{*}(xv)$

A quantum analogue of the Killing form would allow us to view matrix elements
as sitting inside the enveloping algebra. The canonical tensor for such a form
is given by $\mathcal{R}_{21}\mathcal{R}$.

######  Theorem 3.3 ([JL92]).

The map $\kappa:\mathcal{O}\rightarrow\mathcal{U}$ given by

$\kappa(v^{*}\otimes v)=((v^{*}\otimes v)(-)\otimes
1)\mathcal{R}_{21}\mathcal{R}$

is a $\mathcal{U}$-equivariant algebra embedding, where $\mathcal{U}$ is
endowed with the adjoint action. Furthermore, it is an isomorphism onto the
subspace of elements in $\mathcal{U}$ that are locally-finite for the adjoint
action.

Observe that the map $V^{*}\otimes V\otimes W\rightarrow W$ given by

$v^{*}\otimes v\otimes w\mapsto\kappa(v^{*}\otimes v)(w)$

is the morphism in $\mathcal{C}$ depicted by the following diagram:

We will depict $\kappa(\mathcal{O})$ by replacing the strand for $W$ above by
a dotted line oriented upwards—“ghost strand”. Our graphical calculus is still
valid for such strands because it holds for any choice of representation $W$
“filling in” the ghost strand; by the quantum analogue of a theorem of Harish-
Chandra [HC49], any $x\in\mathcal{U}$ that acts trivially on all
$W\in\mathcal{C}$ is necessarily zero. Multiplying left-to-right in
$\kappa(\mathcal{O})$ corresponds to stacking ghost strands top-to-bottom. For
example, that $\kappa$ is an algebra map follows from the manipulation below:

Finally, it can be useful to know that $\kappa\circ\iota$ yields the opposite
crossing:

Theorem 3.3 omits any statement about coalgebra structures because $\kappa$
does not intertwine $\nabla$ with $\Delta$. By considering the action of
$\kappa(v^{*}\otimes v)$ on a tensor product of modules, we see that
$\Delta\circ\kappa$ merely doubles the dotted strand. We can force an instance
of $\kappa$ for each dotted strand using $\nabla$ and some braidings:

Properly comprehending the diagram on the right, we get:

######  Proposition 3.3 (Example 1.3.3(c) of [VV10]).

Let $\left\\{v_{i}\right\\}\subset V$ and $\left\\{v^{i}\right\\}\subset
V^{*}$ be dual bases. For $v^{*}\otimes v\in V^{*}\otimes V$, we have

$(\Delta\circ\kappa)(v^{*}\otimes v)=\kappa(v^{*}\otimes
v_{i})\tensor[]{r}{{}_{s}}\tensor[]{r}{{}_{t}}\otimes\kappa(\tensor[_{s}]{r}{}v^{i}\otimes\tensor[_{t}]{r}{}v)$
(3.18)

In particular, $\kappa(\mathcal{O})$ is a left coideal sublagebra of
$\mathcal{U}$, i.e.

$\displaystyle(\Delta\circ\kappa)(\mathcal{O})$
$\displaystyle\subset\mathcal{U}\otimes\kappa(\mathcal{O})$

###  3.5. Etingof-Kirillov theory

Equipped with a notion of functions on quantum $GL_{n}$, we move on to a
realization of Macdonald polynomials as spherical functions on the quantum
group. This was discovered by Etingof and Kirillov, Jr. [EK94]. We conclude
this section by making contact with
$\mathcal{S}\ddot{\mathcal{H}}_{n}(q,q^{k})$.

####  3.5.1. Traces of intertwiners

Let $U\in\mathcal{C}$. We will be concerned with the space of homomorphisms

$I(V_{\lambda},U):=\mathrm{Hom}_{\mathcal{U}}(V_{\lambda},V_{\lambda}\otimes
U)$

also called intertwiners. Let $v_{\lambda}\in V_{\lambda}$ be the highest
weight vector and $U[0]$ denote the zero weight space. For an intertwiner
$\Phi$, set

$\langle\Phi\rangle=(v_{\lambda}^{*}\otimes 1_{U})\Phi(v_{\lambda})\in U[0]$

######  Proposition 3.4 (cf. 3.1 in [EL05]).

The map $\Phi\mapsto\langle\Phi\rangle$ is a (vector space) isomorphism
$I(V_{\lambda},U)\cong U[0]$.

To extract a Laurent polynomial from this, we take the weighted trace:

$\Phi\mapsto\varphi:=\mathrm{tr}_{V_{\lambda}}(\Phi(q^{2\mu}))$

Viewing this as a function of $\mu\in P$ and setting
$x_{i}=q^{2\langle\epsilon_{i},-\rangle}$, we obtain a Laurent polynomial
valued in $U[0]$. From Proposition 3.4, it follows that this trace map is
injective because the intertwiner is determined by the coefficient of
$x_{1}^{\lambda_{1}}\cdots x_{n}^{\lambda_{n}}$.

We would prefer to work instead with what can be called $U$-spherical
functions on quantum $GL_{n}$. This amounts to applying quantum coevaluation
maps:

$\displaystyle I(V_{\lambda},U)$ $\displaystyle\cong(V_{\lambda}^{*}\otimes
V_{\lambda}\otimes U)^{\mathcal{U}}$ $\displaystyle\Phi$
$\displaystyle\mapsto(1_{V_{\lambda}^{*}}\otimes\Phi)\circ(\mathrm{qcoev}_{V_{\lambda}})=:\tensor[_{\cup}]{\Phi}{}$

Since $q^{2\rho}$ and $q^{2\mu}$ are both grouplike, we have

$\varphi=(\mathrm{ev}_{V_{\lambda}}\otimes 1_{U})\left((q^{-2\mu-2\rho}\otimes
1_{V_{\lambda}\otimes U})\tensor[_{\cup}]\Phi{}\right)$ (3.19)

This has a clear graphical interpretation in terms of closing the loop between
the $V_{\lambda}$ strands, but we will refrain from drawing it as the
insertion of $q^{-2\mu}$ breaks $\mathcal{U}$-equivariance.

####  3.5.2. Macdonald polynomials revisited

Now, fix $k\in\mathbb{Z}_{\geq 0}$. We will consider the case

$U=U_{k}:=S_{q}^{nk}\mathbb{V}\otimes\mathbb{1}_{-k}$

$U_{k}[0]$ is one-dimensional, spanned by the vector whose tensorand in
$S_{q}^{nk}\mathbb{V}$ is the $q$-symmetrization of

$e_{1}^{\otimes{k}}\otimes e_{2}^{\otimes{k}}\otimes\cdots\otimes
e_{n}^{\otimes k}$

Fixing a vector $u_{0}\in U_{k}[0]$, we can identify $U[0]$ with
$\mathbb{C}(q)$. Set $\Phi^{k}_{\lambda}$ to be the intertwiner

$\Phi^{k}_{\lambda}:V_{\lambda+(k-1)\rho}\rightarrow
V_{\lambda+(k-1)\rho}\otimes U_{k-1}$

with $\langle\Phi^{k}_{\lambda}\rangle=u_{0}$ and let $\varphi^{k}_{\lambda}$
be its weighted trace
$\mathrm{tr}_{V_{\lambda}}(\Phi^{k}_{\lambda}(q^{2\mu}))$, which we can now
view as an element of $\mathbb{C}_{q}[\mathbf{x}_{n}^{\pm}]$. Note that
$\varphi_{\lambda}^{1}$ is just the usual character of $V_{\lambda}$.

######  Theorem 3.4 ([EK94]).

$\Phi^{k}_{\lambda}$ is nonzero and satisfies

$P_{\lambda}(q,q^{k})=\varphi^{k}_{\lambda}\big{/}\varphi^{k}_{0}$

####  3.5.3. Spherical DAHA generators

Recall the generators of $\mathcal{S}\ddot{\mathcal{H}}_{n}(q,q^{k})$ given by
Lemma 2.1 (cf. also (2.6) and (2.7)). We have two natural actions of
$\mathcal{O}^{\mathcal{U}}$ on the Macdonald polynomials. Let
$\mathrm{ch}_{V_{\pi}}$ denote the character of $V_{\pi}$. The first action is
through insertion via $\kappa$:

######  Theorem 3.5 ([EK94]).

We have

$\Phi^{k}_{\lambda+k\rho}(\kappa(\mathrm{qcoev}_{V^{*}_{\pi}}(1)))=\mathrm{ch}_{V_{\pi}}(q^{2(\lambda+k\rho)})\Phi_{\lambda+k\rho}^{k}$

This induces the same action on the weighted trace. In terms of spherical
functions, this corresponds to:

$(\kappa(\mathrm{qcoev}_{V_{\pi}}(1))\otimes 1_{V_{\lambda}\otimes
U_{k}})\tensor[_{\cup}]{\Phi}{}_{\lambda}^{k}=\mathrm{ch}_{V_{\pi}}(q^{2(\lambda+k\rho)})\tensor[_{\cup}]{\Phi}{}_{\lambda}^{k}$

Note that the discrepancy between $V_{\pi}$ and $V_{\pi}^{*}$ comes from the
change in orientation on the circle when bending the left leg of
$\tensor[_{\cup}]{\Phi}{}_{\lambda}^{k}$ down to obtain $\Phi_{\lambda}^{k}$.
In particular, we obtain the
$\rho\left(\mathbf{s}e_{r}(\mathbf{Y}_{n}^{-1})\mathbf{s}\right)$ when
$V_{\pi}=\wedge_{q}^{r}\mathbb{V}^{*}$ and
$\rho(\mathbf{s}\mathbf{Y}_{n}\mathbf{s})$ when $V_{\pi}=\mathbb{1}_{1}$.

For the second action, consider the multiplication map:

$m\otimes 1_{U_{k}}:\mathcal{O}\otimes\mathcal{O}\otimes
U_{k}\rightarrow\mathcal{O}\otimes U_{k}$

Since it is a $\mathcal{U}$-homomorphism, it restricts to a map

$\mathcal{O}^{\mathcal{U}}\otimes(\mathcal{O}\otimes
U_{k})^{\mathcal{U}}\rightarrow(\mathcal{O}\otimes
U_{k})^{\mathcal{U}}=\bigoplus_{\lambda\in P^{+}}(V_{\lambda}^{*}\otimes
V_{\lambda}\otimes U_{k})^{\mathcal{U}}$

We can depict $(m\otimes
1_{U_{k}})(\mathrm{qcoev}_{V_{\pi}}(1)\otimes\tensor[_{\cup}]{\Phi}{}_{\lambda}^{k})$
diagrammatically as:

Converting to the weighted trace as in (3.19) yields
$\mathrm{ch}_{V_{\pi}}(x_{1},\ldots x_{n})\varphi_{\lambda}^{k}$. Here, we
obtain $\rho\left(\mathbf{s}e_{r}(\mathbf{X}_{n})\mathbf{s}\right)$ when
$V_{\pi}=\wedge_{q}^{r}\mathbb{V}$ and
$\rho\left(\mathbf{s}\mathbf{X}_{n}^{-1}\mathbf{s}\right)$ when
$V_{\pi}=\mathbb{1}_{-1}$.

##  4\. Quantum differential operators

###  4.1. Varagnolo-Vasserot double

Out of our quantum ring of functions $\mathcal{O}$, we would like to define a
notion of quantum differential operators with very specific equivariance
properties. Let us catalogue the $\mathcal{U}$-actions on $\mathcal{O}$:

  1. (1)

Left coregular action: $g\triangleright(v^{*}\otimes v)=gv^{*}\otimes v$ (this
is an action of the co-opposite $\mathcal{U}_{c}$);

  2. (2)

Right coregular action: $g\triangleleft(v^{*}\otimes v)=v^{*}\otimes gv$;

  3. (3)

Coadjoint action: $g\bowtie(v^{*}\otimes v)=g_{(1)}v^{*}\otimes g_{(2)}v$.

We have been mainly concerned with the coadjoint action, for which the Hopf
algebra structure maps are homomorphisms. Our goal is to construct a smash
product between $\mathcal{O}$ and $\kappa(\mathcal{O})$ utilizing the left
coregular action, but we would like all structure maps to be equivariant with
respect to the coadjoint action on $\mathcal{O}$ and the adjoint action on
$\kappa(\mathcal{O})$.

####  4.1.1. Drinfeld twist

The left and right coregular actions give an action of
$\mathcal{U}_{c}\otimes\mathcal{U}$ on $\mathcal{O}$. However, the braidings
in (3.15) prevent the product from being a
$\mathcal{U}_{c}\otimes\mathcal{U}$-homomorphism. This can be fixed by
altering the coproduct of $\mathcal{U}_{c}\otimes\mathcal{U}$ using Drinfeld’s
twisting procedure [Dd89b]. First, let $\Delta_{2}$ denote the coproduct of
$\mathcal{U}_{c}\otimes\mathcal{U}$:

$\Delta_{2}(g\otimes h)=g_{(2)}\otimes h_{(1)}\otimes g_{(1)}\otimes h_{(2)}$

Let $\widetilde{\mathcal{U}}^{2}$ denote the algebra
$\mathcal{U}_{c}\otimes\mathcal{U}$ endowed with coproduct

$\displaystyle\widetilde{\Delta}_{2}(g\otimes h)$
$\displaystyle:=(\mathcal{R}_{13}\mathcal{R}_{23})^{-1}\Delta_{2}(g\otimes
h)(\mathcal{R}_{13}\mathcal{R}_{23})$
$\displaystyle=\mathcal{R}_{23}^{-1}(g_{(1)}\otimes h_{(1)}\otimes
g_{(2)}\otimes h_{(2)})\mathcal{R}_{23}$

and antipode

$\widetilde{S}_{2}(g\otimes h)=\mathcal{R}_{21}\big{(}S(g)\otimes
S(h)\big{)}\mathcal{R}_{21}^{-1}$

With these structures, $\widetilde{\mathcal{U}}^{2}$ is a Hopf algebra in a
suitably completed sense. We will use an embellished Sweedler notation for
$\widetilde{\Delta}_{2}$:

$\widetilde{\Delta}_{2}(g\otimes
h)=\tilde{g}_{(1)}\otimes\tilde{h}_{(1)}\otimes\tilde{g}_{(2)}\otimes\tilde{h}_{(2)}$

######  Proposition 4.1.

The multiplication map $m$ of $\mathcal{O}$ is a
$\widetilde{\mathcal{U}}^{2}$-homomorphism.

