Foldings of KLR algebras

Ying Ma, Toshiaki Shoji and Zhiping Zhou

###### Abstract.

Let $\mathbf{U}_{q}^{-}$ be the negative half of the quantum group associated
to the Kac-Moody algebra $\mathfrak{g}$, and $\underline{\mathbf{U}}_{q}^{-}$
the quantum group obtained by a folding of $\mathfrak{g}$. Let
$\mathbf{A}=\mathbf{Z}[q,q^{-1}]$. McNamara showed that
$\underline{\mathbf{U}}_{q}^{-}$ is categorified over a certain extension ring
$\widetilde{\mathbf{A}}$ of $\mathbf{A}$, by using the folding theory of KLR
algebras. He posed a question whether $\widetilde{\mathbf{A}}$ coincides with
$\mathbf{A}$ or not. In this paper, we give an affirmative answer for this
problem.

Introduction

Let $X=(I,(\ ,\ ))$ be a Cartan datum, and $\mathbf{U}_{q}^{-}$ the negative
half of the quantum group associated to $X$. Let $Q=\bigoplus_{i\in
I}\mathbf{Z}\alpha_{i}$ be the root lattice of $X$, and set $Q_{+}=\sum_{i\in
I}\mathbf{N}\alpha_{i}$. Set $\mathbf{A}=\mathbf{Z}[q,q^{-1}]$, and let
${}_{\mathbf{A}}\mathbf{U}_{q}^{-}$ be Lusztig’s integral form of
$\mathbf{U}_{q}^{-}$, which is an $\mathbf{A}$-subalgebra of
$\mathbf{U}_{q}^{-}$. In the case where $X$ is of symmetric type, Lusztig
constructed in [L1] the canonical basis of $\mathbf{U}_{q}^{-}$, by using the
geometry of a quiver $\overrightarrow{Q}$ associated to $X$. More precisely,
he constructed, for each $\beta\in Q_{+}$, a category
$\mathcal{Q}_{V_{\beta}}$ consisting of certain semisimple
$\bar{\mathbf{Q}}_{l}$-complexes on the representation space $V_{\beta}$ of
$\overrightarrow{Q}$ associated to $\beta$, and showed that the direct sum
$K(\mathcal{Q})=\bigoplus_{\beta\in Q_{+}}K(\mathcal{Q}_{V_{\beta}})$ of the
Grothendieck group $K(\mathcal{Q}_{V_{\beta}})$ of $\mathcal{Q}_{V_{\beta}}$
has a structure of $A$-algebra, which is isomorphic to
${}_{\mathbf{A}}\mathbf{U}_{q}^{-}$. The canonical basis of
$\mathbf{U}_{q}^{-}$ is obtained from a natural basis of
$K(\mathcal{Q}_{V_{\beta}})$ coming from simple perverse sheaves in
$\mathcal{Q}_{V_{\beta}}$.

Let $\sigma$ be an admissible diagram automorphism on $X$ (see 1.5), which
gives a bijection $\sigma:I\to I$. Let $J$ be the set of $\sigma$-orbits in
$I$. Then $\underline{X}=(J,(\ ,\ ))$ gives a Cartan datum. We denote by
$\underline{\mathbf{U}}_{q}^{-}$ the corresponding quantum group. In [L2],
Lusztig constructed the canonical basis of $\underline{\mathbf{U}}_{q}^{-}$,
by extending the discussion in the symmetric case. By a suitable choice of the
orientation, $\sigma$ acts on the quiver $\overrightarrow{Q}$, and it induces
a periodic functor $\sigma^{*}$ on $\mathcal{Q}_{V_{\beta}}$ whenever $\beta$
is $\sigma$-stable. He defined a new category
$\widetilde{\mathcal{Q}}_{V_{\beta}}$ attached to $\mathcal{Q}_{V_{\beta}}$
and $\sigma^{*}$, and a (modified) Grotehndieck group
$K(\widetilde{\mathcal{Q}}_{V_{\beta}})$. Let $Q_{+}^{\sigma}$ be the set of
$\sigma$-stable elements in $Q_{+}$. Then the direct sum
$K(\widetilde{\mathcal{Q}})=\bigoplus_{\beta\in
Q_{+}^{\sigma}}K(\widetilde{\mathcal{Q}}_{V_{\beta}})$ has a structure of
$\mathbf{Z}[\zeta_{\mathbf{n}},q,q^{-1}]$-algebra, where $\mathbf{n}$ is the
order of $\sigma$, and $\zeta_{\mathbf{n}}$ is a primitive $\mathbf{n}$-th
root of unity in $\bar{\mathbf{Q}}_{l}$. He proved that a certain
$\mathbf{A}$-subalgebra ${}_{\mathbf{A}}K(\widetilde{\mathcal{Q}})$ is
isomorphic to ${}_{\mathbf{A}}\underline{\mathbf{U}}_{q}^{-}$. In that case, a
natural basis of ${}_{\mathbf{A}}K(\widetilde{\mathcal{Q}})$ gives the
canonical basis of $\underline{\mathbf{U}}_{q}^{-}$, up to sign, i.e., gives
the canonical signed basis. The canonical basis of
$\underline{\mathbf{U}}_{q}^{-}$ is obtained from it by suitably fixing the
signs, which was done by an algebraic argument using Kashiwara’s tensor
product rule.

$K(\mathcal{Q})$ (for symmetric $X$) gives a geometric categorification of
$\mathbf{U}_{q}^{-}$. On the other hand, in [KL], [R1], Khovanov-Lauda, and
Rouquier introduced the Khovanov-Lauda-Rouquier algebra $R=\bigoplus_{\beta\in
Q_{+}}R(\beta)$ (KLR algebra in short) over an algebraically closed field
$\mathbf{k}$, and gave a categorification of $\mathbf{U}_{q}^{-}$ (for general
$X$) in terms of KLR algebras. For each $\beta\in Q_{+}$, let
$R(\beta)\operatorname{-{\bf gp}}$ be the category of finitely generated
graded projective $R(\beta)$-modules, and $K_{\operatorname{{\bf gp}}}(\beta)$
the Grothendieck group of $R(\beta)\operatorname{-{\bf gp}}$. They proved that
$K_{\operatorname{{\bf gp}}}=\bigoplus_{\beta\in Q_{+}}K_{\operatorname{{\bf
gp}}}(\beta)$ has a structure of $\mathbf{A}$-algebra, and is isomorphic to
${}_{\mathbf{A}}\mathbf{U}_{q}^{-}$.

It is natural to expect a similar discussion as in Lusztig’s case will hold
also for KLR algebras. McNamara [M] developed the theory of foldings for KLR
algebras. Let $X$ and $\underline{X}$ be as above (for $X$: general type), and
choose $\beta\in Q_{+}^{\sigma}$. By a suitable choice of defining parameters
of $R$, $\sigma$ acts on $R(\beta)$, thus we obtain a periodic functor
$\sigma^{*}$ on $R(\beta)\operatorname{-{\bf gp}}$. One can define a new
category $\widetilde{R}(\beta)\operatorname{-{\bf gp}}=\mathscr{P}_{\beta}$,
and a modified Grothendieck group $K(\mathscr{P}_{\beta})$.
$K(\mathscr{P})=\bigoplus_{\beta\in Q_{+}^{\sigma}}K(\mathscr{P}_{\beta})$ has
a structure of $\mathbf{Z}[\zeta_{\mathbf{n}},q,q^{-1}]$-algebra. McNamara
constructed a natural basis $\underline{\mathbf{B}}$ of $K(\mathscr{P})$ by
using the theory of crystals. Define an extension $\widetilde{\mathbf{A}}$ of
$\mathbf{A}$ as the smallest subring $\mathbf{Z}[\zeta_{\mathbf{n}},q,q^{-1}]$
containing the structure constants of $\underline{\mathbf{B}}$ with respect to
the product in $K(\mathscr{P})$. Hence the $\widetilde{\mathbf{A}}$-span
${}_{\widetilde{\mathbf{A}}}K(\mathscr{P})$ of $\underline{\mathbf{B}}$ gives
an $\widetilde{\mathbf{A}}$-algebra. He proved that
${}_{\widetilde{\mathbf{A}}}K(\mathscr{P})$ is isomorphic to
${}_{\widetilde{\mathbf{A}}}\underline{\mathbf{U}}_{q}^{-}$, where
${}_{\widetilde{\mathbf{A}}}\underline{\mathbf{U}}_{q}^{-}$ is the extension
of ${}_{\mathbf{A}}\underline{\mathbf{U}}_{q}^{-}$ to
$\widetilde{\mathbf{A}}$, thus obtained a categorification of
$\underline{\mathbf{U}}_{q}^{-}$. He showed that
$\mathbf{A}\subset\widetilde{\mathbf{A}}\subset\mathbf{Z}[\zeta_{\mathbf{n}}+\zeta_{\mathbf{n}}^{-1},q,q^{-1}]$,
and posed a question whether $\widetilde{\mathbf{A}}$ coincides with
$\mathbf{A}$ or not ([M, Question 11.2]).

The aim of this paper is to show that this certainly holds (Theorem 3.16). The
main ingredient for the proof is the following result proved in [MSZ]. We
consider the case where the order $\mathbf{n}$ of $\sigma$ is a power of a
prime number $\ell$. Let $\mathbf{F}=\mathbf{Z}/\ell\mathbf{Z}$ be the finite
field of $\ell$-elements. Let
$\mathbf{A}^{\prime}=\mathbf{F}[q,q^{-1}]=\mathbf{A}/\ell\mathbf{A}$, and
consider the $\mathbf{A}^{\prime}$-algebras
${}_{\mathbf{A}^{\prime}}\mathbf{U}_{q}^{-}$ and
${}_{\mathbf{A}^{\prime}}\underline{\mathbf{U}}_{q}^{-}$, by changing the base
ring from $\mathbf{A}$ to $\mathbf{A}^{\prime}$. $\sigma$ acts naturally on
${}_{\mathbf{A}^{\prime}}\mathbf{U}_{q}^{-}$, and let
${}_{\mathbf{A}^{\prime}}\mathbf{U}_{q}^{-,\sigma}$ be the fixed point
subalgebra by $\sigma$. In [MSZ, Thm. 3.4, Thm. 4.27], it is proved that
${}_{\mathbf{A}^{\prime}}\underline{\mathbf{U}}_{q}^{-}$ is realized as the
quotient algebra of ${}_{\mathbf{A}^{\prime}}\mathbf{U}_{q}^{-,\sigma}$, and
that the canonical basis of $\underline{\mathbf{U}}_{q}^{-}$ is obtained from
the $\sigma$-stable canonical basis of $\mathbf{U}_{q}^{-}$, up to sign,
through the quotient map. This gives an alternate approach for Lusztig’s
geometric construction of canonical signed basis of
$\underline{\mathbf{U}}_{q}^{-}$. A similar argument also works for the KLR
algebras. In this scheme, the finite field $\mathbf{F}$ enters in the theory,
but $\zeta_{\mathbf{n}}$ is not involved. By comparing this with McNamara’s
isomorphism, we obtain the required result.

In the case where $X$ is symmetric, and $R$ is a symmetric type KLR algebra
(see 5.1) over a field $\mathbf{k}$ of characteristic zero, Varagnolo-Vasserot
[VV] and Rouquier [R2] proved that the natural basis of $K_{\operatorname{{\bf
gp}}}$ gives the canonical basis (which coincides with Kashiwara’s lower
global basis by [GL]) of $\mathbf{U}_{q}^{-}$, by using the geometric
interpretation of $R$ in terms of the Ext algebra coming from Lusztig’s
category $\mathcal{Q}_{V_{\beta}}$. A similar result holds also for
$K(\mathscr{P})$, namely, the basis $\underline{\mathbf{B}}$ corresponds, via
McNamara’s isomorphism, to the canonical basis of
$\underline{\mathbf{U}}_{q}^{-}$, which coincides with the lower global basis.
This is stated in the last paragraph of [M], with a brief indication for the
proof, which is based on the geometric argument. In Theorem 5.13 and Theorem
5.16, we give a simple proof of this fact, without using the geometry (though
assuming the results in [VV], [R2]).

##  1\.  Quantum groups

1.1. Let $X=(I,(\ ,\ ))$ be a Cartan datum, where $I$ is a finite set, and $(\
,\ )$ is a symmetric bilinear form on the vector space $\bigoplus_{i\in
I}\mathbf{Q}\alpha_{i}$ with basis $\alpha_{i}$, satisfying the properties
that

  1. (i)

$(\alpha_{i},\alpha_{i})\in 2\mathbf{Z}_{>0}$ for any $i\in I$,

  2. (ii)

$\frac{2(\alpha_{i},\alpha_{i^{\prime}})}{(\alpha_{i},\alpha_{i})}\in\mathbf{Z}_{\leq
0}$ for any $i\neq i^{\prime}\in I$.

For $i,i^{\prime}\in I$, set
$a_{ii^{\prime}}=2(\alpha_{i},\alpha_{i^{\prime}})/(\alpha_{i},\alpha_{i})\in\mathbf{Z}$.
$A=(a_{ii^{\prime}})$ is called the Cartan matrix associated to $X$. The
Cartan datum is called symmetric if $(\alpha_{i},\alpha_{i})=2$ for any $i\in
I$.

