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# The Brundan-Kleshchev isomorphism revisited

Fan Kong and Zhi-Wei Li  Fan Kong
School of Mathematics and Statistics
Southwest University
Chongqing 400715
P. R. China<EMAIL_ADDRESS>Zhi-Wei Li
School of Mathematics and Statistics
Jiangsu Normal University
Xuzhou 221116 Jiangsu
P. R. China<EMAIL_ADDRESS>
(Date: September 3, 2024)

###### Abstract.

We give a short and unified proof of the Brundan-Kleshchev isomorphism between
blocks of cyclotomic Hecke algebras and cyclotomic Khovanov-Lauda-Rouquier
algebras of type A.

*Corresponding author

2000 Mathematics Subject Classification: 20C08.

The first author is supported by the NSF of China (No. 11501057) and the
Doctoral Fund of Southwest University (SWU-118003); The second author is
supported by the NSF of China (No. 11671174).

##  1\. Introduction

In 2008, Brundan and Kleshchev [1] gave an explicit isomorphism between blocks
of cyclotomic affine Hecke algebras of symmetric groups, or the corresponding
degenerated cyclotomic affine Hecke algebras, and cyclotomic KLR algebras of
type A. On the other hand, in 1989, Lusztig showed that there is a natural
isomorphism between affine Hecke algebras and their graded versions [5].
Motivated by Lusztig’s work, we introduced the semi-rationalization extensions
of affine Hecke algebras in [4], where an isomorphism between direct sums of
blocks of cyclotomic affine Hecke algebra of all types and their cyclotomic
semi-rationalization algebras was given. The aim of this paper is to give a
shorter proof of Brundan and Kleshchev isomorphism using the machinery of our
general result. Main result (Theorem 5.6) The cyclotomic KLR algebra of type A
and blocks of the cyclotomic $($degenerate$)$ affine Hecke algebra of a
symmetric group can be realized as the same quotient of an algebra. It can be
illustrated by the following diagram of algebras.

$\tilde{\mathcal{L}}$$\mathcal{L}$$\tilde{\mathcal{L}}(\Lambda)$$\mathcal{R}(\Lambda)$$\mathcal{H}(\Lambda)e(\mathcal{C})$$\mathcal{H}_{q}(\Lambda)e(\mathcal{C})$$\cong$$\cong$$\cong$

The algebra $\mathcal{L}$, called the Lusztig algebra, is some extension of a
degenerate affine Hecke algebra or a non-degenerate affine Hecke algebra. This
algebra contains all corresponding rational functions, which allow us to
construct the KLR-basis in a unified way either in the degenerate case or in
the non-degenerate case. The algebra $\tilde{\mathcal{L}}$, called the semi-
rationalization algebra, can be viewed as a subalgebra of $\mathcal{L}$ only
containing part of the rational functions. The semi-rationalization algebra
$\tilde{\mathcal{L}}$ has three bases: the first one is the KLR basis by which
we construct an isomorphism between the cyclotomic semi-rationalization
algebra $\tilde{\mathcal{L}}(\Lambda)$ and the cyclotomic KLR algebra
$\mathcal{R}(\Lambda)$; the second one is the degenerate Bernstein-Zelevinski
basis by which we give an isomorphism between $\tilde{\mathcal{L}}(\Lambda)$
and the cyclotomic degenerate affine Hecke algebra $\mathcal{H}(\Lambda)$; the
third one is the non-degenerate Bernstein-Zelevinski basis by which we give an
isomorphism between $\tilde{\mathcal{L}}(\Lambda)$ and the cyclotomic non-
degenerate affine Hecke algebra $\mathcal{H}_{q}(\Lambda)$.

##  2\. Preliminaries

###  2.1. The Demazure operator

Let $S_{n}$ be the symmetric group with basic transpositions
$\sigma_{1},\cdots,\sigma_{n-1}$. Let $\Bbbk$ be a field. Then $S_{n}$ acts on
the left on the polynomial ring $\Bbbk[X]$ and on the Laurent polynomial ring
$\Bbbk[X^{\pm 1}]$ in $X:=X_{1},\cdots,X_{n}$ by permuting variables. Using
the $S_{n}$-action above, the Demazure operators are defined as $\partial_{r}$
on $\Bbbk[x]$ for all $1\leq r<n$ as

$\partial_{r}(f)=\frac{\sigma_{r}(f)-f}{X_{r}-X_{r+1}}.$

It is well-known that the Demazure operators satisfy the Leibniz rule

$\partial_{r}(fg)=\partial_{r}(f)g+\sigma_{r}(f)\partial_{r}(g),$

for all $f,g\in\Bbbk[X]$ and for $1\leq r<n$, and the relations

$\sigma_{r}(\partial_{r}(f))=\partial_{r}(f),\quad\partial_{r}(\sigma_{r}(f))=-\partial_{r}(f).$

Let $\Bbbk(X)$ be the corresponding rational function field, then the
$S_{n}$-action on $\Bbbk[X]$ above can be extended to an action
$w\colon\tfrac{f}{g}\mapsto\tfrac{w(f)}{w(g)}$ of $S_{n}$ on $\Bbbk(X)$ (by
the field automorphism). This means that the action of the Demazure operators
on $\Bbbk[X]$ also extends to operators on $\Bbbk(X)$.

Let $F$ be the quotient field of the subalgebra

$Z:=\Bbbk[X]^{S_{n}}=\\{f\in\Bbbk[X]|w(f)=f\ \mbox{for every}\ w\in S_{n}\\}$

of $S_{n}$-invariants. By [5, 3.12 (a)], there is a natural $\Bbbk$-algebra
isomorphism

$\Bbbk[X]\otimes_{Z}F\to\Bbbk(X),f\otimes g\mapsto fg.$ (2.1)

###  2.2. Index sets

Let $I=\mathbb{Z}/e\mathbb{Z}=\\{0,\cdots,e-1\\}$ ($e=0\ \mbox{or}\ 2\leq
e\in\mathbb{Z}$) be the vertex set of the quiver of type $A_{\infty}$ if $e=0$
or $A_{e-1}^{(1)}$ if $e\geq 2$:

$A_{\infty}:$$\cdots$$-2$$-1$$0$$1$$2$$\cdots$$A_{e-1}^{(1)}:$$0$$1$$2$$0$$1$$3$$0$$1$$2$$3$$4$$0$$1$$2$$\cdots$

For $1\leq r<n$, we define the map
$\sigma_{r}(\text{\boldmath$i$})_{s}=i_{\sigma_{r}(s)}$. Then $S_{n}$ acts on
the set of $n$-tuples $\text{\boldmath$i$}=(i_{1},\cdots,i_{n})\in I^{n}$ by
the place permutation $:w(\text{\boldmath$i$})_{s}=i_{w^{-1}(s)}$.

Throughout this paper, We fix an $S_{n}$-orbit $\mathcal{C}$ of $I^{n}$.

##  3\. Affine Hecke algebras and their rationalizations

###  3.1. The degenerate affine Hecke algebra

Following [3], the degenerate affine Hecke algebra $\mathcal{H}$ of $S_{n}$ is
defined to be the associated unital $\Bbbk$-algebra with generators
$\\{X_{1},\cdots,X_{n},T_{1},\cdots,T_{n-1}\\}$ subject to the following
relations for all admissible indices:

$\displaystyle X_{r}X_{s}$ $\displaystyle=X_{s}X_{r};$ (3.1) $\displaystyle
T_{r}X_{s}$ $\displaystyle=\sigma_{r}(X_{s})T_{r}+\partial_{r}(X_{s});$ (3.2)
$\displaystyle\quad T_{r}^{2}$ $\displaystyle=1;$ (3.3) $\displaystyle
T_{r}T_{s}$ $\displaystyle=T_{s}T_{r}\hskip 123.48494pt\text{if $|r-s|>2$};$
(3.4) $\displaystyle T_{r}T_{r+1}T_{r}$ $\displaystyle=T_{r+1}T_{r}T_{r+1};$
(3.5)

For $w=\sigma_{r_{1}}\sigma_{r_{2}}\cdots\sigma_{r_{m}}\in S_{n}$ a reduced
expression we put $T_{w}:=T_{r_{1}}T_{r_{2}}\cdots T_{r_{m}}$. Then $T_{w}$ is
a well-defined element in $\mathcal{H}$ and the algebra $\mathcal{H}$ is a
free $\Bbbk[X]$-module with the basis $\\{T_{w}\ |\ w\in S_{n}\\}$.

###  3.2. The non-degenerate affine Hecke algebra

Assume $1\neq q\in\Bbbk^{\times}$. We define the affine Hecke algebra
$\mathcal{H}_{q}$ to be the associated unital $\Bbbk$-algebra with generators
$\\{X_{1}^{\pm 1},\cdots,X_{n}^{\pm 1},T_{1},\cdots,T_{n-1}\\}$ subject to the
following relations for all admissible indices:

$\displaystyle X_{r}^{\pm 1}X_{s}^{\pm 1}$ $\displaystyle=X_{s}^{\pm
1}X_{r}^{\pm 1},\quad X_{r}X_{r}^{-1}=X_{r}^{-1}X_{r}=1;$ (3.6) $\displaystyle
T_{r}X_{s}$
$\displaystyle=\sigma_{r}(X_{s})T_{r}+(q-1)X_{r+1}\partial_{r}(X_{s});$ (3.7)
$\displaystyle T_{r}^{2}$ $\displaystyle=(q-1)T_{r}+q;$ (3.8) $\displaystyle
T_{r}T_{s}$ $\displaystyle=T_{s}T_{r}\hskip 123.48494pt\text{if $|r-s|>2$};$
(3.9) $\displaystyle T_{r}T_{r+1}T_{r}$ $\displaystyle=T_{r+1}T_{r}T_{r+1};$
(3.10)

Similar to the degenerate case, $\mathcal{H}_{q}$ is a free $\Bbbk[X^{\pm
1}]$-module with the basis $\\{T_{w}\ |\ w\in S_{n}\\}$.

