# Yangians, mirabolic subalgebras, and Whittaker vectors

Artem Kalmykov

###### Abstract.

We construct an element, which we call the Kirillov projector, that connects
the topics of the title: on the one hand, it is defined using the Yangian of
$\mathfrak{gl}_{N}$, on the other hand, it gives a canonical projection onto
the space of Whittaker vectors for any Whittaker module over the mirabolic
subalgebra. Using the Kirillov projector, we deduce some categorical
properties of Whittaker modules, for instance, we prove a mirabolic analog of
Skryabin’s theorem. We also show that it quantizes a rational version of the
Cremmer-Gervais $r$-matrix. As application, we construct a universal vertex-
IRF transformation from the standard dynamical $R$-matrix to this constant one
in categorical terms.

## Introduction

Solutions to the _quantum Yang-Baxter equation_ (QYBE shortly)

$R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12},$

the so-called _quantum $R$-matrices_, have many applications in mathematical
physics and theory of quantum groups. The most well-studied example is the
_standard solution_ , associated to any reductive Lie algebra $\mathfrak{g}$,
that leads to Drinfeld-Jimbo quantum groups [Dri87, Jim85].

The starting point for the paper is a particular non-standard solution in type
A which is called the _Cremmer-Gervais $R$-matrix_. Originally, it appeared in
[CG90] and was later studied, for instance, in [Hod95, BRT07]. This solution
is special among other non-standard solutions in at least two ways (which
might be connected to each other).

First, it is related to the so-called _vertex-IRF transformation_. Namely,
there is another important equation in mathematical physics, which is a
generalization of the QYBE, called _quantum dynamical Yang-Baxter equation_ :

$R_{12}(\lambda-\hbar h_{3})R_{13}(\lambda)R_{23}(\lambda-\hbar
h_{1})=R_{23}(\lambda)R_{13}(\lambda-\hbar h_{2})R_{12}(\lambda);$

where $\lambda$ is a parameter on the dual space $\mathfrak{h}^{*}$ to an
abelian Lie algebra $\mathfrak{h}$ and $h_{i}$ are weight operators, see 1.7.
As explained by Felder [Fel95], it is closely related to the star-triangle
relation for face-type statistical models [Bax89]. Moreover, it naturally
appears in the description of the Liouville and Toda field theories [GN84]. As
in the non-dynamical case, it leads to the theory of _dynamical quantum
groups_ , and for any reductive $\mathfrak{g}$, there are canonically defined
standard solutions, see [ES01, Eti02]. It turns out that in type A, it is
possible to gauge the standard dynamical $R$-matrix to a constant one which is
exactly the Cremmer-Gervais $R$-matrix (in fact, this is how it was discovered
in [CG90]). What is even more interesting is that such kind of gauge
transformations (which are called _vertex-IRF_ transformations) is unique,
that is, it exists only in type A and the only constant $R$-matrix we can get
is the Cremmer-Gervais one, see [BDF90]. So, it is natural to ask: what is a
representation-theoretic explanation of this fact? Or, at least, what is a
representation-theoretic meaning of _the_ vertex-IRF transformation?

This brings to the second special feature of the Cremmer-Gervais $R$-matrix:
its relation to W-algebras. In principle, the exchange algebra of the Toda
field theory (which is related to affine W-algebras) is “controlled” by the
standard dynamical $R$-matrix [GN84], but, as we mentioned before, it was
shown in [CG90] that is possible to gauge away the dependence on the dynamical
parameter. Likewise, it was observed in [BRT12] that the Cremmer-Gervais
$R$-matrix can be constructed using the _Sevostyanov characters_ [Sev00] which
are used to define a $q$-analog of the principal finite W-algebra. So, one may
ask: what is a direct relation between the Cremmer-Gervais $R$-matrix and
W-algebras?

This paper is a small attempt to answer these questions in a somewhat simpler
situation of the rational analog of the Cremmer-Gervais solution (or, rather,
a degeneration thereof). It turns out that, intrinsically, it is defined not
for the whole algebra $\mathfrak{gl}_{N}$, but for a certain (almost)
parabolic subgroup of it.

For the rest of the paper, we work over an algebraically closed field $k$ of
characteristic zero. For instance, $k$ can be the field of complex numbers.

### Mirabolic subgroup

This is a summary of Section 2.

Tautologically, the group $\mathrm{GL}_{N}$ is a group of invertible
transformation of the vector space $k^{N}=\mathrm{span}(e_{i}|1\leq i\leq N)$.
Likewise, the Lie algebra $\mathfrak{gl}_{N}$ is the Lie algebra of
endomorphisms of $V$. In what follows, we denote by $E_{ij}$ the matrix unit
elements of $\mathfrak{gl}_{N}$.

Choose a vector $v\in k^{N}$; for instance, we will usually take the last
basis vector $e_{N}$. By definition, the _mirabolic subgroup_ $\mathrm{M}_{N}$
is the subgroup of $\mathrm{GL}_{N}$ preserving $v$. In terms of matrices,

$\mathrm{M}_{N}=\begin{pmatrix}*&\ldots&*&0\\\ \vdots&\ddots&\vdots&\vdots\\\
*&\ldots&*&0\\\ *&\ldots&*&1\end{pmatrix}$

The _mirabolic subalgebra_ $\mathfrak{m}_{N}$ is the Lie algebra of
$\mathrm{M}_{N}$. It can be identified with the elements of
$\mathfrak{gl}_{N}$ acting by zero on $v$. In terms of matrices,

$\mathfrak{m}_{N}=\begin{pmatrix}*&\ldots&*&0\\\
\vdots&\ddots&\vdots&\vdots\\\ *&\ldots&*&0\\\ \end{pmatrix}$

Recall that on the classical level, quantum groups correspond to Poisson-Lie
structures [Dri87]. It turns out that there is a natural family of non-
degenerate Poisson-Lie structures on $\mathrm{M}_{N}$. Namely, let
$e\in\mathfrak{gl}_{N}$ be the so-called _principal nilpotent element_
$e=E_{12}+\ldots+E_{N-1,N}$. Consider a vector of $(N-1)$-parameters
$\vec{u}=(u_{1},\ldots,u_{N-1})$. Define

(1) $e_{\vec{u}}:=e-u_{1}E_{11}-\ldots-u_{N-1}E_{N-1,N-1}.$

While $e_{\vec{u}}$ does not lie in $\mathfrak{m}_{N}$, it defines the
following 2-form:

$\omega(\vec{u})\colon\mathfrak{m}_{N}\wedge\mathfrak{m}_{N}\rightarrow
k,\qquad x\wedge y\mapsto\mathrm{Tr}(e_{\vec{u}}\cdot[x,y]).$

By Proposition 2.2, this form is non-degenerate for all choices of parameters,
so, by general arguments (for instance, see [ES02]), its inverse
$r^{\mathrm{CG}}(\vec{u}):=\omega(\vec{u})^{-1}$ satisfies the _classical
Yang-Baxter equation_ which is a classical analog of QYBE. We call it _a
rational Cremmer-Gervais $r$-matrix_, see 2.3. For zero parameters
$\vec{u}=(0,\ldots,0)$, it becomes _the_ rational Cremmer-Gervais $r$-matrix
which is a degeneration of the classical Cremmer-Gervais solution [EH00]. For
instance, here is an example for $\mathfrak{gl}_{2}$:

(2) $r^{\mathrm{CG}}(u)=-E_{21}\wedge E_{11}.$

So, we have a family of Poisson-Lie structures, and it is natural to ask what
their quantization is. We understand “quantization” in the Tannakian sense.
Namely, let $G$ be an affine algebraic group. Then a tensor structure on the
forgetful functor $\operatorname{Rep}(G)\rightarrow\mathrm{Vect}$ is
equivalent to a natural collection of isomorphisms
$J_{VW}\in\mathrm{End}(V\otimes W)$ satisfying the Drinfeld twist equation
(1.1). In particular, the collection $R_{VW}:=F_{WV}^{-1}F_{VW}$ satisfies the
QYBE. Therefore, one can try to construct a monoidal functor
$\operatorname{Rep}(\mathrm{M}_{N})\rightarrow\mathrm{Vect}$ such that the
semiclassical limit of its tensor structure is a rational Cremmer-Gervais
solution.

It turns out that it is possible to do it using a variant of the so-called
_exchange construction_ [EV99].

### Parabolic restriction

This is a summary of Section 1 that mostly follows [KS22]. We must note that
for simplicity of exposition in the introduction, we use the classical
language of exchange constructions, for instance, as in [EV99], but in the
paper itself, we adopt an interpretation in terms of Harish-Chandra bimodules.
In what follows, for any Lie algebra $\mathfrak{g}$, we consider a certain
$\hbar$-version $\mathrm{U}_{\hbar}(\mathfrak{g})$ of the universal enveloping
algebra, see 1.1 and Section 1.1.

Let $\mathfrak{g}$ be a reductive Lie algebra. Denote by $M_{\lambda}$ the
Verma module with highest weight $\lambda\in\mathfrak{h}^{*}$, and let $V,W$
be finite-dimensional representations of $\mathfrak{g}$. Consider the space of
intertwiners

$\Phi_{V}\colon M_{\lambda}\rightarrow V\otimes M_{\mu}.$

It has a natural tensor structure: if $\Phi_{W}\colon M_{\mu}\rightarrow
M_{\nu}\otimes W$ is an intertwiner for $W$, then the composition

$M_{\lambda}\xrightarrow{\Phi_{V}}V\otimes
M_{\mu}\xrightarrow{\mathrm{id}_{V}\otimes\Phi_{W}}V\otimes W\otimes M_{\nu}$

is an intertwiner for $V\otimes W$. Now assume that $\lambda$ is generic. Then
it follows that there is a natural identification

(3) $\mathrm{Hom}_{\mathfrak{g}}(M_{\lambda},V\otimes M_{\mu})\cong
V[\lambda-\mu]$

of the space of such intertwiners with the weight space of $V$. For $v\in
V[\lambda-\mu]$, denote by $\Phi_{V}^{v}$ the corresponding intertwiner. Then
for $w\in W[\mu-\nu]$, the composition above should correspond to some element
in $V\otimes W$. So, there exists a natural map
$J_{VW}(\lambda)\in\mathrm{End}_{\mathfrak{h}}(V\otimes W)$ such that the
corresponding element is given by $J_{VW}(\lambda)(v\otimes w)$. It turns out
that this map satisfies the dynamical analog of the twist equation (see 1.7)
and $R^{\mathrm{dyn}}_{VW}(\lambda):=J_{WV}(\lambda)^{-1}J_{VW}(\lambda)$ is a
quantum dynamical $R$-matrix. In fact, this $R$-matrix quantizes the standard
solution to the classical dynamical Yang-Baxter equation. For all the
statements and proofs, we refer the reader to [EV99].

Now, let us allow for weights $\lambda$ and $\mu$ to vary. So, essentially, we
consider $\mathfrak{g}$-homomorphisms

(4) $M^{\mathrm{univ}}\rightarrow V\otimes M^{\mathrm{univ}},$

where
$M^{\mathrm{univ}}:=\mathrm{U}_{\hbar}(\mathfrak{g})\otimes_{\mathrm{U}_{\hbar}(\mathfrak{n})}k$
is the _universal Verma module_ (certainly, one needs to take into account
that weights should be generic; we refer the reader to Section 1.5 for more
details). Then the space of such intertwiners can be identified with
$V$-valued functions on $\mathfrak{h}^{*}$ (again, meromorphic ones, but let
us ignore it for simplicity of exposition). Since $\mathfrak{h}$ is
commutative, we have an identification
$\mathcal{O}(\mathfrak{h}^{*})\cong\mathrm{U}_{\hbar}(\mathfrak{h})$. In other
words, we have an assignment $V\mapsto\mathrm{U}_{\hbar}(\mathfrak{h})\otimes
V$ that can be upgraded to a monoidal functor of _parabolic restriction_
$\mathrm{res}_{\mathfrak{g}}\colon\operatorname{Rep}(G)\rightarrow{}_{\mathrm{U}_{\hbar}(\mathfrak{h})}\mathrm{BiMod}_{\mathrm{U}_{\hbar}(\mathfrak{h})}$,
where the target is the category of
$\mathrm{U}_{\hbar}(\mathfrak{h})$-bimodules (in fact, a smaller subcategory
of _Harish-Chandra bimodules_ , where a certain integrability assumption on
the diagonal action is imposed, see Section 1.2).

Here, $\mathfrak{n}$ acts on $k$ trivially. But, in principle, we can take an
arbitrary character of $\mathfrak{n}$. For instance, let us consider a generic
one; this is a natural setup for the _principal W-algebra_ and _Whittaker
modules_ [Kos78], see [Ara17, Los10] for introduction to the subject.

For certain reasons that will be explained in Section 6, let us consider
_right_ $\mathfrak{g}$-modules and the negative nilpotent subalgebra
$\mathfrak{n}_{-}$ of $\mathfrak{g}$. Let
$\psi\colon\mathfrak{n}_{-}\rightarrow k$ be a generic character; for the rest
of the paper, we will assume that it takes value 1 on all simple roots.

Denote by
$Q_{\mathfrak{g}}=\mathfrak{n}_{-}^{\psi}\backslash\mathrm{U}_{\hbar}(\mathfrak{g})$
the quotient by the right ideal generated by the shift

$\mathfrak{n}_{-}^{\psi}:=\mathrm{span}(x-\psi(x)|x\in\mathfrak{n}_{-}).$

Then one can also consider the space of intertwiners

(5) $\Phi_{V}\colon Q_{\mathfrak{g}}\rightarrow Q_{\mathfrak{g}}\otimes V$

for any finite-dimensional $\mathfrak{g}$-representation $V$. Composition
gives a natural tensor structure as well. The only thing, that we need, to
apply the same machinery as in the parabolic restriction case is a Whittaker
analog of the trivialization isomorphism (3). There are several constructions
of the latter, and the one that we can adopt to the Whittaker setting is via
the so-called _extremal projector_ introduced in [AST71], see Section 1.4.

Observe that the space of (4) is the space of $\mathfrak{n}$-invariant vectors
in $V\otimes M^{\mathrm{univ}}$. The extremal projector is an explicitly
constructed element $P$ in a certain completion and localization of
$\mathrm{U}_{\hbar}(\mathfrak{g})$ satisfying

$\displaystyle eP=Pf$ $\displaystyle=0,\ \forall
e\in\mathfrak{n},f\in\mathfrak{n}_{-},$ $\displaystyle P^{2}$
$\displaystyle=P.$

In particular, its defines a projection to the space of
$\mathfrak{n}$-invariant vectors. Here is an example for
$\mathfrak{g}=\mathfrak{sl}_{2}$: let $e_{\alpha},f_{\alpha},h_{\alpha}$ be
the standard generators, then

$P_{\alpha}=\sum_{k\geq
0}\frac{(-\hbar)^{-k}}{k!}\frac{1}{\prod_{j=1}^{k}(h_{\alpha}+\hbar
j+\hbar)}f_{\alpha}^{k}e_{\alpha}^{k}.$

In general, the extremal projector is a certain product of $P_{\alpha}$ over
positive roots $\alpha$, see (1.9).

Likewise, the space of intertwiners (5) can be identified with the vectors
$w\in Q_{\mathfrak{g}}\otimes V$ satisfying $w(x-\psi(x))=0$ for all
$x\in\mathfrak{n}_{-}$ which are called _Whittaker vectors_. We will construct
an analog of the extremal projector in this setting for $\mathfrak{gl}_{N}$;
as the reader will see, it has a totally different flavor.

### Yangian

This is a summary of Section 3.

Certainly, an appearance of Yangians with relation to Whittaker modules is not
surprising: for instance, there is a well-known isomorphism between truncated
shifted Yangians and W-algebras in type A [BK06, RS99]. Here, however, the
relation is somewhat “dual”: for instance, in _loc. cit._ , principal
W-algebras are realized as quotients of the Yangian
$\mathrm{Y}_{\hbar}(\mathfrak{gl}_{1})$, while constructions in this paper
involve $\mathrm{Y}_{\hbar}(\mathfrak{gl}_{N})$.

The _Yangian_ is an algebra introduced by [Dri85] for an arbitrary reductive
$\mathfrak{g}$ which deforms the universal enveloping algebra
$\mathrm{U}_{\hbar}(\mathfrak{g}[t])$ of the polynomial current algebra
$\mathfrak{g}[t]$. In this paper, we restrict to the case
$\mathfrak{g}=\mathfrak{gl}_{N}$; we refer the reader to [Mol07] for all the
results about $\mathrm{Y}_{\hbar}(\mathfrak{gl}_{N})$ that we use in this
paper. Then it can be generated by the coefficients of the power series
$t_{ij}(u)=\delta_{ij}+\sum_{k\geq 1}t_{ij}^{(k)}u^{-k}$ with the _RTT
relation_

$R(u-v)T_{1}(u)T_{2}(v)=T_{2}(v)T_{1}(u)R(u-v),$

where $T(u):=(t_{ij}(u))$ is the generating matrix and $R(u)$ is _Yang’s
$R$-matrix_, see Section 3. It turns out that many familiar constructions from
linear algebra that involve a spectral parameter admit a “quantization” in
terms of $\mathrm{Y}_{\hbar}(\mathfrak{gl}_{N})$. For instance, the one that
we use actively in the paper is _quantum minors_ : for any collections of
indices $\\{a_{1},\ldots,a_{m}\\}$ and $\\{b_{1},\ldots,b_{m}\\}$, they are
defined by

$t_{b_{1}\ldots b_{m}}^{a_{1}\ldots
a_{m}}(u):=\sum\limits_{\sigma}\mathrm{sgn}(\sigma)t_{a_{\sigma(1)}b_{1}}(u)t_{a_{\sigma(2)}b_{2}}(u-\hbar)\ldots
t_{a_{\sigma(m)}b_{m}}(u-\hbar m+\hbar),$

where the sum is over all permutations $\sigma$ of $\\{1,\ldots,m\\}$.

There is the so-called _evaluation homomorphism_
$\mathrm{Y}_{\hbar}(\mathfrak{gl}_{N})\rightarrow\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})$
defined by $t_{ij}(u)\mapsto\delta_{ij}+u^{-1}E_{ij}$. For the rest of the
paper, we will use the same notations for constructions in the Yangian and
their images under the evaluation homomorphism, we hope it will not cause any
confusion. For instance, we will use a polynomial versions of quantum minors:

$L_{b_{1}\ldots b_{m}}^{a_{1}\ldots a_{m}}(u):=u(u-\hbar)\ldots(u-\hbar
m+\hbar)t_{b_{1}\ldots b_{m}}^{a_{1}\ldots
a_{m}}(u)\in\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})[u].$

In particular, the element $L_{1\ldots N}^{1\ldots N}(u)$ quantizes the
characteristic polynomial of a matrix, see 3.3.

### Kirillov projector

This is a summary of Section 4.

Let $N\geq i>j\geq 1$. Consider the following infinite sum depending on a
spectral parameter $u_{j}$:

(6) $P_{ij}^{\psi}(u):=\sum\limits_{k\geq
0}\hbar^{-k}(\psi(E_{ij})-E_{ij})^{k}\frac{\prod_{l=0}^{k-1}(-1)^{i+j}L_{j\ldots
i-1}^{j\ldots i-1}(u_{j}-l\hbar)}{k!}$

Throughout the paper, we will consider it as an operator on _right Whittaker
modules_ , i.e. right $\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})$-modules where

$\mathfrak{n}_{-}^{\psi}=\mathrm{span}(E_{ij}-\psi(E_{ij})|E_{ij}\in\mathfrak{n}_{-})$

acts locally nilpotently. Here is an example for $\mathfrak{gl}_{2}$:

$P_{21}^{\psi}(u_{1})=\sum\limits_{k\geq
0}\hbar^{-k}(E_{21}-1)^{k}\frac{(E_{11}+u_{1})\ldots(E_{11}+u_{1}-k\hbar+\hbar)}{k!}.$

Observe that for any $i,j$, the element $P_{ij}^{\psi}(u_{j})$ lies not just
in $\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})$, but in the universal enveloping
algebra of the mirabolic subalgebra $\mathrm{U}_{\hbar}(\mathfrak{m}_{N})$. In
particular, its action is well-defined on any right
$\mathrm{U}_{\hbar}(\mathfrak{m}_{N})$-modules where $\mathfrak{n}_{-}^{\psi}$
acts locally nilpotently.

Now we are ready to define the main object of the paper that we call the
_Kirillov projector_ , see 4.2:

$P_{\mathfrak{m}_{N}}(\vec{u}):=\left(P_{N,N-1}^{\psi}(u_{N-1})\right)\left(P_{N-1,N-2}^{\psi}(u_{N-2})P_{N,N-2}^{\psi}(u_{N-2})\right)\ldots\left(P_{21}^{\psi}(u_{1})\ldots
P_{N1}^{\psi}(u_{1})\right),$

and formulate Theorem 4.3, the core result of the paper:

######  Theorem.

For any right Whittaker module $W$ over
$\mathrm{U}_{\hbar}(\mathfrak{m}_{N})$, the Kirillov projector
$P_{\mathfrak{m}_{N}}(u_{1},\ldots u_{N-1})$ defines a unique map
$P(\vec{u})\colon W\rightarrow W^{\mathfrak{n}_{-}^{\psi}}$ satisfying

$\displaystyle P(\vec{u})(E_{ij}-\psi(E_{ij}))$ $\displaystyle=0,\ 1\leq
j<i\leq N,$ $\displaystyle(E_{kl}+\delta_{kl}u_{k})P(\vec{u})$
$\displaystyle=0,\ 1\leq k\leq l\leq N-1,$

and the normalization condition that it acts identically on Whittaker vectors.

The proof is technical, it uses various properties of quantum minors that we
list in Section 3. Intuitively, the formula corresponds to a step-by-step
reduction, where we pass from the Whittaker invariants for the _truncated
subalgebra_

$\begin{pmatrix}0&0&\ldots&0&0\\\ 0&*&\ldots&*&0\\\
\vdots&\ddots&\vdots&\vdots\\\ 0&*&\ldots&*&0\end{pmatrix},$

to the ones for $\mathfrak{m}_{N}$ by projecting to the shifted invariants for
the complement

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

via the block $P_{21}^{\psi}(u_{1})\ldots P_{N1}^{\psi}(u_{1})$. In fact, this
was one of the motivations for the name as it resembles the _Kirillov model_
[Kir63, GK75] (at least on a linguistic level).