Let $Q=\bigoplus_{i\in I}\mathbf{Z}\alpha_{i}$ be the root lattice of $X$. We
set $Q_{+}=\sum_{i\in I}\mathbf{N}\alpha_{i}$, and $Q_{-}=-Q_{+}$.

1.2. Let $q$ be an indeterminate, and for an integer $n$, a positive integer
$m$, set

$[n]=\frac{q^{n}-q^{-n}}{q-q^{-1}},\quad[m]^{!}=[1][2]\cdots[m],\quad[0]^{!}=1.$

For each $i\in I$, set $d_{i}=(\alpha_{i},\alpha_{i})/2\in\mathbf{N}$, and
$q_{i}=q^{d_{i}}$. We denote by $[n]_{i}$ the element obtained from $[n]$ by
replacing $q$ by $q_{i}$.

For the Cartan datum $X$, let $\mathbf{U}_{q}^{-}=\mathbf{U}_{q}^{-}(X)$ be
the negative half of the quantum group associated to $X$. $\mathbf{U}_{q}^{-}$
is an associative algebra over $\mathbf{Q}(q)$ generated by $f_{i}\ (i\in I)$
satisfying the $q$-Serre relations

(1.2.1)
$\sum_{k+k^{\prime}=1-a_{ii^{\prime}}}(-1)^{k}f_{i}^{(k)}f_{i^{\prime}}f_{i}^{(k^{\prime})}=0,\quad\text{
for $i\neq i^{\prime}\in I$,}$

where $f_{i}^{(n)}=f_{i}^{n}/[n]_{i}^{!}$.

Set $\mathbf{A}=\mathbf{Z}[q,q^{-1}]$, and let
${}_{\mathbf{A}}\mathbf{U}_{q}^{-}$ be Lusztig’s integral form of
$\mathbf{U}_{q}^{-}$, namely, the $\mathbf{A}$-subalgebra of
$\mathbf{U}_{q}^{-}$ generated by $f_{i}^{(n)}$ for $i\in I,n\in\mathbf{N}$.

We define a $\mathbf{Q}$-algebra automorphism, called the bar-involution,
${}^{-}:\mathbf{U}_{q}^{-}\to\mathbf{U}_{q}^{-}$ by $\overline{q}=q^{-1}$,
$\overline{f}_{i}=f_{i}$ for $i\in I$. We define an anti-algebra automorphism
${}^{*}:\mathbf{U}_{q}^{-}\to\mathbf{U}_{q}^{-}$ by $f_{i}^{*}=f_{i}$ for any
$i\in I$.

1.3. $\mathbf{U}_{q}^{-}$ has a weight space decomposition
$\mathbf{U}_{q}^{-}=\bigoplus_{\beta\in Q_{-}}(\mathbf{U}_{q}^{-})_{\beta}$,
where $(\mathbf{U}_{q}^{-})_{\beta}$ is a subspace of $\mathbf{U}_{q}^{-}$
spanned by elements $f_{i_{1}}\cdots f_{i_{N}}$ such that
$\alpha_{i_{1}}+\cdots+\alpha_{i_{N}}=-\beta$. $x\in\mathbf{U}_{q}^{-}$ is
called homogeneous with $\operatorname{wt}x=\beta$ if
$x\in(\mathbf{U}_{q}^{-})_{\beta}$. We define a multiplication on
$\mathbf{U}_{q}^{-}\otimes\mathbf{U}_{q}^{-}$ by

(1.3.1) $(x_{1}\otimes x_{2})\cdot(x_{1}^{\prime}\otimes
x_{2}^{\prime})=q^{-(\operatorname{wt}x_{2},\operatorname{wt}x_{1}^{\prime})}x_{1}x_{1}^{\prime}\otimes
x_{2}x_{2}^{\prime}$

where $x_{1},x_{1}^{\prime},x_{2},x_{2}^{\prime}$ are homogeneous in
$\mathbf{U}_{q}^{-}$. Then $\mathbf{U}_{q}^{-}\otimes\mathbf{U}_{q}^{-}$
becomes an associative algebra with respect to this twisted product. One can
define a homomorphism
$r:\mathbf{U}_{q}^{-}\to\mathbf{U}_{q}^{-}\otimes\mathbf{U}_{q}^{-}$ by
$r(f_{i})=f_{i}\otimes 1+1\otimes f_{i}$ for each $i\in I$. It is known that
there exists a unique bilinear form $(\ ,\ )$ on $\mathbf{U}_{q}^{-}$
satisfying the following properties; $(1,1)=1$ and

(1.3.2) $\displaystyle(f_{i},f_{j})=\delta_{ij}(1-q_{i}^{2})^{-1},$
$\displaystyle(x,y^{\prime}y^{\prime\prime})=(r(x),y^{\prime}\otimes
y^{\prime\prime}),$
$\displaystyle(x^{\prime}x^{\prime\prime},y)=(x^{\prime}\otimes
x^{\prime\prime},r(y)),$

where the bilinear form on $\mathbf{U}_{q}^{-}\otimes\mathbf{U}_{q}^{-}$ is
defined by $(x_{1}\otimes x_{2},x_{1}^{\prime}\otimes
x_{2}^{\prime})=(x_{1},x_{1}^{\prime})(x_{2},x_{2}^{\prime})$. Thus defined
bilinear form is symmetric and non-degenerate.

Following [L2, 1.2.13], for each $i$, we define a $\mathbf{Q}(q)$-linear map
${}_{i}r:\mathbf{U}_{q}^{-}\to\mathbf{U}_{q}^{-}$ by the condition that

(1.3.3) $r(x)=f_{i}\otimes{}_{i}r(x)+\sum y\otimes z,$

where $y$ runs over homogeneous elements such that
$\operatorname{wt}y\neq-\alpha_{i}$. Since ${}_{i}r$ coincides with the
operator $e_{i}^{\prime}$ defined in [K1, 3.4], (see [L2, Prop. 3.1.6] and
[K1, Lemma 3.4.1]), hereafter we write ${}_{i}r$ as $e_{i}^{\prime}$. Thus, if
we define the action of $f_{i}$ on $\mathbf{U}_{q}^{-}$ by the left
multiplication, then $e_{i}^{\prime},f_{i^{\prime}}$ satisfies the $q$-boson
relations, as operators on $\mathbf{U}_{q}^{-}$,

(1.3.4)
$e_{i}^{\prime}f_{i^{\prime}}=q^{-(\alpha_{i},a_{i^{\prime}})}f_{i^{\prime}}e_{i}^{\prime}+\delta_{ii^{\prime}}.$

1.4. Let $V$ be a $\mathbf{Q}(q)$-subspace of $\mathbf{U}_{q}^{-}$. A basis
$\mathcal{B}$ of $V$ is said to be almost orthonormal if

(1.4.1)
$(b,b^{\prime})\in\begin{cases}1+q\mathbf{Z}[[q]]\cap\mathbf{Q}(q)&\quad\text{
if }b=b^{\prime},\\\ q\mathbf{Z}[[q]]\cap\mathbf{Q}(q)&\quad\text{ if }b\neq
b^{\prime}.\end{cases}$

Recall that $\mathbf{A}=\mathbf{Z}[q,q^{-1}]$. Let
$\mathbf{A}_{0}=\mathbf{Q}[[q]]\cap\mathbf{Q}(q)$. Set

(1.4.2)
$\mathscr{L}_{\mathbf{Z}}(\infty)=\\{x\in{}_{\mathbf{A}}\mathbf{U}_{q}^{-}\mid(x,x)\in\mathbf{A}_{0}\\}.$

Then $\mathscr{L}_{\mathbf{Z}}(\infty)$ is a $\mathbf{Z}[q]$-submodule of
${}_{\mathbf{A}}\mathbf{U}_{q}^{-}$. It is known that if $\mathcal{B}$ is an
$\mathbf{A}$-basis of ${}_{\mathbf{A}}\mathbf{U}_{q}^{-}$, which is almost
orthonormal, then $\mathcal{B}$ gives a $\mathbf{Z}[q]$-basis of
$\mathscr{L}_{\mathbf{Z}}(\infty)$ by [L2, Lemma 16.2.5].

We define a subset $\widetilde{\mathcal{B}}$ of $\mathbf{U}_{q}^{-}$ by

(1.4.3)
$\widetilde{\mathcal{B}}=\\{x\in{}_{\mathbf{A}}\mathbf{U}_{q}^{-}\mid\overline{x}=x,(x,x)\in
1+q\mathbf{Z}[[q]]\\}.$

If there exists a basis $\mathcal{B}^{\prime}$ of $\mathbf{U}_{q}^{-}$ such
that
$\widetilde{\mathcal{B}}=\mathcal{B}^{\prime}\sqcup-\mathcal{B}^{\prime}$,
then $\widetilde{\mathcal{B}}$ is called the canonical signed basis. In that
case, the basis $\mathcal{B}^{\prime}$ is determined uniquely, up to sign, by
the condition (1.4.3). Lusztig proved in [L2], by using the geometric theory
of quivers, that $\widetilde{\mathcal{B}}$ is the canonical signed basis.
Kashiwara’s global crystal basis $\mathcal{B}$ in [K1] also satisfies the
condition that $\widetilde{\mathcal{B}}=\mathcal{B}\sqcup-\mathcal{B}$ (see
[K1, Prop. 5.1.2], but note that the inner products $(\ ,\ )$ on
$\mathbf{U}_{q}^{-}$ defined by Lusztig and by Kashiwara differ by a scalar
factor on each weight space).

1.5. A permutation $\sigma:I\to I$ is called an admissible automorphism on $X$
if it satisfies the property that
$(\alpha_{i},\alpha_{i^{\prime}})=(\alpha_{\sigma(i)},\alpha_{\sigma(i^{\prime})})$
for any $i,i^{\prime}\in I$, and that $(\alpha_{i},\alpha_{i^{\prime}})=0$ if
$i$ and $i^{\prime}$ belong to the same $\sigma$-orbit in $I$. Assume that
$\sigma$ is admissible. We denote by $\mathbf{n}$ the order of $\sigma:I\to
I$. Let $J$ be the set of $\sigma$-orbits in $I$. For each $j\in J$, set
$\alpha_{j}=\sum_{i\in j}\alpha_{i}$, and consider the subspace
$\bigoplus_{j\in J}\mathbf{Q}\alpha_{j}$ of $\bigoplus_{i\in
I}\mathbf{Q}\alpha_{i}$, with basis $\alpha_{j}$. We denote by $|j|$ the size
of the orbit $j$ in $I$. The restriction of the form $(\ ,\ )$ on
$\bigoplus_{j\in J}\mathbf{Q}\alpha_{j}$ is given by

$(\alpha_{j},\alpha_{j^{\prime}})=\begin{cases}(\alpha_{i},\alpha_{i})|j|\quad(i\in
j)&\quad\text{ if }j=j^{\prime},\\\ \sum_{i\in j,i^{\prime}\in
j^{\prime}}(\alpha_{i},\alpha_{i^{\prime}})&\quad\text{ if }j\neq
j^{\prime}.\end{cases}$

Then $\underline{X}=(J,(\ ,\ ))$ turns out to be a Cartan datum, which is
called the Cartan datum induced from $(X,\sigma)$. $\sigma$ acts naturally on
$Q$, and the set $Q^{\sigma}$ of $\sigma$-fixed elements in $Q$ is identified
with the root lattice of $\underline{X}$. We have $Q_{+}^{\sigma}=\sum_{j\in
J}\mathbf{N}\alpha_{j}$.

Let $\sigma$ be an admissible automorphism of $X$. Then $\sigma$ induces an
algebra automorphism on $\mathbf{U}_{q}^{-}\to\mathbf{U}_{q}^{-}$ by
$f_{i}\mapsto f_{\sigma(i)}$, which we also denote by $\sigma$. The action of
$\sigma$ leaves ${}_{\mathbf{A}}\mathbf{U}_{q}^{-}$ invariant. We denote by
${}_{\mathbf{A}}\mathbf{U}_{q}^{-,\sigma}$ the fixed point subalgebra of
${}_{\mathbf{A}}\mathbf{U}_{q}^{-}$.

Let $\underline{X}$ be the Cartan datum induced from $(X,\sigma)$. Let
$\underline{\mathbf{U}}_{q}^{-}=\mathbf{U}_{q}^{-}(\underline{X})$ the quantum
group associated to $\underline{X}$. Thus $\underline{\mathbf{U}}_{q}^{-}$ is
an associative $\mathbf{Q}(q)$-algebra, with generators $f_{j}$ ($j\in J$),
and similar relations as in (1.2.1). The integral form
${}_{\mathbf{A}}\underline{\mathbf{U}}_{q}^{-}$ of
$\underline{\mathbf{U}}_{q}^{-}$ is the $\mathbf{A}$-subalgebra generated by
$f_{j}^{(n)}$ ($j\in J,n\in\mathbf{Z}$).

1.6. Quotient algebras $\mathbf{V}_{q}$. In general, let $V$ be a vector space
on which $\sigma$ acts. For each $x\in V$, we denote by $O(x)$ the orbit sum
of $x$, namely $O(x)=\sum_{0\leq i<k}\sigma^{i}(x)$, where $k$ is the smallest
integer such that $\sigma^{k}(x)=x$. Hence $O(x)$ is a $\sigma$-invariant
element in $V$.