###  3.3. The rationalization

By [5, Proposition 3.11], the center of $\mathcal{H}$ is the commutative
algebra $Z$ of $S_{n}$-invariants. Thus $\mathcal{H}$ can be seen as a
$Z$-subalgebra (identified with the subspace $\mathcal{H}\otimes 1$) of the
$F$-algebra $\mathcal{H}_{F}:=\mathcal{H}\otimes_{Z}F$.

Similarly, the non-degenerate affine Hecke algebra $\mathcal{H}_{q}$ also can
be seen as a $Z$-subalgebra (identified with the subspace
$\mathcal{H}_{q}\otimes 1$) of the $F$-algebra
$\mathcal{H}_{q,F}:=\mathcal{H}_{q}\otimes_{\mathcal{Z}}\mathcal{F}.$ For any
$1\leq r<n$ and $f\in\Bbbk(X)$, as a consequence of [5, 3.12 (d)], we have

$\displaystyle T_{r}f=\begin{cases}\sigma_{r}(f)T_{r}+\partial_{r}(f)&\text{if
$f\in\mathcal{H}_{F}$},\\\
\sigma_{r}(f)T_{r}+(q-1)X_{r+1}\partial_{r}(f)&\text{if
$f\in\mathcal{H}_{q,F}$}.\end{cases}$ (3.11)

In light of the Bernstein and Zelevinski basis of $\mathcal{H}$ and
$\mathcal{H}_{q}$, there are decompositions, see [5, 3.12(c)]:

$\mathcal{H}_{F}(\mbox{resp.},\mathcal{H}_{q,F})=\oplus_{w\in
S_{n}}T_{w}\Bbbk(x)=\oplus_{w\in S_{n}}\Bbbk(x)T_{w}.$ (3.12)

###  3.4. Intertwining elements

For $1\leq r<n$, we define the intertwining element $\kappa_{r}$ as follows:

$\kappa_{r}:=\begin{cases}T_{r}+\tfrac{1}{X_{r}-X_{r+1}}&\text{in
$\mathcal{H}_{F}$},\\\ T_{r}+\tfrac{(q-1)X_{r+1}}{X_{r}-X_{r+1}}&\text{ in
$\mathcal{H}_{q,F}$}.\end{cases}$

The elements $\kappa_{r}$ have the following properties [5, Proposition 5.2]:

$\displaystyle\kappa_{r}f$ $\displaystyle=\sigma_{r}(f)\kappa_{r}\hskip
42.67912pt\text{$\forall f\in\Bbbk(X)$};$ (3.13) $\displaystyle\kappa_{r}^{2}$
$\displaystyle=\begin{cases}1-\tfrac{1}{(X_{r}-X_{r+1})^{2}}&\text{in
$\mathcal{H}_{F}$},\\\
q-\tfrac{(q-1)^{2}X_{r}X_{r+1}}{(X_{r}-X_{r+1})^{2}}&\text{ in
$\mathcal{H}_{q,F}$};\end{cases}$ (3.14) $\displaystyle\kappa_{r}\kappa_{s}$
$\displaystyle=\kappa_{s}\kappa_{r}\hskip 56.9055pt\text{if $s\neq r,r+1$};$
(3.15) $\displaystyle\kappa_{r}\kappa_{r+1}\kappa_{r}$
$\displaystyle=\kappa_{r+1}\kappa_{r}\kappa_{r+1}.$ (3.16)

If $w=\sigma_{r_{1}}\sigma_{r_{2}}\cdots\sigma_{r_{m}}$ is a reduced
expression in $S_{n}$, then we can define
$\kappa_{w}:=\kappa_{r_{1}}\kappa_{r_{2}}\cdots\kappa_{r_{m}}$ in
$\mathcal{H}_{F}$ (resp., $\mathcal{H}_{q,F}$). It is also a well-defined
element by the braid relations (3.15) and (3.16). By the decompositions
(3.12), we know that $\\{\kappa_{w}\ |\ w\in S_{n}\\}$ is a basis of
$\mathcal{H}_{F}$ (resp., $\mathcal{H}_{q,F}$) as free $\Bbbk(X)$-module.

##  4\. The Lusztig extensions of affine Hecke algebras

From now on, we fix $I=\mathbb{Z}/e\mathbb{Z}$, where $e=\mathrm{char}\Bbbk$
for the degenerate affine Hecke algebras, and $e$ is the smallest positive
integer such that $1+q+\cdots+q^{e-1}=0$ and setting $e=0$ if no such integer
exists for the non-degenerate affine Hecke algebras.

###  4.1. The Lusztig extensions

Let $\mathcal{E}$ be the unital $\Bbbk$-algebra with basis
$\\{\epsilon(\text{\boldmath$i$})|\text{\boldmath$i$}\in\mathcal{C}\\}$.
Multiplication is given by

$\epsilon(\text{\boldmath$i$})\epsilon(\text{\boldmath$j$})=\delta^{\text{\boldmath$i$}}_{\text{\boldmath$j$}}\epsilon(\text{\boldmath$i$}).$
(4.1)

The Lusztig extension of $\mathcal{H}$ with respect to $\mathcal{E}$ is the
$\Bbbk$-algebra $\mathcal{L}$ which is equal as $\Bbbk$-space to the tensor
product

$\mathcal{L}:=\mathcal{H}_{F}\otimes_{\Bbbk}\mathcal{E}=\oplus_{w\in
S_{n},\text{\boldmath$i$}\in\mathcal{C}}\kappa_{w}\Bbbk(X)\epsilon(\text{\boldmath$i$})$

of the rationalization algebra $\mathcal{H}_{F}$ and the semi-simple algebra
$\mathcal{E}$. Multiplication is defined so that $\mathcal{H}_{F}$ (identified
with the subspace $\mathcal{H}_{F}\otimes 1$) and $\mathcal{E}$ (identified
with the subspace $1\otimes\mathcal{E}$) are subalgebras of $\mathcal{L}$, and
in addition

$\displaystyle X_{r}\epsilon(\text{\boldmath$i$})$
$\displaystyle=\epsilon(\text{\boldmath$i$})X_{r},$ (4.2)
$\displaystyle\kappa_{r}\epsilon(\text{\boldmath$i$})$
$\displaystyle=\epsilon(\sigma_{r}(\text{\boldmath$i$}))\kappa_{r},$ (4.3)

for all $1\leq r<n$ and $\text{\boldmath$i$}\in\mathcal{C}$.

The Lusztig extension $\mathcal{L}_{q}$ of $\mathcal{H}_{q}$ with respect to
$\mathcal{E}$ is defined similarly. We have the following relations

$T_{r}\epsilon(\text{\boldmath$i$})=\begin{cases}\epsilon(\sigma_{r}(\text{\boldmath$i$}))T_{r}+\tfrac{1}{X_{r}-X_{r+1}}(\epsilon(\sigma_{r}(\text{\boldmath$i$}))-\epsilon(\text{\boldmath$i$}))&\text{in
$\mathcal{L}$},\\\
\epsilon(\sigma_{r}(\text{\boldmath$i$}))T_{r}+\tfrac{(q-1)X_{r+1}}{X_{r}-X_{r+1}}(\epsilon(\sigma_{r}(\text{\boldmath$i$}))-\epsilon(\text{\boldmath$i$}))&\text{in
$\mathcal{L}_{q}$}.\end{cases}$ (4.4)

###  4.2. The Brundan-Kleshchev auxiliary elements

Recall that [1, (3.12), (4.21)], for each $1\leq r\leq n$, Brundan and
Kleshchev introduced the element

$y_{r}:=\begin{cases}\sum_{\text{\boldmath$i$}\in\mathcal{C}}(X_{r}-i_{r})\epsilon(\text{\boldmath$i$})&\text{in
$\mathcal{L}$},\\\
\sum_{\text{\boldmath$i$}\in\mathcal{C}}(1-q^{-i_{r}}X_{r})\epsilon(\text{\boldmath$i$})&\text{in
$\mathcal{L}_{q}$}.\end{cases}$ (4.5)

Then $y_{r}\epsilon(\text{\boldmath$i$})=\epsilon(\text{\boldmath$i$})y_{r}$
by 4.2, and $y_{r}$ is a unit in $\mathcal{L}$ with

$y_{r}^{-1}=\begin{cases}\sum_{\text{\boldmath$i$}\in\mathcal{C}}(X_{r}-i_{r})^{-1}\epsilon(\text{\boldmath$i$})&\text{in
$\mathcal{L}$},\\\
\sum_{\text{\boldmath$i$}\in\mathcal{C}}(1-q^{-i_{r}}X_{r})^{-1}\epsilon(\text{\boldmath$i$})&\text{in
$\mathcal{L}_{q}$}.\end{cases}$

Let $\Bbbk[y]$ be the polynomial ring with $y:=y_{1},y_{2},\cdots,y_{n}$ and
$\Bbbk(y)$ be the rational function field.

######  Lemma 4.6.

$\Bbbk(X)\otimes_{\Bbbk}\mathcal{E}=\Bbbk(y)\otimes_{\Bbbk}\mathcal{E}$ in
$\mathcal{L}$ or $\mathcal{L}_{q}$ as subalgebras.