On the classical level, the Kirillov projector defines a map
$\mathrm{pt}\rightarrow\mathfrak{m}_{N}^{*}$ given by the point $e_{\vec{u}}$
of (1), see Remark 4.4. It would be interesting to interpret the appearance of
the Yangian purely in terms of the mirabolic subalgebra: for instance, to
understand the fact that the Kirillov projector transforms spectral variables
$\vec{u}$ into the ones on the Cartan subalgebra, i.e. dynamical ones.

Since
$\mathrm{U}_{\hbar}(\mathfrak{m}_{N})\subset\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})$,
the Kirillov projector also acts on right Whittaker
$\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})$-modules. In this case, in addition to
the properties above, we have a somewhat weaker, but nicely interpretable
result, see Proposition 4.19 for a general case:

######  Proposition.

Define the coefficients of the quantum characteristic polynomial by

$L_{1\ldots N}^{1\ldots
N}(v)=v^{N}+\sum\limits_{i=0}^{N-1}(-1)^{N-i}A_{i}v^{i}.$

Then for $u_{k}=-k\hbar+\hbar$, we have an equality

$(E_{kN}-A_{k})P_{\mathfrak{m}_{N}}(0,-\hbar,\ldots,-N\hbar+2\hbar)\equiv 0\
\mod(\mathfrak{n}_{-}^{\psi}\mathrm{U}_{\hbar}(\mathfrak{gl}_{N}))$

in the left quotient by the right ideal generated by the shift
$\mathfrak{n}_{-}^{\psi}$.

In particular, for this choice of parameters, the Kirillov projector quantizes
the companion matrix, see Remark 4.20:

$\begin{pmatrix}0&\ldots&0&A_{N}\\\ 1&\ldots&0&A_{N-1}\\\
\vdots&\ddots&\vdots&\vdots\\\ 0&\ldots&1&A_{1}\end{pmatrix}$

Let us also point out some obvious downsides of the construction that,
unfortunately, we were not able to resolve.

  * •

It is clear that there should a definition of the Kirillov projector purely in
terms of the Yangian or, possibly, the Yangian double as in [Ioh96]. However,
we were not able to find a correct replacement for the nilpotent part of (6),
and the proof heavily uses the fact that we work in evaluation
representations;

  * •

The usual extremal projector is ambidextrous in the sense that its action is
well-defined either in left modules with a locally nilpotent
$\mathfrak{n}$-action or in right modules with a locally nilpotent
$\mathfrak{n}_{-}$-action. As for the Kirillov projector, it is not clear
under which conditions on a module it has a well-defined left action.

### Kostant-Whittaker reduction and quantization

This is a summary of Section 5.

It turns out that using the Kirillov projector, one can deduce some non-
trivial results on categorical behavior of Whittaker modules. For instance, we
have a mirabolic analog of _Skryabin’s equivalence_ [Kos78, Pre02], see
Theorem 5.1:

######  Theorem.

The category of Whittaker modules over $\mathrm{U}_{\hbar}(\mathfrak{m}_{N})$
is equivalent to the category $\mathrm{Vect}$ of vector spaces.

Likewise, recall the Whittaker analog of the exchange construction. We can
consider its mirabolic version: define
$Q_{\mathfrak{m}_{N}}:=\mathfrak{n}_{-}^{\psi}\backslash\mathrm{U}_{\hbar}(\mathfrak{m}_{N})$,
then there is a natural tensor structure on the intertwiners

$Q_{\mathfrak{m}_{N}}\rightarrow Q_{\mathfrak{m}_{N}}\otimes V,\
V\in\operatorname{Rep}(\mathrm{M}_{N}).$

Also recall that to construct a Tannakian functor from
$\operatorname{Rep}(\mathrm{M}_{N})$, we needed an analog of the
trivialization (3). It is given by Theorem 5.4:

######  Theorem.

The Kirillov projector defines a natural isomorphism
$V\rightarrow(Q_{\mathfrak{m}_{N}}\otimes V)^{\mathfrak{n}_{-}^{\psi}}$.

In fact, this theorem was another motivation for the name of the Kirillov
projector as this isomorphism bears some similarity to the Kirillov model as
well.

In particular, the exchange construction gives another main result of the
paper, see Theorem 5.5:

######  Theorem.

The monoidal functor
$\operatorname{Rep}(\mathrm{M}_{N})\rightarrow\mathrm{Vect}$, constructed via
the Kostant-Whittaker reduction, quantizes the family of rational Cremmer-
Gervais $r$-matrices.

Likewise, one can reprove known results on Whittaker modules for
$\mathfrak{gl}_{N}$. Denote by $\mathbf{A}_{\mathfrak{gl}_{N}}$ the center of
the universal enveloping algebra $\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})$, by
$Q=\mathfrak{n}_{-}^{\psi}\backslash\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})$ the
Whittaker analog of the universal Verma module, and by
$\mathrm{Wh}_{\hbar}(\mathfrak{gl}_{N})$ the category of
$(\mathbf{A}_{\mathfrak{gl}_{N}},\mathrm{U}_{\hbar}(\mathfrak{gl}_{N}))$-bimodules
that are Whittaker with respect to the right
$\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})$-action. This is an analog of the
_universal category $\mathcal{O}$_ [KS22, Definition 4.2], but without the
integrability assumption.

######  Theorem.

  1. (1)

(Theorem 5.12) The functor of Whittaker invariants defines an equivalence

$\mathrm{Wh}_{\hbar}(\mathfrak{gl}_{N})\xrightarrow{\sim}{}_{\mathbf{A}_{\mathfrak{gl}_{N}}}\mathrm{BiMod}_{\mathbf{A}_{\mathfrak{gl}_{N}}}$

with the category of $\mathbf{A}_{\mathfrak{gl}_{N}}$-bimodules.

  2. (2)

(Theorem 5.15) The Kirillov projector defines a natural isomorphism

$\mathbf{A}_{\mathfrak{gl}_{N}}\otimes
V\xrightarrow{\sim}(Q_{\mathfrak{gl}_{N}}\otimes
V)^{\mathfrak{n}_{-}^{\psi}}.$

  3. (3)

(Theorem 5.17) The exchange construction quantizes the family of (constant)
rational Cremmer-Gervais $r$-matrices.

For instance, for $\mathfrak{gl}_{2}$ the monoidal structure is given by the
action of the universal element

$F(u)=\sum\limits_{k\geq
0}\frac{(-\hbar)^{k}}{k!}E_{21}^{k}\otimes\prod\limits_{i=0}^{k-1}(E_{11}-i),$

and the first $\hbar$-power of the $R$-matrix
$R^{\mathrm{CG}}=R_{WV}(u)^{-1}R_{VW}(u)$ is indeed (2). A version of it
appeared in [GGS92], see also [KST98].

### Vertex-IRF transformation

This is a summary of Section 6. The main result of this section was originally
announced in [Kal21], but not proven; here, we fill in this gap. However, for
the details on all the other statements, we refer the reader to _loc. cit._

Recall that the monoidal structure on the Kostant-Whittaker reduction

$\mathrm{res}_{\mathfrak{gl}_{N}}^{\psi}\colon\operatorname{Rep}(G)\rightarrow{}_{\mathbf{A}_{\mathfrak{gl}_{N}}}\mathrm{BiMod}_{\mathbf{A}_{\mathfrak{gl}_{N}}}$

quantizes the rational Cremmer-Gervais $R$-matrix. Likewise, the monoidal
structure on the parabolic restriction

$\mathrm{res}_{\mathfrak{gl}_{N}}\colon\operatorname{Rep}(G)\rightarrow{}_{\mathrm{U}_{\hbar}(\mathfrak{h})}\mathrm{BiMod}_{\mathrm{U}_{\hbar}(\mathfrak{h})}$

quantizes the standard dynamical $R$-matrix. The Harish-Chandra homomorphism
$\mathbf{A}_{\mathfrak{gl}_{N}}\rightarrow\mathrm{U}_{\hbar}(\mathfrak{h})$
gives, on the one hand, the restriction functor

${}_{U(\mathfrak{h})}\mathrm{BiMod}_{\mathrm{U}_{\hbar}(\mathfrak{h})}\rightarrow{}_{\mathbf{A}_{\mathfrak{gl}_{N}}}\mathrm{BiMod}_{\mathrm{U}_{\hbar}(\mathfrak{h})},$

on the other hand, the extension of scalars functor

$(-)\otimes_{A_{\mathfrak{gl}_{N}}}\mathrm{U}_{\hbar}(\mathfrak{h})\colon{}_{\mathbf{A}_{\mathfrak{gl}_{N}}}\mathrm{BiMod}_{\mathbf{A}_{\mathfrak{gl}_{N}}}\rightarrow{}_{\mathbf{A}_{\mathfrak{gl}_{N}}}\mathrm{BiMod}_{\mathrm{U}_{\hbar}(\mathfrak{h})}.$

In particular, it is possible to compare $\mathrm{res}_{\mathfrak{gl}_{N}}$
with $\mathrm{res}_{\mathfrak{gl}_{N}}^{\psi}$, see Proposition 6.6:

######  Proposition.

There is a generic equivalence
$\mathrm{res}_{\mathfrak{gl}_{N}}\xrightarrow{\sim}\mathrm{res}_{\mathfrak{gl}_{N}}^{\psi}\otimes_{\mathbf{A}_{\mathfrak{gl}_{N}}}\mathrm{U}_{\hbar}(\mathfrak{h})$.

Unfortunately, this equivalence does not respect tensor structures. It turns
out that the right setup to compare them is that of _module categories_ :
namely, given a monoidal category $\mathcal{C}$, a module category
$\mathcal{M}$ over $\mathcal{C}$ is plain category together with a functor
$\mathcal{M}\otimes\mathcal{C}\rightarrow\mathcal{M}$ satisfying an analog of
pentagon axiom. One can naturally define a functor between two module
categories $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$: this is a plain functor
$\mathcal{M}_{1}\rightarrow\mathcal{M}_{2}$ intertwining the corresponding
tensor isomorphisms. One can define an abstract vertex-IRF transformation in
this language, see Theorem 6.5.

Observe that for any algebra $A$, the category of bimodules
${}_{A}\mathrm{BiMod}_{A}$ acts on the category $\mathrm{RMod}_{A}$ of right
modules. In particular, parabolic restriction gives an action of
$\operatorname{Rep}(\mathrm{GL}_{N})$ on
$\mathrm{RMod}_{\mathrm{U}_{\hbar}(\mathfrak{h})}$, whose monoidal
isomorphisms produce the standard dynamical $R$-matrix. Likewise, Kostant-
Whittaker reduction defines an action of $\operatorname{Rep}(G)$ on
$\mathrm{RMod}_{\mathbf{A}_{\mathfrak{gl}_{N}}}$. Using the identification
between two restrictions, we get a categorical interpretation of the vertex-
IRF transformation:

######  Theorem.

The extension of scalars functor
$(-)\otimes_{\mathbf{A}_{\mathfrak{gl}_{N}}}\colon\mathrm{RMod}_{\mathbf{A}_{\mathfrak{gl}_{N}}}\rightarrow\mathrm{RMod}_{\mathrm{U}_{\hbar}(\mathfrak{h})}$
intertwines the corresponding $\operatorname{Rep}(G)$-actions. In particular,
it gives _the_ vertex-IRF transformation between the standard dynamical
$R$-matrix $R^{\mathrm{dyn}}(\lambda)$ and a constant Cremmer-Gervais
$R$-matrix $R^{\mathrm{CG}}(\vec{u})$ for any values of parameters $\vec{u}$.

### Structure of the paper

In Section 1, we give a general setup for the paper and review the categorical
approach to (standard) dynamical quantum groups via the category of Harish-
Chandra bimodules and parabolic restriction functor as in [KS22] . Proofs are
mostly omitted unless there is a version that we need later in the paper and
it is different from _loc. cit._ In Section 2, we define the mirabolic
subgroup and a family of $r$-matrices that deform the degenerate version of
the Cremmer-Gervais $r$-matrix. In Section 3, we review Yangians and related
constructions such as quantum minors. We prove various technical results that
we use later in the paper. In Section 4, we introduce the Kirillov projector
and prove its defining properties. In Section 5, we recall the Kostant-
Whittaker reduction functor, introduce its mirabolic version, and show some
properties thereof. We also obtain quantization of the family of rational
Cremmer-Gervais solutions. In Section 6, we define a categorical version of
the vertex-IRF transformation and show that an equivalence between parabolic
restriction and Kostant-Whittaker reduction gives the vertex-IRF
transformation in the classical sense.

### Acknowledgments

The author would like to thank Roman Bezrukavnikov, Pavel Etingof, Boris
Feigin, Michael Finkelberg, Vasily Krylov, and Leonid Rybnikov for helpful
discussions and explanations. This work was initially supported by the
Ministry of Science and Higher Education of the Russian Federation (agreement
no. 075–15–2022–287); the major part of it was accomplished at the
Massachusetts Institute of Technology. The author is very grateful to the
Department of Mathematics of MIT for hospitality and for the opportunity to
avoid (a form of) politically motivated persecution in Russia.

##  1\. Background: dynamical quantum groups

In this section, we will give a brief overview of the classical extremal
projector and its relation to dynamical quantum groups through the category of
Harish-Chandra bimodules. Note: the cited literature mostly deals with the
case $\hbar=1$; however, the proofs of all the statements we use are easily
generalizable to the case of an arbitrary $\hbar$.

###  1.1. General setup

For the rest of the paper, we work over an algebraically closed field $k$ of
characteristic zero. All categories and functors we will consider are
$k$-linear. Throughout this paper we work with locally presentable categories
(we refer to [AR94] and [BCJ15, Section 2] for more details). We will also use
Deligne’s tensor product of locally presentable categories introduced in
[Del90].

Almost all the constructions involve a parameter $\hbar$: to avoid categorical
complications, we treat $\hbar$ as a _non-formal_ parameter, i.e. $\hbar\in
k^{\times}$ (in particular, we can still deal with $k$-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 $\mathbb{A}^{1}$. We hope it will not lead to any confusion.

###  1.2. Harish-Chandra bimodules

For details and proofs in a more general setting, we refer the reader to
[KS22, Section 2].

Let $G$ be an affine algebraic group and $\mathfrak{g}$ be its Lie algebra
with bracket $[,]$.

######  Definition 1.1.

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

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

######  Remark 1.2.

In principal, such $\hbar$-version is usually called the _asymptotic_
universal enveloping algebra; however, it is usually defined over $k[\hbar]$,
but in our case, parameter $\hbar$ is a number, see Section 1.1. Moreover,
some formulas will use $\hbar^{-1}$. So, we decided to drop the adjective
“asymptotic”.

Denote by $\operatorname{Rep}(G)$ the category of $G$-representations.
Naturally, $\mathrm{U}_{\hbar}(\mathfrak{g})$ is an object in
$\operatorname{Rep}(G)$.

######  Definition 1.3.

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

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

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_ :

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

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

######  Proposition 1.4.

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

###  1.3. Constant and dynamical $R$-matrices

For introduction to the theory of dynamical quantum groups, see [ES01]; for
the interpretation in terms of the category of Harish-Chandra bimodules, see
[KS22].

######  Definition 1.5.

  * •

Let $F_{VW}\colon V\otimes W\rightarrow V\otimes W$ be a collection of linear
maps natural in $V,W\in\operatorname{Rep}(G)$. The _twist_ equation is

(1.1) $F_{U\otimes V,W}\circ(F_{UV}\otimes\mathrm{id}_{W})=F_{U,V\otimes
W}\circ(\mathrm{id}_{U}\otimes F_{VW})\in\mathrm{End}(U\otimes V\otimes W).$

A solution is called a _(Drinfeld) twist_.

  * •

Let $R_{VW}\colon V\otimes W\rightarrow V\otimes W$ be a collection of linear
maps natural in $V,W\in\operatorname{Rep}(G)$. The _quantum Yang-Baxter
equation_ is

(1.2) $R_{UV}R_{UW}R_{VW}=R_{VW}R_{UW}R_{UV}\in\mathrm{End}(U\otimes V\otimes
W).$

A solution is called a _quantum $R$-matrix_.

  * •

Let $r_{VW}\colon V\otimes W\rightarrow V\otimes W$ be a collection of linear
map natural in $V,W\in\operatorname{Rep}(G)$. The _classical Yang-Baxter
equation_ is

(1.3)
$[r_{UV},r_{UW}]+[r_{UV},r_{VW}]+[r_{UW},r_{VW}]=0\in\mathrm{End}(U\otimes
V\otimes W).$

A solution is called a _classical $r$-matrix_.

The following is standard:

######  Proposition 1.6.

  1. (1)

A tensor structure on the forgetful functor
$\operatorname{Rep}(G)\rightarrow\mathrm{Vect}$ is equivalent to the data of a
Drinfeld twist (1.1).

  2. (2)

Using the natural symmetric monoidal structure on $\operatorname{Rep}(G)$, one
can consider $F_{WV}$ as an element of $\mathrm{End}(V\otimes W)$. Define
$R_{VW}=F_{WV}^{-1}F_{VW}$. Then $R_{VW}$ is a quantum $R$-matrix (1.2).

  3. (3)

Assume that there is a family $R_{VW}^{\hbar}$ depending on $\hbar$ such that
$R_{VW}^{\hbar}=\mathrm{id}_{V\otimes W}+\hbar r_{VW}+O(\hbar^{2})$. Then
$r_{VW}$ is a classical $r$-matrix (1.3).

There is a version of these equations involving dynamical parameter
$\lambda\in\mathfrak{h}^{*}$. Let $H\subset G$ be a torus and
$\mathfrak{h}\subset\mathfrak{g}$ be its Lie algebra. Let $U,V,W$ be weight
$\mathfrak{h}$-modules. Define the operator

$h_{V}\colon U\otimes V\otimes W\rightarrow U\otimes V\otimes W,\qquad
h_{V}\cdot u\otimes v\otimes w=\mathrm{wt}(v)\cdot u\otimes v\otimes w,$

similarly for $h_{U},h_{W}$.

######  Definition 1.7.

  * •

Let $J_{VW}(\lambda)\colon V\otimes W\rightarrow V\otimes W$ be a collection
of $\mathrm{End}_{\mathfrak{h}}(V\otimes W)$-valued functions on
$\mathfrak{h}^{*}$ natural in $V,W\in\operatorname{Rep}(G)$. The _dynamical
twist_ equation is

(1.4) $J_{U\otimes
V,W}(\lambda)\circ(J_{UV}(\lambda)\otimes\mathrm{id}_{W})=J_{U,V\otimes
W}\circ(\mathrm{id}_{U}\otimes J_{VW}(\lambda-\hbar h_{U})).$

A solution is called a _dynamical (Drinfeld) twist_.

  * •

Let $R_{VW}(\lambda)\colon V\otimes W\rightarrow V\otimes W$ be a collection
of $\mathrm{End}_{\mathfrak{h}}(V\otimes W)$-valued functions on
$\mathfrak{h}^{*}$ natural in $V,W\in\operatorname{Rep}(G)$. The _quantum
dynamical Yang-Baxter equation_ is

(1.5) $R_{UV}(\lambda-\hbar h_{W})R_{UW}(\lambda)R_{VW}(\lambda-\hbar
h_{U})=R_{VW}(\lambda)R_{UW}(\lambda-\hbar h_{V})R_{UV}(\lambda).$

A solution is called a _quantum dynamical $R$-matrix_.

  * •

Let $r_{VW}\colon V\otimes W\rightarrow V\otimes W$ be a collection of
$\mathrm{End}_{\mathfrak{h}}(V\otimes W)$-valued functions on
$\mathfrak{h}^{*}$ natural in $V,W\in\operatorname{Rep}(G)$. The _classical
dynamical Yang-Baxter equation_ is

(1.6)
$\displaystyle\begin{split}&\sum\limits_{i}\left((x_{i})_{U}\frac{\partial
r_{VW}}{\partial x^{i}}-(x_{i})_{V}\frac{\partial r_{UW}}{\partial
x^{i}}+(x_{i})_{W}\frac{\partial r_{UV}}{\partial x^{i}}\right)+\\\
+&[r_{UV}(\lambda),r_{UW}(\lambda)]+[r_{UV}(\lambda),r_{VW}(\lambda)]+[r_{UW}(\lambda),r_{VW}(\lambda)]=0,\end{split}$

where $\\{x_{i}\\}\subset\mathfrak{h}$ is a basis in $\mathfrak{h}$, and
$\\{x^{i}\\}$ is the dual basis in $\mathfrak{h}^{*}$. A solution is called a
_classical dynamical $r$-matrix_.

It turns out that there is a Tannakian interpretation of dynamical
$R$-matrices in terms of functors to the category of Harish-Chandra bimodules.
Since $\mathfrak{h}$ is commutative, the universal enveloping algebra
$\mathrm{U}_{\hbar}(\mathfrak{h})$ is equal to the space
$\mathcal{O}(\mathfrak{h}^{*})$ of polynomial functions on $\mathfrak{h}^{*}$.
In particular, for any $X\in\operatorname{Rep}(H)$, a map between free Harish-
Chandra bimodules

$\Phi\colon\mathrm{U}_{\hbar}(\mathfrak{h})\otimes
V\rightarrow\mathrm{U}_{\hbar}(\mathfrak{h})\otimes V$

can be equivalently given by an $\mathrm{End}_{\mathfrak{h}}(V)$-valued
function $\Phi(\lambda)$ on $\mathfrak{h}^{*}$. We use this identification for
the rest of this paper.

######  Theorem 1.8.

[KS22] Consider the composition

$\operatorname{Rep}(G)\xrightarrow{\mathrm{forget}}\operatorname{Rep}(H)\xrightarrow{\mathrm{free}}\mathrm{HC}_{\hbar}(H),$

where the first arrow is the forgetful functor.

  1. (1)

A monoidal structure on
$\operatorname{Rep}(G)\rightarrow\mathrm{HC}_{\hbar}(H)$ is equivalent to the
data of a dynamical twist (1.4).

  2. (2)

Using the natural symmetric monoidal structures on $\operatorname{Rep}(G)$,
one can consider $J_{WV}(\lambda)$ as an element of
$\mathrm{End}_{\mathfrak{h}}(V\otimes W)$. Define
$R_{VW}(\lambda)=J_{WV}(\lambda)^{-1}J_{VW}(\lambda)$. Then $R_{VW}(\lambda)$
is a quantum dynamical $R$-matrix (1.5).

  3. (3)

Assume that there is a family $R_{VW}^{\hbar}(\lambda)$ depending on a
parameter $\hbar$ such that

$R_{VW}^{\hbar}(\lambda)=\mathrm{id}_{V\otimes W}+\hbar
r_{VW}(\lambda)+O(\hbar^{2}).$

Then $r_{VW}(\lambda)$ is a classical dynamical $r$-matrix (LABEL:eq::cdybe).

