# Intertwining operators between subregular Whittaker modules for
$\mathfrak{gl}_{N}$ and non-standard quantizations

Artem Kalmykov  Mission San Jose High School,
41717 Palm Avenue, Fremont, CA 94539, United States of America
<EMAIL_ADDRESS>and  Brian Li

###### Abstract.

In this paper, we study intertwining operators between subregular Whittaker
modules of $\mathfrak{gl}_{N}$ generalizing, on the one hand, the classical
exchange construction of dynamical quantum groups, on the other hand, earlier
results for principal W-algebras. We explicitly construct them using the
generators of W-algebras introduced by Brundan-Kleshchev. We interpret the
fusion on intertwining operators in terms of categorical actions and compute
the semi-classical limit of the corresponding monoidal isomorphisms which turn
out to depend on dynamical-like parameters.

##  1\. Introduction

Quantum groups are algebraic objects that arose from the study of the quantum
inverse scattering method for solving quantum integrable systems. By now they
have become a classical subject with connections to many areas of mathematics:
representations in positive characteristic, 3-manifolds and link invariants,
1-dimensional quantum integrable systems.

In general, it is hard to construct non-trivial quantum groups. In this paper,
we adopt a categorical approach via the _Tannakian reconstruction_[Saa72,
Ulb89, Ulb90]: essentially, a quantum group is defined by the collection of
its representations and by how one can “multiply” them; mathematically, it is
formalized by the notion of a _tensor category_ with a tensor structure on the
forgetful functor to vector spaces which is equivalent to a collection of
isomorphisms

(1) $J_{UV}\colon U\otimes V\rightarrow U\otimes V,$

satisfying the _twist equation_ (see [Eti+15]), for all representations $U,V$.
The “quantum” part of quantum groups usually means that these isomorphisms
come in families depending on a quantization parameter, and its first order
defines a Poisson-Lie structure, see [ES02].

One example of such an approach is the _exchange construction_ of Etingof-
Varchenko [EV99] that gives rise to _dynamical quantum groups_ which is a
certain variant of a quantum group depending on additional parameters. Roughly
speaking, the authors of _loc. cit._ consider a finite-dimensional analog of
_vertex operators_ parameterized by a representation of a reductive Lie
algebra: on the one hand, such operators have a natural multiplicative
structure given by composition, on the other hand, following the state-field
correspondence philosophy, they are in bijection with the elements of the
representation itself. In particular, the former induces a non-trivial tensor
structure on the collection of all the representations.

Finite-dimensional vertex operators are essentially maps between _Verma
modules_ ; the latter are defined by a highest-weight vector. It was observed
in [Kal23] that in the case of the general linear group $\mathrm{GL}_{N}$,
there is a version of the state-field correspondence for the _Whittaker_
modules generated by Whittaker vectors. In particular, we can apply a similar
exchange construction; it turns out that it produces a certain _non-standard_
quantum group. In fact, the corresponding solution to the _quantum Yang-Baxter
equation_ , which is closely related to quantum groups, was obtained by
Cremmer-Gervais [CG90] via studying the exchange algebra of the Toda field
theory that can be formulated using _affine W-algebras_ ; at the same time,
the natural setup for Whittaker modules is a _(finite) principal W-algebra_.

In general, a W-algebra is associated to a pair of a reductive Lie algebra and
a nilpotent element in it, see [Los10] for introduction to the subject. So,
one may ask: is there a similar result for other W-algebras? In this paper, we
study the exchange construction for the so-called _subregular_ nilpotent
element $e$ and the corresponding _subregular_ W-algebra $\mathcal{W}$; we
recall the definition in Section 5. We establish a state-field correspondence
in this case in Subsection 5.4 using remarkable generators of W-algebras for
$\mathfrak{gl}_{N}$ introduced by Brundan-Kleshchev [BK06]; we recall the
construction of _loc. cit._ in Section 3.

Unfortunately, in the subregular case, the exchange construction does _not_
give a quantum group. However, there is still a non-trivial tensor structure
analogous to (1). It turns out that the appropriate categorical setup for it
is that of _module categories_ , see Section 2. More precisely, as in [KS22]
and [Kal23], we formulate the exchange construction in terms of the category
of _Harish-Chandra bimodules_ and the _finite Drinfeld-Sokolov reduction_
functor, see Section 4. The finite Drinfeld-Sokolov reduction functor is a
direct generalization of the definition of a W-algebra via quantum Hamiltonian
reduction to the category of Harish-Chandra bimodules and refers to the
quantum Drinfeld-Sokolov reduction, see [FB04]. Then, the subregular analog of
the tensor isomorphisms (1) comes from the induced categorical action of the
category of $\mathrm{GL}_{N}$-representations
$\operatorname{Rep}(\mathrm{GL}_{N})$ on the category of right
$\mathcal{W}$-modules, see Theorem 5.9. Unlike in the regular case of [Kal23],
they depend on a dynamical-like parameters lying on a _non-abelian_ Lie
subalgebra of $\mathfrak{gl}_{N}$, and take the form

(2) $J_{UV}\colon U\otimes V\otimes\mathcal{W}\rightarrow U\otimes
V\otimes\mathcal{W}.$

for $U,V$ representations of $\mathrm{GL}_{N}$.

The main results of the paper are contained in Subsection 5.5. For instance,
we provide an algorithm to compute the monoidal isomorphisms (1). Finally, we
compute its semi-classical limit by explicitly constructing the state-field
correspondence for the vector representation $V=\mathbf{C}^{N}$ using the
W-algebra generators from Brundan-Kleshchev, refer to Theorem 5.12 and Theorem
5.13. Denote by $E_{i,j}\in\mathfrak{gl}_{N}$ the matrix units. Consider the
two-dimensional subalgebra $\mathfrak{l}=\mathrm{span}(E_{2,1},E_{1,1})$. Let
$x_{21},x_{11}\in\mathfrak{l}$ be functions on $\mathfrak{l}^{*}$
corresponding to $E_{2,1},E_{1,1}$. The main result of the paper is Theorem
5.23.

######  Theorem.

The semi-classical limit $\mathbf{j}$ of $J_{UV}$ from (2) is

$\mathbf{j}=\mathbf{j}_{c}+\sum\limits_{j=2}^{N-2}\sum\limits_{i=j+2}^{N}\sum\limits_{r=2}^{i-j}(-1)^{i-j-r}x_{21}x_{11}^{i-j-r}E_{1,r}\otimes
E_{i,j}+\sum\limits_{i=4}^{N}\sum\limits_{r=2}^{i-2}(-1)^{i-r}x_{11}^{i-r-1}E_{1,r}\otimes
E_{i,1}.$

Here $\mathbf{j}_{c}$ is constant, and it comes from a Frobenius-like
structure given by the trace pairing with the subregular nilpotent element $e$
on a certain subspace of $\mathfrak{gl}_{N}$ which we call the _(subregular)
wonderbolic_ subspace:

$\mathfrak{w}=\begin{pmatrix}0&*&\ldots&*&0\\\ 0&*&\ldots&*&0\\\
*&*&\ldots&*&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\ *&*&\ldots&*&0\\\
\end{pmatrix}.$

See Subsection 5.2 for the explicit form for $\mathbf{j}_{c}$. The wonderbolic
subspace plays a similar role to that of the _mirabolic subalgebra_ in the
regular case, which was the main motivation for the name.

It would be interesting to obtain an invariant description of the “dynamical”
part similar to the constant one.

### Organization of the paper

In Section 2, we introduce general notions, including that of a module
category, that we will use in the paper. In Section 3, we recall the Brundan-
Kleshchev’s construction [BK06] of W-algebras for $\mathfrak{gl}_{N}$ in terms
of pyramids. In Section 4, we present the exchange construction in categorical
terms as a finite Drinfeld-Sokolov reduction functor from the category of
Harish-Chandra bimodules. In Section 5, we specialize to the case of
subregular W-algebras: in Subsection 5.4, we compute Whittaker vectors for the
vector representation of $\mathfrak{gl}_{N}$, and in Subsection 5.5, we
compute the corresponding monoidal structure.

### Acknowledgments

This research was conducted while B.L. was a participant in the MIT PRIMES
program; we would like to thank the organizers for making this research
opportunity possible. A.K. would like to thank the Department of Mathematics
of Massachusetts Institute of Technology for hospitality.

##  2\. Background

###  2.1. General setup

In this section, we introduce general notations that we use in the paper.

We work over the field of complex numbers $\mathbf{C}$. Throughout the paper,
we use $\hbar$-versions of constructions in questions. To avoid categorical
complications, we treat $\hbar$ as a _non-formal_ parameter, i.e.
$\hbar\in\mathbf{C}^{\times}$ (for instance, we can still deal with
$\mathbf{C}$-linear categories). The reader may safely assume that $\hbar=1$.
The only purpose of introducing $\hbar$ is to compute classical limits of
certain formulas, and it will be clear from the context how to make sense of
the corresponding $\hbar$-family over $\mathbb{A}^{1}$.

Let $\mathfrak{g}$ be a Lie algebra.

######  Definition 2.1.

The _asymptotic universal enveloping algebra_
$\mathrm{U}_{\hbar}(\mathfrak{g})$ of $\mathfrak{g}$ is a tensor algebra over
$\mathbf{C}$, generated by the vector space $\mathfrak{g}$, with the relations

$xy-yx=\hbar[x,y],\ x,y\in\mathfrak{g}.$

For any $x,y\in\mathrm{U}_{\hbar}(\mathfrak{g})$, the _commutator_ $[x,y]$ is

(2.1) $[x,y]:=\frac{xy-yx}{\hbar}.$

Observe that it is well-defined over $\mathbf{C}[\hbar]$.

######  Remark 2.2.

Usually, asymptotic universal enveloping algebras are defined over the
polynomial ring $\mathbf{C}[\hbar]$. Here, as we mentioned at the beginning of
the section, we treat $\hbar$ just as a complex number.

In this paper, we will be dealing with a general linear Lie algebra. Let
$\mathrm{GL}_{N}$ be the group of invertible $N\times N$-matrices and
$\mathfrak{gl}_{N}$ be its Lie algebra identified with the space of $N\times
N$-matrices. We choose a natural basis $\\{E_{i,j}|1\leq i,j\leq N\\}$ of
matrix units with the commutator

$[E_{i,j},E_{k,l}]=\delta_{jk}E_{i,l}-\delta_{li}E_{k,j}.$

###  2.2. Module categories

In this subsection, we recall the notion of a module category over a tensor
category, for instance, see [Eti+15, Chapter 7].

Recall that a monoidal category is a plain category $\mathcal{C}$ equipped
with a bifunctor
$\otimes\colon\mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$ together
with a unit object $\mathbf{1}_{\mathcal{C}}$ and a natural isomorphism
(associativity constraint)

$a_{X,Y,Z}\colon(X\otimes Y)\otimes Z\xrightarrow{\sim}X\otimes(Y\otimes Z),$

satisfying the _unit_ and _pentagon_ axioms, see [Eti+15, Chapter 2]. Monoidal
categories are categorical analogs of algebras. Likewise, there is a
categorical analog of modules over an algebra.