###### Proof.

For the degenerate case, if $g(y)$ is a polynomial in $\Bbbk[y]$, then

$g(y)=\sum_{\text{\boldmath$i$}\in\mathcal{C}}g(y)\epsilon(\text{\boldmath$i$})=\sum_{\text{\boldmath$i$}\in\mathcal{C}}g(X_{1}-i_{1},\cdots,X_{n}-i_{n})\epsilon(\text{\boldmath$i$})$

in $\mathcal{L}(\mathcal{H},F,\mathcal{C})$. Therefore, if $g(y)\neq 0$, then
$g(X_{1}-i_{1},\cdots,X_{n}-i_{n})\neq 0$ in $\Bbbk[X]$ for all
$\text{\boldmath$i$}\in\mathcal{C}$. Thus
$g(y)^{-1}=\sum_{\text{\boldmath$i$}\in\mathcal{C}}g(X_{1}-i_{1},\cdots,X_{n}-i_{n})^{-1}\epsilon(\text{\boldmath$i$})$
exists in $\mathcal{L}$. So $\Bbbk(y)\otimes_{\Bbbk}\mathcal{E}$ is a
subalgebra of $\mathcal{L}$.

Combining (4.5) with
$X_{r}=\sum_{\text{\boldmath$i$}\in\mathcal{C}}(y_{r}+i_{r})\epsilon(\text{\boldmath$i$})$,
we know that
$\Bbbk(X)\otimes_{\Bbbk}\mathcal{E}=\Bbbk(y)\otimes_{\Bbbk}\mathcal{E}$ in
$\mathcal{L}$.

Imitate the proof above, we can prove the non-degenerate case. ∎

For each $1\leq r<n$, Brundan and Kleshchev defined the element

$Q_{r}(\text{\boldmath$i$})=\begin{cases}1+y_{r+1}-y_{r}&\text{if
$i_{r}=i_{r+1}$},\\\ 1&\text{if $i_{r}-i_{r+1}=1,e\neq 2$},\\\
\tfrac{2+y_{r+1}-y_{r}}{(1+y_{r+1}-y_{r})^{2}}&\text{if
$i_{r}-i_{r+1}=-1,e\neq 2$},\\\ \tfrac{1}{1+y_{r+1}-y_{r}}&\text{if
$i_{r}-i_{r+1}=1,e=2$},\\\ 1-\tfrac{1}{y_{r}-y_{r+1}+i_{r}-i_{r+1}}&\text{if
$i_{r}-i_{r+1}\neq 0,\pm 1$}\end{cases}$

in $\mathcal{L}$, and

$Q_{r}(\text{\boldmath$i$})=\begin{cases}1-q+qy_{r+1}-y_{r}&\text{if
$i_{r}=i_{r+1}$},\\\ q^{i_{r}}&\text{if $i_{r}-i_{r+1}=1,e\neq 2$},\\\
\frac{1-q^{2}+q^{2}y_{r+1}-y_{r}}{q^{i_{r}}(1-q+qy_{r+1}-y_{r})^{2}}&\text{if
$i_{r}-i_{r+1}=-1,e\neq 2$},\\\ \frac{1}{1-q+qy_{r+1}-y_{r}}&\text{if
$i_{r}-i_{r+1}=1,e=2$},\\\
\frac{q^{i_{r}-i_{r+1}}(1-y_{r})-q(1-y_{r+1})}{q^{i_{r}-i_{r+1}}(1-y_{r})-(1-y_{r+1})}&\text{if
$i_{r}-i_{r+1}\neq 0,\pm 1$},\end{cases}$

in $\mathcal{L}_{q}$.

Notice that the symmetric group $S_{n}$ acts on the left on $\Bbbk[y]$ by
permuting variables:

$\sigma_{r}(y_{s})=\begin{cases}\sum_{\text{\boldmath$i$}\in\mathcal{C}}(\sigma_{r}(X_{s})-\text{\boldmath$i$}_{s})\epsilon(\sigma_{r}(\text{\boldmath$i$}))&\text{in
$\mathcal{L}$}\\\
\sum_{\text{\boldmath$i$}\in\mathcal{C}}(1-q^{-i_{r}}\sigma_{r}(X_{s}))\epsilon(\sigma_{r}(\text{\boldmath$i$}))&\text{in
$\mathcal{L}_{q}$}\end{cases}=y_{\sigma_{r}(s)},$

where $1\leq r<n$ and $1\leq s\leq n$.

The following Lemma is proved in [1, (3.27)-(3.29), (4.33)-(4.35)].

######  Lemma 4.7.

Let $1\leq r<n$ and $\text{\boldmath$i$}\in\mathcal{C}$. Then

$\sigma_{r}(Q_{r}(\sigma_{r}(\text{\boldmath$i$})))Q_{r}(\text{\boldmath$i$})=\begin{cases}1-(y_{r+1}-y_{r})^{2}&\text{if
$i_{r}-i_{r+1}=0$};\\\ \frac{2+y_{r}-y_{r+1}}{(1+y_{r}-y_{r+1})^{2}}&\text{if
$i_{r}-i_{r+1}=1,e\neq 2$};\\\
\frac{2+y_{r+1}-y_{r}}{(1+y_{r+1}-y_{r})^{2}}&\text{if $i_{r}-i_{r+1}=-1,e\neq
2$};\\\ \frac{1}{(1+y_{r+1}-y_{r})^{2}}&\text{if $i_{r}-i_{r+1}=1,e=2$};\\\
1-\frac{1}{(y_{r}-y_{r+1}+i_{r}-i_{r+1})^{2}}&\text{if $i_{r}-i_{r+1}\neq
0,\pm 1$}.\end{cases}$

in $\mathcal{L}$, and in $\mathcal{L}_{q}$,

$\displaystyle\sigma_{r}(Q_{r}(\sigma_{r}(\text{\boldmath$i$})))Q_{r}(\text{\boldmath$i$})$
$\displaystyle=\begin{cases}(1-q+qy_{r+1}-y_{r})(1-q+qy_{r}-y_{r+1})&\text{if
$i_{r}=i_{r+1}$};\\\
\tfrac{q(1-q^{2}+q^{2}y_{r}-y_{r+1})}{(1-q+qy_{r}-y_{r+1})^{2}}&\text{if
$i_{r}-i_{r+1}=1,e\neq 2$};\\\
\tfrac{q(1-q^{2}+q^{2}y_{r+1}-y_{r})}{(1-q+qy_{r+1}-y_{r})^{2}}&\text{if
$i_{r}-i_{r+1}=-1,e\neq 2$};\\\
\tfrac{1}{(1-q+qy_{r}-y_{r+1})(1-q+qy_{r+1}-y_{r})}&\text{if
$i_{r}-i_{r+1}=1,e=2$};\\\
q-\tfrac{(1-q)^{2}q^{i_{r}+i_{r+1}}(1-y_{r+1})(1-y_{r})}{[q^{i_{r+1}}(1-y_{r+1})-q^{i_{r}}(1-y_{r})]^{2}}&\text{if
$i_{r}\neq 0,\pm 1$}.\end{cases}$

Surprisingly, either in $\mathcal{L}$ or in $\mathcal{L}_{q}$, we can set the
same named elements

$\theta_{r}:=\kappa_{r}\sum_{\text{\boldmath$i$}\in\mathcal{C}}Q_{r}^{-1}(\text{\boldmath$i$})\epsilon(\text{\boldmath$i$}),$

which share the same relations.

######  Lemma 4.8.

For each $1\leq r<n$ and $\text{\boldmath$i$}\in\mathcal{C}$, we have

$\displaystyle\theta_{r}\epsilon(\text{\boldmath$i$})$
$\displaystyle=\epsilon(\sigma_{r}(\text{\boldmath$i$}))\theta_{r},$ (4.9)
$\displaystyle f\theta_{r}$ $\displaystyle=\theta_{r}\sigma_{r}(f)\hskip
31.29802pt\forall f\in\Bbbk(y),$ (4.10) $\displaystyle\theta_{r}\theta_{s}$
$\displaystyle=\theta_{s}\theta_{r}\hskip 54.06023pt\text{if $s\neq r,r+1$},$
(4.11) $\displaystyle\theta_{r}\theta_{r+1}\theta_{r}$
$\displaystyle=\theta_{r+1}\theta_{r}\theta_{r+1},$ (4.12)
$\theta_{r}^{2}\epsilon(\text{\boldmath$i$})=\begin{cases}-\frac{1}{(y_{r+1}-y_{r})^{2}}\epsilon(\text{\boldmath$i$})&\text{if
$i_{r}=i_{r+1}$},\\\ (y_{r}-y_{r+1})\epsilon(\text{\boldmath$i$})&\text{if
$i_{r}-i_{r+1}=1,e\neq 2$},\\\
(y_{r+1}-y_{r})\epsilon(\text{\boldmath$i$})&\text{if $i_{r}-i_{r+1}=-1,e\neq
2$},\\\ (y_{r}-y_{r+1})(y_{r+1}-y_{r})\epsilon(\text{\boldmath$i$})&\text{if
$i_{r}-i_{r+1}=1,e=2$},\\\ \epsilon(\text{\boldmath$i$})&\text{if
$i_{r}-i_{r+1}\neq 0,\pm 1$}.\end{cases}$ (4.13)

###### Proof.

The assertions follow straightforwardly from (3.13)-(3.16), (4.9), Lemma 4.7
and (3.14). ∎

###  4.3. The KLR basis of the Lusztig extensions

For each $1\leq r<n$, set

$\psi_{r}=\sum_{\text{\boldmath$i$}\in\mathcal{C}}[\theta_{r}-\tfrac{\delta_{i_{r}}^{i_{r+1}}}{y_{r}-y_{r+1}}]\epsilon(\text{\boldmath$i$}).$
(4.14)

######  Proposition 4.15.

The Lusztig extension $\mathcal{L}$ or $\mathcal{L}_{q}$ is generated by

$\\{y_{1},\cdots,y_{n},\psi_{1},\cdots,\psi_{n-1},f^{-1},\epsilon(\text{\boldmath$i$})\
|\ 0\neq f\in\Bbbk[y],\text{\boldmath$i$}\in\mathcal{C}\\}$

subject to the following relations.