From now on, let $G$ be a reductive group. As in the non-dynamical case, there
is a standard solution to classical dynamical Yang-Baxter equation. Namely,
choose a Borel subgroup $B\subset G$ and a maximal torus $H\subset B$. It
induces a triangular decomposition
$\mathfrak{g}=\mathfrak{n}_{-}\oplus\mathfrak{h}\oplus\mathfrak{n}$ with
respect to the set of positive roots $\Delta_{+}$. For $\alpha\in\Delta_{+}$,
denote by $\\{f_{\alpha},h_{\alpha},e_{\alpha}\\}$ the corresponding
$\mathfrak{sl}_{2}$-triple of the Chevalley base. Define (see [EV99])

(1.7) $r(\lambda)=\sum_{\alpha\in\Delta_{+}}\frac{e_{\alpha}\wedge
f_{\alpha}}{(\lambda,\alpha)}.$

One can check that it indeed satisfies (LABEL:eq::cdybe). In what follows, we
will construct its quantization in Section 1.5.

###  1.4. Extremal projector

The original construction can be found in [AST71], see also [Zhe90].

Denote by
$\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}=\mathrm{Frac}(\mathrm{U}_{\hbar}(\mathfrak{h}))$
the ring of fractions of $\mathrm{U}_{\hbar}(\mathfrak{h})$ and by
$\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}=\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}\otimes_{\mathrm{U}_{\hbar}(\mathfrak{h})}\mathrm{U}_{\hbar}(\mathfrak{g})$
the localized version of the universal enveloping algebra. For
$\alpha\in\Delta_{+}$, define the following element lying in a certain
completion of $\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}$:

(1.8) $P_{\alpha}(t)=\sum\limits_{k\geq
0}\frac{(-1)^{k}\hbar^{-k}}{k!}\frac{1}{\prod_{j=1}^{k}(h_{\alpha}+\hbar(t+j))}f_{\alpha}^{k}e_{\alpha}^{k},$

Choose a normal ordering $<$ on $\Delta_{+}$ (i.e. such that $\alpha+\beta$
lies between $\alpha$ and $\beta$). Denote by $\rho$ the half-sum of positive
roots. Define

(1.9) $P:=\prod_{\alpha\in\Delta_{+}}^{<}P_{\alpha}(t_{\alpha}),$

where the product is taken in the normal ordering and
$t_{\alpha}=h_{\alpha}(\rho)$.

######  Theorem 1.9.

[AST71]. The element $P$ satisfies

$e_{\alpha}P=Pf_{\alpha}=0\ \forall\alpha\in\Delta_{+}.$

The action of $P$ is well-defined on left (resp. right)
$\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}$-modules with a locally
nilpotent $\mathfrak{n}$\- (resp. $\mathfrak{n}_{-}$-) action.

For any left $\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}$-module $X$,
denote by $\mathfrak{n}_{-}\backslash
X:=(\mathfrak{n}_{-}\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}\cdot
X)\backslash X$ the quotient by the right
$\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}$ ideal generated by
$\mathfrak{n}_{-}$, and by $X^{\mathfrak{n}}$ the space of left
$\mathfrak{n}$-invariants. Then the extremal projector defines an isomorphism

(1.10) $\mathfrak{n}_{-}\backslash X\xrightarrow{\sim}X^{\mathfrak{n}}$

for any left module with a locally nilpotent $\mathfrak{n}$-action.

###  1.5. Parabolic restriction functor

The main reference for the setup of this paper is [KS22], but the history of
the subject is certainly much richer, for instance, see [ES01].

######  Definition 1.10.

A _universal generic category $\mathcal{O}$_ is the category
$\mathcal{O}^{\mathrm{univ},\mathrm{gen}}$ of
$(\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}},\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}})$-bimodules
such that the diagonal $\mathfrak{b}$-action integrates to a $B$-action. The
_universal generic Verma module_ is

$M^{\mathrm{univ}}:=\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}\otimes_{\mathrm{U}_{\hbar}(\mathfrak{n})}\mathbf{C}.$

Similarly to Section 1.2, we can define a generic version
$\mathrm{HC}_{\hbar}(H)^{\mathrm{gen}}$ of the category of Harish-Chandra
bimodules. We have two functors:

(1.11)
$\mathrm{act}_{H}\colon\mathrm{HC}_{\hbar}(H)^{\mathrm{gen}}\rightarrow\mathcal{O}^{\mathrm{univ},\mathrm{gen}},\qquad
X\mapsto
M^{\mathrm{univ}}\otimes_{\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}}X,$

and

(1.12)
$\mathrm{act}_{G}\colon\mathrm{HC}_{\hbar}(G)\rightarrow\mathcal{O}^{\mathrm{univ},\mathrm{gen}},\qquad
X\mapsto X\otimes_{\mathrm{U}_{\hbar}(\mathfrak{g})}M^{\mathrm{univ}}.$

Denote by
$X/\mathfrak{n}:=X/(X\cdot\mathrm{U}_{\hbar}(\mathfrak{g})\mathfrak{n})$ the
quotient by the left $\mathrm{U}_{\hbar}(\mathfrak{g})$-ideal generated by
$\mathfrak{n}$. Then the latter functor can also be presented as
$X\mapsto(X\mathfrak{n})\otimes_{\mathrm{U}_{\hbar}(\mathfrak{h})}\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}$.

With a help of the extremal projector, one can show the following

######  Proposition 1.11.

The functor $\mathrm{act}_{H}$ is an equivalence.

###### Proof.

We repeat an argument from [KS22, Theorem 4.17].

By [KS22, Proposition 4.6], the functor
$(-)^{\mathfrak{n}}\colon\mathcal{O}^{\mathrm{univ},\mathrm{gen}}\rightarrow\mathrm{HC}_{\hbar}(H)^{\mathrm{gen}}$
is right adjoint to $\mathrm{act}_{H}$. The unit of the adjunction is given by

$X\mapsto(M^{\mathrm{univ}}\otimes_{\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}}X)^{\mathfrak{n}}.$

The extremal projector gives an isomorphism with the quotient functor as in
(1.10):

${}_{\mathfrak{n}_{-}}(-)\colon\mathcal{O}^{\mathrm{univ},\mathrm{gen}}\rightarrow\mathrm{HC}_{\hbar}(H),\qquad
M\mapsto\mathfrak{n}_{-}\backslash M.$

So, the functor $(-)^{\mathfrak{n}}$, is exact. We have

$(M^{\mathrm{univ}}\otimes_{\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}}X)^{\mathfrak{n}}\cong\mathfrak{n}_{-}\backslash(M^{\mathrm{univ}}\otimes_{\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}}X),$

and the composition

$X\mapsto\mathfrak{n}_{-}\backslash(M^{\mathrm{univ}}\otimes_{\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}}X)$

is an isomorphism by the PBW theorem. Therefore, $\mathrm{act}_{H}$ is fully
faithful.

Since the $\mathfrak{n}$-action on
$M\in\mathcal{O}^{\mathrm{univ},\mathrm{gen}}$ is locally nilpotent,
$M^{\mathfrak{n}}=0$ if and only if $M=0$. But $(-)^{\mathfrak{n}}$ is exact,
therefore, it is conservative. Since its left adjoint $\mathrm{act}_{H}$ is
fully faithful, it is an equivalence. ∎

######  Definition 1.12.

The _parabolic restriction_ functor
$\mathrm{res}_{\mathfrak{g}}\colon\mathrm{HC}_{\hbar}(G)\rightarrow\mathrm{HC}_{\hbar}(H)$
is the composition

$\mathrm{HC}_{\hbar}(G)\xrightarrow{\mathrm{act}_{G}}\mathcal{O}^{\mathrm{univ},\mathrm{gen}}\xrightarrow{(-)^{\mathfrak{n}}}\mathrm{HC}_{\hbar}(H)^{\mathrm{gen}}.$

Explicitly, it is given by quantum Hamiltonian reduction and extension of
scalars

$X\mapsto((X/\mathfrak{n})\otimes_{\mathrm{U}_{\hbar}(\mathfrak{h})}\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}})^{\mathfrak{n}}.$

Equivalently, we can extend the scalars on the $\mathrm{HC}_{\hbar}(G)$-side
first and then take the quantum Hamiltonian reduction:

$X\mapsto(\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}\otimes_{\mathrm{U}_{\hbar}(\mathfrak{g})}X/\mathfrak{n})^{\mathfrak{n}}.$

There is a natural lax monoidal structure on $\mathrm{res}_{\mathfrak{g}}$:

(1.13)
$(X/\mathfrak{n})^{\mathfrak{n}}\otimes_{\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}}(Y/\mathfrak{n})^{\mathfrak{n}}\rightarrow(X\otimes_{\mathrm{U}_{\hbar}(\mathfrak{g})}Y/\mathfrak{n})^{\mathfrak{n}},\qquad[x]\otimes[y]\mapsto[x\otimes
y].$

Using abstract categorical techniques, one can deduce from Proposition 1.11
the following

######  Theorem 1.13.

[KS22, Corollary 4.18] The functor $\mathrm{res}_{\mathfrak{g}}$ is colimit-
preserving and monoidal.

Similarly to Section 1.2, one can define a generic version of the functor of
free Harish-Chandra bimodules
$\operatorname{Rep}(H)\rightarrow\mathrm{HC}_{\hbar}(H)^{\mathrm{gen}}$. Then,
on the one hand, we have a monoidal functor
$\operatorname{Rep}(G)\rightarrow\mathrm{HC}_{\hbar}(H)^{\mathrm{gen}}$
obtained by composition with $\mathrm{res}_{\mathfrak{g}}$. On the other hand,
there is the forgetful functor
$\operatorname{Rep}(G)\rightarrow\operatorname{Rep}(H)$ which is also
monoidal. So, we have a diagram

(1.14)
$\textstyle{\operatorname{Rep}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{HC}_{\hbar}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Rep}(H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{HC}_{\hbar}(H)^{\mathrm{gen}}}$

######  Theorem 1.14.

[KS22, Theorem 4.20] The extremal projector defines a natural isomorphism

$P_{V}\colon\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}\otimes
V\rightarrow(\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}\otimes
V/\mathfrak{n})^{\mathfrak{n}}.$

In other words, the diagram (1.14) is commutative.

###### Proof.

We will use a slightly different homomorphism as in the citation. By the PBW
theorem, we have a vector space isomorphism

$\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}\otimes
V\cong\mathrm{U}_{\hbar}(\mathfrak{n}_{-})\otimes
U(\mathfrak{h})^{\mathrm{gen}}\otimes
V\otimes\mathrm{U}_{\hbar}(\mathfrak{n})$

for any $G$-representation $V$. In particular,

$\mathfrak{n}_{-}\backslash\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}\otimes
V/\mathfrak{n}\cong\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}\otimes V.$

Therefore, the extremal projector gives a natural isomorphism as in (1.10):

$(\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}\otimes
V/\mathfrak{n})^{\mathfrak{n}}\xrightarrow{\sim}\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}\otimes
V.$

Explicitly, the inverse is given by

$f(\lambda)\otimes v\mapsto
f(\lambda)Pv\in\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}\otimes
V/\mathfrak{n}$

for any $f(\lambda)\in\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}$ and
$v\in V$. ∎

In particular, one can translate the natural tensor structure on
$\mathrm{res}_{\mathfrak{g}}$ to the free module functor
$\operatorname{Rep}(G)\rightarrow\mathrm{HC}_{\hbar}(H)^{\mathrm{gen}}$.

######  Theorem 1.15.

  1. (1)

There is a collection of maps $J_{VW}(\lambda)$ natural in
$V,W\in\operatorname{Rep}(G)$, such that the diagram

$\textstyle{(\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}\otimes
V)\otimes_{\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}}(\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}\otimes
W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{J_{VW}(\lambda)}$$\scriptstyle{P_{V}\otimes
P_{W}}$$\textstyle{\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}\otimes
V\otimes
W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{V\otimes
W}}$$\textstyle{\mathrm{res}_{\mathfrak{g}}(\mathrm{U}_{\hbar}(\mathfrak{g})\otimes
V)\otimes_{\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}}\mathrm{res}_{\mathfrak{g}}(\mathrm{U}_{\hbar}(\mathfrak{g})\otimes
W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{res}_{\mathfrak{g}}(\mathrm{U}_{\hbar}(\mathfrak{g})\otimes
V\otimes W)}$

is commutative, where the lower arrow is the natural tensor structure on
$\mathrm{res}_{\mathfrak{g}}$. In particular, the collection of
$J_{VW}(\lambda)$ satisfies the dynamical twist equation Eq. 1.4, and
$R^{\mathrm{dyn}}_{VW}(\lambda):=J_{WV}(\lambda)^{-1}J_{VW}(\lambda)$ is a
quantum dynamical $R$-matrix from 1.7.

  2. (2)

The map $J_{VW}(\lambda)$ has the form

$J_{VW}(\lambda)\in\mathrm{id}_{V\otimes
W}+\hbar\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}\otimes\mathrm{U}_{\hbar}(\mathfrak{b}_{-})^{>0}\otimes\mathrm{U}_{\hbar}(\mathfrak{b}_{-})^{>0},$

where the upper subscript $>0$ means the augmentation ideal.

  3. (3)

The coefficient $j_{VW}(\lambda)$ of the first $\hbar$-power of
$J_{VW}(\lambda)$ is given by

$j_{VW}(\lambda)=-\sum\limits_{\alpha\in\Delta_{+}}\frac{1}{h_{\alpha}}f_{\alpha}\otimes
e_{\alpha}=-\frac{f_{\alpha}\otimes e_{\alpha}}{(\alpha,\lambda)}.$

In particular, the dynamical $R$-matrix $R^{\mathrm{dyn}}(\lambda)_{VW}$
quantizes the standard classical dynamical $r$-matrix (1.7):

$r(\lambda):=j(\lambda)-j^{21}(\lambda)=\sum_{\alpha\in\Delta_{+}}\frac{e_{\alpha}\wedge
f_{\alpha}}{(\alpha,\lambda)}.$

###### Proof.

We use the argument from [Kho04]. In what follows, we drop the tensor product
sign, i.e. for any $x\in\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}\otimes
V,y\in\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}\otimes W$, we denote

$xy:=[x\otimes_{\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}}y]\in(\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}\otimes
V)\otimes_{\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}}(\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}\otimes
W)\cong\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}\otimes V\otimes W.$

Observe that by Theorem 1.14, every element in
$(\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}\otimes
V/\mathfrak{n})^{\mathfrak{n}}$ can written as $f(\lambda)PvP$, where
$f(\lambda)\in\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}$ and the right
$P$ multiplier corresponds to taking the quotient by $\mathfrak{n}$. Then the
statement can be presented as follows: for every $v\in V,w\in W$, we have

(1.15) $PvPwP=PJ_{V,W}(\lambda)(v\otimes w)P.$

Since the collection of $J_{VW}(\lambda)$ defines a tensor structure on
$\operatorname{Rep}(G)\rightarrow\mathrm{HC}_{\hbar}(H)^{\mathrm{gen}}$, it
automatically satisfies the dynamical twist equation by Theorem 1.8; the same
is true for the dynamical $R$-matrix $R(\lambda)$.

Let us write $P$ in the PBW basis:

$P=1+\sum_{i}\hbar^{k_{i}}a_{i}(\lambda)f_{i}e_{i},$

where
$a_{i}(\lambda)\in\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}},f_{i}\in\mathrm{U}_{\hbar}(\mathfrak{n}_{-}),e_{i}\in\mathrm{U}_{\hbar}(\mathfrak{n})$,
and $k_{i}$ are some negative numbers. It follows from the construction Eq.
1.8 that

$P\in
1+(\mathfrak{n}_{-}\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}\cap\mathrm{U}_{\hbar}(\mathfrak{g})^{\mathrm{gen}}\mathfrak{n}).$

Therefore, we have $f_{i}=1$ if and only if $e_{i}=1$. Then we proceed as
follows: consider the middle $P$ in (1.15). Using the fact that $P$ commutes
with $\mathrm{U}_{\hbar}(\mathfrak{h})^{\mathrm{gen}}$, we can push
$a_{i}(\lambda)$ to the left. Using Theorem 1.9, we can push $f_{i}$ to the
left until it meets $P$ and becomes zero. Likewise, we can push $e_{i}$ to the
right until it meets $P$ and becomes zero as well.

Let us demonstrate how it works when $f_{i}$ and $e_{i}$ are Lie algebra
elements. Then we are dealing with the term
$\hbar^{-1}a_{i}(\lambda)f_{i}e_{i}$. Let us compute
$\hbar^{-1}Pva_{i}(\lambda)f_{i}e_{i}wP$. First, we push $a_{i}(\lambda)$ to
the left:

$Pva_{i}(\lambda)=Pa_{i}(\lambda-\hbar\mathrm{wt}(v))v=a_{i}(\lambda-\hbar\mathrm{wt}(v))Pv.$

By $Pf_{i}=0$, we have

$Pvf_{i}=Pf_{i}v-\hbar P\operatorname{ad}_{f_{i}}(v)=-\hbar
P\operatorname{ad}_{f_{i}}(v).$

Likewise, by $e_{i}P=0$, we get

$e_{i}wP=\hbar\operatorname{ad}_{e_{i}}(w)P.$

Therefore,

$\hbar^{-1}Pva_{i}(\lambda)f_{i}e_{i}wP=-\hbar
a_{i}(\lambda-\hbar\mathrm{wt}(v))P\operatorname{ad}_{f_{i}}(v)\operatorname{ad}_{e_{i}}(v)P,$

which has the necessary form.

In general, we see that each term $\hbar^{-l}f_{\alpha}^{l}e_{\alpha}^{l}$ in
(1.8) acts with a minimal power of $\hbar^{l}$, and the second part of the
theorem follows. One can easily check that the first power of $\hbar$ is given
by the sum of actions of $k=1$ terms of (1.8) in the product (1.9), which is

$j(\lambda)=\sum\limits_{\alpha\in\Delta_{+}}\frac{1}{h_{\alpha}}f_{\alpha}\otimes
e_{\alpha}=-\sum\limits_{\alpha\in\Delta_{+}}\frac{f_{\alpha}\otimes
e_{\alpha}}{(\lambda,\alpha)}.$

∎

The last part motivates the following definition, see [EV99]:

######  Definition 1.16.

The _standard dynamical twist_ is the collection $J_{VW}(\lambda)$. The
_standard quantum dynamical $R$-matrix_ is $R^{\mathrm{dyn}}_{VW}(\lambda)$.

##  2\. Mirabolic subgroup

Denote by $\mathrm{GL}_{N}$ the group of invertible $N\times N$-matrices and
by $\mathfrak{gl}_{N}$ its Lie algebra. We choose a natural basis
$\\{E_{ij}|1\leq i,j\leq N\\}$ of matrix units in $\mathfrak{gl}_{N}$ with the
commutator

$[E_{ij},E_{kl}]=\delta_{jk}E_{il}-\delta_{li}E_{kj}.$

For the rest of the paper, we will adopt the following notations. For any
$k\leq N$, we consider $\mathfrak{gl}_{k}\subset\mathfrak{gl}_{N}$ embedded as
the _upper left corner_ :

$\mathfrak{gl}_{k}=\begin{pmatrix}*&\ldots&*&0&\ldots&0\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\ *&\ldots&*&0&\ldots&0\\\
0&\ldots&0&0&\ldots&0\\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
0&\ldots&0&0&\ldots&0\end{pmatrix}\begin{array}[]{ c
}\vphantom{\begin{array}[]{ c }b_{N}\\\ \vdots\\\ z_{N}\end{array}}\\\
\left.\kern-1.2pt\vphantom{\begin{array}[]{ c }b_{N}\\\ \vdots\\\
z_{N}\end{array}}\right\\}\text{$N-k$}\end{array}$

Likewise, for any $k\leq N$, we denote by ${}_{k}\mathfrak{gl}_{N}$ the _left
$k$-th truncation_ of $\mathfrak{gl}_{N}$: it is the subalgebra isomorphic to
$\mathfrak{gl}_{N-k}$, but embedded as the _lower right corner_ :

${}_{k}\mathfrak{gl}_{N}=\begin{pmatrix}0&\ldots&0&0&\ldots&0\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\ 0&\ldots&0&0&\ldots&0\\\
0&\ldots&0&*&\ldots&*\\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
0&\ldots&0&*&\ldots&*\end{pmatrix}\begin{array}[]{ c
}\vphantom{\begin{array}[]{ c }b_{N}\\\ \vdots\\\ z_{N}\end{array}}\\\
\left.\kern-1.2pt\vphantom{\begin{array}[]{ c }b_{N}\\\ \vdots\\\
z_{N}\end{array}}\right\\}\text{$N-k$}\end{array}$

We will also use combinations of these notations. For instance, here is an
example of ${}_{1}\mathfrak{gl}_{4}$ inside $\mathfrak{gl}_{5}$:

$\begin{pmatrix}0&0&0&0&0\\\ 0&*&*&*&0\\\ 0&*&*&*&0\\\ 0&*&*&*&0\\\
0&0&0&0&0\end{pmatrix}$

It turns out that almost all the constructions of the paper involve not the
whole group $\mathrm{GL}_{N}$, but its almost parabolic subgroup:

######  Definition 2.1.

The _mirabolic subgroup_ $\mathrm{M}_{N}$ is the subgroup of $\mathrm{GL}_{N}$
preserving the last basis vector:

$\mathrm{M}_{N}=\begin{pmatrix}*&\ldots&*&0\\\ \vdots&\ddots&\vdots&\vdots\\\
*&\ldots&*&0\\\ *&\ldots&*&1\end{pmatrix}$

The _mirabolic subalgebra_ $\mathfrak{m}_{N}$ is the Lie algebra of
$\mathrm{M}_{N}$ identified with the space of matrices whose last column is
zero:

$\mathfrak{m}_{N}=\begin{pmatrix}*&\ldots&*&0\\\
\vdots&\ddots&\vdots&\vdots\\\ *&\ldots&*&0\\\ \end{pmatrix}$

We will also adopt notation ${}_{i}\mathfrak{m}_{j}\subset\mathfrak{m}_{N}$ as
in the case of $\mathfrak{gl}_{N}$. Here is an example of
${}_{1}\mathfrak{m}_{4}\subset\mathfrak{m}_{5}$:

${}_{1}\mathfrak{m}_{4}=\begin{pmatrix}0&0&0&0&0\\\ 0&*&*&0&0\\\ 0&*&*&0&0\\\
0&*&*&0&0\\\ 0&0&0&0&0\end{pmatrix}$

For the rest of the paper, we will use the notations

(2.1) $\displaystyle\begin{split}\mathfrak{n}_{-}&=\mathrm{span}(E_{ij}|N\geq
i>j\geq 1),\\\ \mathfrak{b}&=\mathrm{span}(E_{kl}|1\leq k\leq l\leq
N-1).\end{split}$

unless specified otherwise. In particular, $\mathfrak{b}$ refers to the Borel
subalgebra _inside_ $\mathfrak{m}_{N}$, not $\mathfrak{gl}_{N}$. Likewise, we
will denote by ${}_{k}\mathfrak{n}_{-}$ and ${}_{k}\mathfrak{b}$ the
corresponding truncated subalgebras in ${}_{k}\mathfrak{m}_{N}$.