###### Proof.

This follows from the calculation:

$\displaystyle m\bigg{(}\widetilde{\Delta}_{2}(g\otimes h)(v^{*}\otimes
v\otimes w^{*}\otimes w)\bigg{)}$ $\displaystyle=\beta_{V^{*}\otimes
V,W^{*}}\bigg{(}\mathcal{R}_{23}^{-1}(g_{(1)}v^{*}\otimes
h_{(1)}\tensor[_{s}]{r}{}v\otimes g_{(2)}\tensor[]{r}{{}_{s}}w^{*}\otimes
h_{(2)}w)\bigg{)}$
$\displaystyle=\tensor[]{r}{{}_{t}}g_{(2)}\tensor[]{r}{{}_{s}}w^{*}\otimes\tensor[_{t}]{r}{}g_{(1)}v^{*}\otimes
h_{(1)}\tensor[_{s}]{r}{}v\otimes h_{(2)}w$
$\displaystyle=g_{(1)}\tensor[]{r}{{}_{t}}\tensor[]{r}{{}_{s}}w^{*}\otimes
g_{(2)}\tensor[_{t}]{r}{}v^{*}\otimes h_{(1)}\tensor[_{s}]{r}{}v\otimes
h_{(2)}w$ $\displaystyle=(g\otimes h)m(v^{*}\otimes v\otimes w^{*}\otimes
w)\qed$

####  4.1.2. Three embeddings

Corresponding to the three actions of $\mathcal{U}$ on $\mathcal{O}$, there
are three embeddings of $\mathcal{U}$ into $\widetilde{\mathcal{U}}^{2}$.
First, let us consider the subalgebras $\mathcal{U}\otimes 1$ and
$1\otimes\mathcal{U}$. It is clear that the restrictions of the
$\widetilde{\mathcal{U}}^{2}$-action on $\mathcal{O}$ to these two subalgebras
respectively yield the left and right coregular actions. We have that

$\displaystyle\widetilde{\Delta}_{2}(g\otimes 1)$
$\displaystyle=g_{(1)}\otimes\tensor[_{s}]{r}{}\tensor[_{t}]{r}{}\otimes
S(\tensor[]{r}{{}_{s}})g_{(2)}\tensor[]{r}{{}_{t}}\otimes 1$ (4.1)
$\displaystyle=g_{(1)}\otimes\tensor[_{s}]{r}{}\otimes\mathrm{ad}_{S(\tensor[]{r}{{}_{s}})}(g_{(2)})\otimes
1$

Thus, $\mathcal{U}\otimes 1$ is a left coideal subalgebra, and so by Theorem
3.3 and Proposition 3.3, $\kappa(\mathcal{O})\otimes 1$ is a left coideal
subalgebra as well.

Next, view the coproduct as a map
$\Delta:\mathcal{U}\rightarrow\widetilde{\mathcal{U}}^{2}$. Observe that by
(3.1),

$\displaystyle(\widetilde{\Delta}_{2}\circ\Delta)(g)$
$\displaystyle=g_{(1)}\otimes g_{(2)}\otimes g_{(3)}\otimes g_{(4)}$
$\displaystyle(1\otimes\widetilde{S})(\widetilde{\Delta}_{2}\circ\Delta)(g)$
$\displaystyle=g_{(1)}\otimes g_{(2)}\otimes S(g_{(4)})\otimes S(g_{(3)})$
(4.2)

From these equations, we can see that $\Delta$ is in fact a Hopf algebra
morphism. Moreover, from (4.2), we can see that the adjoint action of
$\Delta(\mathcal{U})$ on $\widetilde{\mathcal{U}}^{2}$ preserves
$\mathcal{U}\otimes 1$, on which it acts by the usual adjoint action on
$\mathcal{U}$.

####  4.1.3. Smash product

Now consider the smash product
$\mathcal{O}\rtimes\widetilde{\mathcal{U}}^{2}$. This is the tensor product
$\mathcal{O}\otimes\widetilde{\mathcal{U}}^{2}$ subject to the relation that
for $v^{*}\otimes v\in\mathcal{O}$ and $g\otimes
h\in\widetilde{\mathcal{U}}^{2}$,

$(g\otimes h)(v^{*}\otimes
v)=(\tilde{g}_{(1)}v^{*}\otimes\tilde{h}_{(1)}v)(\tilde{g}_{(2)}\otimes\tilde{h}_{(2)})$
(4.3)

We will abuse notation and use $\mathcal{O}$ and $\widetilde{\mathcal{U}}^{2}$
to denote $\mathcal{O}\otimes 1\otimes 1$ and
$1\otimes\widetilde{\mathcal{U}}^{2}$, respectively. Since
$\kappa(\mathcal{O})\otimes 1\subset\widetilde{\mathcal{U}}^{2}$ is a left
coideal, the subspace $\mathcal{O}\rtimes(\kappa(\mathcal{O})\otimes 1)$ is in
fact a subalgebra. We denote by $\partial_{\triangleright}$ the embedding
$\mathcal{O}\rightarrow 1\otimes\kappa(\mathcal{O})\otimes
1\subset\mathcal{O}\rtimes\widetilde{\mathcal{U}}^{2}$.

######  Definition 4.1 ([VV10]).

The Varagnolo-Vasserot algebra of quantum differential operators $\mathcal{D}$
is the subalgebra $\mathcal{O}\rtimes\partial_{\triangleright}(\mathcal{O})$
of the smash product $\mathcal{O}\rtimes\widetilde{\mathcal{U}}^{2}$.

The Varagnolo-Vasserot algebra does indeed satisfy our desired equivariance.
Letting $\widetilde{\mathcal{U}}^{2}$ act on itself via the adjoint action,
$\mathcal{O}\rtimes\widetilde{\mathcal{U}}^{2}$ is a
$\widetilde{\mathcal{U}}^{2}$-module-algebra. Restricting this action to
$\Delta(\mathcal{U})\subset\widetilde{\mathcal{U}}^{2}$, we obtain an action
of $\mathcal{U}$ on $\mathcal{D}$ that gives the coadjoint action on
$\mathcal{O}$ and the adjoint action on
$\partial_{\triangleright}(\mathcal{O})$. We will also use $\bowtie$ to denote
this $\mathcal{U}$ action on $\mathcal{D}$.

Finally, we note that by (4.1), commuting
$\partial_{\triangleright}(\mathcal{O})$ past $\mathcal{O}$ in $\mathcal{D}$
does not cleanly incorporate the left coregular action because the right
tensorand of $\mathcal{O}$ is also affected. However, the discrepancy has a
diagrammatic interpretation that is cleaner than the symbolic formula gotten
by combining (3.18) and (4.1):

$\partial_{\triangleright}(v^{*}\otimes v)(w^{*}\otimes
w)=\big{(}\partial_{\triangleright}(v^{*}\otimes
v_{i})\tensor[]{r}{{}_{s}}\tensor[]{r}{{}_{t}}w^{*}\otimes\tensor[_{u}]{r}{}\tensor[_{v}]{r}{}w\big{)}\big{(}S(\tensor[]{r}{{}_{u}})\partial_{\triangleright}(\tensor[_{s}]{r}{}v^{i}\otimes\tensor[_{t}]{r}{}v)\tensor[]{r}{{}_{v}}\big{)}$
(4.4)

####  4.1.4. Basic representation

The smash product $\mathcal{O}\rtimes\widetilde{\mathcal{U}}^{2}$ has an
action on $\mathcal{O}$ where $\mathcal{O}$ acts by multiplication and
$\widetilde{\mathcal{U}}^{2}$ acts via the left and and right coregular
actions. This action is $\widetilde{\mathcal{U}}^{2}$-equivariant. Moreover,
from the definition of the smash product (4.3), we have that $\mathcal{O}$ is
an induced representation:

$\mathcal{O}\cong\mathcal{O}\rtimes\widetilde{\mathcal{U}}^{2}\bigg{/}\mathcal{O}\rtimes\widetilde{\mathcal{U}}^{2}\left(\widetilde{\mathcal{U}}^{2}-\epsilon(\widetilde{\mathcal{U}}^{2})\right)$
(4.5)

We call the representation $\mathcal{O}$ and its restriction to $\mathcal{D}$
the basic representation. $\mathcal{O}$ is then a $\mathcal{U}$-equivariant
$\mathcal{D}$-module (using the $\bowtie$ action). Note that even though the
smash product relations do not cleanly incorporate the left coregular action,
$\partial_{\triangleright}(\mathcal{O})$ does indeed act on $\mathcal{O}$ via
the left coregular action. This follows from (4.1) and (4.5).

###  4.2. Monodromy matrices

Recall the generating matrix $M$ for $\mathcal{O}$ given in 3.4.2. Let
$A=(A^{i}_{j})$ denote a copy of $M$ given by $\mathcal{O}\subset\mathcal{D}$
and let $B=(B^{i}_{j})$ denote another copy given by
$\partial_{\triangleright}(M)$. We set

$\displaystyle A_{j}^{i}$ $\displaystyle=E_{j}^{i}\otimes a_{j}^{i}$
$\displaystyle(A^{-1})^{i}_{j}$
$\displaystyle=E_{j}^{i}\otimes\iota(a^{i}_{j})$ $\displaystyle B_{j}^{i}$
$\displaystyle=E_{j}^{i}\otimes b_{j}^{i}$ $\displaystyle(B^{-1})_{j}^{i}$
$\displaystyle=E_{j}^{i}\otimes\iota(b^{i}_{j}):=E_{j}^{i}\otimes\partial_{\triangleright}(\iota(m^{i}_{j}))$

####  4.2.1. Determinant bigrading

We set

$\displaystyle\textstyle\det_{q}(A)$
$\displaystyle:=\mathrm{qcoev}_{\mathbb{1}_{1}}(1)\otimes
1\in\mathcal{O}\otimes 1\subset\mathcal{D}$
$\displaystyle\textstyle\det_{q}(B)$
$\displaystyle:=\partial_{\triangleright}\mathrm{qcoev}_{\mathbb{1}_{1}}(1)\in
1\otimes\kappa(\mathcal{O})\subset\mathcal{D}$

As in 3.4.2, these elements can be written in terms of the entries of $A$ and
$B$, respectively. Moreover, $\det_{q}(A)$ commutes with $\mathcal{O}\otimes
1$ and $\det_{q}(B)$ commutes with $1\otimes\kappa(\mathcal{O})$. Their
commutation relations with elements from their respective “opposite” tensor
factors are also nice.

First note that, from the factorization (3.6) and the formulas (3.9) for the
determinant representation, we have:

(4.6)

Using these local relations on (4.4), one can see that $\det_{q}(B)$
satisfies:

Similarly, $\det_{q}(A)$ satisfies:

Thus, we can define an internal bigrading on $\mathcal{D}$: $x\in\mathcal{D}$
is homogeneous of degree $(a,b)$ if

$\displaystyle\textstyle\det_{q}(B)x$
$\displaystyle=q^{2a}x\textstyle\det_{q}(B)$
$\displaystyle\textstyle\det_{q}(A)x$
$\displaystyle=q^{-2b}x\textstyle\det_{q}(A)$

The entries of $A$ and $B$ have degrees $(1,0)$ and $(0,1)$, respectively.

####  4.2.2. Double R-matrix presentation

The following is Proposition 1.8.3(b) of [VV10]:

######  Proposition 4.2.

Let $\mathcal{D}_{+}$ be the algebra with generators given by the entries of
$A$ and $B$ and relations

$\displaystyle\begin{split}R_{21}A_{13}R_{12}A_{23}&=A_{23}R_{21}A_{13}R_{12}\\\
R_{21}B_{13}R_{12}B_{23}&=B_{23}R_{21}B_{13}R_{12}\\\
R_{21}B_{13}R_{12}A_{23}&=A_{23}R_{12}B_{13}R_{21}^{-1}\end{split}$ (4.7)

The elements $\\{\det_{q}(A),\det_{q}(B)\\}$ generate an Ore set in
$\mathcal{D}_{+}$ and $\mathcal{D}$ is isomorphic to its localization.

The novel third relation is a rewriting of (4.4), cf. A.5 of [VV10]. This
algebra was defined prior to [VV10] by Alekseev and Schomerus [AS96] as an
intermediate step to constructing their quantized character variety for the
once-punctured torus. There, the $A$\- and $B$-matrices quantize monodromy
matrices along the $a$\- and $b$-cycles, respectively.

###  4.3. Quantum Weyl algebra

A reference for this section is [GZ95]. The quantum Weyl algebra of rank $n$,
denoted $\mathcal{W}$, is the $\mathbb{C}(q)$-algebra with generators

$\left\\{\xi_{i},\partial_{i}\,|\,1\leq i\leq n\right\\}$

and relations

$\displaystyle\begin{split}\xi_{i}\xi_{j}&=q\xi_{j}\xi_{i}\hbox{ for }i>j\\\
\partial_{i}\partial_{j}&=q^{-1}\partial_{j}\partial_{i}\hbox{ for }i>j\\\
\partial_{i}\xi_{j}&=q\xi_{j}\partial_{i}\hbox{ for }i\not=j\\\
\partial_{i}\xi_{i}&=1+q^{2}\xi_{i}\partial_{i}+(q^{2}-1)\sum_{j<i}\xi_{j}\partial_{j}\end{split}$
(4.8)

If we set $\deg\xi_{i}=1$ and $\deg\partial_{i}=-1$, then the relations (4.8)
respect the grading. We denote by $\mathcal{W}_{0}$ the subalgebra of degree 0
elements.