$\displaystyle ff^{-1}$ $\displaystyle=f^{-1}f=1,\forall\quad 0\neq
f\in\Bbbk[y];$ (4.16) $\displaystyle y_{r}\epsilon(\text{\boldmath$i$})$
$\displaystyle=\epsilon(\text{\boldmath$i$})y_{r},\quad
y_{r}y_{s}=y_{s}y_{r};$ (4.17)
$\displaystyle\psi_{r}\epsilon(\text{\boldmath$i$})$
$\displaystyle=\epsilon(\sigma_{r}(\text{\boldmath$i$}))\psi_{r};$ (4.18)
$\displaystyle\psi_{r}y_{s}\epsilon(\text{\boldmath$i$})$
$\displaystyle=[\sigma_{r}(y_{s})\psi_{r}+\delta_{i_{r+1}}^{i_{r}}\partial_{r}(y_{s})]\epsilon(\text{\boldmath$i$});$
(4.19) $\displaystyle\psi_{r}^{2}\epsilon(\text{\boldmath$i$})$
$\displaystyle=\begin{cases}0&\text{if $i_{r}-i_{r+1}=0$},\\\
(y_{r}-y_{r+1})\epsilon(\text{\boldmath$i$})&\text{if $i_{r}-i_{r+1}=1,e\neq
2$},\\\ (y_{r+1}-y_{r})\epsilon(\text{\boldmath$i$})&\text{if
$i_{r}-i_{r+1}=-1,e\neq 2$},\\\
-(y_{r}-y_{r+1})^{2}\epsilon(\text{\boldmath$i$})&\text{if
$i_{r}-i_{r+1}=1,e=2$},\\\ \epsilon(\text{\boldmath$i$})&\text{if $i_{r}\neq
0,\pm 1$}.\end{cases}$ (4.20) $\displaystyle\psi_{r}\psi_{s}$
$\displaystyle=\psi_{s}\psi_{r},\hskip 73.97716pt\text{if $s\neq r,r+1$};$
(4.21)
$\displaystyle(\psi_{r}\psi_{r+1}\psi_{r}-\psi_{r+1}\psi_{r}\psi_{r+1})\epsilon(\text{\boldmath$i$})$
$\displaystyle=\begin{cases}-\epsilon(\text{\boldmath$i$})&\text{if
$i_{r+2}=i_{r},i_{r}-i_{r+1}=1,e\neq 2$},\\\
\epsilon(\text{\boldmath$i$})&\text{if $i_{r+2}=i_{r},i_{r}-i_{r+1}=-1,e\neq
2$},\\\ -(y_{r}+y_{r+2}-2y_{r+1})\epsilon(\text{\boldmath$i$})&\text{if
$i_{r+2}=i_{r},i_{r}-i_{r+1}=1,e=2$},\\\ 0&\text{else}.\end{cases}$ (4.22)

###### Proof.

The relations (4.16) and (4.17) can be proved by straightforward calculations
using Lemma 4.6.

As a result of (4.9) and (4.14), we see that

$\psi_{r}\epsilon(\text{\boldmath$i$})=(\theta_{r}-\tfrac{\delta_{i_{r+1}}^{i_{r}}}{y_{r}-y_{r+1}})\epsilon(\text{\boldmath$i$})=\epsilon(\sigma_{r}(\text{\boldmath$i$}))(\theta_{r}-\tfrac{\delta_{i_{r+1}}^{i_{r}}}{y_{r}-y_{r+1}})=\epsilon(\sigma_{r}(\text{\boldmath$i$}))\psi_{r},$

where the third identity holds since
$\sigma_{r}(\text{\boldmath$i$})=\text{\boldmath$i$}$ whenever
$i_{r}=i_{r+1}$.

By (4.14) and (4.10), we obtain

$\displaystyle\psi_{r}y_{s}\epsilon(\text{\boldmath$i$})$
$\displaystyle=(\theta_{r}-\tfrac{\delta_{i_{r+1}}^{i_{r}}}{y_{r}-y_{r+1}})y_{s}\epsilon(\text{\boldmath$i$})$
$\displaystyle=(\sigma_{r}(y_{s})\theta_{r}-\tfrac{\delta_{i_{r+1}}^{i_{r}}y_{s}}{y_{r}-y_{r+1}})\epsilon(\text{\boldmath$i$})$
$\displaystyle=(\sigma_{r}(y_{s})\psi_{r}+\tfrac{\delta_{i_{r+1}}^{i_{r}}\sigma_{r}(y_{s})}{y_{r}-y_{r+1}}-\tfrac{\delta_{i_{r+1}}^{i_{r}}y_{s}}{y_{r}-y_{r+1}})\epsilon(\text{\boldmath$i$})$
$\displaystyle=(\sigma_{r}(y_{s})\psi_{r}+\delta_{i_{r+1}}^{i_{r}}\partial_{r}(y_{s}))\epsilon(\text{\boldmath$i$}).$

If $s\neq r,r+1$, using (4.18), (4.14), (4.10) and (4.11), then we have

$\displaystyle\psi_{r}\psi_{s}\epsilon(\text{\boldmath$i$})$
$\displaystyle=\psi_{r}\epsilon(\sigma_{s}(\text{\boldmath$i$}))\psi_{s}\epsilon(\text{\boldmath$i$})$
$\displaystyle=(\theta_{r}-\tfrac{\delta_{i_{r+1}}^{i_{r}}}{y_{r}-y_{r+1}})(\theta_{s}-\tfrac{\delta_{i_{s+1}}^{i_{s}}}{y_{s}-y_{s+1}})e(\text{\boldmath$i$})$
$\displaystyle=(\theta_{r}\theta_{s}-\tfrac{\delta_{i_{r+1}}^{i_{r}}}{y_{r}-y_{r+1}}\theta_{s}-\theta_{r}\tfrac{\delta_{i_{s+1}}^{i_{s}}}{y_{s}-y_{s+1}}+\tfrac{\delta_{i_{r+1}}^{i_{r}}\delta_{i_{s+1}}^{i_{s}}}{(y_{r}-y_{r+1})(y_{s}-y_{s+1})})\epsilon(\text{\boldmath$i$})$
$\displaystyle=(\theta_{s}\theta_{r}-\theta_{s}\tfrac{\delta_{i_{r+1}}^{i_{r}}}{y_{r}-y_{r+1}}-\tfrac{\delta_{i_{s+1}}^{i_{s}}}{y_{s}-y_{s+1}}\theta_{r}+\tfrac{\delta_{i_{r+1}}^{i_{r}}\delta_{i_{s+1}}^{i_{s}}}{(y_{r}-y_{r+1})(y_{s}-y_{s+1})})\epsilon(\text{\boldmath$i$})$
$\displaystyle=(\theta_{s}-\tfrac{\delta_{i_{s+1}}^{i_{s}}}{y_{s}-y_{s+1}})(\theta_{r}-\tfrac{\delta_{i_{r+1}}^{i_{r}}}{y_{r}-y_{r+1}})\epsilon(\text{\boldmath$i$})$
$\displaystyle=\psi_{s}\psi_{r}\epsilon(\text{\boldmath$i$})$

So $\psi_{r}\psi_{s}=\psi_{s}\psi_{r}$.

As a result of (4.14), (4.18) and Lemma 4.13, we get

$\displaystyle\psi_{r}^{2}\epsilon(\text{\boldmath$i$})$
$\displaystyle=\psi_{r}\epsilon(\sigma_{r}(\text{\boldmath$i$}))\psi_{r}\epsilon(\text{\boldmath$i$})$
$\displaystyle=(\theta_{r}-\tfrac{\delta_{i_{r}}^{i_{r+1}}}{y_{r}-y_{r+1}})(\theta_{r}-\tfrac{\delta_{i_{r+1}}^{i_{r}}}{y_{r}-y_{r+1}})\epsilon(\text{\boldmath$i$})$
$\displaystyle=(\theta_{r}^{2}+\tfrac{\delta_{i_{r+1}}^{i_{r}}}{(y_{r}-y_{r+1})^{2}})\epsilon(\text{\boldmath$i$})$
$\displaystyle=\begin{cases}0&\text{if $i_{r}-i_{r+1}=0$},\\\
(y_{r}-y_{r+1})\epsilon(\text{\boldmath$i$})&\text{if $i_{r}-i_{r+1}=1,e\neq
2$},\\\ (y_{r+1}-y_{r})\epsilon(\text{\boldmath$i$})&\text{if
$i_{r}-i_{r+1}=-1,e\neq 2$},\\\
(y_{r}-y_{r+1})(y_{r+1}-y_{r})\epsilon(\text{\boldmath$i$})&\text{if
$i_{r}-i_{r+1}=1,e=2$},\\\ \epsilon(\text{\boldmath$i$})&\text{if $i_{r}\neq
0,\pm 1$}.\end{cases}$