Let $e=E_{12}+\ldots+E_{N-1,N}\in\mathfrak{gl}_{N}$ be the so-called
_principal nilpotent element_ and $\vec{u}=(u_{1},\ldots,u_{N-1})$ be a vector
of parameters. Denote by

(2.2) $e_{\vec{u}}:=e-u_{1}E_{11}-\ldots-u_{N-1}E_{N-1,N-1}$

While $e_{\vec{u}}\not\in\mathfrak{m}_{N}$, we can still define the following
2-form on $\mathfrak{m}_{N}$:

(2.3) $\omega(\vec{u})\colon\mathfrak{m}_{N}\wedge\mathfrak{m}_{N}\rightarrow
k,\qquad x\wedge y\mapsto\mathrm{Tr}(e_{\vec{u}}\cdot[x,y]).$

######  Proposition 2.2.

The form $\omega(\vec{u})$ is non-degenerate, and its inverse
$r^{\mathrm{CG}}(\vec{u})$ is uniquely specified by the condition

(2.4)
$r^{\mathrm{CG}}(\vec{u})(E_{i+k,i}^{*})=-E_{i,i+k-1}-(u_{i}-u_{i+k-1})\cdot
r^{\mathrm{CG}}(\vec{u})(E_{i+k-1,i}^{*})+\delta_{i>1}r^{\mathrm{CG}}(\vec{u})(E_{i+k-1,i-1}^{*}).$

for any $1\leq i<i+k\leq N$, where $E_{ij}^{*}$ is the dual basis and we
consider $r^{\mathrm{CG}}(\vec{u})$ as a map
$\mathfrak{m}_{N}^{*}\rightarrow\mathfrak{m}_{N}$.

###### Proof.

Note that both $\mathfrak{n}_{-}$ and $\mathfrak{b}$ are isotropic subspaces
of $\mathfrak{m}_{N}$. Since $\omega(\vec{u})$ is skew-symmetric, it is enough
to construct an inverse only of one map, say
$\omega(\vec{u})\colon\mathfrak{b}\rightarrow\mathfrak{n}_{-}$. Observe that

$\omega(\vec{u})(E_{i,i+k-1})=-E_{i+k,i}^{*}+\delta_{i>1}E_{i+k-1,i-1}^{*}-(u_{i}-u_{i+k-1})\cdot
E_{i+k-1,i}^{*}.$

Hence, to satisfy
$(r^{\mathrm{CG}}(\vec{u})\circ\omega(\vec{u}))(E_{i,i+k-1})=E_{i,i+k-1}$, we
get the equation (2.4). It can be solved inductively which proves the
existence part. ∎

It follows by general arguments (see [ES02]) that $r^{\mathrm{CG}}(\vec{u})$
satisfies the classical Yang-Baxter equation (1.3) for any $\vec{u}$. In
particular, for $\vec{u}=(0,\ldots,0)$, we get

$r^{\mathrm{CG}}(0)=-\sum\limits_{N\geq i>j\geq
1}E_{ij}\wedge\sum\limits_{k=1}^{j}E_{k,k+i-1-j}.$

This is a certain degeneration of the Cremmer-Gervais $r$-matrix mentioned in
the introduction, see [EH00]. This motivates the following

######  Definition 2.3.

A _rational Cremmer-Gervais $r$-matrix_ is $r^{\mathrm{CG}}(\vec{u})$ for
$\vec{u}\in k^{N-1}$.

As any classical $r$-matrix, it defines a Poisson-Lie structure on the
mirabolic subgroup $\mathrm{M}_{N}$ (in this case, even symplectic one);
moreover, by considering $r^{\mathrm{CG}}(\vec{u})$ as an element of
$\mathfrak{gl}_{N}\wedge\mathfrak{gl}_{N}$, it also defines a Poisson-Lie
structure on the whole group $\mathrm{GL}_{N}$. We will construct its
quantization in Section 5.

##  3\. Yangian

In this section we recall some facts about the Yangian of the Lie algebra
$\mathfrak{gl}_{N}$ and prove some results that we use later in the paper. For
details, we refer the reader to [Mol07]; note that we use an $\hbar$-version
of constructions in question (see Section 1.1), while in _loc. cit._ it is
specialized to $\hbar=1$. However, all the proofs can be easily generalized to
the case of an arbitrary $\hbar$.

###  3.1. General definition and properties

The Yangian $\mathrm{Y}_{\hbar}(\mathfrak{gl}_{N})$ is an associative algebra
generated over $k$ by the elements $t_{ij}^{(r)}$, where $1\leq i,j\leq N$ and
$r\geq 1$. In what follows, we also define $t_{ij}^{(0)}:=\delta_{ij}$.
Introduce the formal Laurent series in $u^{-1}$:

$t_{ij}(u):=\sum\limits_{k\geq 0}t_{ij}^{(r)}u^{-r}.$

We combine them into a generating matrix

$T(u):=(t_{ij}(u))\in\mathrm{Y}_{\hbar}(\mathfrak{gl}_{N})\otimes\mathrm{End}(\mathbf{C}^{N})[[u^{-1}]].$

Let $P\in\mathrm{End}(k^{N}\otimes k^{N})$ be the permutation matrix. Consider
the _Yang $R$-matrix_

$R(u-v)=\mathrm{id}-\frac{\hbar P}{u-v}\in\mathrm{End}(k^{N}\otimes k^{N}).$

Then the defining relation of $\mathrm{Y}_{\hbar}(\mathfrak{gl}_{N})$ can be
presented in the form

$R(u-v)T_{1}(u)T_{2}(v)=T_{2}(v)T_{1}(u)R(u-v),$

where the equality takes place in
$\mathrm{Y}_{\hbar}(\mathfrak{gl}_{N})\otimes\mathrm{End}(k^{n}\otimes k^{N})$
and $T_{1}(u):=T(u)\otimes\mathrm{id}$, $T_{2}(v)=\mathrm{id}\otimes T(v)$.
Equivalently, it is given by

(3.1)
$(u-v)[t_{ij}(u),t_{kl}(v)]=\hbar\frac{t_{kj}(u)t_{il}(v)-t_{kj}(v)t_{il}(u)}{u-v}.$

Consider the following $\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})$-valued matrix:

$E=\begin{pmatrix}E_{11}&\ldots&E_{1,n-1}&E_{1N}\\\
\vdots&\ddots&\vdots&\vdots\\\ E_{N-1,1}&\ldots&E_{N-1,N-1}&E_{N-1,N}\\\
E_{N1}&\ldots&E_{N,N-1}&E_{NN}\end{pmatrix},$

One can define a homomorphism
$\mathrm{Y}_{\hbar}(\mathfrak{gl}_{N})\rightarrow\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})$,
called the _evaluation homomorphism_ , by

(3.2) $T(u)\mapsto\mathrm{id}+u^{-1}E.$

In what follows, we will use the same notation $T(u)$ both for the elements of
$\mathrm{Y}_{\hbar}(\mathfrak{gl}_{N})$ and their images in
$\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})$ under the evaluation homomorphism; all
the statements involving $T(u)$ will hold in
$\mathrm{Y}_{\hbar}(\mathfrak{gl}_{N}$) unless specified otherwise. We hope it
will not lead to any confusion.

Actually, we will be mainly dealing with a slightly different version of the
$T$-matrix. Namely, define

(3.3) $L(u)=(L_{ij}(u)):=u\cdot\mathrm{id}+E.$

Obviously, we have $L(u)=uT(u)$. Almost all the properties of $T(u)$ that we
use in the paper are satisfied _mutatis mutandis_ by $L(u)$; we will
explicitly indicate the cases where it is not true or not trivial.

Let $\\{a_{1},\ldots a_{m}\\},\\{b_{1},\ldots,b_{m}\\}$ be two sets of
indices. Define a _quantum minor_ as the sum over all permutations of
$1,2,\ldots,m$, see [Mol07, (1.54),(1.55)]:

$\displaystyle\begin{split}t_{b_{1}\ldots b_{m}}^{a_{1}\ldots
a_{m}}(u):=&\sum\limits_{\sigma}\mathrm{sgn}(\sigma)t_{a_{\sigma(1)}b_{1}}(u)t_{a_{\sigma(2)}b_{2}}(u-\hbar)\ldots
t_{a_{\sigma(m)}b_{m}}(u-\hbar m+\hbar)=\\\
=&\sum\limits_{\sigma}\mathrm{sgn}(\sigma)t_{a_{1}b_{\sigma(1)}}(u)t_{a_{2}b_{\sigma(2)}}(u-\hbar)\ldots
t_{a_{m}b_{\sigma(m)}}(u-\hbar m+\hbar).\end{split}$

Similarly, we can define a quantum minor for $L(u)$:

(3.4) $\displaystyle\begin{split}L_{b_{1}\ldots b_{m}}^{a_{1}\ldots
a_{m}}(u):=&\sum\limits_{\sigma}\mathrm{sgn}(\sigma)L_{a_{\sigma(1)}b_{1}}(u)L_{a_{\sigma(2)}b_{2}}(u-\hbar)\ldots
L_{a_{\sigma(m)}b_{m}}(u-\hbar m+\hbar)=\\\
=&\sum\limits_{\sigma}\mathrm{sgn}(\sigma)L_{a_{1}b_{\sigma(1)}}(u)L_{a_{2}b_{\sigma(2)}}(u-\hbar)\ldots
L_{a_{m}b_{\sigma(m)}}(u-\hbar m+\hbar).\end{split}$

Observe that

(3.5) $L_{b_{1}\ldots b_{m}}^{a_{1}\ldots a_{m}}(u)=u(u-\hbar)\ldots(u-\hbar
m+\hbar)t_{b_{1}\ldots b_{m}}^{a_{1}\ldots a_{m}}(u).$

We will need the following properties of quantum minors:

######  Proposition 3.1.

  1. (1)

For any permutation $\sigma$, we have

$L_{b_{1}\ldots b_{m}}^{a_{\sigma(1)}\ldots
a_{\sigma(m)}}(u)=\mathrm{sgn}(\sigma)L_{b_{1}\ldots b_{m}}^{a_{1}\ldots
a_{m}}=L_{b_{\sigma(1)}\ldots b_{\sigma(m)}}^{a_{1}\ldots a_{m}}(u).$

In particular, if $a_{i}=a_{j}$ or $b_{i}=b_{j}$ for some $i\neq j$, then
$L_{b_{1}\ldots b_{m}}^{a_{1}\ldots a_{m}}(u)=0$.

  2. (2)

[Mol07, Proposition 1.6.8] A quantum minor can be decomposed as

$\displaystyle L_{b_{1}\ldots b_{m}}^{a_{1}\ldots a_{m}}(u)$
$\displaystyle=\sum\limits_{l=1}^{m}(-1)^{m-l}L_{b_{1}\ldots
b_{m-1}}^{a_{1}\ldots\hat{a}_{l}\ldots a_{m}}(u)L_{a_{l}b_{m}}(u-\hbar
m+\hbar)=$
$\displaystyle=\sum\limits_{l=1}^{m}(-1)^{m-l}L_{b_{1}\ldots\hat{b}_{l}\ldots
b_{m}}^{a_{1}\ldots a_{m-1}}(u-\hbar)L_{a_{m}b_{l}}(u)=$
$\displaystyle=\sum\limits_{l=1}^{m}(-1)^{l-1}L_{a_{l}b_{1}}(u)L_{b_{2}\ldots
b_{m}}^{a_{1}\ldots\hat{a}_{l}\ldots a_{m}}(u-\hbar)=$
$\displaystyle=\sum\limits_{l=1}^{m}(-1)^{l-1}L_{a_{1}b_{l}}(u-\hbar
m+\hbar)L_{b_{1}\ldots\hat{b}_{l}\ldots b_{m}}^{a_{2}\ldots a_{m}}(u).$

Quantum minors enjoy the following commutation properties:

######  Proposition 3.2.

For $k\neq l$, we have

$[E_{kl},L_{b_{1}\ldots b_{m}}^{a_{1}\ldots
a_{m}}(u)]=\hbar\left(\sum\limits_{i=1}^{m}\delta_{a_{i},l}L_{b_{1}\ldots
b_{m}}^{a_{1}\ldots k\ldots
a_{m}}(u)-\sum\limits_{i=1}^{m}\delta_{k,b_{i}}L_{b_{1}\ldots l\ldots
b_{m}}^{a_{1}\ldots a_{m}}(u)\right).$

where $k$, resp. $l$, are on the $i$-th position.

###### Proof.

It follows from [Mol07, Proposition 1.7.1] that

$(u-v)[L_{kl}(u),L_{b_{1}\ldots b_{m}}^{a_{1}\ldots
a_{m}}(v)]=\hbar\left(\sum\limits_{i=1}^{m}L_{a_{i}l}(u)L_{b_{1}\ldots
b_{m}}^{a_{1}\ldots k\ldots a_{m}}(v)-\sum\limits_{i=1}^{m}L_{b_{1}\ldots
l\ldots b_{m}}^{a_{1}\ldots a_{m}}(v)L_{kb_{i}}(u)\right),$

where the indices $k$ and $l$ replace the corresponding indices $a_{i}$ and
$b_{i}$ respectively. Since $L_{ij}(u)=\delta_{ij}\cdot u+E_{ij}$, the
statement follows by comparing the coefficients of $u$ on both sides. ∎

A particular case of a quantum minor will be important:

######  Definition 3.3.

The _quantum determinant_ of $L(u)$ (respectively $T(u)$) is

$\mathrm{qdet}(L(u))=L_{1\ldots N}^{1\ldots N}(u)$

(respectively $\mathrm{qdet}(T(u))=T_{1\ldots N}^{1\ldots N}(u)$). In what
follows, we will also $\mathrm{qdet}(L(u))$ the _quantum characteristic
polynomial_.

It follows from Proposition 3.2 that $\mathrm{qdet}(L(u))$ is central:

$[L_{ij}(u),\mathrm{qdet}(L(v))]=0\ \forall i,j.$

Observe the matrix $T(u)$ has the form

$T(u)\in\mathrm{id}+u^{-1}\mathrm{Y}_{\hbar}(\mathfrak{gl}_{N})\otimes\mathrm{End}(\mathbf{C}^{N})[[u^{-1}]],$

hence is invertible. We can explicitly construct its inverse as follows.

######  Definition 3.4.

[Mol07, Definition 1.9.1] Denote by
$\hat{t}_{ij}(u):=(-1)^{i+j}t_{1\ldots\hat{i}\ldots N}^{1\ldots\hat{j}\ldots
N}(u)$ the complementary quantum minor. The _quantum comatrix_ is

$\hat{T}(u):=(\hat{t}_{ij}(u)).$

Likewise, one can define $\hat{L}_{ij}(u):=L_{1\ldots\hat{i}\ldots
N}^{1\ldots\hat{j}\ldots N}(u)$.

######  Proposition 3.5.

[Mol07, Proposition 1.9.2] The quantum comatrix satisfies

$\hat{T}(u)T(u-\hbar N+\hbar)=\mathrm{qdet}(T(u))\cdot\mathrm{id}.$

In particular, the inverse of $T(u)$ is given by

$T(u)^{-1}=\mathrm{qdet}(T(u+\hbar N-\hbar)^{-1}\cdot\hat{T}(u+\hbar
N-\hbar).$

Under the evaluation homomorphism, the inverse of $T(u)$ can be explicitly
presented in terms of $E$ as

(3.6) $T(u)^{-1}=(\mathrm{id}+u^{-1}E)^{-1}=\sum\limits_{k\geq
0}(-1)^{k}u^{-k}E^{k}.$

###  3.2. Technical lemmas

In what follows, we will prove some technical statements that we will use
later in the paper.

######  Proposition 3.6.

For any $k\leq l$, we have

$L_{1\ldots k}^{1\ldots
k}(u)L_{1,\ldots,k-1,k}^{1,\ldots,k-1,l}(u-\hbar)=L_{1,\ldots,k-1,k}^{1,\ldots,k-1,l}(u)L_{1\ldots
k}^{1\ldots k}(u-\hbar).$

###### Proof.

By [Mol07, Theorem 1.12.1], the map

$\mathrm{Y}_{\hbar}(\mathfrak{gl}_{l-k+1})\rightarrow\mathrm{Y}_{\hbar}(\mathfrak{gl}_{l}),\quad
t_{ij}(u)\mapsto t_{1\ldots k-1,k-1+j}^{1\ldots k-1,k-1+i}(u)$

is an algebra homomorphism. The defining commutation relation (3.1) gives

$[t_{11}(u),t_{l-k+1,1}(v)]=\hbar\frac{t_{l-k+1,1}(u)t_{11}(v)-t_{l-k+1,1}(v)t_{11}(u)}{u-v}.$

Substituting $u-v=\hbar$, we conclude. ∎

######  Proposition 3.7.

For any $a>d$, we have

$L_{1\ldots d-1}^{1\ldots d-1}(u)L_{1\ldots
d-1}^{a,2,\ldots,d-1}(u-\hbar)=L_{1\ldots d-1}^{a,2,\ldots d-1}(u)L_{1\ldots
d-1}^{1\ldots d-1}(u-\hbar).$

###### Proof.

We can suppose that the statement takes place in $\mathfrak{gl}_{a}$. Consider
the automorphism of $\mathfrak{gl}_{a}$ cyclically permuting the indices:

$\phi\colon\\{1,2,\ldots,a\\}\mapsto\\{a,1,\ldots,a-1\\},\quad E_{ij}\mapsto
E_{\phi(i),\phi(j)}.$

After some permutation of indices in the minors, the statement of the lemma
becomes

$L_{1,\ldots,d-2,a}^{1,\ldots,d-2,a}(u)L_{1,\ldots,d-2,a}^{1,\ldots,d-2,a-1}(u-\hbar)=L_{1,\ldots,d-2,a}^{1,\ldots,d-2,a-1}(u)L_{1,\ldots,d-2,a}^{1,\ldots,d-2,a}(u-\hbar).$

Then we can proceed as in the proof of Proposition 3.6 reducing to

$\displaystyle[t_{a-d+2,a-d+2}(u),t_{a-d+1,a-d+2}(v)]=$ $\displaystyle=$
$\displaystyle\hbar\frac{t_{a-d+2,a-d+1}(u)t_{a-d+2,a-d+2}(v)-t_{a-d+2,a-d+1}(v)t_{a-d+2,a-d+2}(u)}{u-v},$

the defining commutation relation. ∎

Let $\psi\colon\mathfrak{n}_{-}\rightarrow k$ be the character such that
$\psi(E_{i+1,i})=1$ for all $1\leq i\leq N-1$, and

$\mathfrak{n}_{-}^{\psi}=\mathrm{span}(x-\psi(x)|x\in\mathfrak{n}_{-})\subset\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})$

be the shift of $\mathfrak{n}_{-}$ as in Section 4.

######  Proposition 3.8.

For all $c\leq N$,

$L_{1\ldots c-1}^{1\ldots\hat{l}\ldots c}(u)\equiv L_{1\ldots l-1}^{1\ldots
l-1}(u)\ \mathrm{mod}\
\mathfrak{n}_{-}^{\psi}\mathrm{U}_{\hbar}(\mathfrak{gl}_{N}).$

###### Proof.

For $l=c$ the lemma is obvious. By Proposition 3.1, we can permute the upper
indices:

$L_{1\ldots c-1}^{1\ldots\hat{l}\ldots c}(u)=(-1)^{c}L_{1\ldots
c-1}^{c,1,\ldots,\hat{l},\ldots,c-1}(u).$

By part (2), the right-hand side is equal to

$\sum\limits_{k=1}^{c-1}(-1)^{c+k-1}E_{ck}L_{1\ldots\hat{k}\ldots
c-1}^{1\ldots\hat{l}\ldots c-1}(u).$

But $E_{lk}\equiv 0$ for $k\neq l-1$ and $E_{l,l-1}\equiv 1$. Hence, the
right-hand side is equivalent to $L_{1\ldots c-2}^{1\ldots\hat{l}\ldots
c-1}(u)$. The statement follows from the the obvious induction. ∎

######  Proposition 3.9.

For any $1\leq i,j\leq c$, we have

$\frac{L_{1\ldots\hat{i}\ldots c}^{1\ldots\hat{j}\ldots c}(v)L_{1\ldots
c}^{1\ldots c}(u)-L_{1\ldots\hat{i}\ldots c}^{1\ldots\hat{j}\ldots
c}(v)L_{1\ldots c}^{1\ldots
c}(u)}{u-v}=\sum\limits_{l=1}^{c}L_{1\ldots\hat{i}\ldots
c}^{1\ldots\hat{l}\ldots c}(v)L_{1\ldots\hat{l}\ldots c}^{1\ldots\hat{j}\ldots
c}(u).$

###### Proof.

We can assume that the statement takes place in $\mathfrak{gl}_{c}$. Then, for
any $k,l$:

$\displaystyle L_{1\ldots\hat{k}\ldots c}^{1\ldots\hat{l}\ldots c}(u)$
$\displaystyle=\hat{L}_{kl}(u),$ $\displaystyle L_{1\ldots c}^{1\ldots c}(u)$
$\displaystyle=\mathrm{qdet}(L(u)).$

By (3.5), we have

$\displaystyle\hat{L}_{ab}(u)$ $\displaystyle=u(u-\hbar)\ldots(u-\hbar
N+2\hbar)\hat{t}_{ab},$ $\displaystyle\mathrm{qdet}(L(u))$
$\displaystyle=u(u-\hbar)\ldots(u-\hbar N+\hbar)\mathrm{qdet}(T(u))$

for any $a,b$. Therefore, if we divide both sides by

$\mathrm{qdet}(T(u))\mathrm{qdet}(T(v))\prod\limits_{i=0}^{N-2}(u-\hbar
i)\prod\limits_{i=0}^{N-2}(v-\hbar i),$

(recall that $\mathrm{qdet}(T(u))$ is central), we get

$\displaystyle\frac{(u-\hbar
N+\hbar)\mathrm{qdet}(T(v))^{-1}\hat{t}_{ij}(v)-(v-\hbar
N+\hbar)\mathrm{qdet}(T(u))^{-1}\hat{t}_{ij}(u)}{u-v}=$ $\displaystyle=$
$\displaystyle\sum\limits_{l=1}^{N}(\mathrm{qdet}(T(v))^{-1}\hat{t}_{il}(v))(\mathrm{qdet}(T(u))^{-1}\hat{t}_{lj}(u)).$

Let us shift the variables $u\mapsto u+\hbar N-\hbar,v\mapsto v+\hbar
N-\hbar$:

$\displaystyle\frac{u\cdot\mathrm{qdet}(T(v+\hbar
N-\hbar))^{-1}\hat{t}_{ij}(v+\hbar N-\hbar)-v\cdot\mathrm{qdet}(T(u+\hbar
N-\hbar))^{-1}\hat{t}_{ij}(u+\hbar N-\hbar)}{u-v}=$ $\displaystyle=$
$\displaystyle\sum\limits_{l=1}^{N}\mathrm{qdet}(T(v+\hbar
N-\hbar))^{-1}\hat{t}_{il}(v+\hbar N-\hbar)\cdot\mathrm{qdet}(T(u+\hbar
N-\hbar))^{-1}\hat{t}_{lj}(u+\hbar N-\hbar).$

Denote by $\tilde{t}_{ij}(u)$ the matrix entry of $T(u)^{-1}$. Then by
Proposition 3.5 we have

$\tilde{t}_{ij}(u)=\mathrm{qdet}(T(u+\hbar N-\hbar))^{-1}\hat{t}_{ij}(u+\hbar
N-\hbar).$

Therefore, the equality above is equivalent to

$\frac{u\cdot\tilde{t}_{ij}(v)-v\cdot\tilde{t}_{ij}(u)}{u-v}=\sum\limits_{l=1}^{N}\tilde{t}_{il}(v)\tilde{t}_{lj}(u).$

In terms of series coefficients

$\tilde{t}_{ij}(u)=\sum\limits_{r\geq 0}\tilde{t}_{ij}^{(r)}u^{-r},$

this equality is equivalent to

(3.7)
$\tilde{t}_{ij}^{(r)}=\sum\limits_{l=1}^{N}\tilde{t}_{il}^{(r^{\prime})}\tilde{t}_{lj}^{(s^{\prime})}$

for every $r^{\prime}+s^{\prime}=r$. Now let us recall that the inverse is
given by the series (3.6) under the evaluation homomorphism111I would like to
thank Vasily Krylov for suggesting the argument!. In particular, the element
$\tilde{t}_{ij}^{(r)}$ is the $ij$-entry of the matrix power $(-E)^{r}$.
Therefore, equation (3.7) is just the tautological formula
$(-E)^{r^{\prime}+s^{\prime}}=(-E)^{r^{\prime}}\cdot(-E)^{s^{\prime}}$ in
terms of the matrix entries. ∎

##  4\. Kirillov projector

Recall the notion of the mirabolic subalgebra from Section 2. In this section,
we define the Kirillov projector and show its key properties.