Let $\sigma:X\to X$ be as in 1.5, and $\mathbf{n}$ the order of $\sigma:I\to
I$. In the rest of this section, we assume that $\mathbf{n}$ is a power of a
prime number $\ell$. Let $\mathbf{F}=\mathbf{Z}/\ell\mathbf{Z}$ be the finite
field of $\ell$ elements, and set
$\mathbf{A}^{\prime}=\mathbf{F}[q,q^{-1}]=\mathbf{A}/\ell\mathbf{A}$. We
consider the $\mathbf{A}^{\prime}$-algebra

${}_{\mathbf{A}^{\prime}}\mathbf{U}_{q}^{-,\sigma}=\mathbf{A}^{\prime}\otimes_{\mathbf{A}}{}_{\mathbf{A}}\mathbf{U}_{q}^{-,\sigma}\simeq{}_{\mathbf{A}}\mathbf{U}_{q}^{-,\sigma}/\ell({}_{\mathbf{A}}\mathbf{U}_{q}^{-,\sigma}).$

Let $\mathbf{J}$ be an $\mathbf{A}^{\prime}$-submodule of
${}_{\mathbf{A}^{\prime}}\mathbf{U}_{q}^{-,\sigma}$ generated by $O(x)$ for
$x\in{}_{\mathbf{A}^{\prime}}\mathbf{U}_{q}^{-}$ such that $\sigma(x)\neq x$.
Then $\mathbf{J}$ is a two-sided ideal of
${}_{\mathbf{A}^{\prime}}\mathbf{U}_{q}^{-,\sigma}$. We define an
$\mathbf{A}^{\prime}$-algebra $\mathbf{V}_{q}$ as the quotient algebra of
${}_{\mathbf{A}^{\prime}}\mathbf{U}_{q}^{-,\sigma}$,

(1.6.1)
$\mathbf{V}_{q}={}_{\mathbf{A}^{\prime}}\mathbf{U}_{q}^{-,\sigma}/\mathbf{J}.$

Let $\pi:{}_{\mathbf{A}^{\prime}}\mathbf{U}_{q}^{-,\sigma}\to\mathbf{V}_{q}$
be the natural projection.

For each $j\in J$ and $a\in\mathbf{N}$, set
$\widetilde{f}_{j}^{(a)}=\prod_{i\in j}f_{i}^{(a)}$. Since $\sigma$ is
admissible, $f_{i}^{(a)}$ and $f_{i^{\prime}}^{(a)}$ commute each other for
$i,i^{\prime}\in j$. Hence $\widetilde{f}_{j}^{(a)}$ does not depend on the
order of the product, and
$\widetilde{f}^{(a)}_{j}\in{}_{\mathbf{A}}\widetilde{\mathbf{U}}_{q}^{-,\sigma}$.
We denote its image in ${}_{\mathbf{A}^{\prime}}\mathbf{U}_{q}^{-,\sigma}$
also by $\widetilde{f}^{(a)}_{j}$. Thus we can define
$g_{j}^{(a)}\in\mathbf{V}_{q}$ by

(1.6.2) $g^{(a)}_{j}=\pi(\widetilde{f}_{j}^{(a)}).$

In the case where $a=1$, we set
$\widetilde{f}_{j}^{(1)}=\widetilde{f}_{j}=\prod_{i\in j}f_{i}$ and
$g^{(1)}_{j}=g_{j}$.

We define an ${\mathbf{A}^{\prime}}$-algebra
${}_{\mathbf{A}^{\prime}}\underline{\mathbf{U}}_{q}^{-}$ by
${}_{\mathbf{A}^{\prime}}\underline{\mathbf{U}}_{q}^{-}=\mathbf{A}^{\prime}\otimes_{\mathbf{A}}{}_{\mathbf{A}}\underline{\mathbf{U}}_{q}^{-}$.

The following result was proved in [MSZ].

######  Theorem 1.7 ([MSZ, Thm. 3.4]).

The assignment $f_{j}^{(a)}\mapsto g_{j}^{(a)}$ ($j\in J$) gives an
isomorphism
$\Phi:{}_{\mathbf{A}^{\prime}}\underline{\mathbf{U}}_{q}^{-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,\mathbf{V}_{q}$ of
$\mathbf{A}^{\prime}$-algebras.

Remark 1.8 In Theorem 3.4 in [MSZ], the existence of the canonical signed
basis of $\mathbf{U}_{q}^{-}$ is assumed. This certainly holds, by 1.4. In
[MSZ, Thm. 4.27], the canonical signed basis for
$\underline{\mathbf{U}}_{q}^{-}$ was constructed from the canonical basis in
the symmetric case, by making use of Theorem 1.7.

1.9. The dual space $(\mathbf{U}_{q}^{-})^{*}$. Let
$\mathbf{U}_{q}^{-}=\bigoplus_{\beta\in Q^{-}}(\mathbf{U}_{q}^{-})_{\beta}$ be
the weight space decomposition of $\mathbf{U}_{q}^{-}$. We define the graded
dual $(\mathbf{U}_{q}^{-})^{*}$ of $\mathbf{U}_{q}^{-}$ by

$(\mathbf{U}_{q}^{-})^{*}=\bigoplus_{\beta\in
Q^{-}}(\mathbf{U}_{q}^{-})_{\beta}^{*},$

where
$(\mathbf{U}_{q}^{-})_{\beta}^{*}=\operatorname{Hom}_{\mathbf{Q}(q)}((\mathbf{U}_{q}^{-})_{\beta},\mathbf{Q}(q))$.
$f_{i}^{(n)}$ and $e_{i}^{\prime}$ act on $\mathbf{U}_{q}^{-}$, and we define
the actions of ${e_{i}^{\prime}}^{(n)}$ and $f_{i}$ on
$(\mathbf{U}_{q}^{-})^{*}$ as the transpose of $f_{i}^{(n)},e_{i}^{\prime}$.
We write $e_{i}^{\prime}=e_{i}^{\prime(n)}$ for $n=1$ (as operators on
$(\mathbf{U}_{q}^{-})^{*}$).

${}_{\mathbf{A}}\mathbf{U}^{-}_{q}$ is decomposed as

${}_{\mathbf{A}}\mathbf{U}^{-}_{q}=\bigoplus_{\beta\in
Q_{-}}{}_{\mathbf{A}}(\mathbf{U}_{q}^{-})_{\beta},$

where
${}_{\mathbf{A}}(\mathbf{U}_{q}^{-})_{\beta}={}_{\mathbf{A}}\mathbf{U}_{q}^{-}\cap(\mathbf{U}_{q}^{-})_{\beta}$
is a free $\mathbf{A}$-module. We define
${}_{\mathbf{A}}(\mathbf{U}_{q}^{-})^{*}$ by

${}_{\mathbf{A}}(\mathbf{U}_{q}^{-})^{*}=\bigoplus_{\beta\in
Q_{-}}{}_{\mathbf{A}}(\mathbf{U}_{q}^{-})^{*}_{\beta},$

where
${}_{\mathbf{A}}(\mathbf{U}_{q}^{-})_{\beta}^{*}=\operatorname{Hom}_{\mathbf{A}}({}_{\mathbf{A}}(\mathbf{U}_{q}^{-})_{\beta},\mathbf{A})$.
Since
$\mathbf{Q}(q)\otimes_{\mathbf{A}}{}_{\mathbf{A}}(\mathbf{U}_{q}^{-})^{*}\simeq(\mathbf{U}_{q}^{-})^{*}$,
${}_{\mathbf{A}}(\mathbf{U}_{q}^{-})^{*}$ is regarded as an
$\mathbf{A}$-submodule of $(\mathbf{U}_{q}^{-})^{*}$, and
${}_{\mathbf{A}}(\mathbf{U}_{q}^{-})^{*}$ gives an $\mathbf{A}$-lattice of
$(\mathbf{U}_{q}^{-})^{*}$.

The actions of $f_{i}^{(n)},e_{i}^{\prime}$ preserve
${}_{\mathbf{A}}\mathbf{U}_{q}^{-}$, and the actions of
${e_{i}^{\prime}}^{(n)},f_{i}$ preserve
${}_{\mathbf{A}}(\mathbf{U}_{q}^{-})^{*}$.

Recall that $u\mapsto\overline{u}$ is the bar involution on
$\mathbf{U}_{q}^{-}$. We define the bar involution
$\varphi\mapsto\overline{\varphi}$ on $(\mathbf{U}_{q}^{-})^{*}$ by
$\overline{\varphi}(\overline{u})=\overline{\varphi(u)}$
($\varphi\in(\mathbf{U}_{q}^{-})^{*},u\in\mathbf{U}_{q}^{-}$).

In [K2], Kashiwara introduced the upper global basis $\mathcal{B}^{*}$ of
$(\mathbf{U}_{q}^{-})^{*}$, which is characterized by the following
properties.

######  Theorem 1.10.

There exists a unique $\mathbf{A}$-basis $\mathcal{B}^{*}$ of
${}_{\mathbf{A}}(\mathbf{U}_{q}^{-})^{*}$ satisfying the following properties.
For $b\in\mathcal{B}^{*}$, set

$\varepsilon_{i}(b)=\max\\{k\geq 0\mid e_{i}^{\prime k}b\neq 0\\}.$

  1. (i)

$\mathcal{B}^{*}$ has a weight space decomposition
$\mathcal{B}^{*}=\bigsqcup_{\beta\in Q_{-}}\mathcal{B}^{*}_{\beta}$, where
$\mathcal{B}^{*}_{\beta}=\mathcal{B}^{*}\cap{}_{\mathbf{A}}(\mathbf{U}_{q}^{-})_{\beta}^{*}$.

  2. (ii)

$1^{*}\in\mathcal{B}^{*}$, where
$\\{1^{*}\\}\subset{}_{\mathbf{A}}(\mathbf{U}_{q}^{-})^{*}_{0}$ is the dual
basis of $\\{1\\}\subset{}_{\mathbf{A}}(\mathbf{U}_{q}^{-})_{0}$.

  3. (iii)

Assume that $e_{i}^{\prime}b\neq 0$. Then there exists a unique
$\widetilde{\mathbf{e}}_{i}b\in\mathcal{B}^{*}$ satisfying the following.

$e_{i}^{\prime}b=[\varepsilon_{i}(b)]_{i}\widetilde{\mathbf{e}}_{i}b+\sum_{\begin{subarray}{c}b^{\prime}\in\mathcal{B}^{*}\\\
\varepsilon_{i}(b^{\prime})<\varepsilon_{i}(b)-1\end{subarray}}E_{bb^{\prime}}^{(i)}b^{\prime}\qquad(E_{bb^{\prime}}^{(i)}\in
qq_{i}^{1-\varepsilon_{i}(b)}\mathbf{Z}[q]).$

  4. (iv)

There exists a unique $\widetilde{\mathbf{f}}_{i}b\in\mathcal{B}^{*}$
satisfying the following.

$f_{i}b=q_{i}^{-\varepsilon_{i}(b)}\widetilde{\mathbf{f}}_{i}b+\sum_{\begin{subarray}{c}b^{\prime}\in\mathbf{B}^{*}\\\
\varepsilon_{i}(b^{\prime})<\varepsilon_{i}(b)+1\end{subarray}}F^{(i)}_{bb^{\prime}}b^{\prime}\qquad(F^{(i)}_{bb^{\prime}}\in
qq_{i}^{-\varepsilon_{i}(b)}\mathbf{Z}[q]).$

  5. (v)

$\widetilde{\mathbf{e}}_{i}\widetilde{\mathbf{f}}_{i}b=b$.

  6. (vi)

$\widetilde{\mathbf{f}}_{i}\widetilde{\mathbf{e}}_{i}b=b$ if
$e_{i}^{\prime}b\neq 0$.

  7. (vii)

$\overline{b}=b$.

The $\mathbf{A}$-basis $\mathcal{B}$ of ${}_{\mathbf{A}}\mathbf{U}_{q}^{-}$ is
defined as the dual basis of $\mathcal{B}^{*}$. $\mathcal{B}$ is called the
lower global basis. $\mathcal{B}$ coincides with the global crystal basis
given in 1.4.

1.11. The dual module $\mathbf{V}_{q}^{*}$. We define
${}_{\mathbf{A}^{\prime}}(\mathbf{U}_{q}^{-})^{*}$ similarly to
${}_{\mathbf{A}^{\prime}}\mathbf{U}_{q}^{-}$. Then $e^{\prime(n)}_{i}$ act on
${}_{\mathbf{A}^{\prime}}(\mathbf{U}_{q}^{-})^{*}$.