######  Remark 4.1.

In [Jor14], there is a parameter $t$ that the author then sets to $t=1$ (cf.
Remark 3.11 of loc. cit.). Here, we instead set $t=q^{-1}$. This affects the
formula for the moment map later in 4.4.2.

####  4.3.1. Equivariance

$\mathcal{W}$ has an RTT-type presentation (cf. Example 4.9 of [Jor14]): if we
define the vectors

$\displaystyle\vec{\xi}$
$\displaystyle:=\sum_{i=1}^{n}e^{i}\otimes\xi_{i}=\begin{pmatrix}\xi_{1}&\xi_{2}&\cdots&\xi_{n}\end{pmatrix}$
$\displaystyle\vec{\partial}$
$\displaystyle:=\sum_{i=1}^{n}e_{i}\otimes\partial_{i}=\begin{pmatrix}\partial_{1}\\\
\partial_{2}\\\ \vdots\\\ \partial_{n}\end{pmatrix}$

in $\mathbb{V}\otimes\mathcal{W}$ then relations (4.8) can be rewritten as

$\displaystyle q\vec{\xi}_{13}\vec{\xi}_{23}$
$\displaystyle=\vec{\xi}_{23}\vec{\xi}_{13}R$ $\displaystyle
q\vec{\partial}_{13}\vec{\partial}_{23}$
$\displaystyle=R\vec{\partial}_{23}\vec{\partial}_{13}$ $\displaystyle
q^{-1}\vec{\partial}_{23}\vec{\xi}_{13}$
$\displaystyle=\vec{\xi}_{13}R\vec{\partial}_{23}+q^{-1}\sum_{i=1}^{n}e^{i}\otimes
e_{i}\otimes 1$

This algebra can be written as a quotient of the tensor algebra
$\mathcal{T}(\mathbb{V}\oplus\mathbb{V}^{*})$ by images of the braidings and
evaluations. Here, $\partial_{i}$ corresponds to $e^{i}\in\mathbb{V}^{*}$ and
$\xi_{i}$ corresponds to $e_{i}\in\mathbb{V}$. Consequently, we have:

######  Proposition 4.3.

$\mathcal{W}$ is a $\mathcal{U}$-module-algebra.

We denote the $\mathcal{U}$ action on $\mathcal{W}$ by $\bullet$. Note that
the degree is given by the action of $q^{\omega_{n}}$. Since $q^{\omega_{n}}$
is central in $\mathcal{U}$, it follows that $\mathcal{W}_{0}$ is a
$\mathcal{U}$-submodule.

######  Remark 4.2.

We use a row vector for $\vec{\xi}$ and column vector for $\vec{\partial}$
despite the fact that the $\xi$-variables are associated to $\mathbb{V}$ while
the $\partial$-variables are associated to $\mathbb{V}^{*}$. This is
consistent with Remark 3.2, where we point out that Jordan always uses the
transpose matrix in [Jor14].

####  4.3.2. Functional representation

Let $\mathcal{W}_{\xi}$ be the subalgebra generated by
$\left\\{\xi_{i}\right\\}$. As a $\mathcal{U}$-module, the subalgebra
$\mathcal{W}_{\xi}$ is isomorphic to the quantum symmetric algebra of
$\mathbb{V}$.

$S_{q}\mathbb{V}:=\bigoplus_{m=0}^{\infty}S_{q}^{m}\mathbb{V}$

The ordered monomials

$\xi_{1}^{k_{1}}\cdots\xi_{n-1}^{k_{n-1}}\xi_{n}^{k_{n}}$

form a basis of $S_{q}\mathbb{V}$ (cf. Theorem 5.1 below). We can extend the
natural multiplication action of $\mathcal{W}_{x}$ to one of the entirety of
$\mathcal{W}$ by setting

$\partial_{i}(\xi_{1}^{k_{1}}\xi_{2}^{k_{2}}\cdots\xi_{i}^{k_{i}}\cdots\xi_{n}^{k_{n}})=(q\xi_{1})^{k_{1}}(q\xi_{2})^{k_{2}}\cdots[k_{i}]_{q^{2}}\xi_{i}^{k_{i}-1}\xi_{i+1}^{k_{i+1}}\cdots\xi_{n}^{k_{n}}$

This action is merely the one induced by quotienting
$\mathcal{T}(\mathbb{V}\oplus\mathbb{V}^{*})$ by the left ideal generated by
$\mathbb{V}^{*}$. As such, this action is $\mathcal{U}$-equivariant. The
action of $\mathcal{W}_{0}$ on $S_{q}\mathbb{V}$ preserves each piece
$S_{q}^{m}\mathbb{V}$.

####  4.3.3. Difference operators

It will be useful to interpret the functional representation in terms of
difference operators in commuting variables. Let
$\mathbb{C}(q)[\mathbf{z}_{n}]:=\mathbb{C}(q)[z_{1},\ldots,z_{n}]$ and let
$T_{q,z_{i}}$ denote the $q$-shift operator:

$T_{q,z_{i}}z_{j}=q^{\delta_{i,j}}z_{j}$

For an integer vector
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in\mathbb{Z}^{n}$, we define

$\displaystyle z^{\lambda}$ $\displaystyle:=z_{1}^{\lambda_{1}}\cdots
z_{n}^{\lambda_{n}},$ $\displaystyle T_{\lambda}$
$\displaystyle:=T_{q,z_{1}}^{\lambda_{1}}\cdots T_{q,z_{n}}^{\lambda_{n}}$

Consider the following rings of difference operators:

$\displaystyle\mathbb{D}_{q}(\mathbf{z}_{n})$
$\displaystyle=\left\\{\sum_{\lambda,\mu\in\mathbb{Z}^{n}}a_{\lambda,\mu}z^{\lambda}T_{\mu}\,\middle|\,\begin{array}[]{l}a_{\lambda,\mu}\in\mathbb{C}(q)\\\
\hbox{and only finitely many }a_{\lambda,\mu}\not=0\end{array}\right\\}$
$\displaystyle\mathbb{D}_{q}^{+}(\mathbf{z}_{n})$
$\displaystyle=\left\\{D\in\mathbb{D}_{q}(\mathbf{z}_{n})\,\middle|\,\begin{array}[]{l}Df\in\mathbb{C}(q)[\mathbf{z}_{n}]\\\
\hbox{for all }f\in\mathbb{C}(q)[\mathbf{z}_{n}]\end{array}\right\\}$

######  Proposition 4.4.

$\mathbb{C}(q)[\mathbf{z}_{n}]$ is a faithful representation of
$\mathbb{D}_{q}^{+}(\mathbf{z}_{n})$.

The map

$\xi_{1}^{k_{1}}\cdots\xi_{n}^{k_{n}}\mapsto z_{1}^{k_{1}}\cdots
z_{n}^{k_{n}}$

induces a vector space isomorphism
$S_{q}\mathbb{V}\cong\mathbb{C}(q)[\mathbf{z}_{n}]$. Carrying over the actions
of $\mathcal{U}$ and $\mathcal{W}$, we obtain homomorphisms of both algebras
into $\mathbb{D}_{q}^{+}(\mathbf{z}_{n})$, both of which we denote by
$\mathfrak{qdiff}$:

$\displaystyle\mathfrak{qdiff}(q^{\epsilon_{i}})$ $\displaystyle=T_{q,z_{i}}$
$\displaystyle\mathfrak{qdiff}(\xi_{i})$ $\displaystyle=z_{i}T_{q,z_{1}}\cdots
T_{q,z_{i-1}}$ (4.9) $\displaystyle\mathfrak{qdiff}(E_{i})$
$\displaystyle=\frac{z_{i}}{z_{i+1}}\left(\frac{T_{q,z_{i+1}}-T_{q,z_{i+1}}^{-1}}{q-q^{-1}}\right)$
$\displaystyle\mathfrak{qdiff}(\partial_{i})$
$\displaystyle=z_{i}^{-1}T_{q,z_{1}}\cdots
T_{q,z_{i-1}}\left(\frac{T_{q,z_{i}}^{2}-1}{q^{2}-1}\right)$
$\displaystyle\mathfrak{qdiff}(F_{i})$
$\displaystyle=\frac{z_{i+1}}{z_{i}}\left(\frac{T_{q,z_{i}}-T_{q,z_{i}}^{-1}}{q-q^{-1}}\right)$

Since $\mathbb{C}(q)[\mathbf{z}_{n}]\cong S_{q}\mathbb{V}$ is a faithful
$\mathbb{D}^{+}_{q}(\mathbf{z}_{n})$-module, the fact that $S_{q}\mathbb{V}$
is a $\mathcal{U}$-equivariant $\mathcal{W}$-module implies that
$\mathfrak{qdiff}$ pieces together into an algebra homomorphism out of the
smash product:

$\mathfrak{qdiff}:\mathcal{W}\rtimes\mathcal{U}\rightarrow\mathbb{D}^{+}_{q}(\mathbf{z}_{n})$

###  4.4. Quantum Hamiltonian reduction

To perform quantum Hamiltonian reduction, we will need a notion of quantum
moment maps in our Hopf-algebraic setting. Let $H$ be a Hopf algebra action on
an algebra $A$ such that $A$ is an $H$-module-algebra. If we denote this
action by

$(-)\blacktriangleright(-):H\otimes A\rightarrow A$

then a quantum moment map (in the sense of [VV10]) for the action is an
algebra homomorphism $\mu:H\rightarrow A$ such that for $h\in H$ and $a\in A$,

$\mu(h)a=(h_{(1)}\blacktriangleright a)\mu(h_{(2)})$ (4.10)

More generally, we can define quantum moment maps for the action of a left
coideal subalgebra $H^{\prime}\subset H$, In our case, we will be working with
$H=\mathcal{U}$ and $H^{\prime}=\kappa(\mathcal{O})$.

####  4.4.1. Moment map for $\mathcal{D}$

The following is Proposition 1.8.3(a) of [VV10] and Proposition 7.21 of
[Jor14]:

######  Proposition 4.5.

The map $\mu_{\mathcal{D}}:\kappa(\mathcal{O})\rightarrow\mathcal{D}$ given by

$(1\otimes\mu_{\mathcal{D}}\circ\kappa)(M)=BA^{-1}B^{-1}A$ (4.11)

is a quantum moment map for the $\bowtie$-action restricted to the left
coideal subalgebra $\kappa(\mathcal{O})\subset\mathcal{U}$. Moreover, it is
$\mathcal{U}$-equivariant.

We emphasize that we are viewing $\kappa(\mathcal{O})$ as a left coideal
subalgebra of $\mathcal{U}$. Thus, in (4.10), we are are taking the coproduct
$\Delta$, not $\nabla$.

######  Proposition 4.6.

For $f\in\mathcal{O}$ viewed as an element of the basic representation, we
have

$\mu_{\mathcal{D}}(x)f=\kappa(x)\bowtie f$

###### Proof.

This was shown in the proof of Proposition 1.8.2(c) of [VV10], but we give a
diagrammatic proof. It suffices to consider entries of the generating matrix
$M$. Using (3.16) and (4.4), we calculate the entries of $BA^{-1}B^{-1}A$
images of the following a morphism, working right to left:

(4.12)

Acting on $V^{*}\otimes V\subset\mathcal{O}$, we obtain

upon applying the coend relation (3.13). ∎

####  4.4.2. Moment map for $\mathcal{W}$

The following was proved by Jordan [Jor14]:

######  Proposition 4.7.

The action of the left coideal subalgebra
$\kappa(\mathfrak{R})\subset\mathcal{U}$ on $\mathcal{W}$ has a quantum moment
map

$\mu_{\mathcal{W}}(m^{i}_{j})=\delta_{i,j}+(1-q^{-2})\partial_{i}\xi_{j}$

This map is $\mathcal{U}$-equivariant. The powers of
$\mu_{\mathcal{W}}(\det_{q}(M))$ form an Ore set. Let $\mathcal{W}^{\circ}$
denote the localization at those powers. Then $\mu_{\mathcal{W}}$ extends to a
quantum moment map for $\kappa(\mathcal{O})$ into $\mathcal{W}^{\circ}$.

Note that the image of $\mu_{\mathcal{W}}$ lies in the degree zero part
$\mathcal{W}^{\circ}_{0}$.

######  Proposition 4.8.

We have for $f\in S_{q}\mathbb{V}$,

$\mu_{\mathcal{W}}(m^{i}_{j})f=\kappa(m^{i}_{j})\bullet f$

###### Proof.

We view $S_{q}\mathbb{V}$ as the quotient of $\mathcal{W}$ by the left ideal
$I_{\partial}$ generated by $\left\\{\partial_{i}\right\\}$. By Proposition
3.3, we have

$\mu_{\mathcal{W}}(m^{i}_{j})f=\sum_{k}\left(\kappa(m^{i}_{k})\tensor[]{r}{{}_{s}}\bullet
f\right)\mu_{\mathcal{W}}\big{(}\mathrm{ad}_{\tensor[_{s}]{r}{}}(m^{k}_{j})\big{)}$

The equivariance of $\mu_{\mathcal{W}}$ implies

$\mu_{\mathcal{W}}\big{(}\mathrm{ad}_{\tensor[_{s}]{r}{}}(m^{k}_{j})\big{)}=\tensor[_{s}]{r}{}\bullet\mu_{\mathcal{W}}(m^{k}_{j})$

On the other hand, using relations for $\mathcal{W}$, we have that after
quotienting by $I_{\partial}$,

$\mu_{\mathcal{W}}(m^{k}_{j})+I_{\partial}=\delta_{k,j}+I_{\partial}$

The proposition follows once we note that $I_{\partial}$ is closed under the
$\mathcal{U}$-action. ∎

######  Corollary 4.1.

Recall the map $\mathfrak{qdiff}$ (4.9). We have:

$\mathfrak{qdiff}\circ\kappa=\mathfrak{qdiff}\circ\mu_{\mathcal{W}}$

######  Corollary 4.2.

The functional representation $S_{q}\mathbb{V}$ is preserved under the action
of the localization $\mathcal{W}^{\circ}$.