Denote by $\mathfrak{n}_{-}=\mathrm{span}(E_{ji}|j>i)$ the negative nilpotent
subalgebra of $\mathfrak{m}_{N}$. Let
$\psi\colon\mathfrak{n}_{-}\rightarrow\mathbf{C}$ be a nondegenerate character
of $\mathfrak{n}_{-}$; we will always assume that $\psi(E_{i+1,i})=1$ for any
$i\leq N-1$. For any $x\in\mathfrak{n}_{-}$, denote by $x^{\psi}:=x-\psi(x)$.
Define the shift
$\mathfrak{n}_{-}^{\psi}=\mathrm{span}(x^{\psi}|x\in\mathfrak{n}_{-})\subset\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})$.

######  Definition 4.1.

A right $\mathrm{U}_{\hbar}(\mathfrak{m}_{N})$-module is _Whittaker_ if the
$\mathfrak{n}_{-}^{\psi}$-action is locally nilpotent. A _Whittaker vector_ in
a Whittaker module is an $\mathfrak{n}_{-}^{\psi}$-invariant element.

Recall the notion of a quantum minor (3.4). For $i>j$, consider the following
element in a certain completion of $\mathrm{U}_{\hbar}(\mathfrak{m}_{N})[u]$:

(4.1) $P^{\psi}_{ij}(u):=\sum\limits_{k\geq
0}(-1)^{(i+j)k}\hbar^{-k}(-E_{ij}^{\psi})^{k}\frac{L_{j\ldots i-1}^{j\ldots
i-1}(u)\ldots L_{j\ldots i-1}^{j\ldots i-1}(u-\hbar k+\hbar)}{k!}$

For instance, its action is well-defined on any _right_
$\mathrm{U}_{\hbar}(\mathfrak{m}_{N})$-module where $\mathfrak{n}_{-}^{\psi}$
acts locally nilpotently.

In what follows, we denote by $\vec{u}=(u_{1},\ldots,u_{N-1})$ a vector of
variables. Let us introduce the main object of the paper:

######  Definition 4.2.

The _Kirillov projector_ $P_{\mathfrak{m}_{N}}^{\psi}(\vec{u})$ is the element

(4.2) $\displaystyle\begin{split}&P_{\mathfrak{m}_{N}}^{\psi}(\vec{u})=\\\
=&\left(P^{\psi}_{N,N-1}(u_{N-1})\right)\left(P_{N-1,N-2}^{\psi}(u_{N-2})P_{N,N-2}^{\psi}(u_{N-2})\right)\ldots\left(P_{21}^{\psi}(u_{1})\ldots
P_{N1}^{\psi}(u_{1})\right).\end{split}$

Denote by

(4.3) $\mathfrak{b}^{\vec{u}}=\mathrm{span}(E_{ij}+\delta_{ij}\cdot
u_{i}|1\leq i\leq j\leq N-1).$

This is the main theorem of the paper:

######  Theorem 4.3.

For any right Whittaker module $W$ over $\mathfrak{m}_{N}$, the Kirillov
projector defines a unique linear operator $P(\vec{u})\colon W\rightarrow W$
(acting on the right), satisfying

$\displaystyle P(\vec{u})(x-\psi(x))$ $\displaystyle=0,\
x\in\mathfrak{n}_{-},$ $\displaystyle(E_{ij}+\delta_{ij}\cdot
u_{i})P(\vec{u})$ $\displaystyle=0,\ 1\leq i\leq j\leq N-1,$

and the normalization condition

$wP(\vec{u})=w$

for any Whittaker vector $w\in W^{\mathfrak{n}_{-}^{\psi}}$. In particular, it
defines a canonical isomorphism

(4.4) $W^{\mathfrak{n}_{-}^{\psi}}\rightarrow
W/(W\cdot\mathrm{U}_{\hbar}(\mathfrak{m}_{N})\mathfrak{b}^{\vec{u}}),$

with the quotient
$W/\mathfrak{b}^{\vec{u}}:=W/(W\cdot\mathrm{U}_{\hbar}(\mathfrak{m}_{N})\mathfrak{b}^{\vec{u}})$.

###### Proof.

Uniqueness is clear: assume we have another operator $P_{2}(\vec{u})$
satisfying these assumptions. By Proposition 3.1,

$L_{j\ldots i-1}^{j\ldots
i-1}(u_{j})=(-1)^{i-j}L_{j+1,\ldots,i-1,j}^{j,\ldots,i-1}(u_{j})=\sum\limits_{l=j}^{i-1}(-1)^{l}L_{j\ldots\hat{l}\ldots
i-1}^{j+1\ldots i-1}(u_{j}-\hbar)L_{jl}(u_{j}).$

Observe that $L_{jl}(u_{j})=E_{jl}+\delta_{jl}\cdot u_{j}$, hence $L_{j\ldots
i-1}^{j\ldots i-1}(u_{j})P_{2}(\vec{u})=0$ and

$P_{ij}^{\psi}(u_{j})P_{2}(\vec{u})=P_{2}(\vec{u})$

for every $i>j$. Therefore,

$P_{\mathfrak{m}_{N}}(\vec{u})P_{2}(\vec{u})=P_{2}(\vec{u}).$

Likewise, for any vector $v\in W$, the vector $vP_{\mathfrak{m}_{N}}(\vec{u})$
is Whittaker. Therefore, by the normalization condition on $P_{2}(\vec{u})$,
we have

$vP_{\mathfrak{m}_{N}}(\vec{u})P_{2}(\vec{u})=vP_{\mathfrak{m}_{N}}(\vec{u}).$

In particular, as operators on $W$, we get

$P_{\mathfrak{m}_{N}}(\vec{u})P_{2}(\vec{u})=P_{\mathfrak{m}_{N}}(\vec{u}),$

and so

$P_{2}(\vec{u})=P_{\mathfrak{m}_{N}}(\vec{u}).$

The properties for $P_{\mathfrak{m}_{N}}(\vec{u})$ are proven in Theorem 4.5
and Theorem 4.13. ∎

######  Remark 4.4.

Classically, these properties can be interpreted as follows. The element
$P_{\mathfrak{m}_{N}}(\vec{u})$ defines a projection

$\mathfrak{n}_{-}^{\psi}\backslash\mathrm{U}_{\hbar}(\mathfrak{m}_{N})\rightarrow(\mathfrak{n}_{-}^{\psi}\backslash\mathrm{U}_{\hbar}(\mathfrak{m}_{N}))^{\mathfrak{n}_{-}^{\psi}}.$

It follows from 1.1 and the explicit formula (4.2) that this map is defined
over $k[\hbar]$. By the second property, it induces an isomorphism

$\mathfrak{n}_{-}^{\psi}\backslash\mathrm{U}_{\hbar}(\mathfrak{m}_{N})/\mathfrak{b}^{\vec{u}}\xrightarrow{\sim}(\mathfrak{n}_{-}^{\psi}\backslash\mathrm{U}_{\hbar}(\mathfrak{m}_{N}))^{\mathfrak{n}_{-}^{\psi}}.$

Composing with a natural projection
$\mathrm{U}_{\hbar}(\mathfrak{m}_{N})\rightarrow\mathrm{U}_{\hbar}(\mathfrak{m}_{N})/\mathfrak{b}^{\vec{u}}$,
we get a map

$\mathrm{U}_{\hbar}(\mathfrak{m}_{N})\rightarrow\mathfrak{n}_{-}^{\psi}\backslash\mathrm{U}_{\hbar}(\mathfrak{m}_{N})/\mathfrak{b}^{\vec{u}}.$

Since it is defined over $k[\hbar]$ we can take the limit $\hbar\rightarrow 0$
which gives

$\mathrm{Sym}(\mathfrak{m}_{N})\cong\mathcal{O}(\mathfrak{m}_{N}^{*})\rightarrow\mathfrak{n}_{-}^{\psi}\backslash\mathcal{O}(\mathfrak{m}_{N}^{*})/\mathfrak{b}^{\vec{u}}.$

Observe that the target is the space of functions on the closed subvariety of
$\mathfrak{m}_{N}^{*}$ specified by the equations $E_{ij}=\psi(E_{ij})$ for
$E_{ij}\in\mathfrak{n}_{-}$ and $E_{kl}=-\delta_{kl}\cdot u_{k}$ for $1\leq
k\leq l$, which is just the point $e_{\vec{u}}\in\mathfrak{m}_{N}^{*}$ of
(2.2). Moreover, by the tensor property from Theorem 5.4, this is an algebra
map when $\hbar=0$. Therefore, the classical limit of the Kirillov projector
gives a map of varieties

$\mathrm{pt}\rightarrow\mathfrak{m}_{N}^{*},\qquad\mathrm{pt}\mapsto
e_{\vec{u}}\in\mathfrak{m}_{N}^{*}.$

###  4.1. Kirillov projector: right $\mathfrak{n}_{-}^{\psi}$-invariance

In this subsection, we will prove the following

######  Theorem 4.5.

The Kirillov projector satisfies

$P_{\mathfrak{m}_{N}}^{\psi}(\vec{u})(x-\psi(x))=0\ \forall
x\in\mathfrak{n}_{-}.$

In particular, for any Whittaker module $X$, it defines a canonical projection

$P_{\mathfrak{m}_{N}}^{\psi}(\vec{u})\colon X\mapsto
X^{\mathfrak{n}_{-}^{\psi}}$

to the space of Whittaker vectors.

We will prove it by two-fold induction. In what follows, we will always
identify $u=u_{1}$.

####  4.1.1. Global induction

The first one is the induction on the dimension $N$ of $\mathfrak{m}_{N}$. The
base case is $N=2$.

######  Proposition 4.6.

We have $P_{21}^{\psi}(u)(E_{21}-1)=0$.

###### Proof.

Recall that $L_{1}^{1}(u)=E_{11}+u$. Then

$L_{1}^{1}(u)E_{21}=E_{21}L_{1}^{1}(u-\hbar).$

Therefore,

(4.5) $\displaystyle\begin{split}&P_{21}^{\psi}(u)E_{21}=\\\
=&\sum\limits_{k\geq 0}\hbar^{-k}(E_{21}-1)^{k}\frac{L_{1}^{1}(u)\ldots
L_{1}^{1}(u-\hbar k+\hbar)}{k!}E_{21}=\\\ =&\sum\limits_{k\geq
0}\hbar^{-k}(E_{21}-1)^{k}E_{21}\frac{L_{1}^{1}(u-\hbar)\ldots
L_{1}^{1}(u-\hbar k)}{k!}=\\\ =&\sum\limits_{k\geq
0}\hbar^{-k}(E_{21}-1)^{k+1}\frac{L_{1}^{1}(u-\hbar)\ldots L_{1}^{1}(u-\hbar
k)}{k!}+\\\ +&\sum\limits_{k\geq
0}\hbar^{-k}(E_{21}-1)^{k}\frac{L_{1}^{1}(u-\hbar)\ldots L_{1}^{1}(u-\hbar
k)}{k!}\end{split}$

The constant term (i.e. with no powers of $E_{21}-1$) is 1. Combining the
coefficients of $(E_{21}-1)^{k+1}$, we get

$\displaystyle\hbar^{-k}\frac{L_{1}^{1}(u-\hbar)\ldots L_{1}^{1}(u-\hbar
k)}{k!}\cdot\left(1+\frac{L_{1}^{1}(u-\hbar k-\hbar))}{\hbar(k+1)}\right)=$
$\displaystyle=$ $\displaystyle\hbar^{-k}\frac{L_{1}^{1}(u-\hbar)\ldots
L_{1}^{1}(u-\hbar k)}{k!}\cdot\frac{E_{11}+u}{\hbar(k+1)}=$ $\displaystyle=$
$\displaystyle\hbar^{-k-1}\frac{L_{1}^{1}(u)\ldots L_{1}^{1}(u-\hbar
k)}{(k+1)!}$

Therefore, equation (4.5) becomes

$\displaystyle P_{21}^{\psi}(u)E_{21}=1+\sum\limits_{k\geq
0}\hbar^{-k-1}(E_{21}-1)^{k+1}\frac{L_{1}^{1}(u)\ldots L_{1}^{1}(u-\hbar
k)}{(k+1)!}=P_{21}^{\psi}(u).$

∎

Then we proceed to the induction step. Recall the notation
${}_{1}\mathfrak{m}_{N}$ from Section 2:

(4.6) ${}_{1}\mathfrak{m}_{N}=\begin{pmatrix}0&0&\ldots&0&0\\\
0&*&\ldots&*&0\\\ \vdots&\ddots&\vdots&\vdots\\\ 0&*&\ldots&*&0\end{pmatrix}$

and ${}_{1}\mathfrak{n}_{-}^{\psi}=\mathrm{span}(E_{ij}|i>j\geq 2)$. Denote by
${}_{1}\vec{u}=(u_{2},\ldots,u_{N-1})$ the truncated vector of parameters.
Assume we proved that the element

$P_{{}_{1}\mathfrak{m}_{N}}({}_{1}\vec{u})=\left(P_{N,N-1}^{\psi}(u_{N-1})\right)\ldots\left(P_{32}^{\psi}(u_{2})\ldots
P_{N2}^{\psi}(u_{2})\right)$

is invariant under ${}_{1}\mathfrak{n}_{-}^{\psi}$. Denote by

$\mathcal{P}_{b}(\vec{u})=P_{{}_{1}\mathfrak{m}_{N}}({}_{1}\vec{u})P_{21}^{\psi}\ldots
P_{b1}^{\psi}(u)$

the $b$-truncation of the Kirillov projector for $\mathfrak{m}_{N}$. Then it
would be enough to prove that

$\mathcal{P}_{N}(\vec{u})E_{ij}^{\psi}=0.$

for all $i>j$.

####  4.1.2. Local induction

The second kind of induction is on the truncation $b$:

######  Proposition 4.7.

For any $b\geq i>j$, we have

$\mathcal{P}_{b}(\vec{u})E^{\psi}_{ij}=0.$

Again, the base case $b=2$ is Proposition 4.6. For the rest of the subsection,
we assume that we proved for it for $b$. Obviously, $[E_{b+1,1},E_{ij}]=0$ for
any $b\geq i>j$. By centrality of $L_{1\ldots b}^{1\ldots b}(u)$, we also have
$[L_{1\ldots b}^{1\ldots b}(u),E_{ij}]=0$ for the same $i,j$. Therefore,

$\mathcal{P}_{b+1}(\vec{u})E^{\psi}_{ij}=\mathcal{P}_{b}(\vec{u})P_{b+1,1}^{\psi}(u)E^{\psi}_{ij}=\mathcal{P}_{b}(\vec{u})E_{ij}^{\psi}P_{b+1,1}^{\psi}(u)=0$

by the local induction assumption. Hence, it is enough to prove for $i=b+1$
and $j=b$, since other elements are generated by the corresponding
commutators.

For the proof, we need the following

######  Lemma 4.8.

For any $c\leq d\leq b<a$, we have

$\mathcal{P}_{d}(\vec{u})E_{ac}^{\psi}=(-1)^{c+1}\mathcal{P}_{d}(\vec{u})E_{a1}L_{1\ldots
c-1}^{1\ldots c-1}(u).$

We will prove it by induction on $d$. Base is $d=2$: the case $c=1$ is
trivial, and for $c=2$, we use

$[(E_{21}^{\psi})^{k},E_{a2}^{\psi}]=-k\hbar E_{a1}(E_{21}^{\psi})^{k-1},$

so that

$\displaystyle
P_{{}_{1}\mathfrak{m}_{N}}({}_{1}\vec{u})P_{21}^{\psi}(u_{1})E_{a2}=$
$\displaystyle=$ $\displaystyle
P_{{}_{1}\mathfrak{m}_{N}}({}_{1}\vec{u})\sum\limits_{k\geq
0}(E_{21}^{\psi})^{k}\hbar^{-k}\frac{\prod_{i=0}^{k-1}L_{1}^{1}(u_{1}-i\hbar)}{k!}E_{a2}=$
$\displaystyle=$ $\displaystyle
P_{{}_{1}\mathfrak{m}_{N}}({}_{1}\vec{u})E_{a2}^{\psi}P_{21}^{\psi}(u_{1})-$
$\displaystyle-$ $\displaystyle
P_{{}_{1}\mathfrak{m}_{N}}({}_{1}\vec{u})\sum\limits_{k\geq
0}(E_{21}^{\psi})^{k}E_{a1}\hbar^{-k}\frac{\prod_{i=0}^{k}L_{1}^{1}(u_{1}-i\hbar+\hbar)}{k!}=$
$\displaystyle=$ $\displaystyle-\mathcal{P}_{2}(\vec{u})E_{a1}L_{1}^{1}(u),$

where we used $P_{{}_{1}\mathfrak{m}_{N}}({}_{1}\vec{u})E_{a2}^{\psi}=0$ by
global induction assumption.

######  Lemma 4.9.

For $c\leq d\leq b<a$ and any $k\geq 0$, we have

$\mathcal{P}_{d}(\vec{u})(L_{1,2,\ldots,c}^{a,2,\ldots,c}(v)+(-1)^{c}\psi(E_{ac}))=\mathcal{P}_{d}(\vec{u})E_{a1}\frac{L_{1\ldots
c}^{1\ldots c}(v)-L_{1\ldots c}^{1\ldots c}(u)}{v-u}.$

In particular,

$\mathcal{P}_{d}(\vec{u})\partial_{u}^{k}L_{1\ldots
c}^{a,2,\ldots,c}(u)+(-1)^{c}\delta_{k,0}\psi(E_{ac})\mathcal{P}_{d}(\vec{u})=\frac{1}{k+1}\mathcal{P}_{d}(\vec{u})E_{a1}(\partial_{u}^{k+1}L_{1\ldots
c}^{1\ldots c}(u)),$

where by $\partial_{u}$ we mean the derivative with respect to $u$.