We define an action of $\sigma$ on $(\mathbf{U}_{q}^{-})^{*}$ by
$(\sigma\varphi)(x)=\varphi(\sigma^{-1}(x))$ for
$\varphi\in(\mathbf{U}_{q}^{-})^{*}$ and $x\in\mathbf{U}_{q}^{-}$. Thus the
natural pairing $\operatorname{\langle}\ ,\
\operatorname{\rangle}:\mathbf{U}_{q}^{-}\times(\mathbf{U}_{q}^{-})^{*}\to\mathbf{Q}(q)$
satisfies the property
$\operatorname{\langle}\sigma(x),\sigma(\varphi)\operatorname{\rangle}=\operatorname{\langle}x,\varphi\operatorname{\rangle}$.
By the above discussion, the pairing $\operatorname{\langle}\ ,\
\operatorname{\rangle}$ induces a perfect pairing $\operatorname{\langle}\ ,\
\operatorname{\rangle}:{}_{\mathbf{A}^{\prime}}\mathbf{U}_{q}^{-}\times{}_{\mathbf{A}^{\prime}}(\mathbf{U}_{q}^{-})^{*}\to\mathbf{A}^{\prime}$,
which is compatible with the action of $\sigma$.

By Theorem 1.10, $\sigma$ acts on $\mathcal{B}^{*}$ as a permutation, hence
acts on $\mathcal{B}$ similarly.

Let ${}_{\mathbf{A}^{\prime}}(\mathbf{U}_{q}^{-})^{*,\sigma}$ be the submodule
of ${}_{\mathbf{A}^{\prime}}(\mathbf{U}_{q}^{-})^{*}$ of $\sigma$-fixed
elements. Let $\mathbf{J}^{*}$ be an $\mathbf{A}^{\prime}$-submodule of
${}_{\mathbf{A}^{\prime}}(\mathbf{U}_{q}^{-})^{*,\sigma}$ consisting of $O(x)$
for $x\in{}_{\mathbf{A}^{\prime}}(\mathbf{U}_{q}^{-})^{*}$ such that $O(x)\neq
x$. We define $\mathbf{V}_{q}^{*}$ as the quotient
$\mathbf{A}^{\prime}$-module

(1.11.1)
$\mathbf{V}_{q}^{*}={}_{\mathbf{A}^{\prime}}(\mathbf{U}_{q}^{-})^{*,\sigma}/\mathbf{J}^{*}.$

Since
$\operatorname{\langle}{}_{\mathbf{A}}\mathbf{U}_{q}^{-,\sigma},\mathbf{J}^{*}\operatorname{\rangle}=\operatorname{\langle}\mathbf{J},{}_{\mathbf{A}^{\prime}}(\mathbf{U}_{q}^{-})^{*,\sigma}\operatorname{\rangle}=0$,
the pairing $\operatorname{\langle}\ ,\ \operatorname{\rangle}$ induces a
pairing $\operatorname{\langle}\ ,\
\operatorname{\rangle}:\mathbf{V}_{q}\times\mathbf{V}_{q}^{*}\to\mathbf{A}^{\prime}$.
Let $\mathcal{B}^{\sigma}$ be the set of $\sigma$-fixed elements in
$\mathcal{B}$, and $\mathcal{B}^{*,\sigma}$ the set of $\sigma$-fixed elements
in $\mathcal{B}^{*}$. Then the image of $\mathcal{B}^{\sigma}$ gives a basis
of $\mathbf{V}_{q}$, and the image of $\mathcal{B}^{*,\sigma}$ gives a basis
of $\mathbf{V}_{q}^{*}$. Hence the pairing $\operatorname{\langle}\ ,\
\operatorname{\rangle}:\mathbf{V}_{q}\times\mathbf{V}_{q}^{*}\to\mathbf{A}^{\prime}$
gives a perfect pairing on $\mathbf{A}^{\prime}$.

For each $j\in J$, we define an action of $\widetilde{f}_{j}^{(n)}$ on
$\mathbf{V}_{q}$ by the left multiplication by $g_{j}^{(n)}=\pi(f_{j}^{(n)})$
(see (1.6.2)). We define an operator $\widetilde{e}^{\prime(n)}_{j}$ on
${}_{\mathbf{A}^{\prime}}(\mathbf{U}_{q}^{-})^{*}$ by
$\widetilde{e}^{\prime(n)}_{j}=\prod_{i\in j}e^{\prime(n)}_{i}$. Then
$\widetilde{e}^{\prime(n)}_{j}$ commutes with $\sigma$, and induces an action
on ${}_{\mathbf{A}^{\prime}}(\mathbf{U}_{q}^{-})^{*,\sigma}$. Since
$\widetilde{e}^{\prime(n)}_{j}$ preserves $\mathbf{J}^{*}$,
$\widetilde{e}^{\prime(n)}_{j}$ acts on $\mathbf{V}_{q}^{*}$, which we also
denote by $\widetilde{e}^{\prime(n)}_{j}$. $\widetilde{e}^{\prime(n)}_{j}$
coincides with the transpose of the operation $\widetilde{f}_{j}^{(n)}$ on
$\mathbf{V}_{q}$.

The operators $e^{\prime(n)}_{j}$ on
${}_{\mathbf{A}^{\prime}}(\underline{\mathbf{U}}_{q}^{-})^{*}$ is defined as
in the case of ${}_{\mathbf{A}^{\prime}}(\mathbf{U}_{q}^{-})^{*}$. The
following result is immediate from Theorem 1.7.

######  Proposition 1.12.

Let $\Phi^{*}:\mathbf{V}_{q}^{*}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}^{\prime}}(\underline{\mathbf{U}}_{q}^{-})^{*}$
be the $\mathbf{A}^{\prime}$-module isomorphism obtained as the transpose of
the map $\Phi:{}_{\mathbf{A}^{\prime}}\underline{\mathbf{U}}_{q}^{-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,\mathbf{V}_{q}$. Then $\Phi^{*}$
commutes with the actions $\widetilde{e}^{\prime(n)}_{j}$ on
$\mathbf{V}_{q}^{*}$ and $e^{\prime(n)}_{j}$ on
${}_{\mathbf{A}^{\prime}}(\underline{\mathbf{U}}_{q}^{-})^{*}$.

##  2\.  KLR algebras

2.1. Let $X=(I,(\,\ ))$ be a Cartan datum, and $\sigma$ an admissible
automorphism on $X$. Assume that $\mathbf{k}$ is an algebraically closed
field. We define a KLR algebra associated to $(X,\sigma)$ as follows. For each
$i,i^{\prime}\in I$, choose a polynomial
$Q_{i,i^{\prime}}(u,v)\in\mathbf{k}[u,v]$ satisfying the following properties;

  1. (i)

$Q_{i,i}(u,v)=0$,

  2. (ii)

$Q_{i,i^{\prime}}(u,v)$ is written as
$Q_{i,i^{\prime}}(u,v)=\sum_{p,q}t_{ii^{\prime};pq}u^{p}v^{q}$, where the
coefficients $t_{ii^{\prime};pq}\in\mathbf{k}$ satisfy the condition

$(\alpha_{i},\alpha_{i})p+(\alpha_{i^{\prime}},\alpha_{i^{\prime}})q=-2(\alpha_{i},\alpha_{i^{\prime}})\quad\text{
if $t_{ii^{\prime};pq}\neq 0$. }$

Moreover, $t_{ii^{\prime};-a_{ii^{\prime}},0}\neq
0,t_{ii^{\prime};0,-a_{ii^{\prime}}}\neq 0$.

  3. (iii)

$Q_{i,i^{\prime}}(u,v)=Q_{i^{\prime},i}(v,u)$,

  4. (iv)

$Q_{\sigma(i),\sigma(i^{\prime})}(u,v)=Q_{i,i^{\prime}}(u,v)$.

For $\beta=\sum_{i\in I}n_{i}\alpha_{i}\in Q_{+}$, we set
$|\beta|=\sum_{i}n_{i}$. For $\beta\in Q_{+}$ such that $|\beta|=n$, define
$I^{\beta}$ by

$I^{\beta}=\\{(i_{1},\dots,i_{n})\in I^{n}\mid\sum_{1\leq k\leq
n}\alpha_{i_{k}}=\beta\\}.$

The KLR algebra $R(\beta)$ is an associative $\mathbf{k}$-algebra defined by
the generators, $e(\nu)$ ($\nu=(\nu_{1},\dots,\nu_{n})\in I^{\beta}$), $x_{k}$
$(1\leq k\leq n)$, $\tau_{k}\ (1\leq k<n)$, with relations

(1) $\displaystyle e(\nu)e(\nu^{\prime})$
$\displaystyle=\delta_{\nu,\nu^{\prime}}e(\nu),\quad\sum_{\nu\in
I^{\beta}}e(\nu)=1,$ (2) $\displaystyle x_{k}x_{l}$
$\displaystyle=x_{l}x_{k},\quad x_{k}e(\nu)=e(\nu)x_{k},$ (3)
$\displaystyle\tau_{k}e(\nu)$
$\displaystyle=e(s_{k}\nu)\tau_{k},\quad\tau_{k}\tau_{l}=\tau_{l}\tau_{k},\quad\text{
if }|k-l|>1,$ (4) $\displaystyle\tau_{k}^{2}e(\nu)$
$\displaystyle=Q_{\nu_{k},\nu_{k+1}}(x_{k},x_{k+1})e(\nu),$ (5)
$\displaystyle(\tau_{k}x_{l}$ $\displaystyle-
x_{s_{k}l}\tau_{k})e(\nu)=\begin{cases}-e(\nu)&\quad\text{ if
}l=k,\nu_{k}=\nu_{k+1},\\\ e(\nu)&\quad\text{ if }l=k+1,\nu_{k}=\nu_{k+1},\\\
0&\quad\text{ otherwise, }\end{cases}$ (6) $\displaystyle(\tau_{k+1}$
$\displaystyle\tau_{k}\tau_{k+1}-\tau_{k}\tau_{k+1}\tau_{k})e(\nu)$
$\displaystyle=\begin{cases}\displaystyle\frac{Q_{\nu_{k},\nu_{k+1}}(x_{k},x_{k+1})-Q_{\nu_{k},\nu_{k+1}}(x_{k+2},x_{k+1})}{x_{k}-x_{k+2}}e(\nu)&\quad\text{
if }\nu_{k}=\nu_{k+2},\\\ 0&\quad\text{ otherwise. }\end{cases}$

(Here $s_{k}$ ($1\leq k<n$) is a transvection $(k,k+1)$ in the symmetric group
$S_{n}$. $S_{n}$ acts on $I^{n}$ by
$w:(\nu_{1},\dots,\nu_{n})\mapsto(\nu_{w^{-1}(1)},\dots,\nu_{w^{-1}(n)})$.)

We define the algebra $R_{n}=\bigoplus_{|\beta|=n}R(\beta)$, and the algebra
$R$ by

$R=\bigoplus_{n\geq 0}R_{n}=\bigoplus_{\beta\in Q_{+}}R(\beta).$

Note that the condition (iv) is used in considering the $\sigma$-setup for KLR
algebras. The discussion until 2.9 is independent of $\sigma$, hence the
condition (iv) is redundant.

The algebra $R(\beta)$ is a $\mathbf{Z}$-graded algebra, where the degree is
defined, for $\nu=(\nu_{1},\dots,\nu_{n})\in I^{\beta}$, by

$\deg e(\nu)=0,\quad\deg
x_{k}e(\nu)=(\alpha_{\nu_{k}},\alpha_{\nu_{k}}),\quad\deg\tau_{k}e(\nu)=-(\alpha_{\nu_{k}},\alpha_{\nu_{k+1}}).$

For a graded $R(\beta)$-module $M=\bigoplus_{i\in\mathbf{Z}}M_{i}$, the
grading shift by $-1$ is denoted by $qM$, namely $(qM)_{i}=M_{i-1}$. (Here $q$
is the indeterminate given in Section 1). There exists an anti-involution
$\psi:R(\beta)\to R(\beta)$, which leaves all the generators $e(\nu),x_{k}$
and $\tau_{k}$ invariant.

2.2 For $m,n\in\mathbf{N}$, there exists a $\mathbf{k}$-bilinear map
$R_{m}\times R_{n}\to R_{m+n}$ defined by

$\displaystyle(x_{k},1)\mapsto
x_{k},\quad(\tau_{k},1)\mapsto\tau_{k},\quad(1,x_{k})\mapsto
x_{m+k},\quad(1,\tau_{k})\mapsto\tau_{m+k},$
$\displaystyle(e(\nu),e(\nu^{\prime}))\mapsto e(\nu,\nu^{\prime}),$

where for $\nu\in I^{m},\nu^{\prime}\in I^{n}$, we write their juxtaposition
by $(\nu,\nu^{\prime})\in I^{m+n}$. This defines an injective
$\mathbf{k}$-algebra homomorphism $R_{m}\otimes R_{n}\to R_{m+n}$. By this
map, $R_{m+n}$ has a structure of a right $R_{m}\otimes R_{n}$-module.

For an $R_{m}$-module $M$ and an $R_{n}$-module $N$, we define an
$R_{m+n}$-module $M\circ N$ by

$M\circ N=R_{m+n}\otimes_{R_{m}\otimes R_{n}}M\otimes N.$

$M\circ N$ is called the convolution product of $M$ and $N$. Assume that $M$
is an $R(\beta)$ module, and $N$ is an $R(\gamma)$-module with
$|\beta|=m,|\gamma|=n$. Then $M$ is an $R_{m}$-module by the projection
$R_{m}\to R(\beta)$, and similarly, $N$ is an $R_{n}$-module. The
$R_{m+n}$-module $M\circ N$ has the property that $e(\beta+\gamma)(M\circ
N)=M\circ N$, hence $M\circ N$ is regarded as an $R(\beta+\gamma)$-module. The
$R(\beta+\gamma)$-module $M\circ N$ is defined directly as

$M\circ N=R(\beta+\gamma)e(\beta,\gamma)\otimes_{R(\beta)\otimes
R(\gamma)}M\otimes N,$

where

$e(\beta,\gamma)=\sum_{\begin{subarray}{c}\nu\in I^{\beta},\nu^{\prime}\in
I^{\gamma}\end{subarray}}e(\nu,\nu^{\prime}).$

Let $\beta,\gamma\in Q_{+}$. For an $R(\beta+\gamma)$-module $M$,
$e(\beta,\gamma)M$ has a natural structure of $R(\beta)\otimes
R(\gamma)$-module.