####  4.4.3. Reduction

Finally, we consider the tensor product algebra

$\mathcal{M}:=\mathcal{D}\otimes\mathcal{W}^{\circ}$

in the braided monoidal category of locally-finite $\mathcal{U}$-modules.
Thus, the product involves $\mathcal{R}$: for $a_{1},a_{2}\in\mathcal{D}$ and
$b_{1},b_{2}\in\mathcal{W}^{\circ}$,

$(a_{1}\otimes b_{1})(a_{2}\otimes b_{2})=a_{1}(\tensor[]{r}{{}_{s}}\bowtie
a_{2})\otimes(\tensor[_{s}]{r}{}\bullet b_{1})b_{2}$

With this braiding, $\mathcal{M}$ is a $\mathcal{U}$-module algebra and thus
$\mathcal{M}^{\mathcal{U}}$ is an algebra.

Next, we collate the moment maps into one for $\mathcal{M}$. We define

$\mu_{\mathcal{M}}:=(\mu_{\mathcal{D}}\otimes\mu_{\mathcal{W}})\circ(\kappa\otimes\kappa)\circ\nabla:\mathcal{O}\rightarrow\mathcal{M}$

It is a $\mathcal{U}$-equivariant algebra homomorphism (each of its components
are). The following is Proposition 3.10 of [GJS19], which features a nice
diagrammatic proof:

######  Proposition 4.9.

The map $\mu_{\mathcal{M}}:\mathcal{O}\rightarrow\mathcal{M}$ is a quantum
moment map.

Recall the determinant character $\chi^{k}$ defined in (3.9). We will abuse
notation and use $\chi^{k}$ to also denote the induced character
$\chi^{k}\circ\kappa$ of $\mathcal{O}$. In terms of the generating matrix $M$
of $\mathcal{O}$, we can use the factorization of $\mathcal{R}$ (3.6) to
deduce

$(1\otimes\chi^{k})(M)=q^{-2k}I$

######  Definition 4.2.

Let $\mathcal{I}_{k}\subset\mathcal{M}$ be the following left ideal:

$\mathcal{I}_{k}:=\mathcal{M}\left((1\otimes\mu_{\mathcal{M}})(M)-q^{-2(k-1)}I\right)$

The quantized multiplicative quiver variety is defined to be
$\mathcal{A}_{k}:=\big{(}\mathcal{M}\big{/}\mathcal{I}_{k}\big{)}^{\mathcal{U}}$.

######  Proposition 4.10 ([VV10, GJS19]).

$\mathcal{A}_{k}$ is an algebra.

####  4.4.4. Radial parts map

Denote by

$\left(\mathcal{O}\otimes S_{q}\mathbb{V}\right)^{\chi^{k}}$

the subspace where $\mathcal{U}$ acts by the character $\chi^{k}$. From the
Peter-Weyl decomposition for $\mathcal{O}$ (3.14), we can see that
$q^{\omega_{n}}$ acts trivally on $\mathcal{O}$. Thus,

$\left(\mathcal{O}\otimes
S_{q}\mathbb{V}\right)^{\chi_{k}}=\left(\mathcal{O}\otimes
S_{q}^{nk}\mathbb{V}\right)^{\chi^{k}}$

We can further identify

$\left(\mathcal{O}\otimes
S_{q}^{nk}\mathbb{V}\right)^{\chi^{k}}\cong(\mathcal{O}\otimes
U_{k})^{\mathcal{U}}=\bigoplus_{\lambda\in P^{+}}\left(V_{\lambda}^{*}\otimes
V_{\lambda}\otimes U_{k}\right)^{\mathcal{U}}$

By Theorem 3.4, we can identify the last space with $\Lambda_{n}^{\pm}(q)$ by
taking the weighted trace.

$\mathcal{M}$ acts on the tensor product representation $\mathcal{O}\otimes
S_{q}\mathbb{V}$, where this tensor product representation is taken in the
braided monoidal category of locally-finite $\mathcal{U}$-modules. Therefore,
the action involves the braiding between $\mathcal{W}^{\circ}$ and
$\mathcal{O}$ and the action map is $\mathcal{U}$-equivariant. This action on
$\mathcal{O}\otimes S_{q}\mathbb{V}$ then restricts to an action of
$\mathcal{M}^{\mathcal{U}}$ on $\left(\mathcal{O}\otimes
S_{q}\mathbb{V}\right)^{\chi^{k}}$.

######  Proposition 4.11.

For $k\geq 0$, the action of $\mathcal{M}^{\mathcal{U}}$ on
$\left(\mathcal{O}\otimes S_{q}\mathbb{V}\right)^{\chi^{k}}$ factors to an
action of $\mathcal{A}_{k+1}$.

###### Proof.

We first show that $\mu_{\mathcal{M}}(m^{i}_{j})$ acts on $\mathcal{O}\otimes
S_{q}\mathbb{V}$ as $\kappa(m^{i}_{j})$ (i.e. via the $\mathcal{U}$-action).
Let $\mathfrak{a}_{\mathcal{M}}$ be the action map of $\mathcal{M}$ on
$\mathcal{O}\otimes S_{q}\mathbb{V}$. Using Propositions 4.6 and 4.8, we have:

It follows that the restricted action map

$\mathfrak{a}_{\mathcal{M}}:\mathcal{M}\otimes\left(\mathcal{O}\otimes
S_{q}\mathbb{V}\right)^{\chi^{k}}\rightarrow\mathcal{O}\otimes
S_{q}\mathbb{V}$

factors through $\mathcal{I}_{k+1}$. The result follows from taking invariants
of $\mathcal{M}\big{/}\mathcal{I}_{k+1}$. ∎

######  Definition 4.3.

We denote the representation map by
$\mathfrak{rad}_{k}:\mathcal{A}_{k}\rightarrow\mathrm{End}\left(\Lambda_{n}^{\pm}(q)\right)$
and called it the quantum radial parts map.

######  Proposition 4.12.

The image of $\mathfrak{rad}_{k}$ contains the image of
$\mathcal{S}\ddot{\mathcal{H}}_{n}(q,q^{k})$ under $\rho$:

$\mathfrak{rad}_{k}\left(\mathcal{A}_{k}\right)\supset\rho\left(\mathcal{S}\ddot{\mathcal{H}}_{n}(q,q^{k})\right)$

Moreover, this inclusion respects the bigradings.

###### Proof.

Recall the generators of $\mathcal{S}\ddot{\mathcal{H}}_{n}(q,q^{k})$ from
Lemma 2.1. Our analysis in 3.5.3 shows that

$\displaystyle\mathfrak{rad}_{k}\left(\mathrm{qcoev}_{\wedge_{q}^{r}\mathbb{V}}(1)\right)$
$\displaystyle=\rho\left(\mathbf{s}e_{r}(\mathbf{X}_{n})\mathbf{s}\right)$
(4.13)
$\displaystyle\mathfrak{rad}_{k}\left(\partial_{\triangleright}\mathrm{qcoev}_{\wedge_{q}^{r}\mathbb{V}^{*}}(1)\right)$
$\displaystyle=\rho\left(\mathbf{s}e_{r}(\mathbf{Y}^{-1}_{n})\mathbf{s}\right)$
$\displaystyle\mathfrak{rad}_{k}\left(\textstyle\det_{q}(A)^{-1}\right)$
$\displaystyle=\rho\left(\mathbf{s}\mathbf{X}^{-1}_{n}\mathbf{s}\right)$
$\displaystyle\mathfrak{rad}_{k}\left(\textstyle\det_{q}(B)\right)$
$\displaystyle=\rho\left(\mathbf{s}\mathbf{Y}_{n}\mathbf{s}\right)$

The last two imply the statement about bigradings. ∎

##  5\. Isomorphism

###  5.1. Classical degeneration

Recall that $\mathcal{D}_{+}\subset\mathcal{D}$ is the subalgebra generated by
the entries of $A$ and $B$ (so we exclude those of $A^{-1}$ and $B^{-1}$).
From the relations (4.7) and the formula (3.7) for $R$, we can expect that
$\mathcal{D}_{+}$ becomes the commutative ring
$\mathbb{C}[\mathfrak{gl}_{n}\times\mathfrak{gl}_{n}]$ when $q\mapsto 1$.
Similarly, the quantum Weyl algebra $\mathcal{W}$ (4.8) should degenerate to
the usual Weyl algebra $D(\mathbb{C}^{n})$. Here, we review work of Jordan
[Jor14] that makes this precise and we enhance these results to address
invariants.

####  5.1.1. Lattices

Let

$\mathcal{M}_{+}:=\mathcal{D}_{+}\otimes\mathcal{W}$

(as in 4.4.3, we mean the braided tensor product of algebras). As in [Jor14],
we define a standard monomial in $\mathcal{M}_{+}$ to be a product

$a^{i_{1}}_{j_{1}}\cdots a^{i_{\alpha}}_{j_{\alpha}}b^{k_{1}}_{\ell_{1}}\cdots
b^{k_{\beta}}_{\ell_{\beta}}\xi_{r_{1}}\cdots\xi_{r_{\gamma}}\partial_{s_{1}}\cdots\partial_{s_{\delta}}$
(5.1)

where if $u<v$,

$\displaystyle i_{u}<i_{v}\hbox{ or }\left(i_{u}=i_{v}\hbox{ and }j_{u}\leq
j_{v}\right),$ (5.2) $\displaystyle k_{u}<k_{v}\hbox{ or
}\left(\ell_{r}=\ell_{s}\hbox{ and }j_{u}\leq j_{v}\right)$ $\displaystyle
r_{u}\leq r_{v},$ $\displaystyle s_{u}\leq s_{v}$

whenever such indices are present.

######  Theorem 5.1 ([Jor14]).

The standard monomials form a basis of $\mathcal{M}_{+}$. Denote by
$\mathcal{M}_{+,\mathbb{Z}}\subset\mathcal{M}_{+}$ the $\mathbb{C}[q^{\pm
1}]$-lattice spanned by these monomials. The product in $\mathcal{M}_{+}$
restricts to a product in $\mathcal{M}_{+,\mathbb{Z}}$, and this algebra
structure on $\mathcal{M}_{+,\mathbb{Z}}$ satisfies

$\left[\mathbb{C}[q^{\pm 1}]\bigg{/}(q-1)\right]\otimes_{\mathbb{C}[q^{\pm
1}]}\mathcal{M}_{+,\mathbb{Z}}\cong\mathbb{C}[\mathfrak{gl}_{n}\times\mathfrak{gl}_{n}]\otimes
D(\mathbb{C}^{n})$

where $D(\mathbb{C}^{n})$ is the Weyl algebra.

We emphasize that the $\mathbb{C}[q^{\pm 1}]$-module structure on
$\mathcal{M}_{+,\mathbb{Z}}$ is defined to have basis given by standard
monomials.

For technical reasons, we will need a slight alteration of this result. First,
we consider instead the subalgebra
$\mathcal{D}_{\mathrm{IV}}\subset\mathcal{D}$ generated by the entries of $A$
and $B^{-1}$ and correspondingly set

$\mathcal{M}_{\mathrm{IV}}:=\mathcal{D}_{\mathrm{IV}}\otimes\mathcal{W}$

Next, we will replace $\partial_{i}$ with

$\widetilde{\partial}_{i}:=(1-q^{-2})\partial_{i}$

We then define a standard monomial to be

$a^{i_{1}}_{j_{1}}\cdots
a^{i_{\alpha}}_{j_{\alpha}}\iota(b)^{k_{\beta}}_{\ell_{\beta}}\cdots\iota(b)^{k_{1}}_{\ell_{1}}\xi_{r_{1}}\cdots\xi_{r_{\gamma}}\widetilde{\partial}_{s_{1}}\cdots\widetilde{\partial}_{s_{\delta}}$

where the indices still satisfy (5.2) when $u<v$.