###### Proof.

By Proposition 3.1,

$L_{1\ldots
c}^{a,2,\ldots,c}(v)=\sum\limits_{l=1}^{c}(-1)^{l-1}E_{al}L_{1\ldots\hat{l}\ldots
c}^{2\ldots c}(v).$

Since $E_{al}=E_{al}^{\psi}+\psi(E_{al})$ and $\psi(E_{al})=0$ except for,
possibly, $l=c=a-1$, we can rewrite the sum as

$\sum\limits_{l=1}^{c-1}(-1)^{l-1}E^{\psi}_{al}L_{1\ldots\hat{l}\ldots
c}^{2\ldots c}(v)+(-1)^{c-1}\psi(E_{ac})L_{1\ldots c-1}^{2\ldots c}(v).$

Therefore, by 4.8,

$\mathcal{P}_{d}(\vec{u})L_{1\ldots
c}^{a,2,\ldots,c}(v)=\mathcal{P}_{d}(\vec{u})E_{a1}\sum\limits_{l=1}^{c}L_{1\ldots
l-1}^{1\ldots l-1}(u)L_{1\ldots\hat{l}\ldots c}^{2\ldots
c}(v)+(-1)^{c-1}\mathcal{P}_{d}(\vec{u})\psi(E_{ac})L_{1\ldots c-1}^{2\ldots
c}(v).$

Observe that $[E_{a1},E_{ij}]=0$ for any $c\geq i>j$. By Proposition 3.8, the
classes of $L_{1\ldots l-1}^{1\ldots l-1}(u)$ and $L_{1\ldots
c-1}^{1\ldots\hat{l}\ldots c}(u)$ in the left quotient by the shift
$\mathrm{span}(E_{ij}-\psi(E_{ij})|c\geq i>j\geq 1)$ are equal for every $l$;
therefore, by Proposition 4.7,

$\mathcal{P}_{d}(\vec{u})E_{a1}L_{1\ldots l-1}^{1\ldots
l-1}(u)=\mathcal{P}_{d}(\vec{u})E_{a1}L_{1\ldots c-1}^{1\ldots\hat{l}\ldots
c}(u)$

and the right-hand side is equal to

$\mathcal{P}_{d}(\vec{u})E_{a1}\sum\limits_{l=1}^{c}L_{1\ldots
c-1}^{1\ldots\hat{l}\ldots c}(u)L_{1\ldots\hat{l}\ldots c}^{2\ldots
c}(v)+(-1)^{c-1}\psi(E_{ac})\mathcal{P}_{d}(\vec{u}).$

Recall Proposition 3.9

$\sum\limits_{l=1}^{c}L_{1\ldots c-1}^{1\ldots\hat{l}\ldots
c}(u)L_{1\ldots\hat{l}\ldots c}^{2\ldots c}(v)=\frac{L_{1\ldots c-1}^{2\ldots
c}(v)L_{1\ldots c}^{1\ldots c}(u)-L_{1\ldots c-1}^{2\ldots c}(u)L_{1\ldots
c}^{1\ldots c}(v)}{u-v}$

Again, by Proposition 3.8, the class $L_{1\ldots c-1}^{2\ldots c}(u)$ in the
quotient is just 1, so,

$\mathcal{P}_{d}(\vec{u})(L_{1\ldots
c}^{a,2,\ldots,c}(v)+(-1)^{c}\psi(E_{ac}))=\mathcal{P}_{d}(\vec{u})E_{a1}\frac{L_{1\ldots
c}^{1\ldots c}(v)-L_{1\ldots c}^{1\ldots c}(u)}{v-u}.$

The second part of the lemma follows by setting $v=u+t$ and comparing the
Taylor series in $t$ on both sides. ∎

###### Proof of 4.8.

First, let us show that it will be enough to prove it for $c=d$. Indeed: let
$c<d$, then

$\displaystyle\mathcal{P}_{d}(\vec{u})E_{ac}^{\psi}=\hbar^{-1}(\mathcal{P}_{d}(\vec{u})E_{ad}^{\psi}E_{dc}^{\psi}-\mathcal{P}_{d}(\vec{u})E_{dc}^{\psi}E_{ad}^{\psi})=$
$\displaystyle=$
$\displaystyle\hbar^{-1}\mathcal{P}_{d}(\vec{u})E_{ad}^{\psi}E_{dc}^{\psi}=(-1)^{d+1}\hbar^{-1}\mathcal{P}_{d}(\vec{u})E_{a1}L_{1\ldots
d-1}^{1\ldots d-1}(u)E_{dc}^{\psi}$

where we used Proposition 4.7 in the second equality. By Proposition 3.2

$[L_{1\ldots d-1}^{1\ldots d-1}(u),E_{dc}^{\psi}]=-\hbar L_{1\ldots
d-1}^{1\ldots d\ldots d-1}(u)=(-1)^{d-c}\hbar L_{1\ldots
d-1}^{1\ldots\hat{c}\ldots d}(u).$

so,

$(-1)^{d+1}\hbar^{-1}\mathcal{P}_{d}(\vec{u})E_{a1}L_{1\ldots d-1}^{1\ldots
d-1}(u)E_{dc}^{\psi}=(-1)^{c+1}\mathcal{P}_{d}(\vec{u})E_{a1}L_{1\ldots
d-1}^{1\ldots\hat{c}\ldots d}(u).$

The statement then follows from Proposition 3.8 as in the proof of 4.9.

So, let us assume that $c=d$. Then

$\displaystyle\mathcal{P}_{d-1}(\vec{u})P^{\psi}_{d1}(u)E_{ad}^{\psi}=$
$\displaystyle=$ $\displaystyle\mathcal{P}_{d-1}(\vec{u})\sum\limits_{k\geq
0}\hbar^{-k}(-1)^{(d+1)k}(-E_{d1}^{\psi})^{k}\frac{\prod\limits_{i=0}^{k-1}L_{1\ldots
d-1}^{1\ldots d-1}(u-i\hbar)}{k!}E_{ad}^{\psi}=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{d-1}(\vec{u})\left(E_{ad}^{\psi}P^{\psi}_{d1}(u)+E_{a1}\sum\limits_{k\geq
1}\hbar^{-k+1}(-1)^{(d+1)k}(-E_{d1}^{\psi})^{k-1}\frac{\prod\limits_{i=0}^{k-1}L_{1\ldots
d-1}^{1\ldots d-1}(u-i\hbar)}{(k-1)!}\right)=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{d-1}(\vec{u})\left(E_{ad}^{\psi}P^{\psi}_{d1}(u)+(-1)^{d+1}E_{a1}P^{\psi}_{d1}(u-\hbar)L_{1\ldots
d-1}^{1\ldots d-1}(u)\right),$

where we used

$[(-E_{d1}^{\psi})^{k},E_{ad}^{\psi}]=k\hbar(-E_{d1}^{\psi})^{k-1}E_{a1}.$

It follows from the definition that
$[\mathcal{P}_{d-1}(\vec{u}),E_{ad}^{\psi}]=0$, therefore, the first summand
is zero. Hence,

$\mathcal{P}_{d}(\vec{u})E_{ad}^{\psi}=(-1)^{d+1}\mathcal{P}_{d-1}(\vec{u})E_{a1}P^{\psi}_{d1}(u-\hbar)L_{1\ldots
d-1}^{1\ldots d-1}(u).$

Therefore, the statement of the lemma would follow from

(4.7)
$\mathcal{P}_{d-1}(\vec{u})P_{d1}^{\psi}(u)E_{a1}=\mathcal{P}_{d-1}(\vec{u})E_{a1}P_{d1}^{\psi}(u-\hbar).$

Let us commute $E_{a1}$ past $P_{d1}^{\psi}(u)$. We need three formulas:

  * •

It follows from Proposition 3.2 that

$[L_{1\ldots d-1}^{1\ldots d-1}(u),E_{a1}]=-\hbar L_{1\ldots
d-1}^{a,2,\ldots,d-1}(u).$

  * •

From Proposition 3.7 we get

$L_{1\ldots d-1}^{1\ldots d-1}(v)L_{1\ldots
d-1}^{a,2,\ldots,d-1}(v-\hbar)=L_{1\ldots d-1}^{a,2,\ldots d-1}(v)L_{1\ldots
d-1}^{1\ldots d-1}(v-\hbar).$

  * •

By Proposition 3.2

$[L^{a,2,\ldots d-1}_{1\ldots d-1}(u),E_{a1}]=0.$

Using them, one can show that

$[\prod\limits_{i=0}^{k-1}L_{1\ldots d-1}^{1\ldots d-1}(u),E_{a1}]=-k\hbar
L_{1\ldots d-1}^{a,2,\ldots,d-1}(u)\prod\limits_{i=1}^{k-1}L_{1\ldots
d-1}^{1\ldots d-1}(u-\hbar i)$

Then one uses the induction on $d$. By 4.9, we have

$\displaystyle\mathcal{P}_{d-1}(\vec{u})P_{d1}^{\psi}(u)E_{a1}=$
$\displaystyle=$ $\displaystyle\mathcal{P}_{d-1}(\vec{u})\sum\limits_{k\geq
0}\hbar^{-k}(-1)^{(d+1)k}(-E_{d1}^{\psi})^{k}\frac{\prod L_{1\ldots
d-1}^{1\ldots d-1}(u-i\hbar)}{k!}E_{a1}=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{d-1}(\vec{u})E_{a1}\sum\limits_{k\geq
0}(-1)^{(d+1)k}\hbar^{-k}(-E_{d1}^{\psi})^{k}\frac{\prod_{i=0}^{k-1}L_{1\ldots
d-1}^{1\ldots d-1}(u-i\hbar)}{k!}-$ $\displaystyle-$
$\displaystyle\mathcal{P}_{d-1}(\vec{u})L_{1\ldots
d-1}^{a,2,\ldots,d-1}(u)\sum\limits_{k\geq
1}\hbar^{-k+1}(-1)^{(d+1)k}(-E_{d1}^{\psi})^{k}\frac{\prod_{i=1}^{k-1}L_{1\ldots
d-1}^{1\ldots d-1}(u-i\hbar)}{(k-1)!}=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{d-1}(\vec{u})E_{a1}\sum\limits_{k\geq
0}(-1)^{(d+1)k}\hbar^{-k}(-E_{d1}^{\psi})^{k}\frac{\prod_{i=0}^{k-1}L_{1\ldots
d-1}^{1\ldots d-1}(u-i\hbar)}{k!}-$ $\displaystyle-$
$\displaystyle\mathcal{P}_{d-1}(\vec{u})E_{a1}\partial_{u}L_{1\ldots
d-1}^{1\ldots d-1}(u)\sum\limits_{k\geq
1}\hbar^{-k+1}(-1)^{(d+1)k}(-E_{d1}^{\psi})^{k}\frac{\prod_{i=1}^{k-1}L_{1\ldots
d-1}^{1\ldots d-1}(u-i\hbar)}{(k-1)!}.$

Recall (4.7). By term-by-term comparison, the lemma would follow from

(4.8)
$\displaystyle\begin{split}&\mathcal{P}_{d-1}(\vec{u})\left[E_{a1}(-E_{d1}^{\psi})^{k}\frac{L_{1\ldots
d-1}^{1\ldots d-1}(u)}{k}-\hbar E_{a1}\partial_{u}L_{1\ldots d-1}^{1\ldots
d-1}(u)(-E_{d1}^{\psi})^{k}\right]=\\\
=&\mathcal{P}_{d-1}(\vec{u})E_{a1}(-E_{d1}^{\psi})^{k}\frac{L_{1\ldots
d-1}^{1\ldots d-1}(u-k\hbar)}{k}.\end{split}$

For $d=2$, it is obviously satisfied. Assume $d>2$. Then we can identify
$E_{d1}^{\psi}=E_{d1}$.

Let us study the term $\hbar E_{a1}\partial_{u}L_{1\ldots d-1}^{1\ldots
d-1}(u)(-E_{d1}^{\psi})^{k}$. Taking derivative of Proposition 3.2, we obtain

$[\partial_{u}L_{1\ldots d-1}^{1\ldots d-1}(u),(-E_{d1}^{\psi})^{k}]=\hbar
k\partial_{u}L_{1,2,\ldots,d-1}^{d,2,\ldots,d-1}(u)(-E_{d1}^{\psi})^{k-1}.$

Hence, by 4.9, we get

(4.9)
$\displaystyle\begin{split}&\hbar\mathcal{P}_{d-1}(\vec{u})E_{a1}\partial_{u}L_{1\ldots
d-1}^{1\ldots d-1}(u)(-E_{d1}^{\psi})^{k}=\\\
=&\hbar\mathcal{P}_{d-1}(\vec{u})\left(E_{a1}(-E_{d1}^{\psi})^{k}\partial_{u}L_{1\ldots
d-1}^{1\ldots
d-1}(u)+k\hbar^{2}E_{a1}\partial_{u}L_{1,2,\ldots,d-1}^{d,2,\ldots,d-1}(u)(-E_{d1}^{\psi})^{k-1}\right)=\\\
=&\hbar\mathcal{P}_{d-1}(\vec{u})\left(E_{a1}(-E_{d1}^{\psi})^{k}\partial_{u}L_{1\ldots
d-1}^{1\ldots
d-1}(u)+k\hbar^{2}\partial_{u}L_{1,2,\ldots,d-1}^{d,2,\ldots,d-1}(u)E_{a1}(-E_{d1}^{\psi})^{k-1}\right)=\\\
=&\hbar\mathcal{P}_{d-1}(\vec{u})\left(E_{a1}(-E_{d1}^{\psi})^{k}\partial_{u}L_{1\ldots
d-1}^{1\ldots
d-1}(u)-\frac{k\hbar^{2}}{2}(-E_{d1}^{\psi})\partial^{2}_{u}L_{1\ldots
d-1}^{1\ldots d-1}(u)E_{a1}(-E_{d1}^{\psi})^{k-1}\right)\end{split}$

We will need the following technical statement:

######  Lemma 4.10.

For any $l$, we have

$\displaystyle\mathcal{P}_{d-1}(\vec{u})(-E_{d1}^{\psi})\partial^{l}_{u}L_{1\ldots
d-1}^{1\ldots d-1}(u)(-E_{a1})(-E_{d1}^{\psi})^{k-1}=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{d-1}(\vec{u})\left((-E_{a1})(-E_{d1}^{\psi})^{k}\partial_{u}^{l}L_{1\ldots
d-1}^{1\ldots d-1}(u)-k\hbar(-E_{d1}^{\psi})\frac{\partial_{u}^{l+1}L_{1\ldots
d-1}^{1\ldots d-1}(u)}{l+1}(-E_{a1})(-E_{d1}^{\psi})^{k-1}\right).$

###### Proof.

Here is a series of equalities where we repeatedly use 4.9:

$\displaystyle\mathcal{P}_{d-1}(\vec{u})(-E_{d1}^{\psi})\partial^{l}_{u}L_{1\ldots
d-1}^{1\ldots d-1}(u)(-E_{a1})(-E_{d1}^{\psi})^{k-1}=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{d-1}(\vec{u})(-E_{d1}^{\psi})(-E_{a1})\partial^{l}_{u}L_{1\ldots
d-1}^{1\ldots
d-1}(u)(-E_{d1}^{\psi})^{k-1}+\hbar\partial_{u}^{l}L_{1\ldots,d-1}^{a,2,\ldots,d-1}(u)(-E_{d1}^{\psi})^{k}=$
$\displaystyle=$
$\displaystyle\mathcal{P}_{d-1}(\vec{u})(-E_{a1})(-E_{d1}^{\psi})^{k}\partial^{l}_{u}L_{1\ldots
d-1}^{1\ldots
d-1}(u)+(k-1)\hbar\partial_{u}^{l}L_{1,2\ldots,d-1}^{d,2,\ldots,d-1}(u)(-E_{a1})(-E_{d1}^{\psi})^{k-1}-$
$\displaystyle-$
$\displaystyle\mathcal{P}_{d-1}(\vec{u})\hbar(-E_{a1})\frac{\partial_{u}^{l+1}L_{1\ldots
d-1}^{1\ldots d-1}(u)}{l+1}(-E_{d1}^{\psi})^{k}=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{d-1}(\vec{u})(-E_{a1})(-E_{d1}^{\psi})^{k}\partial^{l}_{u}L_{1\ldots
d-1}^{1\ldots
d-1}(u)-(k-1)\hbar(-E_{d1}^{\psi})\frac{\partial_{u}^{l+1}L_{1\ldots
d-1}^{1\ldots d-1}(u)}{l+1}(-E_{a1})(-E_{d1}^{\psi})^{k-1}-$ $\displaystyle-$
$\displaystyle\mathcal{P}_{d-1}(\vec{u})\hbar(-E_{a1})\frac{\partial_{u}^{l+1}L_{1\ldots
d-1}^{1\ldots d-1}(u)}{l+1}(-E_{d1}^{\psi})^{k}.$

Therefore, the claim would follow from

$\mathcal{P}_{d-1}(\vec{u})(-E_{a1})\partial_{u}^{l+1}L_{1\ldots d-1}^{1\ldots
d-1}(u)(-E_{d1}^{\psi})=\mathcal{P}_{d-1}(\vec{u})(-E_{d1}^{\psi})\partial_{u}^{l+1}L_{1\ldots
d-1}^{1\ldots d-1}(u)(-E_{a1}).$

It can be proven be descending induction on $l$. Indeed, for $l$ large, the
corresponding derivative is just zero and the equation is trivial. Assume it
holds for the $k$-th derivative. Then, on the one hand,

$\displaystyle\mathcal{P}_{d-1}(\vec{u})(-E_{a1})\partial_{u}^{k-1}L_{1\ldots
d-1}^{1\ldots d-1}(u)(-E_{d1}^{\psi})=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{d-1}(\vec{u})\left((-E_{a1})(-E_{d1}^{\psi})\partial_{u}^{k-1}L_{1\ldots
d-1}^{1\ldots d-1}(u)+\hbar\partial_{u}^{k-1}L_{1,2,\ldots
d-1}^{d,2,\ldots,d-1}(u)(-E_{a1})\right)=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{d-1}(\vec{u})\left((-E_{a1})(-E_{d1}^{\psi})\partial_{u}^{k-1}L_{1\ldots
d-1}^{1\ldots d-1}(u)+\hbar(-E_{d1}^{\psi})\frac{\partial_{u}^{k}L_{1\ldots
d-1}^{1\ldots d-1}(u)}{k}(-E_{a1})\right),$

on the other hand,

$\displaystyle\mathcal{P}_{d-1}(\vec{u})(-E_{d1}^{\psi})\partial_{u}^{k-1}L_{1\ldots
d-1}^{1\ldots d-1}(u)(-E_{a1})=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{d-1}(\vec{u})\left((-E_{d1}^{\psi})(-E_{a1})\partial_{u}^{k-1}L_{1\ldots
d-1}^{1\ldots d-1}(u)+\hbar\partial_{u}^{k-1}L_{1,2,\ldots
d-1}^{a,2,\ldots,d-1}(u)(-E_{d1}^{\psi})\right)=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{d-1}(\vec{u})\left((-E_{a1})(-E_{d1}^{\psi})\partial_{u}^{k-1}L_{1\ldots
d-1}^{1\ldots d-1}(u)+\hbar(-E_{a1})\frac{\partial_{u}^{k}L_{1\ldots
d-1}^{1\ldots d-1}(u)}{k}(-E_{d1}^{\psi})\right),$

and we can apply induction. ∎

By repeated application of the lemma, one can see that
(LABEL:eq::derivative_past_k_power) is equal to

$-\frac{1}{k}\mathcal{P}_{d-1}(\vec{u})E_{a1}(-E_{d1}^{\psi})^{k}\sum\limits_{l=1}^{+\infty}\frac{(-\hbar
k)^{l}}{l!}\partial_{u}^{l}L_{1\ldots k-1}^{1\ldots k-1}(u).$

However, this is nothing but

$\frac{1}{k}\mathcal{P}_{d-1}(\vec{u})E_{a1}(-E_{d1}^{\psi})^{k}(L_{1\ldots
d-1}^{1\ldots d-1}(u)-L_{1\ldots d-1}^{1\ldots d-1}(u-k\hbar))$

Substituting it back into (4.8), we conclude.

∎

Finally, we are ready to prove Proposition 4.7.

###### Proof of Proposition 4.7.

As it was mentioned in the discussion after the proposition statement, it is
enough to show

$\mathcal{P}_{b+1}(\vec{u})E_{b+1,b}^{\psi}=0.$

As usual, let us commute $E_{b+1,b}^{\psi}$ past $P_{b+1,1}^{\psi}(u)$. We
need three formulas:

  * •

It follows from Proposition 3.2 that

$[L_{1\ldots b}^{1\ldots b}(u),E_{b+1,b}^{\psi}]=-\hbar
L_{1,\ldots,b-1,b}^{1,\ldots,b-1,b+1}(u).$

  * •

From Proposition 3.6 we get

$L_{1\ldots b}^{1\ldots
b}(v)L_{1,\ldots,b-1,b}^{1,\ldots,b-1,b+1}(v-\hbar)=L_{1,\ldots,b-1,b}^{1,\ldots,b-1,b+1}(v)L_{1\ldots
b}^{1\ldots b}(v-\hbar).$

  * •

By Proposition 3.2

$[L^{1,\ldots,b-1,b+1}_{1,\ldots,b-1,b}(u),E_{b+1,b}^{\psi}]=0.$

It implies that

$\left[\prod_{i=0}^{k-1}L_{1\ldots b}^{1\ldots
b}(u-i\hbar),E_{b+1,b}^{\psi}\right]=-k\hbar
L_{1,\ldots,b-1,b}^{1,\ldots,b-1,b+1}(u)\prod_{i=1}^{k-1}L_{1\ldots
b}^{1\ldots b}(u-i\hbar).$

Hence,

$\displaystyle\mathcal{P}_{b}(\vec{u})P_{b+1,1}^{\psi}(u)E_{b+1,b}^{\psi}=$
$\displaystyle=$ $\displaystyle\mathcal{P}_{b}(\vec{u})\sum\limits_{k\geq
0}(-E_{b+1,1}^{\psi})^{k}(-1)^{(b+2)k}\hbar^{-k}\frac{\prod_{i=0}^{k-1}L_{1\ldots
b}^{1\ldots b}(u-i\hbar)}{k!}E_{b+1,b}^{\psi}=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u})E_{b+1,b}^{\psi}P_{b+1,1}^{\psi}(u)-$
$\displaystyle-$ $\displaystyle\mathcal{P}_{b}(\vec{u})\sum\limits_{k\geq
1}(-E_{b+1,1}^{\psi})^{k}(-1)^{(b+2)k}\hbar^{-k+1}L_{1,\ldots,b-1,b}^{1,\ldots,b-1,b+1}(u)\frac{\prod_{i=1}^{k-1}L_{1\ldots
b}^{1\ldots b}(u-i\hbar)}{(k-1)!}.$

Denote

(4.10)
$\displaystyle\begin{split}A&:=\mathcal{P}_{b}(\vec{u})E_{b+1,b}^{\psi}P_{b+1,1}^{\psi}(u),\\\
B&:=-\mathcal{P}_{b}(\vec{u})\sum\limits_{k\geq
1}(-E_{b+1,1}^{\psi})^{k}(-1)^{(b+2)k}\hbar^{-k+1}L_{1,\ldots,b-1,b}^{1,\ldots,b-1,b+1}(u)\frac{\prod_{i=1}^{k-1}L_{1\ldots
b}^{1\ldots b}(u-i\hbar)}{(k-1)!}\end{split}$

Then, naturally, we need to prove that

$A+B=0.$

Let us study the $A$-term first. By 4.8

$\mathcal{P}_{b}(\vec{u})E_{b+1,b}^{\psi}=(-1)^{b+1}\mathcal{P}_{b}(\vec{u})E_{b+1,1}L_{1\ldots
b-1}^{1\ldots b-1}(u).$

Since we assume $b>2$, we can identify $E_{b+1,1}^{\psi}=E_{b+1,1}$.
Therefore,

$\displaystyle\mathcal{P}_{b}(\vec{u})E_{b+1,b}^{\psi}P_{b+1,1}^{\psi}(u)=$
$\displaystyle=$ $\displaystyle\mathcal{P}_{b}(\vec{u})E_{b+1,1}L_{1\ldots
b-1}^{1\ldots b-1}(u)=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u})(-1)^{b+1}E_{b+1,1}\sum\limits_{k\geq
0}L_{1\ldots b-1}^{1\ldots
b-1}(u)(-E_{b+1,1})^{k}(-1)^{(b+2)k}\hbar^{-k}\frac{\prod_{i=0}^{k-1}L_{1\ldots
b}^{1\ldots b}(u-i\hbar)}{k!}.$