######  Definition 2.3.

Let $\mathcal{C}$ be a monoidal category. A _(right) module category_ over
$\mathcal{C}$ is a plain category $\mathcal{M}$ equipped with a bifunctor

$\otimes\colon\mathcal{M}\times\mathcal{C}\rightarrow\mathcal{M}$

with natural isomorphisms

$m_{M,X,Y}\colon M\otimes(X\otimes Y)\xrightarrow{\sim}(M\otimes X)\otimes Y,$

for all $M\in\mathcal{M},\ X,Y\in\mathcal{C}$, such that the functor
$M\otimes\mathbf{1}_{\mathcal{C}}\mapsto M$ is an autoequivalence of
$\mathcal{M}$ and the associativity constraint $m$ satisfies the _pentagon
axiom_ , shown below.

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
46.82628pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 70.82628pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{M\otimes(X\otimes(Y\otimes
Z))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
164.43646pt\raise-13.49722pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.35834pt\hbox{$\scriptstyle{\mathrm{id}_{M}\otimes
a_{X,Y,Z}^{-1}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern
204.4306pt\raise-29.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
37.91992pt\raise-14.42084pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.43472pt\hbox{$\scriptstyle{m_{M,X,Y\otimes Z}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 30.88477pt\raise-29.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(M\otimes X)\otimes(Y\otimes
Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.43472pt\hbox{$\scriptstyle{m_{M\otimes X,Y,Z}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-69.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
114.65256pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
188.47884pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{M\otimes((X\otimes Y)\otimes
Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
235.30511pt\raise-60.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.43472pt\hbox{$\scriptstyle{m_{M,X\otimes
Y,Z}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern
235.30511pt\raise-69.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{((M\otimes X)\otimes Y)\otimes
Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
97.96564pt\raise-73.49722pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.35834pt\hbox{$\scriptstyle{m_{M,X,Y}\otimes\mathrm{id}_{Z}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 188.47884pt\raise-80.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
114.65256pt\raise-80.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
188.47884pt\raise-80.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{(M\otimes(X\otimes Y))\otimes
Z}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

######  Example 2.4.

Tautologically, any monoidal category $\mathcal{C}$ is a module category over
itself.

There is also a generalization of a homomorphism between algebra modules.

######  Definition 2.5.

Let $\mathcal{C}$ be a monoidal category and $\mathcal{M}_{1},\mathcal{M}_{2}$
be two module categories over $\mathcal{C}$. A _functor of
$\mathcal{C}$-module categories_ is a plain functor
$F\colon\mathcal{M}_{1}\rightarrow\mathcal{M}_{2}$ with a collection of
natural isomorphisms

(2.2) $J_{M,X}\colon F(M\otimes X)\rightarrow F(M)\otimes X$

for all $M\in\mathcal{M},\ X\in\mathcal{C}$, satisfying the compatibility
condition

$\textstyle{F(M\otimes(X\otimes
Y))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F(m_{M,X,Y})}$$\scriptstyle{J_{M,X\otimes
Y}}$$\textstyle{F((M\otimes X)\otimes
Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{J_{M\otimes
X,Y}}$$\textstyle{F(M)\otimes(X\otimes
Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{n_{F(M),X,Y}}$$\textstyle{F(M\otimes
X)\otimes
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{J_{M,X}\otimes\mathrm{id}_{Y}}$$\textstyle{(F(M)\otimes
X)\otimes Y}$

and

$\textstyle{F(M\otimes\mathbf{1}_{\mathcal{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{J_{M,\mathbf{1}_{\mathcal{C}}}}$$\textstyle{F(M)\otimes\mathbf{1}_{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F(M)}$

where the diagonal arrows come from the corresponding unit autoequivalences.

##  3\. W-algebras for $\mathfrak{gl}_{N}$

In general, a _finite W-algebra_ is associated to a pair $(\mathfrak{g},e)$ of
a reductive Lie algebra $\mathfrak{g}$ and a nilpotent element
$e\in\mathfrak{g}$, see [Los10] for a survey of the subject. From now on, we
will be interested in the case $\mathfrak{g}=\mathfrak{gl}_{N}$; according to
[BK06], the W-algebras in this case admit a description in terms of
combinatorial objects called _pyramids_. In what follows, we recall this
description; for the details and proofs, we refer the reader to _loc. cit._

######  Definition 3.1.

[BK06, Section 7] A _pyramid_ $\pi$ is a sequence of positive numbers
$(q_{1},\ldots,q_{l})$, the _column heights_ , such that
$\sum\limits_{i=1}^{l}q_{i}=N$ and

$0<q_{1}\leq\dots\leq q_{k},\quad q_{k+1}\geq\dots\geq q_{l}>0$

for some $k\leq l$. The _maximal height_ $n$ is $\max(q_{1},\ldots,q_{l})$.

We number the blocks starting from top to bottom and from left to right; for
each $i$, we denote by $\mathrm{col}(i)$ (resp. $\mathrm{row}(i)$) the
corresponding column (resp. row) of $i$ counted from left to right (resp. from
top to bottom). Here is an example of a pyramid

with column heights $(q_{1},q_{2},q_{3},q_{4})=(1,3,2,1)$. For instance,
$\mathrm{col}(3)=2$ and $\mathrm{row}(3)=2$. Observe the rows must be in non-
decreasing order.

In what follows, we will need an inductive structure on pyramids.

######  Definition 3.2.

For a pyramid $\pi$, the _k-th truncated pyramid_ ${}_{k}\pi$ is $\pi$ without
the last $k$ rightmost columns. We denote by ${}_{k}N$ the number of blocks in
${}_{k}\pi$. When $k=1$, we also denote $\dot{\pi}:={}_{1}\pi$.

For instance, we have

for $\pi$ as above. Introduce a $\mathbf{Z}$-grading on
$\mathfrak{g}=\bigoplus_{j\in\mathbf{Z}}\mathfrak{g}_{j}$ by declaring that
$\deg(E_{ij})=\mathrm{col}(j)-\mathrm{col}(i)$. Let

(3.1) $\mathfrak{p}=\bigoplus\limits_{j\geq
0}\mathfrak{g}_{j},\qquad\mathfrak{m}=\bigoplus\limits_{j<0}\mathfrak{g}_{j}.$

Similarly to 3.2, we will use an inductive structure on the algebras.

######  Definition 3.3.

The _k-th truncated nilpotent subalgebra_ ${}_{k}\mathfrak{m}$ (resp. _k-th
truncated parabolic subalgebra_) is the nilpotent subalgebra (resp. the
parabolic subalgebra) associated to the truncated pyramid ${}_{k}\pi$.
Alternatively, we denote them by $\mathfrak{m}_{{}_{k}N}$ (resp.
$\mathfrak{p}_{{}_{k}N}$) if the truncation is clear from the context.

To a pyramid $\pi$, we assign a nilpotent element $e$ defined by

(3.2) $e:=\sum\limits_{\begin{subarray}{c}1\leq i,j\leq N\\\
\mathrm{row}(i)=\mathrm{row}(j)\\\
\mathrm{col}(i)=\mathrm{col}(j)-1\end{subarray}}E_{i,j}.$

For instance, for pyramid (3), we have $e=E_{3,5}+E_{1,4}+E_{4,6}+E_{6,7}$.

The nilpotent element $e$ defines a character $\psi$ of $\mathfrak{m}$:

$\psi\colon\mathfrak{m}\rightarrow\mathbf{C},\qquad x\mapsto\mathrm{Tr}(ex).$

Denote by

(3.3)
$Q:=\mathrm{U}_{\hbar}(\mathfrak{g})\otimes_{\mathrm{U}_{\hbar}(\mathfrak{m})}\mathbf{C}^{\psi},$

where $\mathfrak{m}$ acts on $\mathbf{C}^{\psi}$ via the character $\psi$. It
is naturally a left $\mathrm{U}_{\hbar}(\mathfrak{g})$-module. As a vector
space, we can identify

(3.4) $Q\cong\mathrm{U}_{\hbar}(\mathfrak{p})$

by the PBW theorem.

For any $\xi\in\mathfrak{m}$, denote by $\xi^{\psi}=\xi-\psi(\xi)$, and define
the shift

(3.5)
$\mathfrak{m}^{\psi}=\mathrm{span}(\xi^{\psi}|\xi\in\mathfrak{m})\subset\mathrm{U}_{\hbar}(\mathfrak{g}).$

######  Definition 3.4.

[BK06, Section 8] A _finite W-algebra_ $\mathcal{W}$, associated to the
nilpotent element $e$ (3.2), is the space

$\mathcal{W}:=Q^{\mathfrak{m}^{\psi}}=\\{w\in Q|\xi^{\psi}w=0\
\forall\xi\in\mathfrak{m}\\}$

of $\mathfrak{m}^{\psi}$-invariant vectors in $Q$.

In [BK06], the authors introduced explicit generators of $\mathcal{W}$ whose
construction we recall now. Observe that in _loc. cit._ , the authors use the
version with $\hbar=1$; to pass to the “asymptotic” version, one can use the
Rees construction with respect to the _Kazhdan filtration_ , see [BK06,
(8.3)]. We will explicitly indicate how to modify the corresponding
definitions and statements.

Let $\rho_{\pi,r}=n-\sum\limits_{k=r}^{l}q_{k}$. Introduce modified generators

(3.6)
$\widetilde{E}_{i,j}=(-1)^{\mathrm{col}(j)-\mathrm{col}(i)}(E_{i,j}+\delta_{ij}\hbar\rho_{\pi,\mathrm{col}(i)})$

for all $1\leq i,j\leq n$.

######  Definition 3.5.

The _Kazhdan filtration_ on $\mathrm{U}_{\hbar}(\mathfrak{g})$ is defined by
declaring that $\deg(E_{i,j})=\mathrm{col}(j)-\mathrm{col}(i)+1$.

In particular, assigning $\deg(\hbar)=1$, we see that this modification
preserves the filtration.

Let $1\leq x\leq n$.

######  Definition 3.6.

[BK06, Section 9] Consider the set of signs
$\sigma_{1}=\sigma_{2}=\ldots=\sigma_{x}=-,\sigma_{x+1}=\ldots=\sigma_{n}=+$.
For any $1\leq i,j\leq N$, define $T_{ij,x}^{(0)}:=\delta_{ij}\sigma_{i}$ and,
for $r>0$,

(3.7)
$T_{ij;x}^{(r)}=\sum_{s=1}^{r}\sum\limits_{\begin{subarray}{c}i_{1},\ldots,i_{s}\\\
j_{1},\ldots,j_{s}\end{subarray}}\sigma_{\mathrm{row}(j_{1})}\cdots\sigma_{\mathrm{row}(j_{s-1})}\widetilde{E}_{i_{1},j_{1}}\cdots\widetilde{E}_{i_{s},j_{s}},$

with the following conditions on $1\leq
i_{1},\ldots,i_{s},j_{1},\ldots,j_{s}\leq N$:

  1. (1)

$\mathrm{row}(i_{1})=i,\mathrm{row}(j_{s})=j$

  2. (2)

$\mathrm{row}(j_{k})=\mathrm{row}(i_{k+1})\text{ for all }1\leq k\leq s-1$

  3. (3)

$\mathrm{col}(i_{k})\leq\mathrm{col}(j_{k})\text{ for all }1\leq k\leq s$

  4. (4)

$\sum\limits_{k=1}^{s}(\mathrm{col}(j_{k})-\mathrm{col}(i_{k})+1)=r$

  5. (5)

If $\sigma_{\mathrm{row}(j_{k})}=+$, then
$\mathrm{col}(j_{k})<\mathrm{col}(i_{k+1})\text{ for all }1\leq k\leq s-1$.