In the light of (4.18), (4.14) and (4.10), there holds that

$\displaystyle\psi_{r}\psi_{r+1}\psi_{r}\epsilon(\text{\boldmath$i$})$
$\displaystyle=(\theta_{r}-\tfrac{\delta_{i_{r+2}}^{i_{r+1}}}{y_{r}-y_{r+1}})(\theta_{r+1}-\tfrac{\delta_{i_{r+2}}^{i_{r}}}{y_{r+1}-y_{r+2}})(\theta_{r}-\tfrac{\delta_{i_{r+1}}^{i_{r}}}{y_{r}-y_{r+1}})\epsilon(\text{\boldmath$i$})$
$\displaystyle=[\theta_{r}\theta_{r+1}\theta_{r}-\tfrac{\delta_{i_{r+1}}^{i_{r}}}{y_{r+1}-y_{r+2}}\theta_{r}\theta_{r+1}-\tfrac{\delta_{i_{r+2}}^{i_{r+1}}}{y_{r}-y_{r+1}}\theta_{r+1}\theta_{r}+\tfrac{\delta_{i_{r+1}}^{i_{r}}\delta_{i_{r+2}}^{i_{r+1}}}{(y_{r}-y_{r+2})(y_{r+1}-y_{r+2})}\theta_{r}$
$\displaystyle+\tfrac{\delta_{i_{r+1}}^{i_{r}}\delta_{i_{r+2}}^{i_{r+1}}}{(y_{r}-y_{r+1})(y_{r}-y_{r+2})}\theta_{r+1}-\tfrac{\delta_{i_{r+2}}^{i_{r}}}{y_{r}-y_{r+2}}\theta_{r}^{2}-\tfrac{\delta_{i_{r+1}}^{i_{r}}\delta_{i_{r+2}}^{i_{r}}\delta_{i_{r+2}}^{i_{r+1}}}{(y_{r}-y_{r+1})^{2}(y_{r+1}-y_{r+2})}]\epsilon(\text{\boldmath$i$})$

Similarly, we have

$\displaystyle\psi_{s}\psi_{r}\psi_{s}\epsilon(\text{\boldmath$i$})$
$\displaystyle=(\theta_{r+1}-\tfrac{\delta_{i_{r+1}}^{i_{r}}}{y_{r+1}-y_{r+2}})(\theta_{r}-\tfrac{\delta_{i_{r+2}}^{i_{r}}}{y_{r}-y_{r+1}})(\theta_{r+1}-\tfrac{\delta_{i_{r+2}}^{i_{r+1}}}{y_{r+1}-y_{r+2}})\epsilon(\text{\boldmath$i$})$
$\displaystyle=[\theta_{r+1}\theta_{r}\theta_{r+1}-\tfrac{\delta_{i_{r+1}}^{i_{r}}}{y_{r+1}-y_{r+2}}\theta_{r}\theta_{r+1}-\tfrac{\delta_{i_{r+2}}^{i_{r+1}}}{y_{r}-y_{r+1}}\theta_{r+1}\theta_{r}+\tfrac{\delta_{i_{r+1}}^{i_{r}}\delta_{i_{r+2}}^{i_{r+1}}}{(y_{r}-y_{r+2})(y_{r+1}-y_{r+2})}\theta_{r}$
$\displaystyle+\tfrac{\delta_{i_{r+1}}^{i_{r}}\delta_{i_{r+2}}^{i_{r+1}}}{(y_{r}-y_{r+1})(y_{r}-y_{r+2})}\theta_{r+1}-\tfrac{\delta_{i_{r+2}}^{i_{r}}}{y_{r}-y_{r+2}}\theta_{r+1}^{2}-\tfrac{\delta_{i_{r+1}}^{i_{r}}\delta_{i_{r+2}}^{i_{r}}\delta_{i_{r+2}}^{i_{r+1}}}{(y_{r}-y_{r+1})(y_{r+1}-y_{r+2})^{2}}]\epsilon(\text{\boldmath$i$})$

By (4.12) and 4.13, we arrive at

$\displaystyle[\psi_{r}\psi_{r+1}\psi_{r}-\psi_{r+1}\psi_{r}\psi_{r+1}]\epsilon(\text{\boldmath$i$})$
$\displaystyle=\delta_{i_{r+2}}^{i_{r}}[\tfrac{1}{y_{r}-y_{r+2}}(\theta_{r+1}^{2}-\theta_{r}^{2})+\tfrac{\delta_{i_{r+1}}^{i_{r}}(y_{r}+y_{r+2}-2y_{r+1})}{(y_{r}-y_{r+1})^{2}(y_{r+1}-y_{r+2})^{2}}]\epsilon(\text{\boldmath$i$})$
$\displaystyle=\begin{cases}-\epsilon(\text{\boldmath$i$})&\text{if
$i_{r+2}=i_{r},i_{r}-i_{r+1}=1,e\neq 2$},\\\
\epsilon(\text{\boldmath$i$})&\text{if $i_{r+2}=i_{r},i_{r}-i_{r+1}=-1,e\neq
2$},\\\ -(y_{r}+y_{r+2}-2y_{r+1})\epsilon(\text{\boldmath$i$})&\text{if
$i_{r+2}=i_{r},i_{r}-i_{r+1}=1,e=2$},\\\ 0&\text{else}.\end{cases}$

To finish the proof of the theorem, we need to prove the relations
(4.16)-(4.15) generate all relations. In fact, for each $w\in S_{n}$, we fix a
reduced decomposition $w=\sigma_{r_{1}}\sigma_{r_{2}}\cdots\sigma_{r_{m}}$ and
define the element

$\psi_{w}:=\psi_{r_{1}}\psi_{r_{2}}\cdots\psi_{r_{m}}\in\mathcal{L}.$

Note that $\psi_{w}$ in general does depend on the choice of reduced
decomposition of $w$ [2, Proposition 2.5]. Since

$\psi_{r}=\theta_{r}-\sum_{\text{\boldmath$i$}\in\mathcal{C}}\tfrac{\delta_{i_{r}}^{i_{r+1}}}{y_{r}-y_{r+1}}\epsilon(\text{\boldmath$i$})=\kappa_{r}\sum_{\text{\boldmath$i$}\in\mathcal{C}}Q_{r}^{-1}(\text{\boldmath$i$})\epsilon(\text{\boldmath$i$})-\sum_{\text{\boldmath$i$}\in\mathcal{C}}\tfrac{\delta_{i_{r}}^{i_{r+1}}}{y_{r}-y_{r+1}}\epsilon(\text{\boldmath$i$})$

and $\\{\kappa_{w}\ |\ w\in S_{n}\\}$ is a $\Bbbk(x)$-basis of
$\mathcal{H}_{F}$, we can show that $\\{\psi_{w}\ |\ w\in S_{n}\\}$ is a
$\Bbbk(y)\otimes_{\Bbbk}\mathcal{E}$-basis of $\mathcal{L}$ by Lemma 4.6 since
every element in $\mathcal{L}$ can be written as $\sum_{w\in
S_{n},\text{\boldmath$i$}\in\mathcal{C}}\psi_{w}f_{w,\text{\boldmath$i$}}(y)\epsilon(\text{\boldmath$i$})$
with $f_{w,\text{\boldmath$i$}}(y)$ in $\Bbbk(y)$ by the relations
(4.16)-(4.15). Therefore the generating set

$\\{y_{1},\cdots,y_{n},\psi_{1},\cdots,\psi_{n-1},f^{-1},\epsilon(\text{\boldmath$i$})\
|\ 0\neq f\in\Bbbk[y],\text{\boldmath$i$}\in\mathcal{C}\\}$

subject to relations (4.16)-(4.15) is complete.

The same proof above is valid for $\mathcal{L}_{q}$, and then we are done. ∎

According to the Proposition above, we will not distinguish between Lusztig
extensions in the sequel.

###  4.4. The cyclotomic KLR algebras

Recall [1, Subsection 2.2], the KLR algebra of type A is defined to be the
$\Bbbk$-algebra $\mathcal{R}$ generated by

$\\{y_{1},\cdots,y_{n},\psi_{1},\cdots,\psi_{n},\epsilon(\text{\boldmath$i$})\
|\ \text{\boldmath$i$}\in\mathcal{C},\\}$

subject to the relations (4.16)-(4.15) of Proposition 4.15. Thus $\mathcal{R}$
can be viewed as a subalgebra of $\mathcal{L}$.

From now on, we fix an index $\Lambda=(\Lambda_{i})_{i\in I}\in\mathbb{N}^{I}$
(we follow the convention that $\mathbb{N}=\\{0,1,2,\cdots\\}$) with
$\sum_{i\in I}\Lambda_{i}<\infty$. We call the quotient

$\mathcal{R}(\Lambda):=\mathcal{R}/\langle
y_{1}^{\Lambda_{i_{1}}}\epsilon(\text{\boldmath$i$})|\text{\boldmath$i$}\in\mathcal{C}\rangle$

a cyclotomic KLR algebra of type A.

Recall that, Brund and Kleshchev proved the following result in [1, Lemma
2.1].

######  Lemma 4.23.

The elements $y_{r}$ are nilpotent in $\mathcal{R}(\Lambda)$.

###  4.5. The semi-rationalizations

We define the semi-rationalization algebra as the $\Bbbk$-algebra
$\tilde{\mathcal{L}}$ generated by

$\\{y_{1},\cdots,y_{n},\psi_{1},\cdots,\psi_{n-1},f^{-1}(y),\epsilon(\text{\boldmath$i$})\
|\ \text{\boldmath$i$}\in\mathcal{C},f(y)\in\Bbbk[y]\ \mbox{with}\ f(0)\neq
0\\}$

subject to the relations (4.16)-(4.15) of Proposition 4.15.

The following result shows that we can construct a complete generating set
from the degenerate affine Hecke algebras or from the non-degenerate affine
Hecke algebras. This is the key point for proving the BK isomorphism in the
sequel.