2.3. Let $R(\beta)\operatorname{-Mod}$ be the abelian category of graded
$R(\beta)$-modules. Let $R(\beta)\operatorname{-{\bf gm}}$ be the full
subcategory of $R(\beta)\operatorname{-Mod}$ consisting of finite dimensional
graded $R(\beta)$-modules, and $R(\beta)\operatorname{-{\bf gp}}$ the full
subcategory of $R(\beta)\operatorname{-Mod}$ consisting of finitely generated
graded projective $R(\beta)$-modules. Thus $R(\beta)\operatorname{-{\bf gm}}$
is an abelian category, and $R(\beta)\operatorname{-{\bf gp}}$ is an additive
category. Let $K_{\operatorname{{\bf gm}}}(\beta)$ be the Grothendieck group
of the category $R(\beta)\operatorname{-{\bf gm}}$, and $K_{\operatorname{{\bf
gp}}}(\beta)$ the Grothendieck group of the category
$R(\beta)\operatorname{-{\bf gp}}$. Set

(2.3.1) $\displaystyle K_{\operatorname{{\bf gm}}}=\bigoplus_{\beta\in
Q_{+}}K_{\operatorname{{\bf gm}}}(\beta),\qquad K_{\operatorname{{\bf
gp}}}=\bigoplus_{\beta\in Q_{+}}K_{\operatorname{{\bf gp}}}(\beta).$

By the convolution product, $K_{\operatorname{{\bf gm}}}$ and
$K_{\operatorname{{\bf gp}}}$ are equipped with structures of associative
algebras over $\mathbf{Z}$. The shift operation $M\mapsto qM$ on the graded
module $M$ induces an action of $\mathbf{A}$ on the Grothendieck groups. Thus
$K_{\operatorname{{\bf gm}}}$ and $K_{\operatorname{{\bf gp}}}$ have the
structure of $\mathbf{A}$-algebras.

2.4. Let $P$ be a finitely generated projective $R(\beta)$-module. We define a
dual module $\mathbf{D}P$ by

(2.4.1)
$\mathbf{D}P=\operatorname{Hom}_{R(\beta)}(P,R(\beta))=\bigoplus_{n\in\mathbf{Z}}\operatorname{Hom}_{R(\beta)}(q^{n}P,R(\beta))_{0},$

where $\operatorname{Hom}_{R(\beta)}(-,-)_{0}$ is the space of degree
preserving homomorphisms. Hence $\mathbf{D}P$ is a graded $\mathbf{k}$-vector
space. $R(\beta)$ acts on $\mathbf{D}P$ by

(2.4.2) $x(\lambda)(m)=\lambda(\psi(x)m)$

for $x\in R(\beta),\lambda\in\mathbf{D}P,m\in P$. Then $\mathbf{D}P\in
R(\beta)\operatorname{-{\bf gp}}$.

Let $M$ be a finite dimensional $R(\beta)$-module. Define a dual $\mathbf{D}M$
by

(2.4.3)
$\mathbf{D}M=\operatorname{Hom}_{\mathbf{k}}(M,\mathbf{k})=\bigoplus_{n\in\mathbf{Z}}\operatorname{Hom}_{\mathbf{k}}(q^{n}M,\mathbf{k})_{0}.$

Similarly to (2.4.2), $\mathbf{D}M$ has a structure of $R(\beta)$-module, and
$\mathbf{D}M\in R(\beta)\operatorname{-{\bf gm}}$.

Thus $\mathbf{D}$ gives a contravariant functor $R(\beta)\operatorname{-{\bf
gp}}\to R(\beta)\operatorname{-{\bf gp}}$ or $R(\beta)\operatorname{{\bf
gm}}\to R(\beta)\operatorname{{\bf gm}}$ such that
$\mathbf{D}^{2}\simeq\operatorname{Id}$.

The object $M$ in $R(\beta)\operatorname{-{\bf gp}}$ or in
$R(\beta)\operatorname{-{\bf gm}}$ is said to be self-dual if
$\mathbf{D}M\,\raise 2.0pt\hbox{$\underrightarrow{\sim}$}\,M$.

For $\beta\in Q_{+}$, let $\mathbf{B}_{\beta}$ be the set of $[P]$ in
$K_{\operatorname{{\bf gp}}}(\beta)$, where $P$ runs over self-dual finitely
generated projective indecomposable $R(\beta)$-modules. Let
$\mathbf{B}^{*}_{\beta}$ be the set of $[L]$ in $K_{\operatorname{{\bf
gm}}}(\beta)$, where $L$ runs over self-dual finite dimensional simple
$R(\beta)$-modules. Then $\mathbf{B}_{\beta}$ gives an $\mathbf{A}$-basis of
$K_{\operatorname{{\bf gp}}}(\beta)$, and $\mathbf{B}^{*}_{\beta}$ gives an
$\mathbf{A}$-basis of $K_{\operatorname{{\bf gm}}}(\beta)$. We set
$\mathbf{B}=\bigsqcup_{\beta\in Q_{+}}\mathbf{B}_{\beta}$, and
$\mathbf{B}^{*}=\bigsqcup_{\beta\in Q_{+}}\mathbf{B}^{*}_{\beta}$.

2.5. For any $P\in R(\beta)\operatorname{-{\bf gp}},M\in
R(\beta)\operatorname{-{\bf gm}}$, we define

(2.5.1)
$\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}[P],[M]\operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}}=\operatorname{qdim}_{\mathbf{k}}(P^{\psi}\otimes_{R(\beta)}M)=\sum_{d\in\mathbf{Z}}\dim_{\mathbf{k}}(P^{\psi}\otimes_{R(\beta)}M)_{d}q^{d}.$

Then this defines a $\mathbf{A}$-bilinear map $K_{\operatorname{{\bf
gp}}}(\beta)\times K_{\operatorname{{\bf gm}}}(\beta)\to\mathbf{A}$. Since
$\mathbf{k}$ is algebraically closed, an irreducible module $L$ is absolutely
irreducible. Then for a finitely generated projective indecomposable module
$P$ and a finite dimensional simple module $L$, we have

(2.5.2)
$\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}[P],[L]\operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}}=\begin{cases}1&\quad\text{
if $P$ is the projective cover of $\mathbf{D}L$, }\\\ 0&\quad\text{
otherwise.}\end{cases}$

Thus $\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}\ ,\
\operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}}:K_{\operatorname{{\bf
gp}}}(\beta)\times K_{\operatorname{{\bf gm}}}(\beta)\to\mathbf{A}$ gives a
perfect pairing on $\mathbf{A}$, and $\mathbf{B}_{\beta}$ and
$\mathbf{B}^{*}_{\beta}$ are dual to each other.

On the other hand, for any $P,Q\in R(\beta)\operatorname{-{\bf gp}}$, we
define

(2.5.3)
$([P],[Q])=\sum_{d\in\mathbf{Z}}\dim(P^{\psi}\otimes_{R(\beta)}Q)_{d}q^{d}.$

This gives a well-defined symmetric bilinear form $(\ ,\
):K_{\operatorname{{\bf gp}}}(\beta)\times K_{\operatorname{{\bf
gp}}}(\beta)\to\mathbf{Z}((q))$.

2.6. For $i\in I$, let $L_{i}=\mathbf{k}$ be the (graded) simple
$R(\alpha_{i})$-module, and $P_{i}$ the projective cover of $L_{i}$. Then
$P_{i}=\mathbf{k}[x_{1}]e(i)=R(\alpha_{i})$. For any $n\geq 1$, set

(2.6.1) $L_{i}^{(n)}=q_{i}^{\binom{n}{2}}L_{i}^{\circ n},\qquad
P_{i}^{(n)}=q_{i}^{\binom{n}{2}}P_{i}^{\,\circ n},$

where $L_{i}^{\circ n}=\underbrace{L_{i}\circ\cdots\circ
L_{i}}_{n\text{-times}}$, and similarly for $P_{i}^{\circ n}$. Then
$L_{i}^{(n)}$ is a simple $R(n\alpha_{i})$-module lying in
$R(n\alpha_{i})\operatorname{-{\bf gm}}$, and $P_{i}^{(n)}\in
R(n\alpha_{i})\operatorname{-{\bf gp}}$ is the projective cover of
$L_{i}^{(n)}$.

Assume that $\beta,\in Q_{+}$ and $n\in\mathbf{N}$. We further assume that
$\beta-n\alpha_{i}\in Q_{+}$. We define a functor
$E_{i}^{(n)}:R(\beta)\operatorname{-{\bf gm}}\to
R(\beta-n\alpha_{i})\operatorname{-{\bf gm}}$ by

(2.6.2) $M\mapsto\bigl{(}R(\beta-n\alpha_{i})\circ
P_{i}^{(n)}\bigr{)}^{\psi}\otimes_{R(\beta)}M.$

If $n=1$, we write $E_{i}^{(n)}$ as $E_{i}$. $E_{i}^{(n)}M$ is also written as

$\displaystyle E_{i}^{(n)}M$
$\displaystyle=\bigl{(}R(\beta-n\alpha_{i})\otimes_{\mathbf{k}}(P_{i}^{(n)})^{\psi}\bigr{)}\otimes_{R(\beta-n\alpha_{i})\otimes
R(n\alpha_{i})}M$
$\displaystyle=\bigl{(}R(\beta-n\alpha_{i})\otimes_{\mathbf{k}}(P_{i}^{(n)})^{\psi}\bigr{)}\otimes_{R(\beta-n\alpha_{i})\otimes
R(n\alpha_{i})}e(\beta-n\alpha_{i},n\alpha_{i})M.$

The action of $R(\beta-n\alpha_{i})\otimes R(n\alpha_{i})$ on $M$ is equal to
the $R(\beta-n\alpha_{i})\otimes R(n\alpha_{i})$-module
$e(\beta-n\alpha_{i},n\alpha_{i})M$, and it acts as 0 outside. Since
$P_{i}=R(\alpha_{i})$, we have $R(\beta-\alpha_{i})\circ
P_{i}=R(\beta)e(\beta-\alpha_{i},\alpha_{i})$. Hence

(2.6.3)
$E_{i}M=e(\beta-\alpha_{i},\alpha_{i})R(\beta)\otimes_{R(\beta)}M=e(\beta-\alpha_{i},\alpha_{i})M.$

$E_{i}^{(n)}$ is an exact functor, and it induces an $\mathbf{A}$-homomorphism

${e^{\prime}}_{i}^{(n)}:K_{\operatorname{{\bf gm}}}(\beta)\to
K_{\operatorname{{\bf gm}}}(\beta-n\alpha_{i}).$

For $\beta\in Q_{+},n\in\mathbf{N}$, define an additive functor
$F_{i}^{(n)}:R(\beta)\operatorname{-{\bf gp}}\to
R(\beta+n\alpha_{i})\operatorname{-{\bf gp}}$ by

(2.6.4) $P\mapsto P\circ P_{i}^{(n)}.$

The functor $F_{i}^{(n)}$ induces an $\mathbf{A}$-homomorphism on the
Grothendieck groups,

$f_{i}^{(n)}:K_{\operatorname{{\bf gp}}}(\beta)\to K_{\operatorname{{\bf
gp}}}(\beta+n\alpha_{i}).$

It is known that the operators $f_{i}^{(n)}$ on $K_{\operatorname{{\bf gp}}}$,
and $e_{i}^{\prime(n)}$ on $K_{\operatorname{{\bf gm}}}$ satisfy the following
adjunction relation with respect to the pairing
$\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}\ ,\
\operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}}$. (The proof
is also given as the special case of Lemma 3.10).

(2.6.5)
$\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}f_{i}^{(n)}[P],[M]\operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}}=\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}[P],e^{\prime(n)}_{i}[M]\operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}}.$

The following categorification theorem was proved by Khovanov-Lauda [KL], and
Rouquier [R1].

######  Theorem 2.7 ([KL], [R1]).

There exists an isomorphism of $\mathbf{A}$-algebras

(2.7.1) $\widetilde{\varTheta}_{0}:{}_{\mathbf{A}}\mathbf{U}_{q}^{-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,K_{\operatorname{{\bf gp}}},$

which maps $f_{i}^{(n)}$ to $[P_{i}^{(n)}]$. Moreover,
$\widetilde{\varTheta}_{0}$ is an isometry.