  6. (6)

If $\sigma_{\mathrm{row}(j_{k})}=-$, then
$\mathrm{col}(j_{k})\geq\mathrm{col}(i_{k+1})\text{ for all }1\leq k\leq s-1$.

It follows from condition (3) that
$T_{ij;x}^{(r)}\in\mathrm{U}_{\hbar}(\mathfrak{p})$ by (3.4). Condition (4)
can be equivalently reformulated that the degree of $T_{ij;x}^{(r)}$ with
respect to the Kazhdan filtration is $r$. For instance, when $r=1$, it can be
shown that

(3.8) $T_{ij;x}^{(1)}=\sum\limits_{\begin{subarray}{c}1\leq h,k\leq n\\\
\mathrm{col}(h)=\mathrm{col}(k)\\\
\mathrm{row}(h)=i,\mathrm{row}(k)=j\end{subarray}}\widetilde{E}_{h,k}.$

Using these elements, the authors of [BK06] constructed generators of a
W-algebra corresponding to a pyramid $\pi$. We will give a precise statement
for the subregular case in Section 5.

We will also need an inductive structure on these elements. Recall 3.2. Let
$\mathfrak{gl}_{{}_{k}N}$ be the Lie algebra corresponding to the truncated
pyramid ${}_{k}\pi$. Denote by ${}_{k}\widetilde{E}_{i,j}$ the corresponding
modified generators (3.6). Consider a (non-standard) embedding

(3.9)
$\iota\colon\mathrm{U}_{\hbar}(\mathfrak{gl}_{{}_{k}N})\rightarrow\mathrm{U}_{\hbar}(\mathfrak{gl}_{N}),\
\iota({}_{k}\widetilde{E}_{i,j})=\widetilde{E}_{i,j}.$

Define the truncated analog of elements (3.7) as

(3.10) ${}_{k}T_{ij;x}^{(r)}:=\iota(T_{ij;x}^{(r)}).$

##  4\. Finite Drinfeld-Sokolov reduction

###  4.1. Harish-Chandra bimodules

Let $G$ be an affine algebraic group over $\mathbf{C}$ and $\mathfrak{g}$ be
its Lie algebra. Denote by $\mathrm{U}_{\hbar}(\mathfrak{g})$ the universal
enveloping algebra of $\mathfrak{g}$ as in 2.1. Let $\operatorname{Rep}(G)$ be
the category of $G$-representations. Naturally,
$\mathrm{U}_{\hbar}(\mathfrak{g})$ is an object in $\operatorname{Rep}(G)$.

######  Definition 4.1.

A _Harish-Chandra bimodule_ is a left
$\mathrm{U}_{\hbar}(\mathfrak{g})$-module $X$ in the category
$\operatorname{Rep}(G)$. In other words, it has a structure of a
$G$-representation and a left $\mathrm{U}_{\hbar}(\mathfrak{g})$-module such
that the action morphism

$\mathrm{U}_{\hbar}(\mathfrak{g})\otimes X\rightarrow X$

is a homomorphism of $G$-representations. The category of Harish-Chandra
bimodules is denoted by $\mathrm{HC}_{\hbar}(G)$.

There is a natural right $\mathrm{U}_{\hbar}(\mathfrak{g})$-module structure
on any Harish-Chandra bimodule $X$ (justifying the name). Namely, for
$\xi\in\mathfrak{g}$, denote by $\operatorname{ad}_{\xi}\colon X\rightarrow X$
the derivative of the $G$-action on $X$ along $\xi$. Then we can define

(4.1) $x\xi:=\xi x-\hbar\operatorname{ad}_{\xi}(x),\ x\in X,$

and extend it to a right $\mathrm{U}_{\hbar}(\mathfrak{g})$-action. Therefore,
the category $\mathrm{HC}_{\hbar}(G)$ is a subcategory of
$\mathrm{U}_{\hbar}(\mathfrak{g})$-bimodules, hence is equipped with a tensor
structure:

$X\otimes^{\mathrm{HC}_{\hbar}(G)}Y:=X\otimes_{\mathrm{U}_{\hbar}(\mathfrak{g})}Y.$

There is a natural functor of the so-called _free Harish-Chandra bimodules_ :

(4.2)
$\mathrm{free}\colon\operatorname{Rep}(G)\rightarrow\mathrm{HC}_{\hbar}(G),\qquad
V\mapsto\mathrm{U}_{\hbar}(\mathfrak{g})\otimes V.$

One can check that this functor is monoidal. In fact, all Harish-Chandra
bimodules can be “constructed” from the free ones.

######  Proposition 4.2.

[KS22, Proposition 2.7] The category $\mathrm{HC}_{\hbar}(G)$ is generated by
$\mathrm{free}(V)$ for $V\in\operatorname{Rep}(G)$.

###  4.2. Drinfeld-Sokolov reduction

Now let restrict to the case $G=\mathrm{GL}_{N}$ and
$\mathfrak{g}=\mathfrak{gl}_{N}$. We use the notations from Section 3, in
particular, we fix a pyramid $\pi$ and consider the corresponding nilpotent
subalgebra $\mathfrak{m}$ with a character $\psi\in\mathfrak{m}^{*}$.

######  Definition 4.3.

A _Whittaker module_ is a left $\mathrm{U}_{\hbar}(\mathfrak{g})$-module $M$
such that the action of $\mathfrak{m}^{\psi}$ from (3.5) is locally nilpotent.
A _Whittaker vector_ is an $\mathfrak{m}^{\psi}$-invariant vector in $m\in M$,
i.e. satisfying

$\xi^{\psi}m=0\ \text{ for all }\xi\in\mathfrak{m}.$

The space of Whittaker vectors is denoted by $M^{\mathfrak{m}^{\psi}}$.

######  Example 4.4.

In $\mathfrak{gl}_{2}$, the series

$P^{\psi}=\sum_{k=0}^{\infty}(-1)^{k}\frac{E_{1,1}(E_{1,1}+1)\cdots(E_{1,1}+k-1)}{k!}(E_{2,1}-1)^{k}$

is $(E_{21}-1)$-invariant on the left action and generates the Whittaker
vectors; see [Kal21] for a version in the left quotient.

Denote by $\mathrm{Wh}_{\hbar}$ the category of
$(\mathrm{U}_{\hbar}(\mathfrak{g}),\mathcal{W})$-bimodules that are Whittaker
with respect to the $\mathrm{U}_{\hbar}(\mathfrak{g})$-action. Naturally, the
quotient $Q$ from (3.3) is an object of $\mathrm{Wh}_{\hbar}$. In particular,
it defines an action functor

$\mathrm{act}_{\mathfrak{g}}^{\psi}\colon\mathrm{HC}_{\hbar}(G)\rightarrow\mathrm{Wh}_{\hbar},\qquad
X\mapsto X\otimes_{\mathrm{U}_{\hbar}(\mathfrak{g})}Q.$

Likewise, there is an action

(4.3)
$\mathrm{act}_{\mathcal{W}}\colon{}_{\mathcal{W}}\mathrm{BiMod}_{\mathcal{W}}\rightarrow\mathrm{Wh}_{\hbar},\qquad
Y\mapsto Q\otimes_{\mathcal{W}}Y.$

Consider the functor

$(-)^{\mathfrak{m}^{\psi}}\colon\mathrm{Wh}_{\hbar}\rightarrow{}_{\mathcal{W}}\mathrm{BiMod}_{\mathcal{W}}$

of Whittaker invariants sending a Whittaker module to its space of Whittaker
vectors.

The following result is a direct consequence of _Skryabin’s equivalence_
[Pre02].

######  Theorem 4.5.

The functor $(-)^{\mathfrak{m}^{\psi}}$ is an equivalence.

This motivates the following definition.

######  Definition 4.6.

The _(finite) Drinfeld-Sokolov reduction_ is the functor

$\mathrm{res}^{\psi}\colon\mathrm{HC}_{\hbar}(\mathrm{GL}_{N})\rightarrow{}_{\mathcal{W}}\mathrm{BiMod}_{\mathcal{W}},\qquad
X\mapsto(X\otimes_{\mathrm{U}_{\hbar}(\mathfrak{g})}Q)^{\mathfrak{m}^{\psi}}.$

In what follows, for any Harish-Chandra bimodule $X$, we denote

$X/\mathfrak{m}^{\psi}:=X\otimes_{\mathrm{U}_{\hbar}(\mathfrak{g})}Q.$

######  Remark 4.7.

There is an equivalent presentation of the Drinfeld-Sokolov reduction that we
will use later in the paper. Namely, recall the adjoint
$\mathfrak{gl}_{N}$-action from Subsection 4.1. For any Harish-Chandra
bimodule $X$, define

$\operatorname{ad}_{m}([x]):=[\operatorname{ad}_{m}(x)]\in
X/\mathfrak{m}^{\psi},\ m\in\mathfrak{m},[x]\in X/\mathfrak{m}^{\psi}.$

We will also use the notation

$[m,x]:=\operatorname{ad}_{m}([x]),\ m\in\mathfrak{m},x\in X,$

if the quotient is clear from the context. Since $\psi$ is a character, this
action is well-defined. One can easily see that

$\hbar\cdot\operatorname{ad}_{m}([x])=m^{\psi}\cdot[x].$

In particular, the space of Whittaker vectors in $X/\mathfrak{m}^{\psi}$ can
be identified with the space of $\operatorname{ad}_{\mathfrak{m}}$-invariant
vectors in $X/\mathfrak{m}^{\psi}$.

As in [KS22, Corollary 4.18], we obtain the following.

######  Theorem 4.8.

The Drinfeld-Sokolov reduction is colimit-preserving and monoidal.

Explicitly, the monoidal structure is given by the usual product on quantum
Hamiltonian reductions:

(4.4)
$(X/\mathfrak{m}^{\psi})^{\mathfrak{m}^{\psi}}\otimes_{\mathcal{W}}(Y/\mathfrak{m}^{\psi})^{\mathfrak{m}^{\psi}}\xrightarrow{\sim}(X\otimes_{\mathrm{U}_{\hbar}(\mathfrak{g})}Y/\mathfrak{m}^{\psi})^{\mathfrak{m}^{\psi}},\
[x]\otimes[y]\mapsto[x\otimes y].$

In particular, composing with the monoidal functor of free Harish-Chandra
bimodules (4.2), we get a monoidal functor:

(4.5)
$\operatorname{Rep}(G)\rightarrow{}_{\mathcal{W}}\mathrm{BiMod}_{\mathcal{W}}.$

We study its properties in the next section.