Let us look at the term

$\mathcal{P}_{b}(\vec{u})(-E_{b+1,1})L_{1\ldots b-1}^{1\ldots
b-1}(u)(-E_{b+1,1})^{k}.$

Similarly to the proof of 4.8, we need the following technical statement:

######  Lemma 4.11.

For any $l\leq b-1$, we have

$\mathcal{P}_{b}(\vec{u})(-E_{b+1,1})L_{1\ldots l}^{1\ldots
l}(u)(-E_{b+1,1})^{k}=\mathcal{P}_{b}(\vec{u})(-E_{b+1,1})^{k+1}L_{1\ldots
l}^{1\ldots l}(u-k\hbar).$

###### Proof.

Indeed: by 4.8,

$\displaystyle\mathcal{P}_{b}(\vec{u})(-E_{b+1,1})L_{1\ldots l}^{1\ldots
l}(u)(-E_{b+1,1})^{k}=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u})(-E_{b+1,1})^{k+1}L_{1\ldots l}^{1\ldots
l}(u)+k\hbar L_{1,2,\ldots,l}^{b+1,2,\ldots,l}(u)(-E_{b+1,1})^{k}=$
$\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u})(-E_{b+1,1})^{k+1}L_{1\ldots l}^{1\ldots
l}(u)-k\hbar(-E_{b+1,1})\partial_{u}L_{1\ldots l}^{1\ldots
l}(u)(-E_{b+1,1})^{k}.$

Then we continue the process by pushing the derivative to the right. For
instance:

$\displaystyle\mathcal{P}_{b}(\vec{u})(-E_{b+1,1})^{k+1}L_{1\ldots l}^{1\ldots
l}(u)-k\hbar(-E_{b+1,1})\partial_{u}L_{1\ldots l}^{1\ldots
l}(u)(-E_{b+1,1})^{k}=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u})(-E_{b+1,1})^{k+1}L_{1\ldots l}^{1\ldots
l}(u)-k\hbar(-E_{b+1,1})^{k+1}\partial_{u}L_{1\ldots l}^{1\ldots l}(u)-$
$\displaystyle-$
$\displaystyle\mathcal{P}_{b}(\vec{u})(k\hbar)^{2}\partial_{u}L_{1,2,\ldots,l}^{b+1,2,\ldots,l}(u)(-E_{b+1,1})^{k}=$
$\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u})(-E_{b+1,1})^{k+1}L_{1\ldots l}^{1\ldots
l}(u)-(-E_{b+1,1})^{k+1}k\hbar\partial_{u}L_{1\ldots l}^{1\ldots l}(u)+$
$\displaystyle+$
$\displaystyle\mathcal{P}_{b}(\vec{u})(k\hbar)^{2}(-E_{b+1,1})\frac{\partial_{u}^{2}L_{1\ldots
l}^{1\ldots l}(u)}{2}(-E_{b+1,1})^{k}$

In the end, we arrive at

$\displaystyle\mathcal{P}_{b}(\vec{u})(-E_{b+1,1})L_{1\ldots l}^{1\ldots
l}(u)(-E_{b+1,1})^{k}=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u})(-E_{b+1,1})^{k+1}\sum\limits_{j=0}^{+\infty}\frac{(-k\hbar)^{j}}{j!}\partial_{u}^{j}L_{1\ldots
l}^{1\ldots l}(u)=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u})(-E_{b+1,1})^{k+1}L_{1\ldots l}^{1\ldots
l}(u-k\hbar),$

where the sum is finite. ∎

Therefore,

(4.11)
$\displaystyle\begin{split}A=&\mathcal{P}_{b}(\vec{u})E_{b+1,b}^{\psi}P_{b+1,1}^{\psi}(u)=\\\
=&\mathcal{P}_{b}(\vec{u})\sum\limits_{k\geq
0}(-E_{b+1,1})^{k+1}(-1)^{(b+2)(k+1)}\hbar^{-k}L_{1\ldots b-1}^{1\ldots
b-1}(u-k\hbar)\frac{\prod_{i=0}^{k-1}L_{1\ldots b}^{1\ldots
b}(u-i\hbar)}{k!}.\end{split}$

Now let us study $B$ of (4.10). Consider the term

$\mathcal{P}_{b}(\vec{u})(-E_{b+1,1})^{k}L_{1,\ldots,b-1,b}^{1,\ldots,b-1,b+1}(u).$

By Proposition 3.1, we have

$L_{1,\ldots,b-1,b}^{1,\ldots,b-1,b+1}(u)=(-1)^{b+1}\sum\limits_{l=1}^{b}(-1)^{l-1}E_{b+1,l}L_{1\ldots\hat{l}\ldots
b}^{1\ldots b-1}(u).$

Recall that $E_{b+1,,b}=E_{b+1,b}^{\psi}+1$. By 4.8 and 4.11, we get

$\displaystyle\mathcal{P}_{b}(\vec{u})(-E_{b+1,1})^{k}L_{1,\ldots,b-1,b}^{1,\ldots,b-1,b+1}(u)=$
$\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u})\left(\sum\limits_{l=1}^{b}E_{b+1,l}^{\psi}(-1)^{b+l}(-E_{b+1,1})^{k}L_{1\ldots\hat{l}\ldots
b}^{1\ldots b-1}(u)\right)+(-E_{b+1,1})^{k}L_{1\ldots b-1}^{1\ldots b-1}(u)=$
$\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u})(-1)^{b+2}(-E_{b+1,1})\left(\sum\limits_{l=1}^{b}L_{1\ldots
l-1}^{1\ldots l-1}(u)(-E_{b+1,1})^{k}L_{1\ldots\hat{l}\ldots b}^{1\ldots
b-1}(u)\right)+(-E_{b+1,1})^{k}L_{1\ldots b-1}^{1\ldots b-1}(u)=$
$\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u})(-1)^{b+2}(-E_{b+1,1})^{k+1}\left(\sum\limits_{l=1}^{b}L_{1\ldots
l-1}^{1\ldots l-1}(u-k\hbar)L_{1\ldots\hat{l}\ldots b}^{1\ldots
b-1}(u)\right)+(-E_{b+1,1})^{k}L_{1\ldots b-1}^{1\ldots b-1}(u).$

By Proposition 3.8 and Proposition 3.9, we have

$\displaystyle\mathcal{P}_{b}(\vec{u})(-E_{b+1,1})^{k+1}\sum\limits_{l=1}^{b}L_{1\ldots
l-1}^{1\ldots l-1}(u-k\hbar)L_{1\ldots\hat{l}\ldots b}^{1\ldots b-1}(u)=$
$\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u})(-E_{b+1,1})^{k+1}\frac{L_{1\ldots
b-1}^{1\ldots b-1}(u-k\hbar)L_{1\ldots b}^{1\ldots b}(u)-L_{1\ldots
b}^{1\ldots b}(u-k\hbar)L_{1\ldots b-1}^{1\ldots b-1}(u)}{k\hbar}.$

Therefore,

$\displaystyle B=-$ $\displaystyle\mathcal{P}_{b}(\vec{u})\sum\limits_{k\geq
1}(-E_{b+1,1})^{k}(-1)^{(b+2)k}\hbar^{-k+1}L_{1,\ldots,b-1,b}^{1,\ldots,b-1,b+1}(u)\frac{\prod_{i=1}^{k-1}L_{1\ldots
b}^{1\ldots b}(u-i\hbar)}{(k-1)!}=$ $\displaystyle=-$
$\displaystyle\mathcal{P}_{b}(\vec{u})\sum\limits_{k\geq
1}(-E_{b+1,1})^{k+1}(-1)^{(b+2)(k+1)}\frac{\hbar^{-k}}{k!}L_{1\ldots
b-1}^{1\ldots b-1}(u-k\hbar)\prod_{i=0}^{k-1}L_{1\ldots b}^{1\ldots
b}(u-i\hbar)-$ $\displaystyle+$
$\displaystyle\mathcal{P}_{b}(\vec{u})\sum\limits_{k\geq
1}(-E_{b+1,1})^{k+1}(-1)^{(b+2)(k+1)}\frac{\hbar^{-k}}{k!}L_{1\ldots
b-1}^{1\ldots b-1}(u)\prod_{i=1}^{k}L_{1\ldots b}^{1\ldots b}(u-i\hbar)+$
$\displaystyle-$ $\displaystyle\mathcal{P}_{b}(\vec{u})\sum\limits_{k\geq
1}(-E_{b+1,1})^{k}(-1)^{(b+2)k}\frac{\hbar^{-(k-1)}}{(k-1)!}L_{1\ldots
b-1}^{1\ldots b-1}(u)\prod_{i=1}^{k-1}L_{1\ldots b}^{1\ldots b}(u-i\hbar).$

By comparing the second and the third rows, we see that their sum is
$\mathcal{P}_{b}(\vec{u})(-E_{b+1,1})$. Likewise, recall (4.11) that

$A=-\mathcal{P}_{b}(\vec{u})\sum\limits_{k\geq
0}(-E_{b+1,1})^{k+1}(-1)^{(b+2)(k+1)}\hbar^{-k}L_{1\ldots b-1}^{1\ldots
b-1}(u-k\hbar)\frac{\prod_{i=0}^{k-1}L_{1\ldots b}^{1\ldots
b}(u-i\hbar)}{k!}.$

In particular, the sum of $A$ and the first row of $B$ is equal to
$-\mathcal{P}_{b}(\vec{u})(-E_{b+1,1})$. The proposition follows. ∎

######  Remark 4.12.

It follows from the proof that instead of $P_{ij}^{\psi}(u_{j})$, one can
consider its image under the quotient map. Namely, denote by

$[L_{j\ldots i-1}^{j\ldots
i-1}(u_{j})]\in\mathfrak{n}_{-}^{\psi}\backslash\mathrm{U}_{\hbar}(\mathfrak{m}_{j})\cong\mathrm{U}_{\hbar}(\mathfrak{b}_{j})$

where $\mathfrak{b}_{j}\subset\mathfrak{m}_{j}$ is the positive Borel
subalgebra. The image of $L_{j\ldots i-1}^{j\ldots i-1}(u_{j})$ in
$\mathrm{U}_{\hbar}(\mathfrak{b}_{j})$. Consider an analog of (4.1):

$[P^{\psi}_{ij}(u)]:=\sum\limits_{k\geq
0}(-1)^{(i+j)k}\hbar^{-k}(-E_{ij}^{\psi})^{k}\frac{[L_{j\ldots i-1}^{j\ldots
i-1}(u)]\ldots[L_{j\ldots i-1}^{j\ldots i-1}(u-\hbar k+\hbar)]}{k!}$

and $[P_{\mathfrak{m}_{N}}(\vec{u})]$ given by a similar formula as (4.2). It
follows that it defines the same element as $P_{\mathfrak{m}_{N}}(\vec{u})$.
Indeed: as usual, one can show it by two-fold induction. Consider the element

$P_{{}_{1}\mathfrak{m}_{N}}({}_{1}\vec{u})P_{21}^{\psi}(u_{1})\ldots
P_{b1}^{\psi}(u_{1}).$

We know that it is invariant under the right action of $E_{cd}^{\psi}$ for
$1\leq d<c\leq b$. Assume we proved it for $b$. To show the statement for
$b+1$, observe that

$P^{\psi}_{b+1,1}(u):=\sum\limits_{k\geq
0}(-1)^{(b+2)k}\hbar^{-k}(-E_{b+1,1}^{\psi})^{k}\frac{L_{1\ldots b}^{1\ldots
b}(u_{1})\ldots L_{1\ldots b}^{1\ldots b}(u_{1}-\hbar(k-1))}{k!}$

Since $L_{1\ldots b}^{1\ldots b}(v)$ is central in $\mathfrak{gl}_{b}$ and
$E_{b+1,1}^{\psi}$ commutes with all $E_{cd}^{\psi}$ as above, the statement
follows.

In particular, it should be possible to express the Kirillov projector using
the generators from [BK06].

###  4.2. Kirillov projector: left $\mathfrak{b}^{\vec{u}}$-invariance

In this subsection, we will prove the following

######  Theorem 4.13.

The Kirillov projector satisfies

(4.12)
$\displaystyle\begin{split}(E_{ii}+u_{i})P_{\mathfrak{m}_{N}}^{\psi}(\vec{u})&=0,\
1\leq i\leq N-1,\\\ E_{ij}P_{\mathfrak{m}_{N}}^{\psi}(\vec{u})&=0,\ 1\leq
i<j\leq N-1.\end{split}$

We will prove this statement by global induction. The base case is $N=2$.

######  Proposition 4.14.

The Kirillov projector $P_{\mathfrak{m}_{2}}^{\psi}(u_{1})$ satisfies

$\displaystyle(E_{11}+u)P_{21}^{\psi}(u)$ $\displaystyle=0.$

###### Proof.

First, observe that

(4.13) $E_{21}P^{\psi}_{21}(u-\hbar)=P_{21}(u).$

Indeed, we have

$\displaystyle P_{21}^{\psi}(u)E_{21}=$ $\displaystyle=$
$\displaystyle\sum\limits_{k\geq
0}(E_{21}^{\psi})^{k}\hbar^{-k}\frac{L_{1}^{1}(u)\ldots L_{1}^{1}(u-\hbar
k+\hbar)}{k!}E_{21}=$ $\displaystyle=$ $\displaystyle E_{21}\sum\limits_{k\geq
0}(E_{21}^{\psi})^{k}\hbar^{-k}\frac{L_{1}^{1}(u-\hbar)\ldots
L_{1}^{1}(u-\hbar k)}{k!}=$ $\displaystyle=$ $\displaystyle
E_{21}P_{21}^{\psi}(u-\hbar).$

But by Theorem 4.13, we have $P_{21}^{\psi}(u)E_{21}=P_{21}^{\psi}(u)$.

Then, we will use the following commutation relation:

$[E_{11},(E_{21}^{\psi})^{k}]=-k\hbar E_{21}(E_{21}^{\psi})^{k-1}.$

Therefore,

$\displaystyle(E_{11}+u)P_{21}^{\psi}=$ $\displaystyle=$ $\displaystyle
E_{11}\sum(E_{21}^{\psi})^{k}\hbar^{-k}\frac{L_{1}^{1}(u)\ldots
L_{1}^{1}(u-\hbar k+\hbar)}{k!}=$ $\displaystyle=$
$\displaystyle\sum(E_{21}^{\psi})^{k}\hbar^{-k}(E_{11}+u)\frac{L_{1}^{1}(u)\ldots
L_{1}^{1}(u-\hbar k+\hbar)}{k!}-$ $\displaystyle-$ $\displaystyle
E_{21}\sum(E_{21}^{\psi})^{k-1}\hbar^{1-k}\frac{L_{1}^{1}(u)\ldots
L_{1}^{1}(u-\hbar k+\hbar)}{(k-1)!}=$ $\displaystyle=$ $\displaystyle
P_{21}^{\psi}(u)L_{1}^{1}(u)-E_{21}P_{21}^{\psi}(u-\hbar)L_{1}^{1}(u)=0$

by (4.13). ∎

Now we make an induction step. Recall the notation ${}_{1}\mathfrak{m}_{N}$
from (4.6). Assume we proved Theorem 4.13 for
$P_{{}_{1}\mathfrak{m}_{N}}^{\psi}({}_{1}\vec{u})$. Observe that

$P_{\mathfrak{m}_{N}}^{\psi}(\vec{u})=P_{{}_{1}\mathfrak{m}_{N}}^{\psi}({}_{1}\vec{u})P_{21}^{\psi}(u_{1})\ldots
P_{N1}^{\psi}(u_{1}).$

Then the statement is reduced to

$\displaystyle\begin{split}(E_{11}+u_{1})P_{\mathfrak{m}_{N}}^{\psi}(\vec{u})&=0,\\\
E_{1b}P_{\mathfrak{m}_{N}}^{\psi}(\vec{u})&=0,\ 1<k\leq N-1.\end{split}$

The first property follows from Proposition 4.14, as $E_{11}$ commutes with
$P_{{}_{1}\mathfrak{m}_{N}}^{\psi}({}_{1}\vec{u})$. For the second property,
let us show first an a priori weaker statement:

######  Proposition 4.15.

We have

$P_{{}_{1}\mathfrak{m}_{N}}^{\psi}({}_{1}\vec{u})E_{1b}P_{\mathfrak{m}_{N}}^{\psi}(\vec{u})=0.$

In other words, the equality $E_{1b}P_{\mathfrak{m}_{N}}^{\psi}(\vec{u})=0$
holds in the left quotient by ${}_{1}\mathfrak{n}_{-}^{\psi}$.

###### Proof.

By global induction and by Proposition 4.18, we have

$E_{1b}P_{\mathfrak{m}_{b}}(u_{1},u_{2},\ldots,u_{b-1})=P_{\mathfrak{m}_{b}}(u_{1}-\hbar,u_{2},\ldots,u_{b-1})L_{1\ldots
b}^{1\ldots b}(u_{1}).$

We claim that nothing changes in the quotient when we pass to a larger
algebra, namely,

$\displaystyle
P_{{}_{1}\mathfrak{m}_{N}}^{\psi}({}_{1}\vec{u})E_{1b}P_{{}_{1}\mathfrak{m}_{N}}^{\psi}({}_{1}\vec{u})P_{21}^{\psi}(u_{1})\ldots
P_{b1}^{\psi}(u_{1})=$ $\displaystyle=$ $\displaystyle
P_{{}_{1}\mathfrak{m}_{N}}^{\psi}({}_{1}\vec{u})P_{21}^{\psi}(u_{1}-\hbar)\ldots
P_{b1}^{\psi}(u_{1}-\hbar)L_{1\ldots b}^{1\ldots b}(u_{1}).$

It follows from the following technical, but in fact simple lemma:

######  Lemma 4.16.

Assume that $x=\sum_{i=0}^{l-1}E_{1,b-i}\alpha_{i}$ with
$\alpha_{i}\in\mathrm{U}_{\hbar}({}_{b-1-i}\mathfrak{gl}_{b-1})$. Then

$P_{{}_{1}\mathfrak{m}_{N}}^{\psi}({}_{1}\vec{u})xP_{b-l+1,b-l}^{\psi}(u_{b-l})\ldots
P_{b,b-l}(u_{b-l})\in\sum_{i=0}^{l}E_{1,b-i}\cdot({}_{b-1-i}\mathfrak{gl}_{b-1})$

and

$P_{{}_{1}\mathfrak{m}_{N}}^{\psi}({}_{1}\vec{u})xP_{b+1,b-l+1}^{\psi}(u_{b-l+1})\ldots
P_{N,b-l+1}^{\psi}(u_{b-l+1})=P_{{}_{1}\mathfrak{m}_{N}}^{\psi}({}_{1}\vec{u})x.$

###### Proof.

Let us only show the second property. The condition
$\alpha_{i}\in({}_{b-1-i}\mathfrak{gl}_{b-1})$ means that $\alpha_{i}$ is the
sum of products of elements of the form $E_{\alpha\beta}$ with
$b-l+1\leq\alpha,\beta\leq b-1$. It is clear that for any such $\alpha,\beta$,
the commutator $[E_{j,b-l+1},E_{\alpha\beta}]$ for $j>b$ will be a nilpotent
element of the form $E_{jc}$ with $b-l+1\leq c\leq b-1$, in particular, the
value of $\psi$ on it is zero. Therefore, we have

$P_{{}_{1}\mathfrak{m}_{N}}^{\psi}({}_{1}\vec{u})\mathrm{ad}_{E_{j,b-l+1}}(x)=0.$

Since

$P_{{}_{1}\mathfrak{m}_{N}}^{\psi}({}_{1}\vec{u})x\xi^{\psi}=P_{{}_{1}\mathfrak{m}_{N}}^{\psi}({}_{1}\vec{u})\mathrm{ad}_{\xi}(x)$

for any $\xi$ in the negative nilpotent subalgebra of
${}_{1}\mathfrak{m}_{N}$, we conclude that every
$P_{j,b-l+1}^{\psi}(u_{b-l+1})$ for $j>b$ acts on
$P_{{}_{1}\mathfrak{m}_{N}}^{\psi}({}_{1}\vec{u})x$ by identity. Hence,

$P_{{}_{1}\mathfrak{m}_{N}}^{\psi}({}_{1}\vec{u})xP_{b+1,b-l+1}^{\psi}(u_{b-l+1})\ldots
P_{N,b-l+1}^{\psi}(u_{b-l+1})(u_{b-l+1})=P_{{}_{1}\mathfrak{m}_{N}}^{\psi}({}_{1}\vec{u})x.$

The first property follows by similar analysis of commutators and is left to
the reader. ∎

Therefore, the claim of the proposition would follow from

$P_{{}_{1}\mathfrak{m}_{N}}^{\psi}({}_{1}\vec{u})P_{21}^{\psi}(u_{1}-\hbar)\ldots
P_{b1}^{\psi}(u_{1}-\hbar)L_{1\ldots b}^{1\ldots
b}(u_{1})P_{b+1,1}^{\psi}(u_{1})=0.$

For brevity, denote by $\vec{u}-\hbar:=(u_{1}-\hbar,u_{2},\ldots,u_{N-1})$ and
$u=u_{1}$. Adopting notations of Section 4.1, it can be reformulated as

$\mathcal{P}_{b}(\vec{u}-\hbar)L_{1\ldots b}^{1\ldots
b}(u)P_{b+1,1}^{\psi}(u)=0.$

Observe that $b\geq 2$, in particular, $E_{b+1,1}=E_{b+1,1}^{\psi}$. We need
two facts. The first one: by 4.9, we have

$\displaystyle\mathcal{P}_{b}(\vec{u}-\hbar)L_{1\ldots b}^{1\ldots
b}(u_{1})(-E_{b+1,1})=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u}-\hbar)(-E_{b+1,1})L_{1\ldots b}^{1\ldots
b}(u)+\hbar\mathcal{P}_{b}(\vec{u}-\hbar)L_{1,2,\ldots,b}^{b+1,2,\ldots,b}(u)=$
$\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u}-\hbar)(-E_{b+1,1})L_{1\ldots b}^{1\ldots
b}(u)+\mathcal{P}_{b}(\vec{u}-\hbar)\hbar E_{b+1,1}\frac{L_{1\ldots
b}^{1\ldots b}(u)-L_{1\ldots b}^{1\ldots b}(u-\hbar)}{\hbar}-$
$\displaystyle-$ $\displaystyle(-1)^{b+2}\hbar\mathcal{P}_{b}(\vec{u}-\hbar)=$
$\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u}-\hbar)(-E_{b+1,1})L_{1\ldots b}^{1\ldots
b}(u-\hbar)-(-1)^{b+2}\hbar\mathcal{P}_{b}(\vec{u}-\hbar).$