Let

$\mathfrak{M}:=\mathfrak{gl}_{n}\times\mathfrak{gl_{n}}\times\mathbb{C}^{n}\times\mathbb{C}^{n}$

######  Corollary 5.1.

The standard monomials form a basis of $\mathcal{M}_{\mathrm{IV}}$ and the
$\mathbb{C}[q^{\pm 1}]$-lattice $\mathcal{M}_{\mathrm{IV},\mathbb{Z}}$ they
span is closed under the product. Moreover,

$\left[\mathbb{C}[q^{\pm 1}]\bigg{/}(q-1)\right]\otimes_{\mathbb{C}[q^{\pm
1}]}\mathcal{M}_{\mathrm{IV},\mathbb{Z}}\cong\mathbb{C}[\mathfrak{M}]$

###### Proof.

The basis statement is clear because $\iota$ is an algebra anti-
automorphism—we can map $\mathcal{M}_{+}$ isomorphically to
$\mathcal{M}_{\mathrm{IV}}$ within $\mathcal{D}\otimes\mathcal{W}$ in a manner
that sends standard monomials to standard monomials. To see that
$\mathcal{M}_{\mathrm{IV},\mathbb{Z}}$ is closed under the product of
$\mathcal{M}_{\mathrm{IV}}$, first observe that (4.7) yields

$R_{12}A_{23}R_{21}B_{13}^{-1}=B_{13}^{-1}R_{21}^{-1}A_{23}R_{12}$

The formula for $R$ (3.7) and the Hecke condition (3.8) imply that when
multiplying two standard monomials and straightening into larger standard
monomial, the commutation relations will only feature scalars in
$\mathbb{C}[q^{\pm 1}]$ (among entries of $B^{-1}$ we just apply $\iota$ to
the straightening process for $B$). Finally, note that from (4.8), we have:

$\widetilde{\partial}_{i}\xi_{i}=(1-q^{-2})+q^{2}\xi_{i}\widetilde{\partial}_{i}+(q^{2}-1)\sum_{k>i}\xi_{k}\widetilde{\partial}_{k}$

Therefore, $\left[\mathbb{C}[q^{\pm
1}]\big{/}(q-1)\right]\otimes_{\mathbb{C}[q^{\pm
1}]}\mathcal{M}_{\mathrm{IV},\mathbb{Z}}$ is a commutative ring. ∎

Thus, whenever we perform $q\mapsto 1$ degeneration in
$\mathcal{M}_{\mathrm{IV}}$, it will always be with respect to this basis of
standard monomials. Namely, for any subspace
$V\subset\mathcal{M}_{\mathrm{IV}}$, we define:

$V_{q=1}:=\left\\{f_{1}(1)M_{1}+\cdots+f_{m}(1)M_{m}\,\middle|\,\begin{array}[]{l}f_{1}(q)M_{1}+\cdots+f_{m}(q)M_{m}\in
V\cap\mathcal{M}_{\mathrm{IV},\mathbb{Z}}\\\ \hbox{and
}M_{1},\ldots,M_{m}\hbox{ are standard monomials}\end{array}\right\\}$

######  Proposition 5.1.

For a finite-dimensional subspace $V\subset\mathcal{M}_{\mathrm{IV}}$, we
have:

$\dim_{\mathbb{C}(q)}V\geq\dim_{\mathbb{C}}V_{q=1}$ (5.3)

###### Proof.

Since $V$ is finite-dimensional, it lies in the span of finitely many standard
monomials. Given a basis of $V_{q=1}$, lift this to a subset of vectors in
$V$. Write these vectors in terms of the finite set of spanning standard
monomials and put the coefficients into a matrix. The maximal minors of this
matrix are nonzero since they have nonzero evaluation at $q=1$. ∎

####  5.1.2. Invariants

We endow
$\mathbb{C}[\mathfrak{M}]=\mathbb{C}[\mathfrak{gl}_{n}\times\mathfrak{gl}_{n}\times\mathbb{C}^{n}\times\mathbb{C}^{n}]$
with the following $GL_{n}$-action: for the natural coordinates $(X,Y,i,j)$ on
$\mathfrak{M}$ and $g\in GL_{n}$, let

$g(X,Y,i,j)=(gXg^{-1},gYg^{-1},gi,(g^{-1})^{T}j)$

It is thus more appropriate to view the last factor as $(\mathbb{C}^{n})^{*}$.
$U(\mathfrak{gl}_{n})$ then acts on $\mathbb{C}[\mathfrak{M}]$ via derivations
induced by the coadjoint and vector representations. By a classic theorem of
Weyl [Wey39], $\mathbb{C}[\mathfrak{M}]^{GL_{n}}$ is generated by classical
trace functions:

$\mathrm{tr}(X^{a_{1}}Y^{b_{1}}(ij^{T})^{c_{1}}\cdots
X^{a_{m}}Y^{b_{m}}(ij^{T})^{c_{m}})$ (5.4)

for
$a_{1},b_{1},c_{1},a_{2},b_{2},c_{2},\ldots,a_{m},b_{m},c_{m}\in\mathbb{Z}_{\geq
0}$.

It is at this point that we emphasize a critical consequence of Remark 3.2,
wherein we stated that all matrices should be transposed. Specifically, we
identify $(A,B^{-1},\vec{\xi},(1-q^{-2})\vec{\partial})$ with $(X,Y,i,j)$ in
the $q\mapsto 1$ degeneration as follows:

$A\mapsto X^{T},B^{-1}\mapsto Y^{T},\vec{\xi}\mapsto
i^{T},(1-q^{-2})\vec{\partial}\mapsto j$ (5.5)

This is not merely cosmetic—it allows the $\mathcal{U}$-action to degenerate
into the $U(\mathfrak{gl}_{n})$-action.

With this set, let us define natural quantum versions of (5.4). For $k\geq 0$,
we define the quantum trace of $M^{k}$ as:

$\mathrm{tr}_{q}(M^{k}):=\sum_{1\leq i_{1},\ldots,i_{k}\leq n}q^{-\langle
2\rho,\epsilon_{i_{k}}\rangle}m^{i_{k}}_{i_{1}}m^{i_{1}}_{i_{2}}\cdots
m^{i_{k-1}}_{i_{k}}$ (5.6)

Similarly, for
$a_{1},b_{1},c_{1},a_{2},b_{2},c_{2},\ldots,a_{m},b_{m},c_{m}\in\mathbb{Z}_{\geq
0}$, we define

$\mathrm{tr}_{q}\left((\mu_{\mathcal{W}}(M)-I)^{c_{1}}B^{-b_{1}}A^{a_{1}}\cdots(\mu_{\mathcal{W}}(M)-I)^{c_{m}}B^{-b_{m}}A^{a_{m}}\right)\in\mathcal{M}_{\mathrm{IV}}$
(5.7)

as (5.6) with $m_{j}^{i}$ replaced with $a^{i}_{j}$, $\iota(b)^{i}_{j}$, or

$\left(\mu_{\mathcal{W}}(M)-I\right)^{i}_{j}=\widetilde{\partial}_{i}\xi_{j}$

depending on the location. We call these quantum trace elements.

######  Proposition 5.2.

The quantum traces are elements of
$\mathcal{M}_{\mathrm{IV}}^{\mathcal{U}}\cap\mathcal{M}_{\mathrm{IV,\mathbb{Z}}}$
that are sent to their corresponding classical traces when $q\mapsto 1$.

###### Proof.

As a $\mathbb{C}[q^{\pm 1}]$-linear combination of a product of standard
monomials, Corollary 5.1 implies that they sit inside
$\mathcal{M}_{\mathrm{IV},\mathbb{Z}}$ and are sent to (5.4) when $q\mapsto
1$. To see that they are $\mathcal{U}$-invariants, observe that they are the
image of $1\in\mathbb{1}$ under a composition of $\mathcal{U}$-morphisms:

  1. (1)

the map $\mathbb{1}\rightarrow(\mathbb{V}^{*}\otimes\mathbb{V})^{\otimes k}$
induced by

$1\mapsto\sum_{1\leq i_{1},\ldots,i_{k}\leq n}q^{-\langle
2\rho,\epsilon_{i_{k}}\rangle}m^{i_{k}}_{i_{1}}\otimes
m^{i_{1}}_{i_{2}}\otimes\cdots\otimes m^{i_{k-1}}_{i_{k}}$

which is depicted diagrammatically as

  2. (2)

the antipode $\iota$ on tensorands that will correspond to $B^{-1}$ matrix
elements;

  3. (3)

the inclusion into $\mathcal{M}_{\mathrm{IV}}$ (identity for $A$-matrix
elements and $\partial_{\triangleright}$ for $B^{-1}$-matrix elements, and
$\mu_{W}-\mathrm{ev}_{\mathbb{V}}$);

  4. (4)

the product in $\mathcal{M}_{\mathrm{IV}}$.∎

######  Lemma 5.1.

For any $\mathcal{U}$-submodule $V\subset\mathcal{M}_{\mathrm{IV}}$, $V_{q=1}$
is a $GL_{n}$-module. Furthermore, we have

$\left[V^{\mathcal{U}}\right]_{q=1}=\left[V_{q=1}\right]^{GL_{n}}$

###### Proof.

Observe that $\mathcal{U}$ acts on standard monomials in a way that preserves
the lattice $\mathcal{M}_{\mathrm{IV},\mathbb{Z}}$. Moreover, because of
(5.5), the action of $E_{i}$ and $F_{i}$ degenerate to the action of their
corresponding generators $e_{i},f_{i}\in U(\mathfrak{gl}_{n})$ on monomials in
$\mathbb{C}[\mathfrak{M}]$. Finally, the degeneration preserves the weight.
Thus, $V_{q=1}$ is a $GL_{n}$-module and we have the containment

$\left[V^{\mathcal{U}}\right]_{q=1}\subset\left[V_{q=1}\right]^{GL_{n}}$

For the other containment, let $v\in
V\cap\mathcal{M}_{\mathrm{IV,\mathbb{Z}}}$ be a lift of
$\mathfrak{v}\in\left[V_{q=1}\right]^{GL_{n}}$. We can write $\mathfrak{v}$ in
terms of classical traces. Let
$\mathrm{tr}_{q}^{\mathfrak{v}}\in\mathcal{M}_{\mathrm{IV},\mathbb{Z}}$ be the
element where every classical trace is replaced with its quantum version. We
can write

$v=\mathrm{tr}_{q}^{\mathfrak{v}}+(q-1)w$

for some $w\in\mathcal{M}_{\mathrm{IV,\mathbb{Z}}}$. Since
$\mathrm{tr}_{q}^{\mathfrak{v}}$ is $\mathcal{U}$-invariant, we have that

$\displaystyle E_{i}v$ $\displaystyle=(q-1)E_{i}w,$ $\displaystyle F_{i}v$
$\displaystyle=(q-1)F_{i}w$

for all $i$. If $v\not\in V^{\mathcal{U}}$, then one of the expressions above
is nonzero and $w\in V$ due to complete reducibility
($\mathcal{M}_{\mathrm{IV}}$ is locally-finite). It then follows that
$\mathrm{tr}_{q}^{\mathfrak{v}}$ is an element of
$V^{\mathcal{U}}\cap\mathcal{M}_{\mathrm{IV},\mathbb{Z}}$ that lifts
$\mathfrak{v}$. ∎

For a finite-dimensional $\mathcal{U}$-submodule
$V\subset\mathcal{M}_{\mathrm{IV}}$, we can combine (5.3) with Lemma 5.1 to
obtain

$\dim_{\mathbb{C}(q)}V^{\mathcal{U}}\geq\dim_{\mathbb{C}}\left[V_{q=1}\right]^{GL_{n}}$
(5.8)

######  Proposition 5.3.

Let $\mathcal{D}_{\mathrm{IV}}[a,b]$ be the degree $(a,b)$ piece of
$\mathcal{D}_{\mathrm{IV}}$ under the determinant bigrading. We have

$\dim_{\mathbb{C}(q)}\mathcal{D}_{\mathrm{IV}}[a,b]^{\mathcal{U}}=\dim_{\mathbb{C}}\left(\mathcal{D}_{\mathrm{IV}}[a,b]_{q=1}\right)^{GL_{n}}$

###### Proof.

$\mathcal{D}_{\mathrm{IV}}[a,b]$ is a $\mathcal{U}$-module spanned by standard
monomials. By Lemma 5.1, $\mathcal{D}_{\mathrm{IV}}[a,b]_{q=1}$ is a
$GL_{n}$-module spanned by the corresponding classical monomials. The weights
under the Cartan are the same, so the result follows from the equality of
quantum and classical character formulas for finite-dimensional modules
[Lus88]. ∎

###  5.2. The case $\mathbf{t=q^{k}}$

Proposition 4.12 shows that $\mathcal{A}_{k}$ is at least as large as
$\mathcal{S}\ddot{\mathcal{H}}_{n}(q,q^{k})$. Our strategy to handle the
$t=q^{k}$ case is to show that $\mathcal{A}_{k}$ is also no bigger than
$\mathcal{S}\ddot{\mathcal{H}}_{n}(q,q^{k})$. Of course, this is but an
impressionistic statement—we will need to make it precise.

####  5.2.1. Mise en place

We begin with a little result allows us to focus on $\mathcal{D}$:

######  Lemma 5.2.

The image of the composition

$\mathcal{D}^{\mathcal{U}}\hookrightarrow\mathcal{M}^{\mathcal{U}}\rightarrow\mathcal{A}_{k}$

becomes all of $\mathcal{A}_{k}$ after performing a localization.

###### Proof.

First note that by considering the action of $q^{\omega_{n}}$, any
$\mathcal{U}$-invariant of $\mathcal{M}$ must have degree zero in its
$\mathcal{W}$ component, i.e.

$\displaystyle\mathcal{M}^{\mathcal{U}}$
$\displaystyle=\left(\mathcal{D}\otimes\mathcal{W}_{0}^{\circ}\right)^{\mathcal{U}}$
$\displaystyle\mathcal{I}_{k}^{\mathcal{U}}$
$\displaystyle=\left(\mathcal{I}_{k}\cap\mathcal{D}\otimes\mathcal{W}_{0}^{\circ}\right)^{\mathcal{U}}$

From multiplication on the left, we have

$\mathcal{D}\otimes\mathcal{W}_{0}\bigg{(}\delta_{i,j}+(q-q^{-1})\partial_{i}\xi_{j}-q^{-2k}\sum_{i}(A^{-1}BAB^{-1})_{j}^{i}\bigg{)}\subset\mathcal{I}_{k}\cap\mathcal{D}\otimes\mathcal{W}_{0}$
(5.9)

On the other hand, $\mathcal{W}_{0}$ is generated by
$\\{\partial_{i}\xi_{j}\\}$. For any $w\in\mathcal{W}_{0}$ and
$x\in\mathcal{D}$, we can use the R-matrix to move $w$ to the right:

$wx=(\tensor[]{r}{{}_{s}}\bowtie x)(\tensor[_{s}]{r}{}\bullet w)$

We can then use (5.9) to write $\tensor[_{s}]{r}{}\bullet w$ in terms of
$\mathcal{D}$ in the quotient
$\mathcal{D}\otimes\mathcal{W}_{0}\big{/}(\mathcal{I}_{k}\cap\mathcal{D}\otimes\mathcal{W}_{0})$.
Therefore, the map

$\mathcal{D}\hookrightarrow\mathcal{D}\otimes\mathcal{W}_{0}\rightarrow\mathcal{D}\otimes\mathcal{W}_{0}\big{/}(\mathcal{I}_{k}\cap\mathcal{D}\otimes\mathcal{W}_{0})$

is surjective. Since all the $\mathcal{U}$-modules involved are locally-
finite, taking $\mathcal{U}$-invariants is exact. After doing so,
$\det_{q}(M)$ can be written in terms of elements of
$\mathcal{D}^{\mathcal{U}}$. We localize the image at that latter expression.
∎

######  Corollary 5.2.

$\mathcal{A}_{k}$ is a localization of
$\mathcal{A}_{k,\mathrm{IV}}:=\mathcal{D}_{\mathrm{IV}}^{\mathcal{U}}\big{/}(\mathcal{D}_{\mathrm{IV}}\cap\mathcal{I}_{k})^{\mathcal{U}}$.