2.8. Let $[\mathbf{k}]$ be the isomorphism class of the trivial representation
$\mathbf{k}$ of $R(0)$, which gives a basis of $K_{\operatorname{{\bf
gm}}}(0)$ and of $K_{\operatorname{{\bf gp}}}(0)$. Let
$1^{*}\in{}_{\mathbf{A}}(\mathbf{U}_{q}^{-})^{*}_{0}$ be the dual basis of
$1\in{}_{\mathbf{A}}(\mathbf{U}_{q}^{-})_{0}$. As a corollary to Theorem 2.7,
we have the following result.

######  Proposition 2.9.

  1. (i)

There exists an isomorphism of $\mathbf{A}$-modules (actually, an anti-algebra
isomorphism),

$\widetilde{\varTheta}:{}_{\mathbf{A}}\mathbf{U}_{q}^{-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,K_{\operatorname{{\bf gp}}},$

which maps $1\in{}_{\mathbf{A}}(\mathbf{U}_{q}^{-})_{0}$ to $[\mathbf{k}]\in
K_{\operatorname{{\bf gp}}}(0)$. $\widetilde{\varTheta}$ commutes with the
actions of $f_{i}^{(n)}$. $\widetilde{\varTheta}$ is an isometry with respect
to the inner product $(\ ,\ )$ on ${}_{\mathbf{A}}\mathbf{U}_{q}^{-}$ and on
$K_{\operatorname{{\bf gp}}}$.

  2. (ii)

There exists an isomorphism of $\mathbf{A}$-modules,

$\widetilde{\varTheta}^{*}:K_{\operatorname{{\bf gm}}}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}}(\mathbf{U}_{q}^{-})^{*}$

which maps $[\mathbf{k}]$ to $1^{*}$. Moreover, $\widetilde{\varTheta}^{*}$
commutes with the actions of $e^{\prime(n)}_{i}$.

###### Proof.

We define $\widetilde{\varTheta}:{}_{\mathbf{A}}\mathbf{U}_{q}^{-}\to
K_{\operatorname{{\bf gp}}}$ by
$\widetilde{\varTheta}=\widetilde{\varTheta}_{0}\circ*$, where
$*:{}_{\mathbf{A}}\mathbf{U}_{q}\to{}_{\mathbf{A}}\mathbf{U}_{q}^{-}$ is the
anti-involution on ${}_{\mathbf{A}}\mathbf{U}_{q}^{-}$ (see 1.2). Since the
action of $f_{i}^{(n)}$ on $K_{\operatorname{{\bf gp}}}$ is defined by the
functor $F_{i}^{(n)}:R(\beta)\operatorname{-{\bf gp}}\to
R(\beta+n\alpha_{i})\operatorname{-{\bf gp}},P\mapsto P\circ P_{i}^{(n)}$, it
is clear from (2.7.1) that $\widetilde{\varTheta}$ commutes with
$f_{i}^{(n)}$. Since the inner product on $\mathbf{U}_{q}^{-}$ is invariant
under the action of $*$, $\widetilde{\varTheta}$ is an isometry. Hence (i)
holds. By using
$\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}\ ,\
\operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}}$, we have an
isomorphism $K_{\operatorname{{\bf
gm}}}(\beta)\simeq\operatorname{Hom}_{\mathbf{A}}(K_{\operatorname{{\bf
gp}}}(\beta),\mathbf{A})$. Thus one can define a map
$\widetilde{\varTheta}^{*}$ as the transpose of $\widetilde{\varTheta}$. Then
by (2.6.5), the map $\widetilde{\varTheta}^{*}$ commutes with
$e_{i}^{\prime(n)}$. Hence (ii) holds. ∎

2.10. From now on , we consider the diagram automorphism $\sigma:I\to I$.
$\sigma$ acts on $\nu=(\nu_{1},\dots,\nu_{n})\in I^{n}$ by
$\sigma\nu=(\sigma(\nu_{1}),\dots,\sigma(\nu_{n}))$. In view of the condition
(iv) in 2.1, the assignment $e(\nu)\mapsto e(\sigma(\nu)),x_{i}\mapsto
x_{i},\tau_{i}\mapsto\tau_{i}$ induces an algebra isomorphism
$R(\beta)\,\raise 2.0pt\hbox{$\underrightarrow{\sim}$}\,R(\sigma(\beta))$.
Thus $\sigma$ gives a functor
$\sigma^{*}:R(\sigma(\beta))\operatorname{-Mod}\to
R(\beta)\operatorname{-Mod}$, where, for an $R(\sigma(\beta))$-module $M$,
$\sigma^{*}M$ is the pull-back of $M$ by the homomorphism $R(\beta)\to
R(\sigma(\beta))$. We consider the functor
$\tau=(\sigma^{-1})^{*}:R(\beta)\operatorname{-Mod}\to
R(\sigma(\beta))\operatorname{-Mod}$ for any $\beta\in Q_{+}$. Then $\tau$
gives functors $R(\beta)\operatorname{-{\bf gm}}\to
R(\sigma(\beta))\operatorname{-{\bf gm}}$, and $R(\beta)\operatorname{-{\bf
gp}}\to R(\sigma(\beta))\operatorname{-{\bf gp}}$. Thus $\tau$ induces
automorphisms on $K_{\operatorname{{\bf gp}}}$ and on $K_{\operatorname{{\bf
gm}}}$, which we also denote by $\tau$. Note that $\tau$ permutes the bases
$\mathbf{B}$ and $\mathbf{B}^{*}$.

Since
$\tau(L_{i}^{(n)})=L_{\sigma(i)}^{(n)},\tau(P_{i}^{(n)})=P_{\sigma(i)}^{(n)}$,
we see that

(2.10.1) $\tau\circ f_{i}^{(n)}=f_{\sigma(i)}^{(n)}\circ\tau\quad\text{ on }\
K_{\operatorname{{\bf gp}}},\qquad\tau\circ
e^{\prime(n)}_{i}=e^{\prime(n)}_{\sigma(i)}\circ\tau\quad\text{ on }\
K_{\operatorname{{\bf gm}}}.$

It follows that the map
$\widetilde{\varTheta}:{}_{\mathbf{A}}\mathbf{U}_{q}^{-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,K_{\operatorname{{\bf gp}}}$ is
compatible with the actions of $\sigma$ and $\tau$. Hence the map
$\widetilde{\varTheta}^{*}:K_{\operatorname{{\bf gm}}}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}}(\mathbf{U}_{q}^{-})^{*}$
is also compatible with the actions of $\tau$ and $\sigma$.

2.11. We follow the setup in 1.6. Hence we assume that $\mathbf{n}$ is a power
of a prime number $\ell$, and set $\mathbf{A}^{\prime}=\mathbf{F}[q,q^{-1}]$.
We define ${}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf
gp}}}=\mathbf{A}^{\prime}\otimes_{\mathbf{A}}K_{\operatorname{{\bf gp}}}$,
${}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf
gm}}}=\mathbf{A}^{\prime}\otimes_{\mathbf{A}}K_{\operatorname{{\bf gm}}}$.
Then $\mathbf{B}$ and $\mathbf{B}^{*}$ give the $\mathbf{A}^{\prime}$-bases of
${}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf gp}}}$, and
${}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf gm}}}$, respectively. The
pairing $\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}\
,\ \operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}}$ is
extended to a perfect $\mathbf{A}^{\prime}$-pairing
${}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf
gp}}}\times{}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf
gm}}}\to\mathbf{A}^{\prime}$.

Let ${}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf gp}}}^{\tau}$ be the set of
$\tau$-fixed elements in ${}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf
gp}}}$. Then ${}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf gp}}}^{\tau}$ is
an $\mathbf{A}^{\prime}$-submodule of
${}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf gp}}}$, and $\\{O(b)\mid
b\in\mathbf{B}\\}$ gives an $\mathbf{A}^{\prime}$-basis of
${}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf gp}}}^{\tau}$. Let
$\mathcal{J}$ be the subset of ${}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf
gp}}}^{\tau}$ consisting of $O(x)$ for
$x\in{}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf gp}}}$ such that $O(x)\neq
x$. Then $\mathcal{J}$ is an $\mathbf{A}^{\prime}$-submodule of
${}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf gp}}}^{\tau}$. We define the
quotient module $\mathcal{V}_{q}$ by

(2.11.1) $\mathcal{V}_{q}={}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf
gp}}}^{\tau}/\mathcal{J}.$

Since the isomorphism
$\widetilde{\varTheta}:{}_{\mathbf{A}}\mathbf{U}_{q}^{-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,K_{\operatorname{{\bf gp}}}$ is
compatible with the actions of $\sigma$ and $\tau$ by 2.10, it induces an
isomorphism ${}_{\mathbf{A}^{\prime}}\mathbf{U}_{q}^{-,\sigma}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}^{\prime}}K^{\tau}_{\operatorname{{\bf
gp}}}$, which maps $\mathbf{J}$ onto $\mathcal{J}$. Thus
$\widetilde{\varTheta}$ induces an isomorphism

(2.11.2) $\varTheta:\mathbf{V}_{q}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,\mathcal{V}_{q}.$

For each $j\in J$, let $j=\\{i_{1},\dots,i_{t}\\}$. We define a functor
$\widetilde{F}^{(n)}_{j}:R\operatorname{-{\bf gp}}\to R\operatorname{-{\bf
gp}}$ by $\widetilde{F}^{(n)}_{j}=F_{i_{t}}^{(n)}\cdots F_{i_{1}}^{(n)}$. This
induces an $\mathbf{A}^{\prime}$-homomorphism
$\widetilde{f}_{j}^{(n)}=f_{i_{t}}^{(n)}\cdots f_{i_{1}}^{(n)}$ on
${}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf gp}}}$. Since $f^{(n)}_{i_{k}}$
commute each other by Theorem 2.7, and $\tau$ permutes $f_{i_{k}}^{(n)}$,
$\widetilde{f}_{j}^{(n)}$ commutes with $\tau$, and acts on $\mathcal{V}_{q}$.
The map $\varTheta:\mathbf{V}_{q}\to\mathcal{V}_{q}$ is compatible with the
action of $\widetilde{f}_{j}^{(n)}$.

Next consider the $\mathbf{A}^{\prime}$-submodule
${}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf gm}}}^{\tau}$ consisting of
$\tau$-fixed elements. We define $\mathcal{J}^{*}$ as the
$\mathbf{A}^{\prime}$-submodule of
${}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf gm}}}^{\tau}$ consisting of
$O(x)$ for $x\in{}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf gm}}}$ such that
$O(x)\neq x$. We define a quotient module $\mathcal{V}_{q}^{*}$ by

(2.11.3) $\mathcal{V}_{q}^{*}={}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf
gm}}}^{\tau}/\mathcal{J}^{*}.$

Similarly to the case of $\widetilde{\varTheta}$, the isomorphism
$\widetilde{\varTheta}^{*}:K_{\operatorname{{\bf gm}}}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}}(\widetilde{\mathbf{U}}_{q}^{-})^{*}$
induces an isomorphism ${}_{\mathbf{A}^{\prime}}K_{\operatorname{{\bf
gm}}}^{\tau}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}^{\prime}}(\widetilde{\mathbf{U}}_{q}^{-})^{*,\sigma}$,
which maps $\mathcal{J}^{*}$ onto $\mathbf{J}^{*}$. Thus
$\widetilde{\varTheta}^{*}$ induces an isomorphism

(2.11.4) $\varTheta^{*}:\mathcal{V}_{q}^{*}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,\mathbf{V}_{q}^{*}.$

For $j\in J$, we define a functor
$\widetilde{E}^{(n)}_{j}:R\operatorname{-{\bf gm}}\to R\operatorname{-{\bf
gm}}$ by $\widetilde{E}_{j}^{(n)}=E_{i_{1}}^{(n)}\cdots E_{i_{t}}^{(n)}$. This
induces an $\mathbf{A}^{\prime}$-homomorphism
$\widetilde{e}^{\prime(n)}_{j}={e^{\prime}}_{i_{1}}^{(n)}\cdots{e^{\prime}}_{i_{t}}^{(n)}$,
commuting with $\tau$. Thus, it induces an action of
$\widetilde{e}^{\prime(n)}_{j}$ on $\mathcal{V}_{q}^{*}$. The map
$\varTheta^{*}$ commutes with the action of
$\widetilde{e^{\prime}}_{j}^{(n)}$.

Let $\underline{\mathbf{B}}_{\bullet}$ be the image of $\mathbf{B}^{\tau}$
onto $\mathcal{V}_{q}$, and $\underline{\mathbf{B}}^{*}_{\bullet}$ the image
of $\mathbf{B}^{*,\tau}$ onto $\mathcal{V}_{q}^{*}$. Then
$\underline{\mathbf{B}}_{\bullet}$ gives a basis of $\mathcal{V}_{q}$, and
$\underline{\mathbf{B}}^{*}_{\bullet}$ gives a basis of $\mathcal{V}_{q}^{*}$,
which are dual to each other.

Summing up the above discussion, we obtain the following.

######  Proposition 2.12.

There exist isomorphisms of $\mathbf{A}^{\prime}$-modules,

$\varTheta:\mathbf{V}_{q}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,\mathcal{V}_{q},\qquad\varTheta^{*}:\mathcal{V}_{q}^{*}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,\mathbf{V}_{q}^{*}.$

The map $\varTheta$ commutes with the actions of $\widetilde{f}_{j}^{(n)}$,
and the map $\varTheta^{*}$ commutes with the actions of
$\widetilde{e^{\prime}}_{j}^{(n)}$.