##  5\. Subregular case

In this section, we apply the finite Drinfeld-Sokolov reduction to subregular
W-algebras and study its tensor properties.

###  5.1. Pyramid

Recall from Section 3 that W-algebras for $\mathfrak{gl}_{N}$ are described by
pyramids. In the subregular case, the corresponding pyramid is

(5.1)

and by (3.2), the subregular nilpotent is given by

(5.2) $e=E_{2,3}+\ldots+E_{N-1,N}.$

The nilpotent algebra $\mathfrak{m}$ is

$\mathfrak{m}=\mathrm{span}(E_{i,j}|3\leq i\leq N,j<i),$

namely,

(5.3) $\mathfrak{m}=\begin{pmatrix}0&0&0&\ldots&0&0\\\ 0&0&0&\ldots&0&0\\\
*&*&0&\ldots&0&0\\\ *&*&*&\ldots&0&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ *&*&*&\ldots&*&0\\\
\end{pmatrix}.$

The parabolic subalgebra $\mathfrak{p}$ is

(5.4) $\mathfrak{p}=\begin{pmatrix}*&*&*&\ldots&*&*\\\ *&*&*&\ldots&*&*\\\
0&0&*&\ldots&*&*\\\ 0&0&0&\ldots&*&*\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&0&\ldots&0&*\\\
\end{pmatrix}.$

###  5.2. Semi-classical limit

It turns out that the semi-classical limits of the tensor structure on (4.5)
is intrinsically related not to the whole Lie algebra $\mathfrak{gl}_{N}$, but
to its subspace of a certain almost parabolic subalgebra.

######  Definition 5.1.

The _subregular wonderbolic subspace_ $\mathfrak{w}$ (for the rest of the
paper, simply _wonderbolic subspace_) is the subspace of matrices of the form

$\mathfrak{w}=\begin{pmatrix}0&*&\ldots&*&0\\\ 0&*&\ldots&*&0\\\
*&*&\ldots&*&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\ *&*&\ldots&*&0\\\
\end{pmatrix}.$

While the subregular nilpotent element $e$ from (5.2) does not lie in
$\mathfrak{w}$, it defines the following 2-form on $\mathfrak{w}$:

$\omega\colon\mathfrak{w}\wedge\mathfrak{w}\rightarrow\mathbf{C},\ x\wedge
y\mapsto\mathrm{Tr}(e\cdot[x,y]).$

Observe that the nilpotent subalgebra $\mathfrak{m}$ from (5.3) lies in
$\mathfrak{w}$, moreover, it is isotropic with respect to $\omega$. A natural
complement is given by the Borel subalgebra

$\mathfrak{b}=\mathrm{span}(E_{k,l}|1\leq k\leq l\leq N-1,2\leq l).$

Namely,

(5.5) $\mathfrak{b}=\begin{pmatrix}0&*&*&\ldots&*&0\\\ 0&*&*&\ldots&*&0\\\
0&0&*&\ldots&*&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&\ldots&*&0\\\ 0&0&0&\ldots&0&0\\\ \end{pmatrix}.$

Similarly to 3.3, we will use the following.

######  Definition 5.2.

The _k-th truncated Borel subalgebra_ ${}_{k}\mathfrak{b}$ is the Borel
subalgebra as in (5.5) associated to the truncated pyramid ${}_{k}\pi$ from
3.2. Alternatively, we denote it by $\mathfrak{b}_{{}_{k}N}$, if the
truncation is clear from the context.

It turns out that $\omega$ is symplectic and both spaces are Lagrangian.

######  Proposition 5.3.

The form $\omega$ is non-degenerate with inverse
$r_{\mathfrak{w}}=\mathbf{j}_{c}-\mathbf{j}_{c}^{21}$, where
$\mathbf{j}_{c}^{21}$ is uniquely defined by

(5.6) $\mathbf{j}_{c}^{21}(E_{i,j}^{*})=\begin{cases}\delta_{i>2}E_{j,i-1},\
j=1,2\\\ E_{j,i-1}-\mathbf{j}_{c}^{21}(E_{i-1,j-1}^{*}),\ j\geq 3\end{cases}$

for any $E_{i,j}^{*}\in\mathfrak{m}^{*}$, where $E_{i,j}^{*}$ is the dual
basis and we consider $\mathbf{j}_{c}^{21}$ as a map
$\mathfrak{m}^{*}\rightarrow\mathfrak{b}$.

###### Proof.

Observe that both $\mathfrak{m}$ and $\mathfrak{b}$ are isotropic subspaces
with respect to $\omega$. Since the latter is skew-symmetric, it is enough to
construct an inverse $-\mathbf{j}_{c}^{21}$ only of one map, say
$\omega\colon\mathfrak{b}\rightarrow\mathfrak{m}$. Observe that

$\omega(E_{j,i-1})=\begin{cases}-\delta_{i>2}E_{i,j}^{*},\ j=1,2\\\
-E_{i,j}^{*}+E_{i-1,j-1}^{*},\ j\geq 3\end{cases}$

for $i\geq j+1$. Then (5.6) follows. Note that these equations allow to
construct $\mathbf{j}_{c}^{21}$ inductively, starting from $j=1$ and $j=2$. In
particular, they define the inverse. ∎

######  Remark 5.4.

The subregular wonderbolic subspace is an analog of the _mirabolic subalgebra_
in the case of the regular nilpotent element in the same way
$r_{\mathfrak{w}}$ is an analog of the rational Cremmer-Gervais $r$-matrix,
see [Kal23]. One main difference is that it is not a subalgebra, thus
$r_{\mathcal{W}}$ does not satisfy the classical Yang-Baxter equation.
However, it turns out $\mathbf{j}_{c}$ is the constant part of the semi-
classical limit of the tensor structure on Whittaker vectors. As the reader
will see in Subsection 5.5, in addition to the constant part
$r_{\mathfrak{w}}$, the semi-classical limit of the tensor structure also
involves certain “dynamical” parameters lying on the subalgebra spanned by
$\\{E_{11},E_{21}\\}$.

###  5.3. Whittaker vectors: general setup

In this subsection, we show that the Drinfeld-Sokolov reduction functor (4.5)
admits a canonical “trivialization.”

Recall the elements $T_{ij;x}^{(r)}$ from (3.7) and their truncated analogs
${}_{k}T_{ij;x}^{(r)}$ from (3.10). As we mentioned in Section 3, the authors
of [BK06] considered the case $\hbar=1$, however, all the proofs can be
translated _mutatis mutandis_ to their $\hbar$-versions and will not be
mentioned explicitly here.

Recall also that a W-algebra is defined as the quantum Hamiltonian reduction
$(\mathrm{U}_{\hbar}(\mathfrak{g})/\mathfrak{m}^{\psi})^{\mathfrak{m}^{\psi}}$.
Since $T_{ij;x}^{(r)}\in\mathrm{U}_{\hbar}(\mathfrak{p})$ by construction, we
may treat them as elements in the quotient
$\mathrm{U}_{\hbar}(\mathfrak{g})/\mathfrak{m}^{\psi}$. By combination of a
particular case of the fundamental result [BK06, Theorem 10.1], identifying
W-algebras with _truncated shifted Yangians_ , and [BK06, Corollary 6.3],
subregular W-algebras admit an explicit presentation.

######  Theorem 5.5.

The monomials in the elements

$T_{11;0}^{(1)},\qquad T_{21;1}^{(1)},\qquad
T_{12;1}^{(N-1)},\qquad\\{T_{22;1}^{(r)}\\}_{1\leq r\leq N-1},$

taken in any fixed order, form a basis of the subregular W-algebra
$\mathcal{W}$.

Observe that $T_{11;0}^{(1)}=E_{1,1}-(N-2)\hbar$ and
$T_{21;1}^{(1)}=-E_{2,1}$. We will consider the following subalgebra of
$\mathfrak{p}$:

(5.7) $\mathfrak{l}:=\mathrm{span}(E_{2,1},E_{1,1}).$

For any left $\mathrm{U}_{\hbar}(\mathfrak{g})$-module $X$, denote by
$\mathfrak{b}\backslash
X:=\mathbf{C}\otimes_{\mathrm{U}_{\hbar}(\mathfrak{b})}\mathrm{U}_{\hbar}(\mathfrak{g})$,
where $\mathfrak{b}$ acts on $\mathbf{C}$ trivially. Consider the composition

(5.8)
$\mathcal{W}=(\mathrm{U}_{\hbar}(\mathfrak{g})/\mathfrak{m}^{\psi})^{\mathfrak{m}^{\psi}}\hookrightarrow\mathrm{U}_{\hbar}(\mathfrak{g})/\mathfrak{m}^{\psi}\rightarrow\mathfrak{b}\backslash\mathrm{U}_{\hbar}(\mathfrak{g})/\mathfrak{m}^{\psi}.$

######  Proposition 5.6.

The map (5.8) is an isomorphism of right $\mathcal{W}$-modules.

###### Proof.

Consider the map between the associated graded spaces with respect to the
filtration induced from the Kazhdan grading on both sides (recall that
$\hbar\in\mathbf{C}^{*}$). It is clear from the formula (3.7) that it sends

$T_{11;0}^{(1)}\mapsto E_{1,1},\qquad T_{21;1}^{(1)}\mapsto E_{2,1},\qquad
T_{22;1}^{(N-1)}\mapsto E_{N,N}.$

It is also clear that it sends

$T_{22;1}^{(r)}\mapsto E_{r+1,N}+x,\ 1\leq r<N-1,$

where $x$ is expressible in terms of $E_{11},E_{21},E_{s+1,N}$ for $s>r$.
Likewise,

$T_{12;1}^{(N-1)}=E_{1,N}+x,$

where $x$ is expressible in terms of $E_{1,1},E_{2,1},E_{s+1,N}$ for $s\geq
1$. In particular, we see that this map sends generators to generators. Since
this is an algebra homomorphism, we conclude by Theorem 5.5 that it is an
isomorphism. In particular, the map (5.8) is an isomorphism as well. ∎

Recall the setting of Section 4.

######  Corollary 5.7.

For any Harish-Chandra bimodule $X$, there is a natural isomorphism of right
$\mathcal{W}$-modules

$\mathrm{res}^{\psi}(X)=(X/\mathfrak{m}^{\psi})^{\mathfrak{m}^{\psi}}\xrightarrow{\sim}\mathfrak{b}\backslash
X/\mathfrak{m}^{\psi}.$

###### Proof.

Recall that by Skryabin’s theorem, the natural action map

$\mathrm{U}_{\hbar}(\mathfrak{g})/\mathfrak{m}^{\psi}\otimes_{\mathcal{W}}(X/\mathfrak{m}^{\psi})^{\mathfrak{m}^{\psi}}\rightarrow
X/\mathfrak{m}^{\psi}$

is an isomorphism. Therefore, by Proposition 5.6 we have

$\mathfrak{b}\backslash
X/\mathfrak{m}^{\psi}\cong\mathfrak{b}\backslash\mathrm{U}_{\hbar}(\mathfrak{g})/\mathfrak{m}^{\psi}\otimes_{\mathcal{W}}(X/\mathfrak{m}^{\psi})^{\mathfrak{m}^{\psi}}\cong\mathcal{W}\otimes_{\mathcal{W}}(X/\mathfrak{m}^{\psi})^{\mathfrak{m}^{\psi}}=(X/\mathfrak{m}^{\psi})^{\mathfrak{m}^{\psi}}$

as required. ∎

In particular, it implies that we can “trivialize” the Drinfeld-Sokolov
reduction (4.5) on free Harish-Chandra bimodules.