The second one: by the same lemma,

$\displaystyle\mathcal{P}_{b}(\vec{u}-\hbar)(-E_{b+1,1})L_{1\ldots b}^{1\ldots
b}(u-\hbar)(-E_{b+1,1})^{k}=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u}-\hbar)(-E_{b+1,1})^{k+1}L_{1\ldots
b}^{1\ldots
b}(u-\hbar)+k\hbar\mathcal{P}_{b}(\vec{u}-\hbar)L_{1,2,\ldots,b}^{b+1,2,\ldots,b}(u-\hbar)(-E_{b1})^{k}=$
$\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u}-\hbar)(-E_{b+1,1})^{k+1}L_{1\ldots
b}^{1\ldots b}(u-\hbar)-k\hbar(-E_{b+1,1})\partial_{u}L_{1\ldots b}^{1\ldots
b}(u-\hbar)(-E_{b+1,1})^{k}-$ $\displaystyle-$
$\displaystyle(-1)^{b+2}k\hbar\mathcal{P}_{b}(\vec{u}-\hbar)(-E_{b+1,1})^{k}.$

Proceeding as in the proof of 4.11, we conclude that

$\displaystyle\mathcal{P}_{b}(\vec{u}-\hbar)(-E_{b+1,1})L_{1\ldots b}^{1\ldots
b}(u-\hbar)(-E_{b+1,1})^{k}=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u}-\hbar)(-E_{b+1,1})^{k+1}L_{1\ldots
b}^{1\ldots
b}(u-k\hbar-\hbar)-(-1)^{b+2}k\hbar\mathcal{P}_{b}(\vec{u}-\hbar)(-E_{b+1,1})^{k}$

Combining the two facts, we conclude that

$\displaystyle\mathcal{P}_{b}(\vec{u}-\hbar)L_{1\ldots b}^{1\ldots
b}(u)(-E_{b+1,1})^{k+1}=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u}-\hbar)(-E_{b1})^{k+1}L_{1\ldots
b}^{1\ldots
b}(u-k\hbar-\hbar)-(-1)^{b+2}(k+1)\hbar\mathcal{P}_{b}(\vec{u}-\hbar)(-E_{b+1,1})^{k}.$

Therefore,

$\displaystyle\mathcal{P}_{b}(\vec{u}-\hbar)L_{1\ldots b}^{1\ldots
b}(u)P_{b+1,1}^{\psi}(u)=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u}-\hbar)\left(1+\sum\limits_{k\geq
0}L_{1\ldots b}^{1\ldots
b}(u)(-E_{b+1,1})^{k+1}(-1)^{(b+2)(k+1)}\hbar^{-k-1}\frac{\prod_{i=0}^{k}L_{1\ldots
b}^{1\ldots b}(u-i\hbar)}{k!}\right)=$ $\displaystyle=$
$\displaystyle\mathcal{P}_{b}(\vec{u}-\hbar)L_{1\ldots b}^{1\ldots b}(u)+$
$\displaystyle+$ $\displaystyle\sum\limits_{k\geq
0}(-E_{b+1,1})^{k+1}L_{1\ldots b}^{1\ldots
b}(u-k\hbar-\hbar)(-1)^{(b+2)(k+1)}\hbar^{-k-1}\frac{\prod_{i=0}^{k}L_{1\ldots
b}^{1\ldots b}(u-i\hbar)}{(k+1)!}-$ $\displaystyle-$
$\displaystyle\sum\limits_{k\geq
0}(-E_{b+1,1})^{k}(-1)^{(b+2)k}\hbar^{-k}\frac{\prod_{i=0}^{k}L_{1\ldots
b}^{1\ldots b}(u-i\hbar)}{k!}.$

Term-by-term comparison gives that this sum is indeed zero, i.e.

$\mathcal{P}_{b}(\vec{u}-\hbar)L_{1\ldots b}^{1\ldots
b}(u)P_{b+1,1}^{\psi}(u)=0,$

and the proposition follows. ∎

It turns out that the weaker version implies the stronger one:

######  Proposition 4.17.

For $2\leq b\leq N-1$, we have

$E_{1b}P_{\mathfrak{m}_{N}}(\vec{u})=0,$

where the equality is understood as of linear operators on any Whittaker
module.

###### Proof.

Let $W$ be a Whittaker module and $w\in W$. Observe that
$\mathrm{span}(E_{1i}|2\leq i\leq N-1)$ can be naturally identified with the
dual of the vector representation of ${}_{1}\mathfrak{m}_{N}$. Therefore, we
can apply Remark 5.6 and conclude that

$\displaystyle
wE_{1b}P_{{}_{1}\mathfrak{m}_{N}}({}_{1}\vec{u})=\sum\limits_{i=2}^{N-1}w_{i}P_{{}_{1}\mathfrak{m}_{N}}({}_{1}\vec{u})E_{1i}P_{{}_{1}\mathfrak{m}_{N}}({}_{1}\vec{u})$

for some $w_{i}\in W$. Hence,

$\displaystyle wE_{1b}P_{\mathfrak{m}_{N}}(\vec{u})=$ $\displaystyle=$
$\displaystyle
wE_{1b}P_{{}_{1}\mathfrak{m}_{N}}({}_{1}\vec{u})P_{21}^{\psi}(u_{1})\ldots
P_{N1}^{\psi}(u_{1})=$ $\displaystyle=$
$\displaystyle\sum\limits_{i=2}^{N-1}w_{i}P_{{}_{1}\mathfrak{m}_{N}}({}_{1}\vec{u})E_{1i}P_{{}_{1}\mathfrak{m}_{N}}({}_{1}\vec{u})P_{21}^{\psi}(u_{1})\ldots
P_{N1}^{\psi}(u_{1})=$ $\displaystyle=$
$\displaystyle\sum\limits_{i=2}^{N-1}w_{i}P_{{}_{1}\mathfrak{m}_{N}}({}_{1}\vec{u})E_{1i}P_{\mathfrak{m}_{N}}(\vec{u})$

for some $\\{w_{i}\\}\subset W$. Therefore, to prove that it is zero, it is
enough to show that

$P_{{}_{1}\mathfrak{m}_{N}}({}_{1}\vec{u})E_{1i}P_{\mathfrak{m}_{N}}(\vec{u})=0$

for every $2\leq i\leq N-1$. But this is exactly Proposition 4.15. ∎

###  4.3. Kirillov projector: action on the last column

In this section, we will study the action of $P_{\mathfrak{m}_{N}}(\vec{u})$
on the complement $\mathfrak{u}_{N}$ of $\mathfrak{m}_{N}$ inside
$\mathfrak{gl}_{N}$:

$\mathfrak{u}_{N}=\begin{pmatrix}0&\ldots&0&*\\\
\vdots&\ddots&\vdots&\vdots\\\ 0&\ldots&0&*\end{pmatrix}$

######  Proposition 4.18.

We have

$E_{1N}P_{\mathfrak{m}_{N}}(u_{1},u_{2},\ldots,u_{N-1})=P_{\mathfrak{m}_{N}}(u_{1}-\hbar,u_{2},\ldots,u_{N-1})L_{1\ldots
N}^{1\ldots N}(u_{1}).$

###### Proof.

Recall the notation $\vec{u}-\hbar:=(u_{1}-\hbar,u_{2},\ldots,u_{N-1})$. Since
$L_{1\ldots N}^{1\ldots N}(u_{1})$ is central, we have

$P_{\mathfrak{m}_{N}}(\vec{u}-\hbar)L_{1\ldots N}^{1\ldots
N}(u_{1})=L_{1\ldots N}^{1\ldots
N}(u_{1})P_{\mathfrak{m}_{N}}(\vec{u}-\hbar).$

By Proposition 3.1, we have

$L_{1\ldots N}^{1\ldots
N}(u_{1})=\sum\limits_{l=1}^{N}(-1)^{l-1}L_{1\ldots\hat{l}\ldots N}^{2\ldots
N}(u_{1}-\hbar)L_{1l}(u_{1}).$

By Theorem 4.13, the only terms that act non-trivially are for $i=1$ and
$i=N$. The former gives

$\displaystyle L_{2\ldots N}^{2\ldots
N}(u_{1}-\hbar)L_{11}(u_{1})P_{\mathfrak{m}_{N}}(\vec{u}-\hbar)=$
$\displaystyle=$ $\displaystyle L_{2\ldots N}^{2\ldots
N}(u_{1}-\hbar)(L_{11}(u_{1}-\hbar)+\hbar)P_{\mathfrak{m}_{N}}(\vec{u}-\hbar)=$
$\displaystyle=$ $\displaystyle\hbar L_{2\ldots N}^{2\ldots
N}(u_{1}-\hbar)P_{\mathfrak{m}_{N}}(\vec{u}-\hbar),$

while the latter

$(-1)^{N-1}L_{1\ldots N-1}^{2\ldots
N}(u_{1}-\hbar)E_{1N}P_{\mathfrak{m}_{N}}(\vec{u}-\hbar).$

We also have

$L_{1\ldots N-1}^{2\ldots N}(u_{1}-\hbar)E_{1N}=E_{1N}L_{1\ldots N-1}^{2\ldots
N}(u_{1}-\hbar)+\hbar(-1)^{N-1}L_{1\ldots N-1}^{1\ldots
N-1}(u_{1}-\hbar)-\hbar(-1)^{N-1}L_{2\ldots N}^{2\ldots N}(u_{1}-\hbar).$

Since

$L_{1\ldots N-1}^{1\ldots
N-1}(u_{1}-\hbar)P_{\mathfrak{m}_{N}}(\vec{u}-\hbar)=0,$

we have

$\displaystyle L_{1\ldots N}^{1\ldots
N}(u_{1})P_{\mathfrak{m}_{N}}(\vec{u}-\hbar)=$ $\displaystyle=$
$\displaystyle\big{[}\hbar L_{2\ldots N}^{2\ldots
N}(u_{1}-\hbar)+(-1)^{N-1}E_{1N}L_{1\ldots N-1}^{2\ldots N}(u_{1}-\hbar)+$
$\displaystyle+$ $\displaystyle\hbar L_{1\ldots N-1}^{1\ldots
N-1}(u_{1}-\hbar)-\hbar L_{2\ldots N}^{2\ldots
N}(u_{1}-\hbar)\big{]}P_{\mathfrak{m}_{N}}(\vec{u}-\hbar)=$ $\displaystyle=$
$\displaystyle(-1)^{N-1}E_{1N}L_{1\ldots N-1}^{2\ldots
N}(u_{1}-\hbar)P_{\mathfrak{m}_{N}}(\vec{u}-\hbar).$

Therefore, the claim of the proposition would follow from

$L_{1\ldots N-1}^{2\ldots
N}(u_{1}-\hbar)P_{\mathfrak{m}_{N}}(\vec{u}-\hbar)=P_{\mathfrak{m}_{N}}(\vec{u}).$

Now recall Theorem 4.3: the operator $P_{\mathfrak{m}_{N}}(\vec{u})$,
satisfying

$\displaystyle P_{\mathfrak{m}_{N}}(\vec{u})(x-\psi(x))$ $\displaystyle=0,\
x\in\mathfrak{n}_{-},$ $\displaystyle(E_{ij}+\delta_{ij}\cdot
u_{j})P_{\mathfrak{m}_{N}}(\vec{u})$ $\displaystyle=0,\ 1\leq i\leq j\leq N-1$

together with the normalization condition
$P_{\mathfrak{m}_{N}}(\vec{u})\big{|}_{W^{\mathfrak{n}_{-}^{\psi}}}=\mathrm{id}$,
is unique. Therefore, it would be enough to prove that the left-hand side
satisfies these properties. The first property is clear, and the normalization
condition follows from Proposition 3.8. For $2\leq i\leq j\leq N-1$, the
second property follows from

$[E_{ij},L_{1\ldots N-1}^{2\ldots N}(u_{1}-\hbar)]=0.$

Therefore, it is enough to consider the case $i=1$. Let $j=1$. Then

$[E_{11},L_{1\ldots N-1}^{2\ldots N}(u_{1}-\hbar)]=-\hbar L_{1\ldots
N-1}^{2\ldots N}(u_{1}-\hbar),$

hence

$\displaystyle(E_{11}+u_{1})L_{1\ldots N-1}^{2\ldots
N}(u_{1}-\hbar)P_{\mathfrak{m}_{N}}(\vec{u}-\hbar)=$ $\displaystyle=$
$\displaystyle L_{1\ldots N-1}^{2\ldots
N}(u_{1}-\hbar)(E_{11}+u_{1}-\hbar)P_{\mathfrak{m}_{N}}(\vec{u}-\hbar)=0.$

Let $j>1$, then

$[E_{1j},L_{1\ldots N-1}^{2\ldots N}(u_{1}-\hbar)]=(-1)^{j}\hbar L_{1\ldots
N-1}^{1\ldots\hat{j}\ldots N}(u_{1}-\hbar).$

Since

$L_{1\ldots N-1}^{1\ldots\hat{j}\ldots
N}(u_{1}-\hbar)=\sum\limits_{l=1}^{N-1}(-1)^{l-1}L_{1\ldots\hat{l}\ldots
N-1}^{2\ldots\hat{j}\ldots N}(u_{1}-2\hbar)L_{1l}(u_{1}-\hbar)$

and each term $L_{1l}(u_{1}-\hbar)$ acts by zero on
$P_{\mathfrak{m}_{N}}(\vec{u}-\hbar)$, we conclude that

$E_{ij}L_{1\ldots N-1}^{2\ldots
N}(u_{1}-\hbar)P_{\mathfrak{m}_{N}}(\vec{u}-\hbar)=0.$

Therefore, the element

$L_{1\ldots N-1}^{2\ldots N}(u_{1}-\hbar)P_{\mathfrak{m}_{N}}(\vec{u}-\hbar)$

satisfies the conditions of Theorem 4.3, thus coincides with
$P_{\mathfrak{m}_{N}}(\vec{u})$. ∎

It seems that there is no closed expression for the action of
$P_{\mathfrak{m}_{N}}(\vec{u})$ on $E_{kN}$. However, it turns out that in the
quotient by the ideal
$\mathfrak{n}_{-}^{\psi}\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})$, their images
are essentially given by the coefficients of the quantum characteristic
polynomial:

######  Proposition 4.19.

Let

(4.14) $L_{1\ldots N}^{1\ldots
N}(v)=v^{N}+\sum_{i=0}^{N-1}(-1)^{N-i}A_{i}v^{i}.$

Then

$E_{kN}P_{\mathfrak{m}_{N}}(u_{1},u_{2},\ldots,u_{N-1})\equiv(A_{k}-\beta_{kk})+\sum_{j>k}\alpha_{kj}(A_{j}-\beta_{kj})\
(\mathrm{mod}\ \mathfrak{n}_{-}^{\psi}\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})),$

for some functions $\alpha_{kj}=\alpha_{kj}(\vec{u})$ and
$\beta_{kj}=\beta_{kj}(\vec{u})$. In particular, the map

(4.15)
$\mathfrak{n}_{-}^{\psi}\backslash\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})/\mathfrak{b}^{\vec{u}}\rightarrow\mathbf{A}_{\mathfrak{gl}_{N}}.$

induced by the Kirillov projector, is an isomorphism of vector spaces. Here
$\mathbf{A}_{\mathfrak{gl}_{N}}$ is the center of the universal enveloping
algebra $\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})$.

###### Proof.

In what follows, we use “$\equiv$” to denote equivalence modulo the ideal
$\mathfrak{n}_{-}^{\psi}\mathrm{U}_{\hbar}(\mathfrak{m}_{N})$.

Since $L_{1\ldots N}^{1\ldots N}(v)$ is central,

$L_{1\ldots N}^{1\ldots N}(v)\equiv P_{\mathfrak{m}_{N}}(\vec{u})L_{1\ldots
N}^{1\ldots N}(v)=L_{1\ldots N}^{1\ldots N}(v)P_{\mathfrak{m}_{N}}(\vec{u}).$

By Proposition 3.1, we have

$L_{1\ldots N}^{1\ldots N}(v)=\sum\limits_{l=1}^{N}(-1)^{N-l}L_{1\ldots
N-1}^{1\ldots\hat{l}\ldots N}(v)L_{lN}(v-\hbar N+\hbar).$

By Proposition 3.8, we get

$\displaystyle L_{1\ldots N}^{1\ldots N}(v)\equiv$
$\displaystyle\sum\limits_{l=1}^{N}(-1)^{N-l}L_{1\ldots l-1}^{1\ldots
l-1}(v)L_{lN}(v-\hbar N+\hbar)=$ $\displaystyle=$
$\displaystyle\sum\limits_{l=1}^{N}(-1)^{N-l}L_{lN}(v-\hbar N+\hbar)L_{1\ldots
l-1}^{1\ldots l-1}(v).$

Using Proposition 3.1, one can show that

$\displaystyle L_{1\ldots l-1}^{1\ldots l-1}(v)P_{\mathfrak{m}_{N}}(\vec{u})=$
$\displaystyle=$ $\displaystyle(v-u_{1})(v-u_{2}-\hbar)\ldots(v-u_{l-1}-\hbar
l+2\hbar)P_{\mathfrak{m}_{N}}(\vec{u}),$

therefore,

$\displaystyle L_{1\ldots N}^{1\ldots
N}(v)P_{\mathfrak{m}_{N}}(\vec{u})\equiv$ $\displaystyle\equiv$
$\displaystyle\sum_{l=1}^{N}\left((-1)^{N-l}\prod_{i=1}^{l-1}(v-u_{i}-\hbar
i+\hbar)\right)E_{lN}P_{\mathfrak{m}_{N}}(\vec{u})+$ $\displaystyle+$
$\displaystyle(v-N\hbar+\hbar)\prod_{i=1}^{N-1}(v-u_{i}-\hbar
i+\hbar)P_{\mathfrak{m}_{N}}(\vec{u})$

The first part of the proposition then follows by comparing the coefficients
of $v$ on both sides. As for the second part, it follows from Remark 5.6 that
for any $x,y\in\mathrm{U}_{\hbar}(\mathfrak{u}_{N})$,

$\displaystyle xyP_{\mathfrak{m}_{N}}(\vec{u})=$ $\displaystyle=$
$\displaystyle
xP_{\mathfrak{m}_{N}}(\vec{u})yP_{\mathfrak{m}_{N}}(\vec{u})+(K_{(1)}\cdot
x)P_{\mathfrak{m}_{N}}(\vec{u})(K_{(2)}\cdot y)P_{\mathfrak{m}_{N}}(\vec{u}),$

where $K=K_{(1)}\otimes K_{(2)}$ is some nilpotent matrix acting on
$\mathrm{U}_{\hbar}(\mathfrak{u}_{N})\otimes\mathrm{U}_{\hbar}(\mathfrak{u}_{N})$,
where we identify $\mathrm{U}_{\hbar}(\mathfrak{u}_{N})$ with the double
quotient
$\mathfrak{n}_{-}^{\psi}\backslash\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})/\mathfrak{b}^{\vec{u}}$.
In particular, we can invert $1+K$. It follows from Theorem 5.11 that
$\mathbf{A}_{\mathfrak{gl}_{N}}$ is a free commutative algebra on generators
$\\{A_{i}\\}$. Since we know that the transformation induced by the Kirillov
projector is invertible on generators, the second part follows by induction on
the degree of generators. ∎

######  Remark 4.20.

It follows from the proof that for a special choice of parameters
$u_{i}=-\hbar i+\hbar$, the images
$E_{kN}P_{\mathfrak{m}_{N}}(0,-\hbar,\ldots,-\hbar N+2\hbar)$ in the quotient
$\mathfrak{n}_{-}^{\psi}\backslash\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})$ are
_literally_ given by the coefficients $A_{k}$ of the quantum characteristic
polynomial (4.14). Classically, it can be interpreted as follows.

Denote $P_{\hbar}:=P_{\mathfrak{m}_{N}}(0,-\hbar,\ldots,-\hbar N+2\hbar)$. The
Kirillov projector gives a linear map

$P_{\hbar}\colon\mathfrak{n}_{-}^{\psi}\backslash\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})\rightarrow(\mathfrak{n}_{-}^{\psi}\backslash\mathrm{U}_{\hbar}(\mathfrak{gl}_{N}))^{\mathfrak{n}_{-}^{\psi}}.$

Recall that by Kostant’s theorem [Kos78], the target is isomorphic to the
center $\mathbf{A}_{\mathfrak{gl}_{N}}$. Composing with the quotient map
$\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})\rightarrow\mathfrak{n}_{-}^{\psi}\backslash\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})$,
we get a homomorphism

$\mathrm{U}_{\hbar}(\mathfrak{gl}_{N})\rightarrow\mathbf{A}_{\mathfrak{gl}_{N}}.$

Taking limit $\hbar\rightarrow 0$ as in Remark 4.4 and identifying
$\mathfrak{g}\cong\mathfrak{g}^{*}$, we obtain a linear map

$P_{0}\colon\mathcal{O}(\mathfrak{g})\rightarrow\mathcal{O}(\mathbb{A}^{N}),$

where we identified $\mathcal{O}(\mathbb{A}^{N})$ with the classical limit of
$\mathbf{A}_{\mathfrak{gl}_{N}}$ using the coefficients $A_{k}$ as the
generators. Moreover, it follows from Remark 5.6 that this is an algebra map,
since

$xyP_{\hbar}=xP_{\hbar}yP_{\hbar}+O(\hbar).$

In particular, it corresponds to a map between varieties
$\mathbb{A}^{N}\rightarrow\mathfrak{g}$. It is not hard to determine which
one: since $P_{0}$ acts as

$\displaystyle(E_{ij}-\psi(E_{ij}))P_{0}$ $\displaystyle=0,\ i>j,$
$\displaystyle E_{kl}P_{0}$ $\displaystyle=0,\ k\leq l,$ $\displaystyle
E_{kN}P_{0}$ $\displaystyle=A_{k},1\leq k\leq N,$

this map is given by

$(A_{1},\ldots,A_{N})\mapsto\begin{pmatrix}0&\ldots&0&A_{N}\\\
1&\ldots&0&A_{N-1}\\\ \vdots&\ddots&\vdots&\vdots\\\
0&\ldots&1&A_{1}\end{pmatrix}$

(up to transposition induced by the isomorphism
$\mathfrak{gl}_{N}\cong\mathfrak{gl}_{N}^{*}$). Therefore, the Kirillov