######  Theorem 4.24.

$(1)$ Denote by
$1\epsilon(\text{\boldmath$i$})=\epsilon(\text{\boldmath$i$})$. Then the
algebra $\tilde{\mathcal{L}}$ is generated by

$\\{X_{1},\cdots,X_{n},T_{1},\cdots,T_{n-1},f^{-1}(X)\epsilon(\text{\boldmath$i$})\
|\ \text{\boldmath$i$}\in\mathcal{C},f(X)\in\Bbbk[X]\ \mbox{with}\
f(\text{\boldmath$i$})\neq 0\\}$

subject to relations (3.1)-(3.5), (4.1), (4.2), (4.4) and for $f\in\Bbbk[X]$
with $f(\text{\boldmath$i$})\neq 0$

$\epsilon(\text{\boldmath$j$})\cdot
f^{-1}\epsilon(\text{\boldmath$i$})=\delta_{\text{\boldmath$i$}}^{\text{\boldmath$j$}}\epsilon(\text{\boldmath$i$})=f^{-1}\epsilon(\text{\boldmath$i$})\cdot\epsilon(\text{\boldmath$j$}),\
f\cdot
f^{-1}\epsilon(\text{\boldmath$i$})=\epsilon(\text{\boldmath$i$})=f^{-1}\epsilon(\text{\boldmath$i$})\cdot
f.$ (4.25)

$(2)$ Denote by
$1\epsilon(\text{\boldmath$i$})=\epsilon(\text{\boldmath$i$})$. Then the
algebra $\tilde{\mathcal{L}}$ is generated by

$\\{X_{1},\cdots,X_{n},T_{1},\cdots,T_{n-1},f^{-1}(X)\epsilon(\text{\boldmath$i$})\
|\ \text{\boldmath$i$}\in\mathcal{C},f(X)\in\Bbbk[X]\ \mbox{with}\
f(q^{\text{\boldmath$i$}})\neq 0\\}$

subject to relations (3.6)-(3.10), (4.1), (4.2), (LABEL:Tepsi) and for
$f\in\Bbbk[X]$ with $f(q^{\text{\boldmath$i$}})\neq 0$

$\epsilon(\text{\boldmath$j$})\cdot
f^{-1}\epsilon(\text{\boldmath$i$})=\delta_{\text{\boldmath$i$}}^{\text{\boldmath$j$}}\epsilon(\text{\boldmath$i$})=(f^{-1}\epsilon(\text{\boldmath$i$})\cdot\epsilon(\text{\boldmath$j$}),\
f\cdot
f^{-1}\epsilon(\text{\boldmath$i$})=\epsilon(\text{\boldmath$i$})=f^{-1}\epsilon(\text{\boldmath$i$})\cdot
f.$ (4.26)

###### Proof.

(1) There is an obvious homomorphism
$\alpha\colon\tilde{\mathcal{L}}\to\mathcal{L}$ by sending generators to the
same named generators. This homomorphism is injective since using relations
(4.16)-(4.15), every element in $\tilde{\mathcal{L}}$ can be written as

$\sum_{w\in
S_{n},\text{\boldmath$i$}\in\mathcal{C}}\psi_{w}\tfrac{f_{w,\text{\boldmath$i$}}(y)}{g_{w,\text{\boldmath$i$}}(y)}\epsilon(\text{\boldmath$i$})$

with $f_{w,\text{\boldmath$i$}}(y),g_{w,\text{\boldmath$i$}}(y)$ in $\Bbbk[y]$
and $g_{w,\text{\boldmath$i$}}(0)\neq 0$, and $\\{\psi_{w}\ |\ w\in S_{n}\\}$
is a $\Bbbk(y)\otimes_{\Bbbk}\mathcal{E}$-basis of $\mathcal{L}$. Thus
$\mathrm{Im}\alpha=\oplus_{w\in S_{n}}\psi_{w}\mathcal{P}(y,\mathcal{E})$,
where $\mathcal{P}(y,\mathcal{E})$ is the commutative algebra $\\{fg^{-1}\ |\
f,g\in\Bbbk[y],g(0)\neq 0\\}\otimes_{\Bbbk}\mathcal{E}$.

Assume $\mathcal{A}$ is the $\Bbbk$-algebra generated by

$\\{X_{1},\cdots,X_{n},T_{1},\cdots,T_{n-1},f^{-1}(X)\epsilon(\text{\boldmath$i$})\
|\ \text{\boldmath$i$}\in\mathcal{C},f(X)\in\Bbbk[X]\ \mbox{with}\
f(\text{\boldmath$i$})\neq 0\\}$

subject to relations (3.1)-(3.5), (4.1), (4.2), (4.4) and (4.25). Then, using
these relations, every element in $\mathcal{A}$ can be written as

$\sum_{w\in
S_{n},\text{\boldmath$i$}\in\mathcal{C}}T_{w}f_{w,\text{\boldmath$i$}(X)}\cdot
g_{w,\text{\boldmath$i$}}^{-1}(X)\epsilon(\text{\boldmath$i$})$

with $f_{w,\text{\boldmath$i$}}(X),g_{w,\text{\boldmath$i$}}(X)$ in $\Bbbk[X]$
and $g_{w,\text{\boldmath$i$}}(\text{\boldmath$i$})\neq 0$. Thus there also
has an injective homomorphism $\alpha^{\prime}\colon\mathcal{A}\to\mathcal{L}$
by sending generators to the same named generators. Therefore
$\mathrm{Im}\alpha^{\prime}=\oplus_{w\in
S_{n}}T_{w}\mathcal{P}(X,\mathcal{E})$, where $\mathcal{P}(X,\mathcal{E})$ is
the commutative algebra $\\{f\cdot g^{-1}\epsilon(\text{\boldmath$i$})\ |\
\text{\boldmath$i$}\in\mathcal{C},f,g\in\Bbbk[X],g(\text{\boldmath$i$})\neq
0\\}$.

We claim that $\mathrm{Im}\alpha=\mathrm{Im}\alpha^{\prime}$. In fact, by
Lemma 4.6, $\mathcal{P}(X,\mathcal{E})=\mathcal{P}(y,\mathcal{E})$ in
$\mathcal{L}$. Notice that

$\psi_{r}=\sum\limits_{\begin{subarray}{1}\text{\boldmath$i$}\in\mathcal{C}\\\
i_{r}\neq
i_{r+1}\end{subarray}}(T_{r}+\tfrac{1}{y_{r}-y_{r+1}+i_{r}-i_{r+1}})Q_{r}^{-1}(\text{\boldmath$i$})\epsilon(\text{\boldmath$i$})+\sum\limits_{\begin{subarray}{1}\text{\boldmath$i$}\in\mathcal{C}\\\
i_{r}=i_{r+1}\end{subarray}}(T_{r}+1)Q^{-1}_{r}(\text{\boldmath$i$})\epsilon(\text{\boldmath$i$})$

and
$Q_{r}(\text{\boldmath$i$}),Q_{r}^{-1}(\text{\boldmath$i$})\in\mathcal{P}(y,\mathcal{E})$
in $\mathcal{L}$, we get $\mathrm{Im}\alpha=\mathrm{Im}\alpha^{\prime}$, and
then $\mathcal{A}=\tilde{\mathcal{L}}$.

(2) Imitate the proof of the statement $(1)$. ∎

######  Lemma 4.27.

$\tilde{\mathcal{L}}/\langle
y_{1}^{\Lambda_{i_{1}}}\epsilon(\text{\boldmath$i$})\ |\
\text{\boldmath$i$}\in\mathcal{C}\rangle=\begin{cases}\tilde{\mathcal{L}}/\langle\prod_{i\in
I}(X_{1}-i)^{\Lambda_{i}}\rangle&\text{for $\mathcal{H}$},\\\
\tilde{\mathcal{L}}/\langle\prod_{i\in
I}(X_{1}-q^{i})^{\Lambda_{i}}\rangle&\text{for $\mathcal{H}_{q}$}.\end{cases}$

###### Proof.

For the degenerate case, by the relation of $y_{1}$ and $X_{1}$, we have

$\displaystyle\prod_{i\in I}(X_{1}-i)^{\Lambda_{i}}$
$\displaystyle=\sum_{\text{\boldmath$j$}\in\mathcal{C}}\prod_{i\in
I}(y_{1}+j_{1}-i)^{\Lambda_{i}}\epsilon(\text{\boldmath$j$})$
$\displaystyle=\sum_{\text{\boldmath$j$}\in\mathcal{C}}\prod_{i\in I,i\neq
j_{1}}(y_{1}+j_{1}-i)^{\Lambda_{i}}y_{1}^{\Lambda_{j_{1}}}\epsilon(\text{\boldmath$j$})$

is in $\langle y_{1}^{\Lambda_{j_{1}}}\epsilon(\text{\boldmath$j$})\ |\
\text{\boldmath$j$}\in\mathcal{C}\rangle$. By Theorem (4.24) (1), $\prod_{i\in
I,i\neq j_{1}}[(X_{1}-i)^{\Lambda_{i}}]^{-1}\epsilon(\text{\boldmath$j$})$ is
in $\tilde{\mathcal{L}}$, thus

$\displaystyle y_{1}^{\Lambda_{j_{1}}}\epsilon(\text{\boldmath$j$})$
$\displaystyle=(X_{1}-j_{1})^{\Lambda_{j_{1}}}\epsilon(\text{\boldmath$j$})$
$\displaystyle=\prod_{i\in I}(X_{1}-i)^{\Lambda_{i}}\prod_{i\in I,i\neq
j_{1}}[(X_{1}-i)^{\Lambda_{i}}]^{-1}\epsilon(\text{\boldmath$j$})$

is in $\langle\prod_{i\in I}(X_{1}-i)^{\Lambda_{i}}\rangle$. Hence $\langle
y_{1}^{\Lambda_{i_{1}}}\epsilon(\text{\boldmath$i$})\ |\
\text{\boldmath$i$}\in\mathcal{C}\rangle=\langle\prod_{i\in
I}(X_{1}-i)^{\Lambda_{i}}\rangle$ in $\tilde{\mathcal{L}}.$