######  Proposition 5.8.

For any $V\in\operatorname{Rep}(G)$, there is a natural isomorphism of right
$\mathcal{W}$-modules

(5.9) $\mathrm{triv}_{V}\colon
V\otimes\mathcal{W}\xrightarrow{\sim}\mathfrak{b}\backslash\mathrm{U}_{\hbar}(\mathfrak{g})\otimes
V/\mathfrak{m}^{\psi},$

i.e. for every $v\in V$, there exists a unique Whittaker vector $v^{\psi}$ of
the form

(5.10) $v^{\psi}=1\otimes v+\sum x_{i}\otimes v_{i},\
x_{i}\in\mathfrak{b}\cdot\mathrm{U}_{\hbar}(\mathfrak{g}).$

In particular, together with the isomorphism

$\mathfrak{b}\backslash\mathrm{U}_{\hbar}(\mathfrak{g})/\mathfrak{m}^{\psi}\rightarrow\mathrm{res}^{\psi}(\mathrm{U}_{\hbar}(\mathfrak{g})\otimes
V),$

we have a commutative diagram

$\textstyle{\operatorname{Rep}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{free}_{\mathcal{W}}}$$\scriptstyle{\mathrm{res}^{\psi}\circ\mathrm{free}}$$\textstyle{{}_{\mathcal{W}}\mathrm{BiMod}_{\mathcal{W}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{forget}}$$\textstyle{\mathrm{RMod}_{\mathcal{W}}}$

where $\mathrm{RMod}_{\mathcal{W}}$ is the category of right
$\mathcal{W}$-modules, the vertical arrow is the forgetful functor, and

$\mathrm{free}_{\mathcal{W}}\colon\operatorname{Rep}(G)\rightarrow\mathrm{RMod}_{\mathcal{W}},\qquad
V\mapsto V\otimes\mathcal{W}$

is the functor of free right $\mathcal{W}$-modules.

###### Proof.

Follows from the PBW theorem and Proposition 5.6 ∎

Observe that $\mathrm{RMod}_{\mathcal{W}}$ is naturally a right module
category over ${}_{\mathcal{W}}\mathrm{BiMod}_{\mathcal{W}}$, see 2.3. Since
$\mathrm{res}^{\psi}$ is a monoidal functor, it becomes a right module
category over $\operatorname{Rep}(G)$ as well. Likewise, the category
$\operatorname{Rep}(G)$ is tautologically a right module category over itself.
Also, using Skryabin’s theorem, we obtain a natural isomorphism

$X/\mathfrak{m}^{\psi}\otimes_{\mathcal{W}}(Y/\mathfrak{m}^{\psi})^{\mathfrak{m}^{\psi}}\cong
X\otimes_{\mathrm{U}_{\hbar}(\mathfrak{g})}\mathrm{U}_{\hbar}(\mathfrak{g})/\mathfrak{m}^{\psi}\otimes_{\mathcal{W}}(Y/\mathfrak{m}^{\psi})^{\mathfrak{m}^{\psi}}\xrightarrow{\sim}X\otimes_{\mathrm{U}_{\hbar}(\mathfrak{g})}Y/\mathfrak{m}^{\psi}.$

In particular, for every $U,V\in\operatorname{Rep}(G)$, we have

$\displaystyle\mathrm{free}_{\mathcal{W}}(U)\otimes_{\mathcal{W}}\mathrm{res}^{\psi}(\mathrm{U}_{\hbar}(\mathfrak{g})\otimes
V)\xrightarrow{\mathrm{triv}_{U}\otimes\mathrm{id}}\mathfrak{b}\backslash(\mathrm{U}_{\hbar}(\mathfrak{g})\otimes
U)/\mathfrak{m}^{\psi}\otimes_{\mathcal{W}}(\mathrm{U}_{\hbar}(\mathfrak{g})\otimes
V/\mathfrak{m}^{\psi})^{\mathfrak{m}^{\psi}}\cong$
$\displaystyle\cong\mathfrak{b}\backslash\mathrm{U}_{\hbar}(\mathfrak{g})\otimes
U\otimes V/\mathfrak{m}^{\psi}\xrightarrow{\mathrm{triv}_{U\otimes
V}^{-1}}U\otimes V\otimes\mathcal{W}.$

canonically. At the same time, since $\mathrm{free}_{\mathcal{W}}(U)$ is a
free $\mathcal{W}$-module, we also have a canonical isomorphism

$\mathrm{free}_{\mathcal{W}}(U)\otimes_{\mathcal{W}}\mathrm{res}^{\psi}(\mathrm{U}_{\hbar}(\mathfrak{g})\otimes
V)=(U\otimes\mathcal{W})\otimes_{\mathcal{W}}\mathrm{res}^{\psi}(\mathrm{U}_{\hbar}(\mathfrak{g})\otimes
V)\cong U\otimes V\otimes\mathcal{W}$

of right $\mathcal{W}$-modules. Combining it with Proposition 5.8 and Theorem
4.8, we get a “matrix” form of the monoidal structure on the Drinfeld-Sokolov
reduction.

######  Theorem 5.9.

The functor
$\mathrm{free}_{\mathcal{W}}\colon\operatorname{Rep}(G)\rightarrow\mathrm{RMod}_{\mathcal{W}}$
is a functor of right $\operatorname{Rep}(G)$-module categories in the sense
of 2.5. In particular, there is a collection of natural isomorphisms

$J_{UV}\colon U\otimes V\otimes\mathcal{W}\rightarrow U\otimes
V\otimes\mathcal{W},$

for all $U,V\in\operatorname{Rep}(G)$.

In what follows, we will compute its semi-classical limit.

###  5.4. Whittaker vectors: vector representation

We explicitly compute the generating Whittaker vectors for
$\mathrm{U}_{\hbar}(\mathfrak{g})\otimes\mathbf{C}^{N}/\mathfrak{m}^{\psi}$,
where

$\mathbf{C}^{N}=\mathrm{span}(v_{i}|1\leq i\leq
N),\qquad\operatorname{ad}_{E_{ij}}(v_{k})=\delta_{jk}v_{i}$

(we use the notation $\operatorname{ad}_{E_{ij}}$ from Subsection 4.1).

Recall the truncated generators (3.10). The next proposition gives a relation
between ${}_{k}T$ for different values of $k$.

######  Proposition 5.10.

[BK06, Lemma 10.4] Suppose that $r>0$. Then

(5.11)
${}_{1}T_{i,2;1}^{(r)}={}_{2}T_{i,2;1}^{(r)}+{}_{2}T_{i,2;1}^{(r-1)}\widetilde{E}_{N-1,N-1}+[{}_{2}T_{i,2;1}^{(r-1)},\widetilde{E}_{N-2,N-1}]$

for $i=1,2$, where $[,]$ refers to the adjoint action from Remark 4.7.

We will need the following lemma.

######  Lemma 5.11.

For $i=1,2$, we have

(5.12) $[E_{N,N-1},{}_{1}T_{i,2;1}^{(r)}]={}_{2}T_{i,2;1}^{(r-1)}.$

###### Proof.

By Proposition 5.10, it suffices to compute

(5.13) $\left[E_{N,N-1},\text{
}{}_{2}T_{i2;1}^{(r)}+{}_{2}T_{i2;1}^{(r-1)}\tilde{E}_{N-1,N-1}+[{}_{2}T_{i2;1}^{(r-1)},\tilde{E}_{N-2,N-1}]\right].$

Since ${}_{2}T_{i,2;x}^{(s)}\in\mathfrak{gl}_{{}_{2}N}$ and ${}_{2}N<N-1$, we
have $[E_{N,N-1},{}_{2}T_{i,2;x}^{(s)}]=0$ for any $s$. Thus, (5.13) becomes

$\left[E_{N,N-1},\text{
}{}_{2}T_{i,2;1}^{(r-1)}\widetilde{E}_{N-1,N-1}+[{}_{2}T_{i,2;1}^{(r-1)},\widetilde{E}_{N-2,N-1}]\right]={}_{2}T_{i,2;1}^{(r-1)}+\left[E_{N,N-1},[{}_{2}T_{i,2;1}^{(r-1)},\widetilde{E}_{N-2,N-1}]\right]$

since $\widetilde{E}_{N-1,N-1}=E_{N-1,N-1}+\hbar\cdot c$ for some constant $c$
by (3.6). From Jacobi’s identity, we have

$\left[E_{N,N-1},[{}_{2}T_{i,2;1}^{(r-1)},\widetilde{E}_{N-2,N-1}]\right]=-\left[{}_{2}T_{i,2;1}^{(r-1)},[E_{N,N-1},\tilde{E}_{N-2,N-1}]\right]-\left[\tilde{E}_{N-2,N-1},[E_{N,N-1},{}_{2}T_{i,2;1}^{(r-1)}]\right]=0,$

which proves the proposition. ∎

We go on to the main theorem.

######  Theorem 5.12.

For $N-j\neq 1$, the following vectors in
$\mathrm{U}_{\hbar}(\mathfrak{g})\otimes\mathbf{C}^{N}/\mathfrak{m}^{\psi}$

(5.14) $\tilde{v}_{N-j}^{\psi}=1\otimes
v_{N-j}+\sum_{i=0}^{j-1}(-1)^{j-i}{}_{i+1}T_{22;1}^{(j-i)}\otimes v_{N-i}$

are Whittaker.

###### Proof.

We proceed with strong induction on the subregular pyramid $\pi$ from (5.1).
The base case is

and the corresponding nilpotent element with the nilpotent subalgebra are
trivial. In particular,

$1\otimes v_{2}\in\mathrm{U}_{\hbar}(\mathfrak{gl}_{2})\otimes\mathbf{C}^{2}$

is automatically Whittaker.