We can thus focus on $\mathcal{D}_{IV}$ and $\mathcal{A}_{k,\mathrm{IV}}$. The
left ideal $\mathcal{I}_{k}$ is generated by elements with homogeneous
determinant bigrading, and so $\mathcal{A}_{k}$ and
$\mathcal{A}_{k,\mathrm{IV}}$ are also bigraded. Let
$\mathcal{A}_{k,\mathrm{IV}}[a,b]$ denote its bidegree $(a,b)$ piece. Recall
our notation from 2.2.1. Proposition 4.12 implies that

$\dim_{\mathbb{C}(q)}\mathcal{A}_{k,\mathrm{IV}}[a,-b]\geq\dim_{\mathbb{C}(q)}\mathcal{S}_{q^{k},IV}[a,-b]=\dim_{\mathbb{C}}\mathbb{C}\left[\mathbf{x}_{n},\mathbf{y}_{n}\right]^{\Sigma_{n}}_{a,b}$
(5.10)

We will use classical degeneration, as covered in 5.1, to obtain the opposite
bound.

####  5.2.2. Almost commuting variety

Let
$\mathfrak{M}_{\mathrm{ac}}\subset\mathfrak{M}=\mathbb{C}[\mathfrak{gl}_{n}\times\mathfrak{gl}_{n}\times\mathbb{C}^{n}\times\mathbb{C}^{n}]$
be the closed subscheme with ideal generated by the entries of

$[X,Y]-ij^{T}$ (5.11)

where $(X,Y,i,j)$ are the usual coordinates of $\mathfrak{M}$. The following
results are proved by Gan-Ginzburg [GG06]:

######  Theorem 5.2.

We have the following:

  1. (1)

$\mathfrak{M}_{\mathrm{ac}}$ is reduced.

  2. (2)

The projection map
$p:\mathfrak{M}\rightarrow\mathfrak{gl}_{n}\times\mathfrak{gl_{n}}$ induces an
isomorphism:

$\mathbb{C}[\mathfrak{M}_{\mathrm{ac}}]^{GL_{n}}\cong\left(\mathbb{C}[\mathfrak{gl}_{n}\times\mathfrak{gl}_{n}]\bigg{/}I\right)^{GL_{n}}\cong\mathbb{C}[\mathbf{x}_{n},\mathbf{y}_{n}]^{\Sigma_{n}}$
(5.12)

where $I$ is the ideal generated by the equations $[X,Y]=0$ and the second
isomorphism is induced by restriction to diagonal matrices.

$\mathfrak{M}_{\textrm{ac}}$ is called the almost commuting variety.

We can endow $\mathbb{C}[\mathfrak{M}]$ with a bigrading where the entries of
$X$ have bidegree $(1,0)$ and those of $Y$ have bidegree $(0,1)$. The
generators (5.11) are homogeneous, so $\mathbb{C}[\mathcal{M}_{\mathrm{ac}}]$
inherits this bigrading. Let $\mathbb{C}\langle
X,Y\rangle\subset\mathbb{C}[\mathfrak{M}_{\mathrm{ac}}]$ denote the subalgebra
generated by the entries of $X$ and $Y$ and let $\mathbb{C}\langle
X,Y\rangle^{GL_{n}}_{a,b}$ denote the bidegree $(a,b)$ piece of the invariant
subalgebra. Tracing through the isomorphism (5.12), we have:

######  Corollary 5.3.

For each bidegree $(a,b)$,

$\dim_{\mathbb{C}}\mathbb{C}\langle
X,Y\rangle_{a,b}^{GL_{n}}=\dim_{\mathbb{C}}\mathbb{C}[\mathbf{x}_{n},\mathbf{y}_{n}]^{\Sigma_{n}}_{a,b}$

######  Lemma 5.3.

There is an algebra homomorphism

$\psi:\mathbb{C}[\mathfrak{M}_{\mathrm{ac}}]\rightarrow\left(\mathcal{M}_{\mathrm{IV}}\right)_{q=1}\bigg{/}\left(\mathcal{I}_{k}\cap\mathcal{M}_{\mathrm{IV}}\right)_{q=1}$

that restricts to a surjection

$\psi:\mathbb{C}\langle
X,Y\rangle\twoheadrightarrow\left(\mathcal{D}_{\mathrm{IV}}\right)_{q=1}\bigg{/}\left(\mathcal{I}_{k}\cap\mathcal{D}_{\mathrm{IV}}\right)_{q=1}$

which respects bigradings.

###### Proof.

From left multiplication on the generators of $\mathcal{I}_{k}$, we can see
that

$\mathcal{M}_{\mathrm{IV}}\bigg{(}(B^{-1}A)^{i}_{j}+(B^{-1}A)^{i}_{\ell}\widetilde{\partial}_{\ell}\xi_{j}-q^{-2k}(AB^{-1})^{i}_{j}\bigg{)}\subset\mathcal{I}_{k}\cap\mathcal{M}_{\mathrm{IV}}$
(5.13)

Let $(A,B^{-1},\vec{\xi},\vec{\partial})$ denote the coordinates of
$\mathrm{Spec}\left(\mathcal{M}_{\mathrm{IV}}\right)_{q=1}\cong\mathfrak{M}$
and $(X,Y,i,j)$ denote the coordinates for another copy of $\mathfrak{M}$.
Upon setting $q=1$, the containments (5.13) imply that the pullback of
coordinates

$\left(X^{T},Y^{T},i^{T},j\right)\mapsto(A,B^{-1},\vec{\xi},B^{-1}A\vec{\partial})$

yields the desired homomorphism $\psi$. ∎

######  Theorem 5.3.

The radial parts map $\mathfrak{rad}_{k}$ gives an isomorphism

$\mathcal{A}_{k}\cong\mathcal{S}\ddot{\mathcal{H}}_{n}(q,q^{k})$

###### Proof.

By Lemma 2.1, equations (4.13), and Corollary 5.2, it suffices to show that
$\mathfrak{rad}_{k}$ restricts to an isomorphism

$\mathcal{A}_{k,\mathrm{IV}}\cong\mathcal{S}\ddot{\mathcal{H}}_{n}(q,q^{k})_{\mathrm{IV}}$

because localization is exact. To that end, we just need to provide the
opposite bound to (5.10). Applying Propositions 5.1 and 5.3 and Lemma 5.1, we
have

$\displaystyle\dim_{\mathbb{C}(q)}\mathcal{A}_{k,\mathrm{IV}}[a,b]$
$\displaystyle=\dim_{\mathbb{C}(q)}\mathcal{D}_{\mathrm{IV}}^{\mathcal{U}}[a,b]-\dim_{\mathbb{C}(q)}\mathcal{I}_{k}\cap\mathcal{D}_{\mathrm{IV}}^{\mathcal{U}}[a,b]$
$\displaystyle\leq\dim_{\mathbb{C}}\left(\mathcal{D}_{\mathrm{IV}}[a,b]_{q=1}\right)^{GL_{n}}-\dim_{\mathbb{C}}\left(\mathcal{I}_{k}\cap\mathcal{D}_{\mathrm{IV}}[a,b]_{q=1}\right)^{GL_{n}}$
$\displaystyle=\dim_{\mathbb{C}}\left[\left(\mathcal{D}_{\mathrm{IV}}\right)_{q=1}\bigg{/}\left(\mathcal{I}_{k}\cap\mathcal{D}_{\mathrm{IV}}\right)_{q=1}\right]^{GL_{n}}_{a,b}$
(5.14)

where in the final line, the subscript denotes the bidegree $(a,b)$ piece.
Finally, we can combine Corollary 5.3 and Lemma 5.3 to bound (5.14) above by
$\dim_{\mathbb{C}}\mathbb{C}[\mathbf{x}_{n},\mathbf{y}_{n}]_{a,b}^{\Sigma_{n}}$.
∎

###  5.3. Generic parameters

Now we introduce the variable $t$ and establish the isomorphism over
$K=\mathbb{C}(q,t)$. First, we will work over the ring
$R_{q}:=\mathbb{C}(q)[t^{\pm 1}]$. We will base change $\mathcal{U}$ and all
its modules to $R_{q}$ but still denote them by the same symbols to avoid
notational clutter. Let $\alpha$ be a parameter such that $t=q^{\alpha}$; we
introduce it for stylistic/notational reasons so as to neatly replace the
integer $k$. All constructions below can be described fully in terms of $t$
rather than $\alpha$.

####  5.3.1. Etingof-Kirillov theory at $t$

In order to extend 3.5, we need a suitable generalization of
$S_{q}^{nk}\mathbb{V}$, and moreover it should be a representation of
$\mathcal{W}^{\circ}_{0}$. Recall the notation from 4.3.3, wherein we
translated the actions of $\mathcal{U}$ and $\mathcal{W}$ on $S_{q}\mathbb{V}$
to ones on the polynomial ring
$\mathbb{C}(q)[\mathbf{z}_{n}]:=\mathbb{C}(q)[z_{1},\ldots,z_{n}]$. From this
we constructed an algebra homomorphism

$\mathfrak{qdiff}:\mathcal{W}\rtimes\mathcal{U}\rightarrow\mathbb{D}_{q}(\mathbf{z}_{n})$

where $\mathbb{D}_{q}(\mathbf{z}_{n})$ is a ring of difference operators. By
Corollary 4.1 and the $\mathcal{U}$-equivariance of $\mu_{\mathcal{W}}$, we
can extend $\mathfrak{qdiff}$ to $\mathcal{W}^{\circ}\rtimes\mathcal{U}$ by
setting

$\mathfrak{qdiff}(\mu_{\mathcal{M}}(\textstyle\det_{q}(M))^{-1})=(\mathfrak{qdiff}\circ\kappa)(\textstyle\det_{q}(M)^{-1})$

Let $W_{\alpha}$ be the free $R_{q}$-module spanned by “monomials” of the form

$\left\\{z_{1}^{k_{1}+\alpha-1}z_{2}^{k_{2}+\alpha-1}\cdots
z_{n}^{k_{n}+\alpha-1}\,\middle|\,(k_{1},\ldots,k_{n})\in\mathbb{Z}^{n}\right\\}$

We emphasize that the $k_{i}$ are now allowed to be negative.
$\mathbb{D}_{q}(\mathbf{z}_{n})$ naturally acts on this space by

$\displaystyle z_{i}\left(z_{1}^{k_{1}+\alpha-1}\cdots
z_{n}^{k_{n}+\alpha-1}\right)$ $\displaystyle=z_{1}^{k_{1}+\alpha-1}\cdots
z_{i}^{k_{i}+1+\alpha-1}\cdots z_{n}^{k_{n}+\alpha-1}$ $\displaystyle
T_{q,z_{i}}\left(z_{1}^{k_{1}+\alpha-1}\cdots z_{n}^{k_{n}+\alpha-1}\right)$
$\displaystyle=q^{k_{i}-1}tz_{1}^{k_{1}+\alpha-1}\cdots
z_{n}^{k_{n}+\alpha-1}$

Therefore, we obtain an action of $\mathcal{W}^{\circ}\rtimes\mathcal{U}$ on
$W_{\alpha}$. In particular, the $\mathcal{W}^{\circ}$-action is
$\mathcal{U}$-equivariant. From the formulas (4.9) for $\mathfrak{qdiff}$, it
is clear that $\mathcal{W}^{\circ}_{0}\rtimes\mathcal{U}$ preserves the
subspace

$W^{0}_{\alpha}:=\mathrm{span}_{R_{q}}\left\\{z_{1}^{k_{1}+\alpha-1}z_{2}^{k_{2}+\alpha-1}\cdots
z_{n}^{k_{n}+\alpha-1}\,\middle|\,\begin{array}[]{l}(k_{1},\ldots,k_{n})\in\mathbb{Z}^{n}\\\
k_{1}+\cdots+k_{n}=0\end{array}\right\\}$

$W^{0}_{\alpha}$ will be our replacement for $S_{q}^{n(k-1)}\mathbb{V}$.

Next, we will need a $t$-version of the determinant representation. Let
$\chi^{\pm\alpha}:\mathcal{U}\rightarrow R_{q}$ be the characters given by

$\begin{gathered}\chi(E_{i})=\chi(F_{i})=0\\\ \chi(q^{h})=t^{\pm\langle
h,\omega_{n}\rangle}\end{gathered}$

We denote by $\mathbb{1}_{-\alpha}\cong R_{q}$ the rank 1 representation
defined using $\chi^{-\alpha}$. It is natural to interpret integer shifts of
$\alpha$ as tensoring by $\mathbb{1}_{k}$. The zero weight space
$W_{\alpha}\otimes\mathbb{1}_{1-\alpha}[0]$ is free of rank 1, spanned by
$z_{1}^{\alpha-1}\cdots z_{n}^{\alpha-1}\otimes 1$; we thus identify it with
$R_{q}$.

Finally, $V_{\lambda+(k-1)\rho}$ is replaced with the Verma module

$M_{\lambda}^{\alpha}:=M_{\lambda+(\alpha-1)\rho}$

which is simple when $t$ is left as a parameter. With this set, the case of
general $t$ is in many ways quite similar to that of $t=q^{k}$.