For the non-degenerate case, by the relation of $y_{1}$ and $X_{1}$, we get
that

$\displaystyle\prod_{i\in I}(X_{1}-q^{i})^{\Lambda_{i}}$
$\displaystyle=\sum_{\text{\boldmath$j$}\in\mathcal{C}}\prod_{i\in
I}(q^{j_{1}}-q^{i}-q^{j_{1}}y_{1})^{\Lambda_{i}}\epsilon(\text{\boldmath$j$})$
$\displaystyle=\sum_{\text{\boldmath$j$}\in\mathcal{C}}\prod_{i\in I,i\neq
j_{1}}(-q^{j_{1}})^{\Lambda_{j_{1}}}(q^{j_{1}}-q^{i}-q^{j_{1}}y_{1})^{\Lambda_{i}}y_{1}^{\Lambda_{j_{1}}}\epsilon(\text{\boldmath$j$})$

is in $\langle y_{1}^{\Lambda_{j_{1}}}\epsilon(\text{\boldmath$j$})\ |\
\text{\boldmath$j$}\in\mathcal{C}\rangle$. By Theorem (4.24) (1), $\prod_{i\in
I,i\neq j_{1}}[(X_{1}-q^{i})^{\Lambda_{i}}]^{-1}\epsilon(\text{\boldmath$j$})$
is in $\tilde{\mathcal{L}}$, thus

$\displaystyle y_{1}^{\Lambda_{j_{1}}}\epsilon(\text{\boldmath$j$})$
$\displaystyle=-q^{-j_{1}}(X_{1}-q^{j_{1}})^{\Lambda_{j_{1}}}\epsilon(\text{\boldmath$j$})$
$\displaystyle=-q^{-j_{1}}\prod_{i\in I}(X_{1}-q^{i})^{\Lambda_{i}}\prod_{i\in
I,i\neq j_{1}}[(X_{1}-q^{i})^{\Lambda_{i}}]^{-1}\epsilon(\text{\boldmath$j$})$

is in $\langle\prod_{i\in I}(X_{1}-q^{i})^{\Lambda_{i}}\rangle$. So $\langle
y_{1}^{\Lambda_{i_{1}}}\epsilon(\text{\boldmath$i$})\ |\
\text{\boldmath$i$}\in\mathcal{C}\rangle=\langle\prod_{i\in
I}(X_{1}-q^{i})^{\Lambda_{i}}\rangle$ in $\tilde{\mathcal{L}}$, and then we
are done. ∎

######  Remark 4.28.

Denote by

$\tilde{\mathcal{L}}(\Lambda):=\tilde{\mathcal{L}}/\langle
y_{1}^{\Lambda_{i_{1}}}\epsilon(\text{\boldmath$i$})|\text{\boldmath$i$}\in\mathcal{C}\rangle.$

Similar to the proof of Lemma 4.23, the elements $y_{r}$ are nilpotent in
$\tilde{\mathcal{L}}(\Lambda)$. Therefore the elements $\prod_{i\in
I}(X_{r}-i)$ (resp., $\prod_{i\in I}(X_{r}-q^{i})$) are also nilpotent in
$\tilde{\mathcal{L}}(\Lambda)$.

The following result is important for us.

######  Lemma 4.29.

We have $\Bbbk$-algebra isomorphsim
$\mathcal{R}(\Lambda)\cong\tilde{\mathcal{L}}(\Lambda)$.

###### Proof.

First note that if $f(y)\in\Bbbk[y]$ with $f(0)\neq 0$, the polynomial
$f(y)-f(0)$ is nilpotent in $\tilde{\mathcal{L}}(\Lambda)$ by the nilpotency
of $y_{r}$’s. So there exists some $g(y)\in\Bbbk[y]$ and $m\in\mathbb{N}$ such
that $g(y)^{m}=0$ and $f^{-1}(y)=f(0)^{-1}\sum_{l=0}^{m}g(y)^{l}$ in
$\mathcal{R}(\Lambda)$. Thus the homomorphism
$\mathcal{R}\hookrightarrow\tilde{\mathcal{L}}\twoheadrightarrow\tilde{\mathcal{L}}(\Lambda)$
is surjective and induces a surjective homomorphism
$\pi\colon\mathcal{R}(\Lambda)\to\tilde{\mathcal{L}}(\Lambda)$. Let
$\tilde{F}$ be the localization of the commutative ring $\Bbbk[y]^{S_{n}}$ of
$S_{n}$-invariants in $\Bbbk[y]$ with respect to $\\{f\in\Bbbk[y]^{S_{n}}\ |\
f(0)\neq 0\\}$. Similar to the proof of (2.1), we have
$\tilde{\mathcal{L}}=\mathcal{R}\otimes_{\Bbbk[y]^{S_{n}}}\tilde{F}$ by
Theorem 4.24. Since the elements $y_{r}$ are nilpotent in
$\mathcal{R}(\Lambda)$, similar to the proof above, we know that if $f(y)\in
K[y]^{S_{n}}$ with $f(0)\neq 0$, then it is a unit in $\mathcal{R}(\Lambda)$.
Thus the homomorphism
$K[y]^{S_{n}}\hookrightarrow\mathcal{R}\stackrel{{\scriptstyle\pi_{1}}}{{\twoheadrightarrow}}\tilde{\mathcal{R}}(\Lambda)$
induces a morphism $\pi_{2}\colon\tilde{F}\to\mathcal{R}(\Lambda)$. Therefore
we have an induced algebra homomorphism
$\pi_{1}\otimes\pi_{2}\colon\tilde{\mathcal{L}}\to\mathcal{R}(\Lambda)$. The
homomorphism $\pi_{1}\otimes\pi_{2}$ induces an algebra homomorphism
$\pi^{\prime}\colon\tilde{\mathcal{L}}(\Lambda)\to\mathcal{R}(\Lambda)$. It is
easy to check that $\pi$ and $\pi^{\prime}$ are two-sided inverses. So
$\mathcal{R}(\Lambda)\cong\tilde{\mathcal{L}}(\Lambda)$. ∎

##  5\. The Brundan-Kleshchev isomorphism

###  5.1. Cyclotomic degenerate affine Hecke algebras

Recall that the cyclotomic degenerate affine Hecke algebra is defined as

$\mathcal{H}(\Lambda):=\mathcal{H}/\langle\prod_{i\in
I}(X_{1}-i)^{\Lambda_{i}}\rangle.$

By [1, Subsection 3.1], there is a system $\\{e(\text{\boldmath$i$})\ |\
\text{\boldmath$i$}\in\mathcal{C}\\}$ of mutually orthogonal idempotents in
$\mathcal{H}(\Lambda)$ such that $1=\sum_{\text{\boldmath$i$}\in
I^{n}}e(\text{\boldmath$i$})$ and

$e(\text{\boldmath$i$})\mathcal{H}(\Lambda)=\\{h\in\mathcal{H}(\Lambda)\ |\
(X_{r}-i_{r})^{m}h=0\ \mbox{for all}\ 1\leq r\leq n\ \mbox{and}\ m\gg 0\\}.$

It is easy to check that
$X_{r}e(\text{\boldmath$i$})=e(\text{\boldmath$i$})X_{r}$ for all $1\leq r\leq
n$ and $\text{\boldmath$i$}\in\mathcal{C}$, and the element
$f(X)e(\text{\boldmath$i$})$ with $f(X)\in\Bbbk[X]$ is a unit in
$e(\text{\boldmath$i$})\mathcal{H}(\Lambda)$ if and only if
$f(\text{\boldmath$i$})\neq 0$. In this case, we write
$f(X)^{-1}e(\text{\boldmath$i$})$ for the inverse.

######  Lemma 5.1.

For $1\leq r<n$ and $\text{\boldmath$i$}\in I^{n}$, we have that

$T_{r}e(\text{\boldmath$i$})=\begin{cases}e(\text{\boldmath$i$})T_{r}&\text{if
$i_{r+1}=i_{r}$},\\\
e(\sigma_{r}(\text{\boldmath$i$}))T_{r}+(X_{r}-X_{r+1})^{-1}(e(\sigma_{r}(\text{\boldmath$i$}))-e(\text{\boldmath$i$}))&\text{if
$i_{r+1}\neq i_{r}$}.\end{cases}$ (5.2)

###### Proof.

For any $1\leq s\leq n$, the element

$(\sigma_{r}(X_{s})-\sigma(\text{\boldmath$i$})_{s})e(\text{\boldmath$i$})=\begin{cases}(X_{s}-i_{s})e(\text{\boldmath$i$})&\text{if
$s\neq r,r+1$,}\\\ (X_{r+1}-i_{r+1})e(\text{\boldmath$i$})&\text{if $s=r,$}\\\
(X_{r}-i_{r})e(\text{\boldmath$i$})&\text{if $s=r+1.$}\end{cases}$

is nilpotent in $\mathcal{H}(\Lambda)e(\text{\boldmath$i$})$. Similarly, we
can show that $\partial_{r}((X_{s}-i_{s})^{m})e(\text{\boldmath$i$})=0$ by the
binomial theorem for an integer $m\gg 0$ whenever $i_{r}=i_{r+1}$. Therefore,
if $i_{r}=i_{r+1}$, we get from (3.11) that