For the step, let assume that $N>2$ and we proved the statement for the
truncated pyramid ${}_{1}\pi$. It is clear that $1\otimes v_{N}$ is Whittaker.
Also, the vector

(5.15) ${}_{1}\tilde{v}_{N-j}^{\psi}=1\otimes
v_{N-j}+\sum\limits_{i=1}^{j-1}(-1)^{j-i}{}_{i+1}T_{22;1}^{(j-i)}\otimes
v_{N-i}$

is invariant under the truncated subalgebra ${}_{1}\mathfrak{m}$, recall 3.3.
Indeed: while the coefficients of (5.15) are different from the ones of (5.14)
for $\mathfrak{gl}_{{}_{1}N}$ by definition of the truncated generators
(3.10), the non-standard embeddings
$\mathrm{U}_{\hbar}(\mathfrak{gl}_{{}_{k}N})\rightarrow\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})$
from (3.9) are homomorphisms for all $1\leq k\leq N-1$, so the Whittaker
property is preserved. Hence, let us rewrite equation (5.14) in a recursive
form:

(5.16)
$\widetilde{v}_{N-j}^{\psi}={}_{1}\widetilde{v}_{N-j}^{\psi}+(-1)^{j}\cdot{}_{1}T_{22;1}^{(j)}\otimes
v_{N}.$

To show that this vector is Whittaker, it suffices to check

$(E_{N,N-1}-1)\cdot\widetilde{v}_{N-j}^{\psi}=\hbar[E_{N,N-1},\widetilde{v}_{N-j}^{\psi}]=0.$

(recall Remark 4.7). Indeed,

  * •

For all $x\in{}_{1}\mathfrak{m}$, we have

$x^{\psi}\cdot{}_{1}\widetilde{v}_{N-j}^{\psi}=0$

by induction hypothesis, and

$x^{\psi}\cdot{}_{1}T_{22;1}^{(j)}\otimes v_{N}=0$

by Theorem 5.5 and because any element of ${}_{1}\mathfrak{m}$ commutes with
$v_{N}$.

  * •

For any $1\leq k<N-1$, there exists $x\in{}_{1}\mathfrak{m}$, such that
$E_{N,N-k}=[E_{N,N-1}^{\psi},x^{\psi}]$. Therefore, assuming we proved
invariance under $E_{N,N-1}^{\psi}$, we have

$E_{N,N-k}^{\psi}\cdot\tilde{v}_{N-j}^{\psi}=\hbar^{-1}(E_{N,N-1}^{\psi}x^{\psi}-x^{\psi}E_{N,N-1}^{\psi})\tilde{v}_{N-j}^{\psi}=0.$

By construction,

${}_{1}\widetilde{v}_{N-j}^{\psi}={}_{2}\widetilde{v}_{N-j}^{\psi}+(-1)^{j-1}\cdot{}_{2}T_{22;1}^{(j-1)}\otimes
v_{N-1}.$

Since
${}_{2}\tilde{v}_{N-j}^{\psi}\in\mathrm{U}_{\hbar}(\mathfrak{gl}_{N-2})\otimes\mathbf{C}^{N-2}$,
we have $[E_{N,N-1},{}_{2}\tilde{v}_{N-j}^{\psi}]=0$. Likewise,

$[E_{N,N-1},{}_{2}T_{22;1}^{(j-1)}\otimes
v_{N-1}]={}_{2}T_{22;1}^{(j-1)}\otimes v_{N},$

and therefore,

$[E_{N,N-1},{}_{1}\widetilde{v}_{N-j}^{\psi}]=(-1)^{j-1}\cdot{}_{2}T_{22;1}^{(j-1)}\otimes
v_{N}.$

By 5.11, we get

$[E_{N,N-1},(-1)^{j}\cdot{}_{1}T_{22;1}^{(j)}\otimes
v_{N}]=(-1)^{j}{}_{2}T_{22;1}^{(j-1)}\otimes v_{N}.$

Summing up these equalities and recalling (5.16), we conclude that

$[E_{N,N-1},\widetilde{v}_{N-j}^{\psi}]=0,$

and the induction is complete. ∎

######  Theorem 5.13.

The remaining vector

(5.17) $\tilde{v}_{1}^{\psi}=1\otimes
v_{1}+\sum_{i=0}^{N-3}(-1)^{N-i-2}\cdot{}_{i+1}T_{12;1}^{(N-i-2)}\otimes
v_{N-i}$

is also Whittaker in
$\mathrm{U}_{\hbar}(\mathfrak{g})\otimes\mathbf{C}^{N}/\mathfrak{m}^{\psi}$.

###### Proof.

Similar to Theorem 5.12. ∎

Observe that these vectors do not quite satisfy the assumption of (5.10).
However, they are not far away from the canonical form. Recall the definition
of the algebra $\mathfrak{l}$ from (5.7). Observe that
$\mathfrak{l}\subset\mathcal{W}$.

######  Definition 5.14.

Consider the natural PBW basis of $\mathrm{U}_{\hbar}(\mathfrak{g})$ induced
from the basis $\\{E_{ij}\\}$ of $\mathfrak{g}$. An element
$x\in\mathrm{U}_{\hbar}(\mathfrak{g})$ is called _$\mathfrak{l}$ -constant_,
if $x\in\mathrm{U}_{\hbar}(\mathfrak{l})$.

Thanks to upper-triangular form of (5.14) and (5.17), we can apply some
strictly upper-triangular (hence invertible) matrix with coefficients in
$\mathrm{U}_{\hbar}(\mathfrak{l})$ to the constructed generators to bring it
to the necessary form. Namely, denote by $c_{N-j}^{N-i}$ the
$\mathfrak{l}$-constant term of $(-1)^{j-i}\cdot{}_{i+1}T_{22;1}^{(j-i)}$.
Then we can perform the following inductive operation:

(5.18)
$\widetilde{v}_{N-j}^{\psi}\mapsto\widetilde{v}_{N-j}^{\psi}-\sum\limits_{i=0}^{j-1}v_{N-i}^{\psi}c_{N-j}^{N-i}.$

Removing step-by-step all the $\mathfrak{l}$-constant terms, we eventually get
the canonical generators, that we denote by

(5.19) $v_{i}^{\psi}=1\otimes v_{i}+\sum\limits_{j>i}x_{i}^{j}\otimes v_{j},\
x_{i}^{j}\in\mathfrak{b}\cdot\mathrm{U}_{\hbar}(\mathfrak{g}),$

for $1\leq i\leq N$.

Moreover, this form is actually more refined: we have

(5.20) $v_{i}^{\psi}=1\otimes v_{i}+\sum\limits_{j>i}x_{i}^{j}\otimes v_{j},\
x_{i}^{j}\in\mathfrak{b}_{j-1}\cdot\mathrm{U}_{\hbar}(\mathfrak{g}),$

where $\mathfrak{b}_{j-1}\subset\mathfrak{gl}_{j-1}$ is the truncated Borel
subalgebra as in 5.2.

###  5.5. Tensor structure

The goal of this section is to compute the semi-classical limit of the
monoidal isomorphism from Theorem 5.9 for the tensor product of
$\mathbf{C}^{N}$ with itself. From now on, we will treat $\hbar$ as a
_variable_ , in particular, we consider the asymptotic universal enveloping
algebra $\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})$ over $\mathbf{C}[\hbar]$. We
will need some definitions regarding “asymptotic” behavior of elements of
$\mathrm{U}_{\hbar}(\mathfrak{g})$.

######  Definition 5.15.

Consider the natural PBW basis of $\mathrm{U}_{\hbar}(\mathfrak{g})$ induced
from the basis $\\{E_{ij}\\}$ of $\mathfrak{g}$. We call an element
$x\in\mathrm{U}_{\hbar}(\mathfrak{g})$ _constant_ if it has degree zero with
respect to this basis. It is called _asymptotically linear_ if the PBW degree
of $x$ is one and it is constant in $\hbar$. We call $x$ _asymptotically
$\mathfrak{l}$-linear_ if it constant in $\hbar$ and has the form $x\in
y\cdot\mathrm{U}_{\hbar}(\mathfrak{l})^{>0}$, where $y\in\mathfrak{b}$.

Theorem 5.9 in this particular case can be reformulated as follows. Let
$\\{v_{i}\otimes v_{j}\\}$ be a natural basis of
$\mathbf{C}^{N}\otimes\mathbf{C}^{N}$. We have two natural choices of
generating vectors in
$(\mathrm{U}_{\hbar}(\mathfrak{g})\otimes\mathbf{C}^{N}\otimes\mathbf{C}^{N}/\mathfrak{m}^{\psi})^{\mathfrak{m}^{\psi}}$:
one is provided by Proposition 5.8, and we denote it by

(5.21) $(v_{i}\otimes v_{j})^{\psi}=v_{i}\otimes
v_{j}+\sum\limits_{k,l}x_{ij}^{kl}\otimes v_{k}\otimes v_{l},\
x_{ij}^{kl}\in\mathfrak{b}\cdot\mathrm{U}_{\hbar}(\mathfrak{p}).$

Another is given by the monoidal structure (4.4) on the Drinfeld-Sokolov
reduction: under canonical trivialization (5.19), we set

$\displaystyle v_{i}^{\psi}\otimes
v_{j}^{\psi}:=v_{i}^{\psi}\otimes_{\mathrm{U}_{\hbar}(\mathfrak{g})}v_{j}^{\psi}\in$
$\displaystyle(\mathrm{U}_{\hbar}(\mathfrak{g})\otimes\mathbf{C}^{N}/\mathfrak{m}^{\psi})^{\mathfrak{m}^{\psi}}\otimes_{\mathcal{W}}(\mathrm{U}_{\hbar}(\mathfrak{g})\otimes\mathbf{C}^{N}/\mathfrak{m}^{\psi})^{\mathfrak{m}^{\psi}}\cong$
$\displaystyle\cong$
$\displaystyle(\mathrm{U}_{\hbar}(\mathfrak{g})\otimes\mathbf{C}^{N}\otimes\mathbf{C}^{N}/\mathfrak{m}^{\psi})^{\mathfrak{m}^{\psi}}.$

######  Proposition 5.16.

The monoidal isomorphisms $J_{\mathbf{C}^{N},\mathbf{C}^{N}}$ from Theorem 5.9
are of the form

$J_{\mathbf{C}^{N},\mathbf{C}^{N}}\in\mathrm{id}_{\mathbf{C}^{N}\otimes\mathbf{C}^{N}\otimes\mathcal{W}}+\hbar\mathrm{U}_{\hbar}(\mathfrak{b})^{>0}\otimes\mathrm{U}_{\hbar}(\mathfrak{m})^{>0}\otimes\mathrm{U}_{\hbar}(\mathfrak{l}),$

where $\mathrm{U}_{\hbar}(\mathfrak{l})\subset\mathcal{W}$.