##  3\. Periodic functors associated to KLR algebras

3.1. First we recall the general theory of periodic functors due to [L2,
11.1]. Let $\mathcal{C}$ be a linear category over a field $\mathbf{k}$,
namely, a category where the space of morphisms between any two objects has
the structure of $\mathbf{k}$-vector space such that the composition of
morphisms is bilinear, and that finite direct sums exist. A functor between
two linear categories is said ro be linear if it preserves the
$\mathbf{k}$-vector space structure.

A linear functor $\sigma^{*}:\mathcal{C}\to\mathcal{C}$ is said to be periodic
if there exists $\mathbf{n}\geq 1$ such that
$(\sigma^{*})^{\mathbf{n}}=\operatorname{id}_{\mathcal{C}}$. Assume that
$\sigma^{*}$ is a periodic functor on $\mathcal{C}$. We define a new category
$\widetilde{\mathcal{C}}$ as follows. The objects of $\widetilde{\mathcal{C}}$
are the pairs $(A,\phi)$, where $A\in\mathcal{C}$, and
$\phi:\sigma^{*}A\,\raise 2.0pt\hbox{$\underrightarrow{\sim}$}\,A$ is an
isomorphism in $\mathcal{C}$ such that the composition satisfies the relation

(3.1.1)
$\phi\circ\sigma^{*}\phi\circ\cdots\circ(\sigma^{*})^{\mathbf{n}-1}\phi=\operatorname{id}_{A}.$

A morphism from $(A,\phi)$ to $(A^{\prime},\phi^{\prime})$ in
$\widetilde{\mathcal{C}}$ is a morphism $f:A\to A^{\prime}$ in $\mathcal{C}$
satisfying the following commutative diagram,

(3.1.2)
$\textstyle{\sigma^{*}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{*}f}$$\scriptstyle{\phi}$$\textstyle{\sigma^{*}A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\prime}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{A^{\prime}.}$

The category $\widetilde{\mathcal{C}}$ is a linear category, hence is an
additive category. $\widetilde{\mathcal{C}}$ is called the category associated
to the periodic functor $\sigma^{*}$ on $\mathcal{C}$

3.2. Let $\widetilde{\mathcal{C}}$ be as in 3.1. Let $\mathbf{n}$ be the
smallest integer such that
$(\sigma^{*})^{\mathbf{n}}=\operatorname{id}_{\mathcal{C}}$. Assume that
$\mathbf{k}$ is an algebraically closed field such that
$\operatorname{ch}\mathbf{k}$ does not divide $\mathbf{n}$. Let
$\zeta_{\mathbf{n}}$ be a primitive $\mathbf{n}$-th root of unity in
$\mathbf{C}$, and fix, once for all, a ring homomorphism
$\mathbf{Z}[\zeta_{\mathbf{n}}]\to\mathbf{k}$, which maps $\zeta_{\mathbf{n}}$
to a primitive $\mathbf{n}$-th root of unity in $\mathbf{k}$. If
$(A,\phi)\in\widetilde{\mathcal{C}}$, then
$(A,\zeta_{\mathbf{n}}\phi)\in\widetilde{\mathcal{C}}$, which we denote by
$\zeta_{\mathbf{n}}(A,\phi)$.

An object $(A,\phi)\in\widetilde{\mathcal{C}}$ is said to be traceless if
there exists an object $B\in\mathcal{C}$, an integer $t\geq 2$ which is a
divisor of $\mathbf{n}$ such that $(\sigma^{*})^{t}B\simeq B$, and an
isomorphism

$A\simeq B\oplus\sigma^{*}B\oplus\cdots\oplus(\sigma^{*})^{t-1}B,$

where, $\phi:\sigma^{*}A\,\raise 2.0pt\hbox{$\underrightarrow{\sim}$}\,A$ is
given by the identity maps $(\sigma^{*})^{k}B\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,(\sigma^{*})^{k}B$ ($1\leq k\leq t-1$)
and an isomorphism $(\sigma^{*})^{t}B\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,B$ on each direct summand, under the
above isomorphism.

For the category $\widetilde{\mathcal{C}}$, $K(\widetilde{\mathcal{C}})$ is
defined as a $\mathbf{Z}[\zeta_{\mathbf{n}}]$-module generated by symbols
$[A,\phi]$ associated to the isomorphism class of the object
$(A,\phi)\in\widetilde{\mathcal{C}}$, subject to the relations

  1. (i)

$[X]=[X^{\prime}]+[X^{\prime\prime}]$ if $X\simeq X^{\prime}\oplus
X^{\prime\prime}$,

  2. (ii)

$[A,\zeta_{\mathbf{n}}\phi]=\zeta_{\mathbf{n}}[A,\phi]$,

  3. (iii)

$[X]=0$ if $X$ is traceless.

In the case where $\widetilde{\mathcal{C}}$ is an abelian category, the
condition (i) is replaced by the condition

(i′) $[X]=[X^{\prime}]+[X^{\prime\prime}]$ if there exists a short exact
sequence

$0\to X^{\prime}\to X\to X^{\prime\prime}\to 0.$

$K(\widetilde{\mathcal{C}})$ is an analogue of the Grothendieck group. In
fact, if $\mathbf{n}=1$, $\widetilde{\mathcal{C}}=\mathcal{C}$, then
$K(\widetilde{\mathcal{C}})$ coincides with the Grothendieck group
$K(\mathcal{C})$. We call $K(\widetilde{\mathcal{C}})$ the Grothendieck group
of $\widetilde{\mathcal{C}}$.

3.3. We consider the KLR algebra $R=\bigoplus_{\beta\in Q_{+}}R(\beta)$
associated to $(X,\sigma)$ as in 2.1. Let $\mathbf{n}$ be the order of
$\sigma$, and assume that $\operatorname{ch}\mathbf{k}$ does not divide
$\mathbf{n}$ (here we don’t give any restriction on $\mathbf{n}$). Assume that
$\beta$ is $\sigma$-stable. Then by 2.10, we obtain a functor
$\sigma^{*}:R(\beta)\operatorname{-Mod}\to R(\beta)\operatorname{-Mod}$. We
apply the discussion in 3.1, and let $\mathscr{C}_{\beta}$ be the category
$\widetilde{\mathcal{C}}$ associated to the periodic functor $\sigma^{*}$ on
$\mathcal{C}=R(\beta)\operatorname{-Mod}$. Then as remarked in [M, Lemma 2.2],
the category $\mathscr{C}_{\beta}$ is equivalent to the category of graded
representations of the algebra

(3.3.1)
$R(\beta)\sharp(\mathbf{Z}/\mathbf{n}\mathbf{Z})=R(\beta)\otimes_{\mathbf{k}}\mathbf{k}[\mathbf{Z}/\mathbf{n}\mathbf{Z}],$

where $\mathbf{k}[\mathbf{Z}/\mathbf{n}\mathbf{Z}]$ is the group algebra of
$\mathbf{Z}/\mathbf{n}\mathbf{Z}$, and
$\mathbf{k}[\mathbf{Z}/\mathbf{n}\mathbf{Z}]$ acts on $R(\beta)$ through the
action of $\sigma^{*}$. Hence $\mathscr{C}_{\beta}$ is an abelian category. We
denote by $\mathscr{P}_{\beta}$ the full subcategory of $\mathscr{C}_{\beta}$
consisting of finitely generated projective objects in $\mathscr{C}_{\beta}$.
Let $\mathscr{L}_{\beta}$ be the full subcategory of $\mathscr{C}_{\beta}$
consisting of $(M,\phi)$ such that $M$ is a finite dimensional
$R(\beta)$-module. Then $\mathscr{P}_{\beta}$ is an additive category and
$\mathscr{L}_{\beta}$ is an abelian category.

Let $K(\mathscr{P}_{\beta})$ (resp. $K(\mathscr{L}_{\beta})$) be the
Grothendieck group of the category $\mathscr{P}_{\beta}$ (resp.
$\mathscr{L}_{\beta}$) as defined in 3.2. $K(\mathscr{P}_{\beta})$ and
$K(\mathscr{L}_{\beta})$ have structures of
$\mathbf{Z}[\zeta_{\mathbf{n}},q,q^{-1}]$-modules, where $q$ acts as the
grading shift as in the case of $K_{\operatorname{{\bf gp}}}(\beta)$ or
$K_{\operatorname{{\bf gm}}}(\beta)$. We set

$K(\mathscr{P})=\bigoplus_{\beta\in
Q_{+}^{\sigma}}K(\mathscr{P}_{\beta}),\qquad
K(\mathscr{L})=\bigoplus_{\beta\in Q_{+}^{\sigma}}K(\mathscr{L}_{\beta}).$

3.4. The contravariant functor $\mathbf{D}$ on $R(\beta)\operatorname{-{\bf
gm}}$ or on $R(\beta)\operatorname{-{\bf gp}}$ given in 2.4 can be extended to
the case of $\mathscr{L}_{\beta}$ or $\mathscr{P}_{\beta}$ as follows. For
$(M,\phi)$ in $\mathscr{P}_{\beta}$ or in $\mathscr{L}_{\beta}$, we define the
dual object by

$\mathbf{D}(M,\phi)=(\mathbf{D}M,\mathbf{D}(\phi)^{-1}),$

where $\mathbf{D}(\phi):\mathbf{D}M\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,\mathbf{D}(\sigma^{*}M)=\sigma^{*}\mathbf{D}M$.
It is clear that $\mathbf{D}$ gives a functor
$\mathscr{L}_{\beta}\to\mathscr{L}_{\beta}$. It was shown in [M, 5] that
$\mathbf{D}$ gives a functor $\mathscr{P}_{\beta}\to\mathscr{P}_{\beta}$.

$(M,\phi)$ in $\mathscr{L}_{\beta}$ or in $\mathscr{P}_{\beta}$ is said to be
self-dual if $\mathbf{D}(M,\phi)\simeq(M,\phi)$ in that category. Note that if
$(M,\phi)$ is a self-dual object, then $\phi$ is uniquely determined by $M$ in
the case where $\mathbf{n}$ is odd, and unique up to $\pm 1$ in the case where
$\mathbf{n}$ is even. In particular,
$(\mathbf{k},\operatorname{id})\in\mathscr{L}_{0}$ is a unique self-dual
simple object if $\mathbf{n}$ is odd, and there exist two self-dual simple
object $(\mathbf{k},\pm\operatorname{id})\in\mathscr{L}_{0}$. We write $[\pm
1]=[\mathbf{k},\pm\operatorname{id}]\in K(\mathscr{L}_{0})$.

Let $\widetilde{\underline{\mathbf{B}}}^{*}_{\beta}$ be the set of elements in
$K(\mathscr{L}_{\beta})$ consisting of isomorphism classes of self-dual
objects $(M,\phi)$ such that $M$ is a finite dimensional simple
$R(\beta)$-module. It was proved in [M, Thm.10.8], by using the crystal
structure of
$\widetilde{\underline{\mathbf{B}}}^{*}=\bigsqcup_{\beta}\widetilde{\underline{\mathbf{B}}}^{*}_{\beta}$,
that there exists a $\mathbf{Z}[\zeta_{\mathbf{n}},q,q^{-1}]$-basis
$\underline{\mathbf{B}}^{*}_{\beta}$ of $K(\mathscr{L}_{\beta})$ such that
$\underline{\mathbf{B}}^{*}_{\beta}\subset\widetilde{\underline{\mathbf{B}}}^{*}_{\beta}$
and that $[1]\in\underline{\mathbf{B}}^{*}_{0}$. Also there exists a
$\mathbf{Z}[\zeta_{\mathbf{n}},q,q^{-1}]$-basis
$\underline{\mathbf{B}}_{\beta}$ of $K(\mathscr{P}_{\beta})$ consisting of
isomorphism classes of finitely generated projective indecomposable self-dual
objects in $\mathscr{P}_{\beta}$ which are projective cover of elements in
$\underline{\mathbf{B}}_{\beta}^{*}$. We set
$\underline{\mathbf{B}}=\bigsqcup_{\beta\in
Q_{+}^{\sigma}}\underline{\mathbf{B}}_{\beta}$, and
$\underline{\mathbf{B}}^{*}=\bigsqcup_{\beta\in
Q_{+}^{\sigma}}\underline{\mathbf{B}}_{\beta}^{*}$.

3.5. We extend the pairing
$\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}\ ,\
\operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}}:K_{\operatorname{{\bf
gp}}}(\beta)\times K_{\operatorname{{\bf gm}}}(\beta)\to\mathbf{A}$ given in
2.5 to our situation. We define a bilinear map
$\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}\ ,\
\operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}}:K(\mathscr{P}_{\beta})\times
K(\mathscr{L}_{\beta})\to\mathbf{Z}[\zeta_{\mathbf{n}},q,q^{-1}]$ by

(3.5.1)
$\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}[P,\phi],[M,\phi^{\prime}]\operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}}=\sum_{d\in\mathbf{Z}}\operatorname{Tr}\,\bigl{(}\phi\otimes\phi^{\prime},(P^{\psi}\otimes_{R(\beta)}M)_{d}\bigr{)}q^{d}.$

Here for given $\phi:\sigma^{*}P\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,P,\phi^{\prime}:\sigma^{*}M\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,M$, $\phi\otimes\phi^{\prime}$ is the
induced map $\sigma^{*}P^{\psi}\otimes\sigma^{*}M\to P^{\psi}\otimes M$. Since
$\sigma^{*}P^{\psi}\otimes\sigma^{*}M$ is canonically isomorphic to
$P^{\psi}\otimes M$, $\phi\otimes\phi^{\prime}$ is regarded as a linear
transformation on $P^{\psi}\otimes M$.