$\displaystyle(X_{s}-i_{s})^{m}T_{r}e(\text{\boldmath$i$})=T_{r}(\sigma_{r}(X_{s})-i_{s})^{m}e(\text{\boldmath$i$})+{\partial}_{r}((X_{s}-i_{s})^{m})e(\text{\boldmath$i$})=0$

whenever $m\gg 0$. Thus $T_{r}e(\text{\boldmath$i$})\in
e(\text{\boldmath$i$})\mathcal{H}(\Lambda)$ and then
$T_{r}e(\text{\boldmath$i$})=e(\text{\boldmath$i$})T_{r}e(\text{\boldmath$i$})=e(\text{\boldmath$i$})T_{r}.$

If $i_{r+1}\neq i_{r}$, as a result of (3.11), it holds that

$\displaystyle(X_{s}-\sigma_{r}(\text{\boldmath$i$})_{s})^{m}[T_{r}(X_{r}-X_{r+1})+1]e(\text{\boldmath$i$})$
$\displaystyle=[T_{r}(X_{r}-X_{r+1})+1](\sigma_{r}(X_{s})-\sigma_{r}(\text{\boldmath$i$})_{s})^{m}e(\text{\boldmath$i$})$
$\displaystyle=0$

whenever $m\gg 0$. Therefore

$\displaystyle
T_{r}(X_{r}-X_{r+1})e(\text{\boldmath$i$})+e(\text{\boldmath$i$})=e(\sigma_{r}(\text{\boldmath$i$}))[T_{r}(X_{r}-X_{r+1})+1]$

Then right-multiplying by $(X_{r}-X_{r+1})^{-1}e(\text{\boldmath$i$})$, we
obtain

$T_{r}e(\text{\boldmath$i$})=e(\sigma_{r}(\text{\boldmath$i$}))T_{r}e(\text{\boldmath$i$})-(X_{r}-X_{r+1})^{-1}e(\text{\boldmath$i$}).$

Similarly, we can prove that

$e(\sigma_{r}(\text{\boldmath$i$}))T_{r}=e(\sigma_{r}(\text{\boldmath$i$}))T_{r}e(\text{\boldmath$i$})-(X_{r}-X_{r+1})^{-1}e(\sigma_{r}(\text{\boldmath$i$})).$

Therefore

$T_{r}e(\text{\boldmath$i$})=e(\sigma_{r}(\text{\boldmath$i$}))T_{r}+(X_{r}-X_{r+1})^{-1}(e(\sigma_{r}(\text{\boldmath$i$}))-e(\text{\boldmath$i$})).$

∎

###  5.2. Cyclotomic non-degenerate affine Hecke algebras

Recall that the non-degenerate cyclotomic affine Hecke algebra
$\mathcal{H}_{q}(\Lambda)$ is

$\mathcal{H}_{q}(\Lambda):=\mathcal{H}_{q}/\langle\prod_{i\in
I}(X_{1}-q^{i})^{\Lambda_{i}}\rangle.$

Similar to [1, Subsection 4.1], there is a system $\\{e(\text{\boldmath$i$})\
|\ \text{\boldmath$i$}\in\mathcal{C}\\}$ of mutually orthogonal idempotents in
$\mathcal{H}_{q}(\Lambda)$ such that $1=\sum_{\text{\boldmath$i$}\in
I^{n}}e(\text{\boldmath$i$})$ and

$\displaystyle
e(\text{\boldmath$i$})\mathcal{H}_{q}(\Lambda)=\\{h\in\mathcal{H}_{q}(\Lambda)\
|\ (X_{r}-q^{i_{r}})^{m}h=0\ \mbox{for all}\ 1\leq r\leq n\ \mbox{and}\ m\gg
0\\}$

. Accordingly, it holds that
$X_{r}e(\text{\boldmath$i$})=e(\text{\boldmath$i$})X_{r}$ for all $1\leq r\leq
n$ and $\text{\boldmath$i$}\in\mathcal{C}$, and the element
$f(X)e(\text{\boldmath$i$})$ with $f(X)\in\Bbbk[X]$ is a unit in
$e(\text{\boldmath$i$})\mathcal{H}_{q}(\Lambda)$ if and only if
$f(q^{\text{\boldmath$i$}})\neq 0$. We write $f(X)^{-1}e(\text{\boldmath$i$})$
for the inverse in this case.

Similar to the proof of Lemma 5.1, we can show that

######  Lemma 5.3.

For $1\leq r<n$ and $\text{\boldmath$i$}\in I^{n}$, we have

$T_{r}e(\text{\boldmath$i$})=\begin{cases}e(\text{\boldmath$i$})T_{r}&\text{if
$i_{r}=i_{r+1}$},\\\
e(\sigma_{r}(\text{\boldmath$i$}))T_{r}+\tfrac{(1-q)X_{r+1}}{X_{r+1}-X_{r}}(e(\sigma_{r}(\text{\boldmath$i$}))-e(\text{\boldmath$i$}))&\text{if
$i_{r}\neq i_{r+1}$}.\end{cases}$ (5.4)

Let

$e(\mathcal{C}):=\sum_{\text{\boldmath$i$}\in\mathcal{C}}e(\text{\boldmath$i$}).$

Then $e(\mathcal{C})$ is a primitive central idempotent by (5.2) and (5.4).

###  5.3. The BK isomorphism

By Lemma 4.27, Theorem (4.24) and 5.1, (5.4)there is a homomorphism

$\rho\colon\tilde{\mathcal{L}}(\Lambda)\to\mathcal{H}(\Lambda)e(\mathcal{C})(\mbox{resp}.\
\mathcal{H}_{q}(\Lambda)e(\mathcal{C}))$

sending the generators $X_{r},T_{r}$ to the same named elements, and
$f^{-1}(X)\epsilon(\text{\boldmath$i$})$ with $f(\text{\boldmath$i$})\neq 0$
(resp. $f^{-1}(X)\epsilon(\text{\boldmath$i$})$ with
$f(q^{\text{\boldmath$i$}})\neq 0$) to $f^{-1}(X)e(\text{\boldmath$i$})$. We
have the following important result.

######  Lemma 5.5.

$\rho$ is an algebra isomorphism.

###### Proof.

We only prove the claim for degenerate case since the proof of non-degenerate
case is similar. Apparently, $\rho$ is surjective, so we only need to
construct a left-inverse of $\rho$. By Lemma 4.27, there is a homomorphism

$\tau\colon\mathcal{H}(\Lambda)\to\tilde{\mathcal{L}}(\Lambda)$

sending the generators $X_{r},T_{r}$ to the same named elements. Let
$\text{\boldmath$i$}\in\mathcal{C}$ and $\text{\boldmath$j$}\in I^{n}$. If
$\text{\boldmath$i$}\neq\text{\boldmath$j$}$, then there is some $1\leq r\leq
n$ such that $j_{r}\neq i_{r}$. We claim that
$\epsilon(\text{\boldmath$i$})\tau(e(\text{\boldmath$j$}))=0$. In fact, by the
construction of $e(\text{\boldmath$j$})$, there is an integer $m\gg 0$ such
that $(X_{r}-j_{r})^{m}e(\text{\boldmath$j$})=0$. Thus we have

$(X_{r}-j_{r})^{m}\epsilon(\text{\boldmath$i$})\tau(e(\text{\boldmath$j$}))=\epsilon(\text{\boldmath$i$})\tau((X_{r}-j_{r})^{m}e(\text{\boldmath$j$}))=0.$

The assumption $j_{r}\neq i_{r}$ implies that the element
$(X_{r}-j_{r})^{-1}\epsilon(\text{\boldmath$i$})\in\tilde{\mathcal{L}}(\Lambda)$.
Thus we get

$\epsilon(\text{\boldmath$i$})\tau(e(\text{\boldmath$j$}))=(X_{r}-j_{r})^{-m}(X_{r}-j_{r})^{m}\epsilon(\text{\boldmath$i$})\tau(e(\text{\boldmath$j$}))=(X_{r}-j_{r})^{-m}\epsilon(\text{\boldmath$i$})0=0.$

Therefore, if $\text{\boldmath$j$}\in I^{n}\setminus\mathcal{C}$ we have
$\tau(e(\text{\boldmath$j$}))=\sum_{\text{\boldmath$i$}\in\mathcal{C}}\epsilon(\text{\boldmath$i$})\tau(e(\text{\boldmath$j$}))=0.$
Therefore, if $\text{\boldmath$j$}\in\mathcal{C}$, we obtain

$\tau(e(\text{\boldmath$j$}))=\sum_{\text{\boldmath$i$}\in\mathcal{C}}\epsilon(\text{\boldmath$i$})\tau(e(\text{\boldmath$j$}))=\epsilon(\text{\boldmath$j$})\tau(e(\text{\boldmath$j$}))=\epsilon(\text{\boldmath$j$})\sum_{\text{\boldmath$i$}\in
I^{n}}\tau(e(\text{\boldmath$i$}))=\epsilon(\text{\boldmath$j$})\tau(1)=\epsilon(\text{\boldmath$j$}).$

These show that
$\tau|_{\mathcal{H}(\Lambda)e(\mathcal{C})}\colon\mathcal{H}(\Lambda)e(\mathcal{C})\to\tilde{\mathcal{L}}(\Lambda)$
is an algebra homomorphism. It is easy to check that $\tau\rho$ is the
identity on each generator of $\tilde{\mathcal{L}}(\Lambda)$. Thus $\rho$ is
an isomorphism. ∎

By Lemma 4.29 and Lemma 5.5, we have arrive at our main result.

######  Theorem 5.6.

The cyclotomic KLR algebra of type A and blocks of the cyclotomic $($
degenerate $)$ affine Hecke algebra of a symmetric group can be realized as
the same quotient of an algebra.

## References

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