###### Proof.

Recall that under identification
$\mathrm{U}_{\hbar}(\mathfrak{g})/\mathfrak{m}^{\psi}\cong\mathrm{U}_{\hbar}(\mathfrak{p})$,
the generating vectors have the form (5.19), so,

(5.22) $v_{i}^{\psi}\otimes v_{j}^{\psi}=\left(1\otimes
v_{i}+\sum\limits_{k>i}x_{i}^{k}\otimes v_{k}\right)\otimes\left(1\otimes
v_{j}+\sum\limits_{l>j}x_{j}^{l}\otimes v_{l}\right).$

It follows from construction that for all $j,l$,

$x_{j}^{l}=(x_{(1)})_{j}^{l}\cdot(x_{(2)})_{j}^{l}$

for some $(x_{(1)})_{j}^{l}\in\mathrm{U}_{\hbar}(\mathfrak{b})$ and
$(x_{(2)})_{j}^{l}\in\mathrm{U}_{\hbar}(\mathfrak{l})$ (here, we use
Sweedler’s sum notation). Moreover, observe that

$(x_{(2)})_{j}^{l}\otimes v_{l}=(1\otimes v_{l})(x_{(2)})_{j}^{l}$

(recall the right action from Subsection 4.1). In particular, we get

$(1\otimes v_{k})\cdot x_{j}^{l}=\sum\limits_{a\leq
k}((x_{(1)})_{jk}^{la}\otimes v_{a})(x_{(2)})_{j}^{l}$

for some $(x_{(1)})_{jk}^{la}\in\mathrm{U}_{\hbar}(\mathfrak{b})$. Therefore,

(5.23) $\displaystyle\begin{split}v_{i}^{\psi}\otimes v_{j}^{\psi}&=1\otimes
v_{i}\otimes v_{j}+\sum\limits_{k>i}x_{i}^{k}\otimes v_{k}\otimes
v_{j}+\sum\limits_{\begin{subarray}{c}l>j\\\
k>i\end{subarray}}\sum\limits_{b\leq k}(x_{i}^{k}(x_{(1)})_{jk}^{lb}\otimes
v_{b}\otimes v_{l})(x_{(2)})_{j}^{l}+\\\
+&\sum\limits_{\begin{subarray}{c}a\leq i\\\
l>j\end{subarray}}((x_{(1)})_{ji}^{la}\otimes v_{a}\otimes
v_{l})(x_{(2)})_{j}^{l}.\end{split}$

Observe that the second line already has the form (5.10), and the third line
almost satisfies this condition as well except for the case when
$(x_{(1)})_{ji}^{la}$ is constant; in this case, denote
$(x_{(1)})_{ji}^{la}\cdot(x_{(2)})_{j}^{l}=:c_{ij}^{al}\in\mathrm{U}_{\hbar}(\mathfrak{l})$.
We see that the map

$v_{i}\otimes v_{j}\otimes 1\mapsto v_{i}\otimes v_{j}\otimes
1+\sum_{\begin{subarray}{c}a\leq i\\\ l>j\end{subarray}}v_{a}\otimes
v_{l}\otimes c_{ij}^{al}$

is strictly upper-triangular. In particular, the canonical generators
$(v_{i}\otimes v_{j})^{\psi}$ of (5.21) can be constructed inductively by
taking

$(v_{i}\otimes v_{j})^{\psi}\otimes 1:=v_{i}^{\psi}\otimes v_{j}^{\psi}\otimes
1-\sum(v_{a}\otimes v_{l})^{\psi}\otimes c_{ij}^{al}.$

Then the tensor isomorphism is given by a
$\mathrm{U}_{\hbar}(\mathfrak{l})$-valued matrix

(5.24)
$J_{\mathbf{C}^{N},\mathbf{C}^{N}}=\mathrm{id}_{\mathbf{C}^{N}\otimes\mathbf{C}^{N}\otimes\mathcal{W}}+(c_{ij}^{al}).$

Moreover, since every commutation produces a power of $\hbar$ by (4.1), the
second part of the theorem follows. ∎

We will compute the first $\hbar$-power $\mathbf{j}$ of the matrix (5.24). In
other words, we need to compute the first $\hbar$-powers of the coefficients
$(x_{j}^{l})_{i}^{a}$ of (5.23).

######  Proposition 5.17.

The only non-trivial contribution to the first $\hbar$-power from
$(x_{j}^{l})_{i}^{a}$ comes from asymptotically linear and
$\mathfrak{l}$-linear terms of $x_{j}^{l}$ in (5.22).

###### Proof.

Indeed: in a PBW basis, the element $x_{j}^{l}$ is the sum of products of the
form $\hbar^{b}y_{1}\cdots y_{a}\cdot x$ with $a\geq 1$ for some
$\\{y_{m}\\}\subset\mathfrak{g}$, some $\hbar$-power $b$, and
$x\in\mathrm{U}_{\hbar}(\mathfrak{l})$. If it is not asymptotically
($\mathfrak{l}$)-linear, then there are two cases:

  1. (1)

It is not linear, i.e. $a\geq 2$; for simplicity, we demonstrate it for $a=2$,
but the general argument is the same:

$(1\otimes v_{i})\cdot\hbar^{b}y_{1}y_{2}x=\hbar^{b}(y_{1}\otimes
v_{i})y_{2}x-(\hbar^{b+1}\otimes[y_{1},v_{i}])y_{2}x$
$=\hbar^{b}(y_{2}y_{1}\otimes v_{i})x-\hbar^{b+1}([y_{2},y_{1}]\otimes
v_{i})x-\hbar^{b+1}(y_{1}\otimes[y_{2},v_{i}])x-\hbar^{b+1}(y_{2}\otimes[y_{1},v_{i}])x+(\hbar^{b+2}\otimes[y_{2},[y_{1},v_{i}]])x.$

So, we see that there is no contribution to the first $\hbar$-power of
constant terms for $a\geq 2$.

  2. (2)

It is linear, but divisible by $\hbar$, i.e. $b\geq 1$. Then

$(1\otimes v_{i})\hbar^{b}y_{1}x=\hbar^{b}(y_{1}\otimes
v_{i})x-(\hbar^{b+1}\otimes[y_{1},v_{i}])x,$

and there is also no contribution to the first $\hbar$-power of constant
terms.

Thus, if $x_{j}^{l}$ is not asymptotically ($\mathfrak{l}$)-linear, then it
must be asymptotically linear for there to be non-trivial contribution to the
first $\hbar$ power. At the same time, the calculation above shows that for
$b=0$, $\mathfrak{l}$-linear terms may contribute to the first power, and the
proposition follows. ∎

Unfortunately, the only explicit form of Whittaker vectors in
$\mathrm{U}_{\hbar}(\mathfrak{g})\otimes\mathbf{C}^{N}/\mathfrak{m}^{\psi}$
available so far is (5.14) or (5.17) which is not canonical; however, thanks
to the next lemmas, it does not affect calculations too much.

######  Lemma 5.18.

For any parameters, the $\mathfrak{l}$-constant part of ${}_{a}T_{ij;x}^{(r)}$
is divisible by $\hbar$.

###### Proof.

From 3.6, we note that a constant term exists in $T_{ij;x}^{(r)}$ if and only
if $i_{k}=j_{k}$ for all $1\leq k\leq s$. But from condition (4), we note that
$\text{col}(j_{k})-\text{col}(i_{k})+1=1$ for all $k$. Thus, $s=r$ and
constant terms only come from summands of the form

$\widetilde{E}_{i_{1}i_{1}}\cdot\ldots\cdot\widetilde{E}_{i_{r}i_{r}}.$

By (3.6), it is clear that the constant term is proportional to $\hbar^{r}$.

As for general $\mathfrak{l}$-constant terms, it follows from the formula
(3.7) that $T_{22;1}^{(r)}$ is the sum of elements of the form

$\widetilde{E}_{2,1}\widetilde{E}_{1,1}^{k}\cdot x,\
x\in\mathrm{U}_{\hbar}(\mathfrak{b})$

for some $k$. Note $x$ must commute with $E_{11}$ from condition (2). Thus,
commuting $x$ to the left produces some elements of
$\mathrm{U}_{\hbar}(\mathfrak{l})$, but they are divisible by $\hbar$ because
of commutation. The same is true for $T_{12;1}^{(r)}$, where the terms have
the form $\widetilde{E}_{1,1}^{k}\cdot x$ for some $k$ and
$x\in\mathrm{U}_{\hbar}(\mathfrak{b})$. ∎

Recall the coefficients $x_{N-j}^{N-i}$ from (5.19).

######  Lemma 5.19.

The asymptotically $\mathfrak{l}$-linear terms in
$(-1)^{j-i}\cdot{}_{i+1}T_{22;1}^{(j-i)}$ are the same as in $x_{N-j}^{N-i}$
for $N-j\neq 1$. Likewise, the asymptotically $\mathfrak{l}$-linear terms in
${}_{i+1}T_{12;1}^{(N-i-1)}$ are the same as $x_{1}^{N-l}$.

###### Proof.

Recall (5.19) that the canonical generators can be constructed from
$\\{\widetilde{v}_{i}^{\psi}\\}$ by inductively removing
$\mathfrak{l}$-constant terms. But, according to 5.18, they are all divisible
by $\hbar$, and the statement follows. ∎

Therefore, by Proposition 5.17, it is enough to consider only the
asymptotically linear terms of $T$-generators.

######  Proposition 5.20.

The explicit forms for the asymptotically linear and $\mathfrak{l}$-linear
terms are given below.

  * •

The asymptotically linear part of $T_{ij;x}^{(r)}$ is

$\sum\limits_{\begin{subarray}{c}\mathrm{row}(i_{1})=i,\\\
\mathrm{row}(j_{1})=j,\\\
\mathrm{col}(j_{1})-\mathrm{col}(i_{1})+1=r\end{subarray}}(-1)^{r-1}E_{i_{1},j_{1}}.$

  * •

The asymptotically $\mathfrak{l}$-linear terms of ${}_{i+1}T_{22;1}^{(j-i)}$
are

$\sum\limits_{r=2}^{j-i}(-1)^{r}E_{1,r}E_{2,1}E_{1,1}^{j-i-r}.$

  * •

The asymptotically $\mathfrak{l}$-linear terms of ${}_{i+1}T_{12;1}^{(N-i-1)}$
are

$\sum\limits_{r=2}^{N-i-2}(-1)^{r}E_{1,r}E_{1,1}^{N-i-r-1}.$

###### Proof.

Recall the formula (3.7) of $T$-generators. Observe that in order to have a
(not necessarily asymptotically) linear term in a summand with $s$ terms, we
need at least $s-1$ of those $\widetilde{E}_{i_{l},j_{l}}$ to carry a constant
term. Hence, we must have $i_{l}=j_{l}$ for at least $s-1$ values of $l$ where
$1\leq l\leq s$. But then $\hbar$ divides each of these constant terms by
(3.6), so we require $s-1=0$. Therefore, only the linear part

$\sum\limits_{\begin{subarray}{c}\mathrm{row}(i_{1})=i,\\\
\mathrm{row}(j_{1})=j,\\\
\mathrm{col}(j_{1})-\mathrm{col}(i_{1})+1=r\end{subarray}}\tilde{E}_{i_{1},j_{1}}$

can contribute to the first power of $\hbar$, which is precisely the formula
from the statement.

Now let us study the asymptotically $\mathfrak{l}$-linear terms of
${}_{i+1}T_{22;1}^{(j-i)}$. Consider a summand of (3.7). By condition (1),
$\mathrm{row}(i_{1})=2$. Assume that $\mathrm{col}(i_{1})>1$. Then by
condition (3), $\mathrm{col}(j_{1})>1$, and so $\mathrm{row}(j_{1})=2$. In
particular, $\sigma_{\mathrm{row}(j_{1})}=+$ meaning that
$\mathrm{col}(i_{2})>1$ by condition (5). Continuing, we obtain that this
summand cannot contain $E_{1,1}$ or $E_{2,1}$.