######  Theorem 5.4 ([EK94]).

We have the following:

  1. (1)

For $\lambda\in P^{+}$, there is a nonzero, unique up to constant intertwiner

$\widetilde{\Phi}_{\lambda}^{\alpha}:M_{\lambda}^{\alpha}\rightarrow
M_{\lambda}^{\alpha}\otimes W^{0}_{\alpha}\otimes\mathbb{1}_{1-\alpha}$

  2. (2)

Upon picking a consistent normalizations, the weighted trace
$\widetilde{\varphi}^{\alpha}_{\lambda}$ satisfies

$P_{\lambda}(q,t)=\widetilde{\varphi}_{\lambda}^{\alpha}\big{/}\widetilde{\varphi}_{0}^{\alpha}$

We note in passing that $\widetilde{\varphi}_{\lambda}^{\alpha}$ is no longer
a polynomial but rather a power series.

####  5.3.2. Admissible diagrams

Since $M_{\lambda}^{\alpha}$ is infinite-dimensional, we can no longer convert
the intertwiner $\widetilde{\Phi}^{\alpha}_{\lambda}$ into an invariant vector
as we we did in 3.5. More generally, the graphical calculus covered in 3.3
still applies to $M_{\lambda}$ except that it no longer has classical and
quantum coevaluations. This leads to some awkwardness in defining the action
of non-invariant elements of $\mathcal{M}$. Our goal here is to be able to
turn elements of $\mathcal{M}$ “upside-down”.

We have already made use of diagrammatic calculus for $\mathcal{D}$—let us
give a similar construction for $\mathcal{W}^{\circ}$. The inclusion of the
generators $\\{\partial_{i}\\}$ and $\\{\xi_{i}\\}$ come from morphisms
$\mathbb{V}^{*}\rightarrow\mathcal{W}$ and $\mathbb{V}\rightarrow\mathcal{W}$,
respectively, and likewise the inclusion of
$\mu_{\mathcal{M}}(\det_{q}(M))^{-1}$ comes from a morphism
$\mathbb{1}\rightarrow\mathcal{W}^{\circ}$. Since the product
$\mathfrak{m}_{\mathcal{W}}$ of $\mathcal{W}^{\circ}$ is a
$\mathcal{U}$-morphism, we can write any element of $\mathcal{W}^{\circ}$ as a
linear combination of images of morphisms
$\mathfrak{d}:X\rightarrow\mathcal{W}^{\circ}$.

Altogether, we have a way to present elements of
$\mathcal{M}\cong\mathcal{O}\otimes\kappa(\mathcal{O})\otimes\mathcal{W}^{\circ}$
as a linear combination of morphisms of the form

(5.15)

evaluated at various inputs. We can define a product $*$ of two such morphisms
by

This yields the product in $\mathcal{M}$ upon evaluation.

More generally, the diagram of a morphism $\mathfrak{D}:W_{1}\otimes
W_{2}\rightarrow\mathcal{M}$ is called admissible if it is of the form

(5.16)

where the ghost strand only undergoes braidings. For such a $\mathfrak{D}$, we
define $\overline{\mathfrak{D}}$ to be the morphism

$\overline{\mathfrak{D}}:\mathcal{O}\rightarrow\mathcal{W}^{\circ}\otimes
W_{2}^{*}\otimes\mathcal{U}\otimes W_{1}^{*}$

given by

where every morphism inside $\mathfrak{D}$ is replaced with its adjoint and
the orientation of the ghost strand is reversed. While $\mathcal{W}^{\circ}$
is infinite-dimensional, it is locally finite and thus the coevaluation is
done on a finite-dimensional subrepresentation. Nonetheless, this operation is
only given in terms of playing with pictures and it needs to be checked that
it is well-defined.

######  Proposition 5.4.

If two admissible diagrams $\mathfrak{D}$ and $\mathfrak{D}^{\prime}$
correspond to the same morphism $W_{1}\otimes W_{2}\rightarrow\mathcal{M}$,
then $\overline{\mathfrak{D}}$ and $\overline{\mathfrak{D}^{\prime}}$ give the
same morphism $\mathcal{O}\rightarrow\mathcal{W}^{\circ}\otimes
W_{2}^{*}\otimes\mathcal{U}\otimes W_{1}^{*}$. Moreover, this morphism only
involves braidings on the ghost strand.

###### Proof.

Specializing $\mathcal{U}$ to act on a finite-dimensional module $U$,
$\overline{\mathfrak{D}}$ and $\overline{\mathfrak{D}^{\prime}}$ become
partial adjoints of the same morphism $W_{1}\otimes U\otimes
W_{2}\rightarrow\mathcal{O}\otimes U\otimes\mathcal{W}^{\circ}$ and thus must
be equal. The statement about braidings follows by definition. ∎

####  5.3.3. Action on linear maps

Let

$\displaystyle\mathfrak{tHom}_{\alpha}$
$\displaystyle:=\bigoplus_{\begin{subarray}{c}M\hbox{ \tiny{highest
weight}}\\\ U\hbox{ \tiny{finite-
dimensional}}\end{subarray}}\mathrm{Hom}_{\mathcal{U}}\left(M,M\otimes
W_{\alpha}\otimes U^{*}\right)\bigg{/}$ $\displaystyle\left\langle(f\otimes
1\otimes 1)\circ\psi-\psi\circ
f\,\middle|\,\begin{array}[]{l}\psi\in\mathrm{Hom}_{\mathcal{U}}\left(M^{\prime},M\otimes
W_{\alpha}\otimes U^{*}\right),\\\
f\in\mathrm{Hom}_{\mathcal{U}}(M,M^{\prime})\end{array}\right\rangle$ (5.19)

The relations (5.19) are analogous to the coend relations (3.13)—the “t”
stands for homomorphisms identified if they have the same trace. One should
view the extra tensorand $U^{*}$ as something we will contract away to yield
an $R_{q}$-linear map $M\rightarrow M\otimes W_{\alpha}$. Since the Verma
module $M_{\lambda}^{\alpha}$ is simple, the intertwiners
$\\{\widetilde{\Phi}_{\lambda}^{\alpha}\\}$ are linearly independent in
$\mathfrak{tHom}_{\alpha}$.

Let $\mathfrak{a}_{\mathcal{W}}:\mathcal{W}^{\circ}\otimes
W_{\alpha}\rightarrow W_{\alpha}$ be the action map. For an admissible diagram
$\mathfrak{D}:W_{1}\otimes W_{2}\rightarrow\mathcal{M}$ as in $(\ref{Admiss})$
and

$\phi\in\mathrm{Hom}_{\mathcal{U}}\left(M,M\otimes W_{\alpha}\otimes
U^{*}\right)\subset\mathfrak{tHom}_{\alpha}$

we define

$\mathfrak{D}\star\phi:V\otimes M\rightarrow(V\otimes M)\otimes
W_{\alpha}\otimes(W_{1}\otimes W_{2}\otimes U)^{*}$

to be the class of the morphism:

(5.20)

Such a morphism is well-defined for infinite-dimensional $M$ because by
Proposition 5.4, $M$ only undergoes braidings. Moreover, any $f$ as in the
relations (5.19) can pass through such braidings. This implies that
$\mathfrak{D}\star\phi$ is independent of the representative of the class of
$\phi$. We leave it as a drawing exercise to see that for two diagrams
$\mathfrak{D}_{1},\mathfrak{D}_{2}$ of the form (5.15), we have

$\mathfrak{D}_{1}\star\left(\mathfrak{D}_{2}\star\phi\right)=\left(\mathfrak{D}_{1}*\mathfrak{D}_{2}\right)\star\phi$
(5.21)

Consider now the space of $R_{q}$-linear maps:

$\displaystyle\mathrm{tHom}_{\alpha}$ $\displaystyle:=\bigoplus_{M\hbox{
\tiny{is highest weight}}}\mathrm{Hom}_{R_{q}}\left(M,M\otimes
W_{\alpha}\otimes\mathbb{1}_{1-\alpha}\right)\bigg{/}$
$\displaystyle\left\langle(f\otimes 1)\circ\psi-\psi\circ
f\,\middle|\,\begin{array}[]{l}\psi\in\mathrm{Hom}_{R_{q}}\left(M^{\prime},M\otimes
W_{\alpha}\otimes\mathbb{1}_{1-\alpha}\right),\\\
f\in\mathrm{Hom}_{\mathcal{U}}(M,M^{\prime})\end{array}\right\rangle$ (5.24)

For the class of $\phi:M\rightarrow M\otimes
W_{\alpha}\otimes\mathbb{1}_{1-\alpha}$ in $\mathrm{tHom}_{\alpha}$, we define
an operation by each of the three tensor components of
$\mathcal{M}\cong\mathcal{O}\otimes\partial_{\triangleright}(\mathcal{O})\otimes\mathcal{W}^{\circ}$:

  * •

for $v^{*}\otimes v\in V^{*}\otimes V\subset\mathcal{O}\otimes
1\subset\mathcal{D}$, define

$\begin{gathered}(v^{*}\otimes v)\star\phi:V\otimes M\rightarrow V\otimes
M\otimes W_{\alpha}\otimes\mathbb{1}_{1-\alpha}\\\ \big{(}(v^{*}\otimes
v)\star\phi\big{)}(x\otimes
m)=v^{*}(\tensor[_{s}]{r}{}x)\big{[}S(\tensor[_{t}]{r}{})v\otimes\phi\left(\tensor[]{r}{{}_{t}}\tensor[]{r}{{}_{s}}m\right)\big{]}\end{gathered}$
(5.25)

  * •

for $\kappa(v^{*}\otimes v)\in
1\otimes\partial_{\triangleright}(\mathcal{O})\subset\mathcal{D}$, define

$\begin{gathered}\kappa(v^{*}\otimes v)\star\phi:M\rightarrow M\otimes
W_{\alpha}\otimes\mathbb{1}_{1-\alpha}\\\ \big{(}\kappa(v^{*}\otimes
v)\star\phi\big{)}(m)=v^{*}\left(S(\tensor[_{t}]{r}{}\tensor[]{r}{{}_{s}})v\right)\phi\left(\tensor[]{r}{{}_{t}}\tensor[_{s}]{r}{}m\right)\end{gathered}$
(5.26)

  * •

for $w\in\mathcal{W}^{\circ}$ define

$\begin{gathered}w\star\phi:M\rightarrow M\otimes
W_{\alpha}\otimes\mathbb{1}_{1-\alpha}\\\
\big{(}w\star\phi\big{)}(m)=\left(\tensor[]{r}{{}_{t}}\otimes(\tensor[_{t}]{r}{}S(\tensor[_{s}]{r}{})\bullet
w)\otimes
1\right)\blacktriangleright\phi\left(\tensor[]{r}{{}_{s}}m\right)\end{gathered}$
(5.27)

where the $\blacktriangleright$ means acting on the output of $\phi$ via the
$\mathcal{U}$\- and $\mathcal{W}^{\circ}$-actions.

Because the input of $\phi$ and $M$-tensorand of the output of $\phi$ is only
acted on by elements of $\mathcal{U}$, these operations are well-defined on
the quotient (5.24). A priori, we do not know that they piece together to form
an action of $\mathcal{M}$. To that end, let:

  * •

$\mathrm{tHom}_{\alpha}^{+}$ be the subspace of $\mathrm{tHom}_{\alpha}$
generated by these operations from the intertwiners
$\\{\widetilde{\Phi}_{\lambda}^{\alpha}\\}_{\lambda\in P^{+}}$;

  * •

$\mathrm{Int}_{\alpha}\subset\mathrm{tHom}_{\alpha}^{+}$ be the span of
$\\{\widetilde{\Phi}_{\lambda}^{\alpha}\\}_{\lambda\in P^{+}}$.

######  Lemma 5.4.

The operations (5.25)-(5.27) define an action of $\mathcal{M}$ on
$\mathrm{tHom}_{\alpha}^{+}$. Under this action, $\mathcal{M}^{\mathcal{U}}$
preserves $\mathrm{Int}_{\alpha}$.

######  Remark 5.1.

The formulas (5.25)-(5.27) should define an action of $\mathcal{M}$ on the
entire space $\mathrm{tHom}_{\alpha}$. We leave the proof to a more skillful
scholar of the Yang-Baxter equation.