Consider $\mathrm{col}(i_{1})=1$, i.e. $\widetilde{E}_{i_{1},j_{1}}=E_{2,1}$.
By condition (2), $\mathrm{row}(j_{1})=1=\mathrm{row}(i_{2})$. If
$\mathrm{row}(j_{2})>1$ and we can apply previous arguments to conclude that
the corresponding summand is $E_{2,1}\widetilde{E}_{1,r}$ (recall that we are
interested only in $\mathfrak{l}$-linear terms). Otherwise, by condition (6),
we see that $i_{3}=1$, and we can repeat the argument. Summing all cases, we
obtain a summand of the form (observe that we drop all the $\hbar$-factors)

$(-1)^{r}E_{2,1}E_{1,1}^{k}E_{1,r}.$

Now we commute the $\mathfrak{l}$-part to the right. Observe that it produces
powers of $\hbar$, so,

$(-1)^{r}E_{2,1}E_{1,1}^{k}E_{1,r}=(-1)^{r}E_{1,r}E_{2,1}E_{1,1}^{k}+O(\hbar).$

The relation between $r$ and $k$ follows from the degree condition (4).

The analysis for ${}_{i+1}T_{12;1}^{(N-i-1)}$ is similar and will be omitted.
∎

Combining all preliminary results, we can compute the tensor isomorphism
Theorem 5.9 for the vector representation.

######  Proposition 5.21.

The monoidal isomorphism $J_{\mathbf{C}^{N},\mathbf{C}^{N}}$ from Proposition
5.16 has the form

$J_{\mathbf{C}^{N},\mathbf{C}^{N}}=\mathrm{id}_{\mathbf{C}^{N}\otimes\mathbf{C}^{N}\otimes\mathcal{W}}+\hbar\mathbf{j}_{\mathbf{C}^{N},\mathbf{C}^{N}}+O(\hbar^{2}),$

where $\mathbf{j}_{\mathbf{C}^{N},\mathbf{C}^{N}}$ is

$\displaystyle\mathbf{j}_{\mathbf{C}^{N},\mathbf{C}^{N}}=\mathbf{j}_{c}+\sum\limits_{j=2}^{N-2}\sum\limits_{i=j+2}^{N}\sum\limits_{r=2}^{i-j}(-1)^{i-j-r}x_{21}x_{11}^{i-j-r}E_{1,r}\otimes
E_{i,j}+\sum\limits_{i=4}^{N}\sum\limits_{r=2}^{i-2}(-1)^{i-r}x_{11}^{i-r-1}E_{1,r}\otimes
E_{i,1}$

with $x_{21},x_{11}$ the coordinate functions on $\mathfrak{l}^{*}$
corresponding to $E_{21},E_{11}\in\mathfrak{l}$. Here,

$\mathbf{j}_{c}=\sum\limits_{j=2}^{N-1}\sum\limits_{i=j+1}^{N}\left(\sum\limits_{l=2}^{j}E_{l,l+i-j-1}\right)\otimes
E_{i,j}+\sum\limits_{i=3}^{N}E_{1,i-1}\otimes E_{i,1}$

is a map $\mathfrak{b}^{*}\rightarrow\mathfrak{m}$ from Proposition 5.3.

###### Proof.

Denote by $L_{N-j}^{N-k}$ the asymptotically linear part of

$\displaystyle(-1)^{j-k}{}_{k+1}T_{22;1}^{(j-k)},\ $ $\displaystyle N-j\neq
1,$ $\displaystyle(-1)^{N-k-2}\cdot{}_{k+1}T_{12;1}^{(N-k-2)},\ $
$\displaystyle N-j=1.$

It follows from Proposition 5.20 that

(5.25)
$\displaystyle\begin{split}L_{N-j}^{N-k}&=-\sum\limits_{l=2}^{N-j}E_{l,l+j-k-1},\\\
L_{1}^{N-k}&=-E_{1,N-k-1}.\end{split}$

Combing Eq. 5.23, 5.18, and 5.19, the constant part of the first $\hbar$-power
of $J_{\mathbf{C}^{N},\mathbf{C}^{N}}$ is given by the action of

$\sum\limits_{j=1}^{N-2}\sum\limits_{k=0}^{j-1}\left(\sum\limits_{l=2}^{N-j}E_{l,l+j-k-1}\right)\otimes
E_{N-k,N-j}+\sum\limits_{k=0}^{N-3}E_{1,N-k-1}\otimes E_{N-k,1}.$

Changing coefficients:

$\sum\limits_{j=2}^{N-1}\sum\limits_{i=j+1}^{N}\left(\sum\limits_{l=2}^{j}E_{l,l+i-j-1}\right)\otimes
E_{i,j}+\sum\limits_{i=3}^{N}E_{1,i-1}\otimes E_{i,1}.$

Now let us consider the “dynamical” part. It follows from Proposition 5.20
that the asymptotically $\mathfrak{l}$-linear part of the coefficient
$(-1)^{j-i}\cdot{}_{i+1}T_{22;1}^{(j-i)}$ of $v_{N-i}$ in
$\widetilde{v}_{N-j}^{\psi}$ is

$\sum\limits_{r=2}^{j-i}(-1)^{j-i+r}E_{1,r}E_{2,1}E_{1,1}^{j-i-r}.$

Recall from the construction of the tensor structure from Proposition 5.16
that we need to compute the first $\hbar$-power of the right action

$\sum\limits_{r=2}^{j-i}(-1)^{j-i+r-1}(1\otimes v_{k})\cdot
E_{1,r}E_{21}E_{11}^{j-i-r}$

modulo $\mathfrak{b}$ for every $k$. It follows that it is equal to

$\sum\limits_{r=2}^{j-i}\left((-1)^{j-i+r}\otimes\operatorname{ad}_{E_{1,r}}(v_{k})\right)E_{2,1}E_{1,1}^{j-i-r},$

which for all admissible $i,j$, gives the contribution

$\sum\limits_{j=2}^{N-2}\sum\limits_{i=j+2}^{N}\sum\limits_{r=2}^{i-j}(-1)^{i-j-r}x_{21}x_{11}^{i-j-r}E_{1,r}\otimes
E_{i,j}.$

Likewise, by considering the coefficients of $v_{1}^{\psi}$, we get the
contribution

$\sum\limits_{i=4}^{N}\sum\limits_{r=2}^{i-1}(-1)^{i-r}x_{11}^{i-r}E_{1,r}\otimes
E_{i,1},$

and the theorem follows. ∎

In fact, the constant part $\mathbf{j}_{c}$ is related to the form $\omega$
from Proposition 5.3.

######  Proposition 5.22.

The inverse $\omega^{-1}$ is equal to $\mathbf{j}_{c}-\mathbf{j}_{c}^{21}$.

###### Proof.

One can easily see that the conditions (5.6) are satisfied. ∎

Finally, by the Schur-Weyl duality, any representation of $\mathrm{GL}_{N}$
can be canonically obtained as a subrepresentation of
$(\mathbf{C}^{N})^{\otimes k}\otimes\det^{l}$ for some $k,l$, where $\det$ is
the one-dimensional determinant representation. By naturality of construction,
we obtain the main result of the paper.

######  Theorem 5.23.

The semi-classical limit of the monoidal isomorphisms $J_{UV}$ from Theorem
5.9 is given by the action of the universal element

$\mathbf{j}=\mathbf{j}_{c}+\sum\limits_{j=2}^{N-2}\sum\limits_{i=j+2}^{N}\sum\limits_{r=2}^{i-j}(-1)^{i-j-r}x_{21}x_{11}^{i-j-r}E_{1,r}\otimes
E_{i,j}+\sum\limits_{i=4}^{N}\sum\limits_{r=2}^{i-2}(-1)^{i-r}x_{11}^{i-r-1}E_{1,r}\otimes
E_{i,1},$

where

$\mathbf{j}_{c}=\sum\limits_{j=2}^{N-1}\sum\limits_{i=j+1}^{N}\left(\sum\limits_{l=2}^{j}E_{l,l+i-j-1}\right)\otimes
E_{i,j}+\sum\limits_{i=3}^{N}E_{1,i-1}\otimes E_{i,1}$

defines an inverse of $\omega$ from Proposition 5.3.

## References

  * [BK06] Jonathan Brundan and Alexander Kleshchev  “Shifted Yangians and finite $W$-algebras”  In _Adv. Math._ 200.1, 2006, pp. 136–195  DOI: 10.1016/j.aim.2004.11.004
  * [CG90] Eugène Cremmer and Jean-Loup Gervais  “The quantum group structure associated with nonlinearly extended Virasoro algebras”  In _Comm. Math. Phys._ 134.3, 1990, pp. 619–632  URL: http://projecteuclid.org/euclid.cmp/1104201824
  * [ES02] Pavel Etingof and Olivier Schiffmann  “Lectures on quantum groups”, Lectures in Mathematical Physics  International Press, Somerville, MA, 2002, pp. xii+242
  * [Eti+15] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik  “Tensor categories” 205, Mathematical Surveys and Monographs  American Mathematical Society, Providence, RI, 2015, pp. xvi+343  DOI: 10.1090/surv/205
  * [EV99] P. Etingof and A. Varchenko  “Exchange dynamical quantum groups”  In _Comm. Math. Phys._ 205.1, 1999, pp. 19–52  DOI: 10.1007/s002200050665
  * [FB04] Edward Frenkel and David Ben-Zvi  “Vertex algebras and algebraic curves” 88, Mathematical Surveys and Monographs  American Mathematical Society, Providence, RI, 2004, pp. xiv+400  DOI: 10.1090/surv/088
  * [Kal21] Artem Kalmykov  “Geometric and categorical approaches to dynamical representation theory”  Zürich: Universität Zürich, 2021
  * [Kal23] Artem Kalmykov  “Yangians, mirabolic subalgebras, and Whittaker vectors”, 2023  arXiv:2310.06669 [math.RT]
  * [KS22] Artem Kalmykov and Pavel Safronov  “A categorical approach to dynamical quantum groups”  In _Forum Math. Sigma_ 10, 2022, pp. Paper No. e7657  DOI: 10.1017/fms.2022.68
  * [Los10] Ivan Losev  “Finite W-algebras”  In _Proceedings of the International Congress of Mathematicians. Volume III_ Hindustan Book Agency, New Delhi, 2010, pp. 1281–1307
  * [Pre02] Alexander Premet  “Special transverse slices and their enveloping algebras” With an appendix by Serge Skryabin  In _Adv. Math._ 170.1, 2002, pp. 1–55  DOI: 10.1006/aima.2001.2063
  * [Saa72] Neantro Saavedra Rivano  “Catégories Tannakiennes”, Lecture Notes in Mathematics, Vol. 265  Springer-Verlag, Berlin-New York, 1972, pp. ii+418
  * [Ulb89] K.-H. Ulbrich  “Fibre functors of finite-dimensional comodules”  In _Manuscripta Math._ 65.1, 1989, pp. 39–46  DOI: 10.1007/BF01168365
  * [Ulb90] K.-H. Ulbrich  “On Hopf algebras and rigid monoidal categories” Hopf algebras  In _Israel J. Math._ 72.1-2, 1990, pp. 252–256  DOI: 10.1007/BF02764622