###### Proof.

On $\mathrm{tHom}_{\alpha}^{+}$, the operations (5.25)-(5.27) are obtained by
contracting the diagram actions on $\mathfrak{tHom}_{\alpha}$ with $\phi$ set
equal to an intertwiner $\widetilde{\Phi}_{\lambda}^{\alpha}$. Equation (5.21)
implies that the diagram actions yield an $\mathcal{M}$-action upon
contraction. From the diagram action (5.20) with $W_{1}=W_{2}=\mathbb{1}$, it
is evident that $\mathcal{M}^{\mathcal{U}}$ sends
$\widetilde{\Phi}_{\lambda}^{\alpha}$ to some other intertwiner. As pointed
out in the proof of Lemma 5.2,
$\mathcal{M}^{\mathcal{U}}\subset\mathcal{D}\otimes\mathcal{W}^{\circ}_{0}$,
and thus acting with an element of $\mathcal{M}^{\mathcal{U}}$ sends
$\widetilde{\Phi}_{\lambda}^{\alpha}$ to some intertwiner

$V\otimes M_{\lambda}^{\alpha}\rightarrow V\otimes M_{\lambda}^{\alpha}\otimes
W_{\alpha}^{0}\otimes\mathbb{1}_{1-\alpha}$

for some finite-dimensional $V$. Using the modified coend relation (5.24),
this can be rewritten as a linear combination of
$\\{\widetilde{\Phi}_{\lambda}^{\alpha}\\}_{\lambda\in P^{+}}$. ∎

####  5.3.4. Radial parts at generic parameters

We are now able to make contact with quantum Hamiltonian reduction:

######  Lemma 5.5.

The $\mathcal{M}^{\mathcal{U}}$ action on $\mathrm{Int}_{\alpha}$ factors
through the subspace $\mathcal{I}_{\alpha}^{\mathcal{U}}$, where
$\mathcal{I}_{\alpha}$ is the left ideal

$\mathcal{I}_{\alpha}:=\mathcal{M}\left(\mu_{\mathcal{M}}(M)-q^{2}t^{-2}I\right)\subset\mathcal{M}$

###### Proof.

We approach this similarly to how we proved Proposition 4.11. Namely, we prove
that the action map

$\mathcal{M}\otimes\mathrm{Int}_{\alpha}\rightarrow\mathrm{tHom}_{\alpha}^{+}$

factors through $\mathcal{I}_{\alpha}$. To that end, we compute the action of
$\mu_{\mathcal{M}}(M)$ on $\widetilde{\Phi}_{\lambda}^{\alpha}$. Using
$(\ref{DMomentDia})$, we get:

For the first equality, we applied the modified coend relation (5.24). From
(3.6), one can see that the final diagram is equal to the action of the
diagram for $q^{2}t^{-2}\mathrm{ev}_{\mathbb{V}}$, which yields the entries of
$q^{2}t^{-2}I$ upon contraction. ∎

We define

$\displaystyle\mathcal{A}_{\alpha}^{R}$
$\displaystyle:=\left(\mathcal{M}\big{/}\mathcal{I}_{\alpha}\right)^{\mathcal{U}}$
$\displaystyle\mathcal{A}_{\alpha}$
$\displaystyle:=K\otimes_{R_{q}}\mathcal{A}_{\alpha}$

Like in Proposition 4.10, $\mathcal{A}_{\alpha}^{R}$ is in fact an algebra.
The action on intertwiners yields an algebra homomorphism

$\mathfrak{rad}_{\alpha}:\mathcal{A}^{R}_{\alpha}\rightarrow\mathrm{End}_{K}(\Lambda^{\pm}_{n}(q,t))$

that we also call the radial parts map. Observe that

$\mathcal{A}_{k}=\left[R_{q}\big{/}(t-q^{k})\right]\otimes_{R_{q}}\mathcal{A}_{\alpha}^{R}$

Through analysis similar to what was done in 3.5.3, the analogue of
Proposition 4.12 and its proof holds in this case as well:

######  Proposition 5.5.

Upon base change to $K$, the image of $\mathfrak{rad}_{\alpha}$ contains the
image of $\mathcal{S}\ddot{\mathcal{H}}_{n}(q,t)$ under $\rho$:

$\mathfrak{rad}_{k}\left(\mathcal{A}_{\alpha}\right)\supset\rho\left(\mathcal{S}\ddot{\mathcal{H}}_{n}(q,t)\right)$

This inclusion respects the bigradings and

$\displaystyle\mathfrak{rad}_{\alpha}\left(\mathrm{qcoev}_{\wedge_{q}^{r}\mathbb{V}}(1)\right)$
$\displaystyle=\rho\left(\mathbf{s}e_{r}(\mathbf{X}_{n})\mathbf{s}\right)$
$\displaystyle\mathfrak{rad}_{\alpha}\left(\partial_{\triangleright}\mathrm{qcoev}_{\wedge_{q}^{r}\mathbb{V}^{*}}(1)\right)$
$\displaystyle=\rho\left(\mathbf{s}e_{r}(\mathbf{Y}^{-1}_{n})\mathbf{s}\right)$
$\displaystyle\mathfrak{rad}_{\alpha}\left(\textstyle\det_{q}(A)^{-1}\right)$
$\displaystyle=\rho\left(\mathbf{s}\mathbf{X}^{-1}_{n}\mathbf{s}\right)$
$\displaystyle\mathfrak{rad}_{\alpha}\left(\textstyle\det_{q}(B)\right)$
$\displaystyle=\rho\left(\mathbf{s}\mathbf{Y}_{n}\mathbf{s}\right)$

######  Theorem 5.5.

Upon base change to $K$, the radial parts map is an isomorphism onto
$\rho\left(\mathcal{S}\ddot{\mathcal{H}}_{n}(q,t)\right)$.

###### Proof.

Recall the notation $\mathcal{D}_{\mathrm{IV}}$ introduced in 5.1. As we did
in 5.2, we can reduce the problem to showing that $\mathfrak{rad}_{\alpha}$
restricts to an isomorphism

$\mathcal{A}^{R}_{\alpha,\mathrm{IV}}:=\mathcal{D}_{\mathrm{IV}}^{\mathcal{U}}\big{/}(\mathcal{D}_{\mathrm{IV}}\cap\mathcal{I}_{\alpha})^{\mathcal{U}}\rightarrow\mathcal{S}\ddot{\mathcal{H}}_{n}(q,t)_{\mathrm{IV}}$

after base change to $K$. First, we show that
$\mathcal{A}_{\alpha,\mathrm{IV}}:=K\otimes_{R_{q}}\mathcal{A}_{\alpha,\mathrm{IV}}$
is generated by

$\left\\{\mathrm{qcoev}_{\wedge_{q}^{r}\mathbb{V}}(1),\,\partial_{\triangleright}\mathrm{qcoev}_{\wedge_{q}^{r}\mathbb{V}^{*}}(1)\right\\}_{r=1,\ldots,n}$
(5.28)

Theorem 5.3 implies that the bigraded piece
$\mathcal{A}^{R}_{\alpha,\mathrm{IV}}[a,b]$, is generated by such elements at
$t=q^{k}$. Since $\mathcal{A}_{\alpha,\mathrm{IV}}^{R}[a,b]$ is finitely
generated, they must generate over $K$ by Nakayama’s lemma. This implies that

$\mathfrak{rad}_{\alpha}\left(\mathcal{A}_{\alpha,\mathrm{IV}}\right)=\rho\left(\mathcal{S}\ddot{\mathcal{H}}_{n}(q,t)_{\mathrm{IV}}\right)$

On the generators (5.28), it follows from Proposition 2.3 that specializing
$\mathfrak{rad}_{\alpha}$ to $t=q^{k}$ yields $\mathfrak{rad}_{k}$, and thus
this is true for the entirety of $\mathcal{A}_{\alpha,\mathrm{IV}}$.
Therefore, a torsion-free element of
$\mathcal{A}^{R}_{\alpha,\mathrm{IV}}[a,b]$ in the kernel of
$\mathfrak{rad}_{\alpha}$ must vanish at infinitely many specializations
$t=q^{k}$. Since $\mathcal{A}_{\alpha,\mathrm{IV}}^{R}[a,b]$ is finitely
generated and $R_{q}$ is a PID, this implies that the kernel is torsion and
hence disappears upon base change to $K$. ∎

## References

  * [AS96] Anton Yu. Alekseev and Volker Schomerus.  Representation theory of Chern-Simons observables.  Duke Math. J., 85(2):447–510, 1996.
  * [BFG06] Roman Bezrukavnikov, Michael Finkelberg, and Victor Ginzburg.  Cherednik algebras and Hilbert schemes in characteristic $p$.  Represent. Theory, 10:254–298, 2006.  With an appendix by Pavel Etingof.
  * [BJ18] Martina Balagovic and David Jordan.  The Harish-Chandra isomorphism for quantum $GL_{2}$.  J. Noncommut. Geom., 12(3):1161–1197, 2018.
  * [BK01] Bojko Bakalov and Alexander Kirillov, Jr.  Lectures on tensor categories and modular functors, volume 21 of University Lecture Series.  American Mathematical Society, Providence, RI, 2001.
  * [BZBJ18a] David Ben-Zvi, Adrien Brochier, and David Jordan.  Integrating quantum groups over surfaces.  J. Topol., 11(4):874–917, 2018.
  * [BZBJ18b] David Ben-Zvi, Adrien Brochier, and David Jordan.  Quantum character varieties and braided module categories.  Selecta Math. (N.S.), 24(5):4711–4748, 2018.
  * [Che95] Ivan Cherednik.  Double affine Hecke algebras and Macdonald’s conjectures.  Ann. of Math. (2), 141(1):191–216, 1995.
  * [Che05] Ivan Cherednik.  Double affine Hecke algebras, volume 319 of London Mathematical Society Lecture Note Series.  Cambridge University Press, Cambridge, 2005.
  * [Dd89a] V. G. Drinfel d.  Almost cocommutative Hopf algebras.  Algebra i Analiz, 1(2):30–46, 1989.
  * [Dd89b] V. G. Drinfel d.  Quasi-Hopf algebras.  Algebra i Analiz, 1(6):114–148, 1989.
  * [DM03] J. Donin and A. Mudrov.  Reflection equation, twist, and equivariant quantization.  Israel J. Math., 136:11–28, 2003.
  * [EG02] Pavel Etingof and Victor Ginzburg.  Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism.  Invent. Math., 147(2):243–348, 2002.
  * [EGGO07] Pavel Etingof, Wee Liang Gan, Victor Ginzburg, and Alexei Oblomkov.  Harish-Chandra homomorphisms and symplectic reflection algebras for wreath-products.  Publ. Math. Inst. Hautes Études Sci., (105):91–155, 2007.
  * [EK94] Pavel I. Etingof and Alexander A. Kirillov, Jr.  Macdonald’s polynomials and representations of quantum groups.  Math. Res. Lett., 1(3):279–296, 1994.
  * [EL05] Pavel Etingof and Frédéric Latour.  The dynamical Yang-Baxter equation, representation theory, and quantum integrable systems, volume 29 of Oxford Lecture Series in Mathematics and its Applications.  Oxford University Press, Oxford, 2005.
  * [EV00] Pavel Etingof and Alexander Varchenko.  Traces of intertwiners for quantum groups and difference equations. I.  Duke Math. J., 104(3):391–432, 2000.
  * [FFJ+11] B. Feigin, E. Feigin, M. Jimbo, T. Miwa, and E. Mukhin.  Quantum continuous $\germ{gl}_{\infty}$: semiinfinite construction of representations.  Kyoto J. Math., 51(2):337–364, 2011.
  * [GG06] Wee Liang Gan and Victor Ginzburg.  Almost-commuting variety, $\scr D$-modules, and Cherednik algebras.  IMRP Int. Math. Res. Pap., pages 26439, 1–54, 2006.  With an appendix by Ginzburg.
  * [GJS19] Iordan Ganev, David Jordan, and Pavel Safronov.  The quantum Frobenius for character varieties and multiplicative quiver varieties.  arXiv preprint arXiv:1901.11450, 2019.
  * [GS06] I. Gordon and J. T. Stafford.  Rational Cherednik algebras and Hilbert schemes. II. Representations and sheaves.  Duke Math. J., 132(1):73–135, 2006.
  * [GZ95] Anthony Giaquinto and James J. Zhang.  Quantum Weyl algebras.  J. Algebra, 176(3):861–881, 1995.
  * [Hai01] Mark Haiman.  Hilbert schemes, polygraphs and the Macdonald positivity conjecture.  J. Amer. Math. Soc., 14(4):941–1006, 2001.
  * [HC49] Harish-Chandra.  On representations of Lie algebras.  Ann. of Math. (2), 50:900–915, 1949.
  * [HC64] Harish-Chandra.  Invariant differential operators and distributions on a semisimple Lie algebra.  Amer. J. Math., 86:534–564, 1964.
  * [JL92] Anthony Joseph and Gail Letzter.  Local finiteness of the adjoint action for quantized enveloping algebras.  J. Algebra, 153(2):289–318, 1992.
  * [Jor14] David Jordan.  Quantized multiplicative quiver varieties.  Adv. Math., 250:420–466, 2014.
  * [JW20] David Jordan and Noah White.  The center of the reflection equation algebra via quantum minors.  J. Algebra, 542:308–342, 2020.
  * [Kir97] Alexander A. Kirillov, Jr.  Lectures on affine Hecke algebras and Macdonald’s conjectures.  Bull. Amer. Math. Soc. (N.S.), 34(3):251–292, 1997.
  * [KR08] Masaki Kashiwara and Raphaël Rouquier.  Microlocalization of rational Cherednik algebras.  Duke Math. J., 144(3):525–573, 2008.
  * [KS97] Anatoli Klimyk and Konrad Schmüdgen.  Quantum groups and their representations.  Texts and Monographs in Physics. Springer-Verlag, Berlin, 1997.
  * [LS95] T. Levasseur and J. T. Stafford.  Invariant differential operators and an homomorphism of Harish-Chandra.  J. Amer. Math. Soc., 8(2):365–372, 1995.
  * [LS96] T. Levasseur and J. T. Stafford.  The kernel of an homomorphism of Harish-Chandra.  Ann. Sci. École Norm. Sup. (4), 29(3):385–397, 1996.
  * [Lus88] G. Lusztig.  Quantum deformations of certain simple modules over enveloping algebras.  Adv. in Math., 70(2):237–249, 1988.
  * [Mac15] I. G. Macdonald.  Symmetric functions and Hall polynomials.  Oxford Classic Texts in the Physical Sciences. The Clarendon Press, Oxford University Press, New York, second edition, 2015.  With contribution by A. V. Zelevinsky and a foreword by Richard Stanley, Reprint of the 2008 paperback edition.
  * [Maj93] Shahn Majid.  Braided groups.  J. Pure Appl. Algebra, 86(2):187–221, 1993.
  * [Obl04] Alexei Oblomkov.  Double affine Hecke algebras and Calogero-Moser spaces.  Represent. Theory, 8:243–266, 2004.
  * [RT90] N. Yu. Reshetikhin and V. G. Turaev.  Ribbon graphs and their invariants derived from quantum groups.  Comm. Math. Phys., 127(1):1–26, 1990.
  * [SV11] O. Schiffmann and E. Vasserot.  The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials.  Compos. Math., 147(1):188–234, 2011.
  * [VV10] M. Varagnolo and E. Vasserot.  Double affine Hecke algebras at roots of unity.  Represent. Theory, 14:510–600, 2010.
  * [Wal93] Nolan R. Wallach.  Invariant differential operators on a reductive Lie algebra and Weyl group representations.  J. Amer. Math. Soc., 6(4):779–816, 1993.
  * [Wey39] Hermann Weyl.  The Classical Groups. Their Invariants and Representations.  Princeton University Press, Princeton, NJ, 1939.