Note that in [M, Lemma 3.1], instead of
$\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}\ ,\
\operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}}$, a sesqui-
linear form $\operatorname{\langle}\ ,\ \operatorname{\rangle}$ is used, which
is defined by

$\operatorname{\langle}\,[P,\phi],[M,\phi^{\prime}]\operatorname{\rangle}=\sum_{d\in\mathbf{Z}}\operatorname{Tr}\,(\sigma_{\phi,\phi^{\prime}},\operatorname{Hom}_{R(\beta)}(P,M)_{d})q^{d}.$

The relation between these forms is given by the isomorphism of graded vector
spaces,

(3.5.2)
$\operatorname{Hom}_{R(\beta)}(P,M)\simeq\mathbf{D}P^{\psi}\otimes_{R(\beta)}M.$

By [M, Thm.11.1], we have the following.

(3.5.3) The pairing
$\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}\ ,\
\operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}}:K(\mathscr{P}_{\beta})\times
K(\mathscr{L}_{\beta})\to\mathbf{Z}[\zeta_{\mathbf{n}},q,q^{-1}]$ gives a
perfect pairing. $\underline{\mathbf{B}}$ and $\underline{\mathbf{B}}^{*}$ are
dual to each other.

The symmetric bilinear form $(\ ,\ ):K_{\operatorname{{\bf gp}}}(\beta)\times
K_{\operatorname{{\bf gp}}}(\beta)\to\mathbf{Z}((q))$ is also extended to our
situation as follows. For $(P,\phi),(Q,\phi^{\prime})\in\mathscr{P}_{\beta}$,
set

(3.5.4)
$([P,\phi],[Q,\phi^{\prime}])=\sum_{d\in\mathbf{Z}}\operatorname{Tr}\,(\phi\otimes\phi^{\prime},(P^{\psi}\otimes_{R(\beta)}Q)_{d})q^{d}.$

Here $\phi\otimes\phi^{\prime}$ gives an isomorphism
$\sigma^{*}P^{\psi}\otimes_{R(\beta)}\sigma^{*}Q\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,P^{\psi}\otimes_{R(\beta)}Q$. By using
the canonical isomorphism
$\sigma^{*}P^{\psi}\otimes_{R(\beta)}\sigma^{*}Q\simeq
P^{\psi}\otimes_{R(\beta)}Q$, we regard $\phi\otimes\phi^{\prime}$ as a linear
transformation on $P^{\psi}\otimes_{R(\beta)}Q$. (3.5.3) induces a symmetric
bilinear pairing $(\ ,\ ):K(\mathscr{P}_{\beta})\times
K(\mathscr{P}_{\beta})\to\mathbf{Z}[\zeta_{\mathbf{n}}]((q))$.

3.6. We extend the convolution product defined in 2.2 to the case of
$\mathscr{C}_{\beta}$. Take
$(M,\phi)\in\mathscr{C}_{\beta},(N,\phi^{\prime})\in\mathscr{C}_{\gamma}$. Set

(3.6.1) $(M,\phi)\circ(N,\phi^{\prime})=(M\circ N,\phi\circ\phi^{\prime}),$

where $\phi\circ\phi^{\prime}$ is given by the composite of the map
$\sigma^{*}M\circ\sigma^{*}N\to M\circ N$ and the canonical isomorphism
$\sigma^{*}(M\circ N)\simeq\sigma^{*}M\circ\sigma^{*}N$.

For $\beta,\gamma\in Q_{+}^{\sigma}$, the automorphisms $\sigma$ on $R(\beta)$
and $R(\gamma)$ induces an automorphism $\sigma$ on $R(\beta)\otimes
R(\gamma)$ by $\sigma(u\otimes w)=\sigma(u)\otimes\sigma(w)$. Thus one can
consider the category of graded representations of $(R(\beta)\otimes
R(\gamma))\sharp(\mathbf{Z}/\mathbf{n}\mathbf{Z})$, which we denote by
$\mathscr{C}_{\beta\sqcup\gamma}$. We can define full subcategories
$\mathscr{L}_{\beta\sqcup\gamma},\mathscr{P}_{\beta\sqcup\gamma}$ of
$\mathscr{C}_{\beta\sqcup\gamma}$ similarly.

The convolution product (3.6.1) gives an induction functor
$\operatorname{Ind}_{\beta,\gamma}:\mathscr{C}_{\beta\sqcup\gamma}\to\mathscr{C}_{\beta+\gamma}$.
On the other hand, for each $(M,\phi)\in\mathscr{C}_{\beta+\gamma}$, we define

(3.6.2)
$\operatorname{Res}_{\beta,\gamma}(M,\phi)=(e(\beta,\gamma)M,\phi^{\prime}),$

where $\operatorname{Res}M=e(\beta,\gamma)M$ is an $R(\beta)\otimes
R(\gamma)$-module given in 2.2. We have the canonical isomoprhism
$\sigma^{*}(\operatorname{Res}M)\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,\operatorname{Res}(\sigma^{*}M)$ since
$e(\beta,\gamma)$ is $\sigma$-invariant, which induces the isomorphism
$\phi^{\prime}:\sigma^{*}(\operatorname{Res}M)\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,\operatorname{Res}M$. Thus
$\operatorname{Res}_{\beta,\gamma}(M,\phi)$ belongs to
$\mathscr{C}_{\beta\sqcup\gamma}$, and we obtain the restriction functor
$\operatorname{Res}_{\beta,\gamma}:\mathscr{C}_{\beta+\gamma}\to\mathscr{C}_{\beta\sqcup\gamma}$.

It is known by [M, Thm. 4.3] that the induction functor and the restriction
functor form an adjoint pair of exact functors between
$\mathscr{C}_{\beta\sqcup\gamma}$ and $\mathscr{C}_{\beta+\gamma}$.

We have isomorphisms of $\mathbf{Z}[\zeta_{\mathbf{n}},q,q^{-1}]$-modules

$K(\mathscr{P}_{\beta})\otimes_{\mathbf{Z}[\zeta_{\mathbf{n}},q^{\pm
1}]}K(\mathscr{P}_{\gamma})\simeq K(\mathscr{P}_{\beta\sqcup\gamma}),\quad
K(\mathscr{L}_{\beta})\otimes_{\mathbf{Z}[\zeta_{\mathbf{n}},q^{\pm
1}]}K(\mathscr{L}_{\gamma})\simeq K(\mathscr{L}_{\beta\sqcup\gamma}),$

by [M, Prop. 4.1], and the induction functors
$\operatorname{Ind}_{\beta,\gamma}$ induces homomorphisms of Grothendieck
groups,

$K(\mathscr{P}_{\beta})\otimes_{\mathbf{Z}[\zeta_{\mathbf{n}},q^{\pm
1}]}K(\mathscr{P}_{\gamma})\to K(\mathscr{P}_{\beta+\gamma}),\quad
K(\mathscr{L}_{\beta})\otimes_{\mathbf{Z}[\zeta_{\mathbf{n}},q^{\pm
1}]}K(\mathscr{L}_{\gamma})\to K(\mathscr{L}_{\beta+\gamma}).$

It is shown that $K(\mathscr{P})$ and $K(\mathscr{L})$ turn out to be
associative algebras over $\mathbf{Z}[\zeta_{\mathbf{n}},q^{\pm 1}]$ with
respect to this product.

3.7. Let $j=\\{i_{1},\dots,i_{t}\\}\in J$, and set
$\alpha_{j}=\alpha_{i_{1}}+\cdots+\alpha_{i_{t}}$, then $\alpha_{j}\in
Q_{+}^{\sigma}$. We consider $R(\alpha_{j})$. Let $\nu=(i_{1},\dots,i_{t})$.
Then $I^{\alpha_{j}}=\\{w\nu\mid w\in S_{t}\\}$, and in particular, for any
$\nu^{\prime}=(\nu_{1}^{\prime},\dots,\nu_{t}^{\prime})\in I^{\alpha_{j}}$,
$\nu_{1}^{\prime},\dots,\nu_{t}^{\prime}$ are all distinct. Moreover,
$Q_{i,i^{\prime}}(u,v)\in\mathbf{k}^{*}$ since
$(\alpha_{i},\alpha_{i^{\prime}})=0$ if $i\neq i^{\prime}\in j$. Hence by the
defining relations on the KLR algebra $R(\alpha_{j})$ im 2.1, we have, for
$1\leq k<t$,

(3.7.1)
$\displaystyle\tau_{k}^{2}e(\nu)\in\mathbf{k}^{*}e(\nu),\qquad\tau_{k}x_{\ell}=x_{s_{k}(\ell)}\tau_{k},\qquad\tau_{k}\tau_{k+1}\tau_{k}=\tau_{k+1}\tau_{k}\tau_{k+1}.$

In particular, $\tau_{w}$ is well-defined for $w\in S_{t}$. Let

$\mathbf{P}[x_{1},\dots,x_{t}]=\bigoplus_{\nu^{\prime}\in
I^{\alpha_{j}}}\mathbf{k}[x_{1},\dots,x_{t}]e(\nu^{\prime})=\bigoplus_{w\in
S_{t}}\mathbf{k}[x_{1},\dots,x_{t}]e(w\nu)$

be the polynomial ring of $R(\alpha_{j})$. By the basis theorem,
$R(\alpha_{j})$ is a free $\mathbf{P}[x_{1},\dots,x_{t}]$-module with basis
$\\{\tau_{w}\mid w\in S_{t}\\}$, namely,

(3.7.2) $R(\alpha_{j})=\bigoplus_{w\in
S_{t}}\mathbf{P}[x_{1},\dots,x_{t}]\tau_{w}.$

For any $f\in\mathbf{k}[x_{1},\dots,x_{t}]$, we have
$\tau_{w}f=r_{w}(f)\tau_{w}$, where $r_{w}\in\mathbf{k}[S_{t}]$ is the element
corresponding to $w\in S_{t}$. Thus
$\tau_{w}\mathbf{P}[x_{1},\dots,x_{t}]=\mathbf{P}[x_{1},\dots,x_{t}]\tau_{w}$.
The left multiplication of $\tau_{w}$ sends
$\mathbf{P}[x_{1},\dots,x_{t}]\tau_{w^{\prime}}$ to
$\mathbf{P}[x_{1},\dots,x_{t}]\tau_{ww^{\prime}}$.

Recall that $R(\alpha_{i_{k}})=P_{i_{k}}=\mathbf{k}[x_{i_{k}}]e(i_{k})$ as
$R(\alpha_{i_{k}})$-modules. Then

$\displaystyle P_{i_{1}}\circ\cdots\circ P_{i_{t}}$
$\displaystyle=R(\alpha_{j})\otimes_{R(\alpha_{i_{1}})\otimes\cdots\otimes
R(\alpha_{i_{t}})}P_{i_{1}}\otimes\cdots\otimes P_{i_{t}}$
$\displaystyle\simeq\bigoplus_{w\in
S_{t}}\tau_{w}\mathbf{k}[x_{1},\dots,x_{t}]e(i_{1},\dots,i_{t})$
$\displaystyle=\bigoplus_{w\in
S_{t}}\mathbf{k}[x_{1},\dots,x_{t}]\tau_{w}e(i_{1},\dots,i_{t}).$

Set $P_{j}=P_{i_{1}}\circ\cdots\circ P_{i_{t}}$. Then $P_{j}$ is written as
$P_{j}=\bigoplus_{w\in S_{t}}\mathbf{k}[x_{1},\dots,x_{t}][w]$, where
$[w]=\tau_{w}e(i_{1},\dots,i_{t})$. Thus $P_{j}$ is a free
$\mathbf{k}[x_{1},\dots x_{t}]$-module with basis $[w]$, where
$e(\nu^{\prime})$ acts on $[w]$ by 1 if $\nu^{\prime}=w\nu$, and by 0
otherwise, $\tau_{i}$ acts on $[w]$ by $[s_{i}(w)]$.

We define a quotient module $L_{j}$ of $P_{j}$ by

$L_{j}=P_{j}/\operatorname{\langle}x_{k}P_{j}\mid 1\leq k\leq
t\operatorname{\rangle}.$

Then $L_{j}\simeq\bigoplus_{w\in S_{t}}\mathbf{k}[w]\simeq
L_{i_{1}}\circ\cdots\circ L_{i_{t}}$, where $x_{k}$ acts on $[w]$ as 0. Thus
$P_{j}$ and $L_{j}$ are isomorphic to the ones defined in [M, 7]. $L_{j}$ is a
simple $R(\alpha_{j})$-module lying in $R(\alpha_{j})\operatorname{-{\bf
gm}}$, and $P_{j}$ is the projective cover of $L_{j}$ lying in
$R(\alpha_{j})\operatorname{-{\bf gp}}$.

We have $P_{j}=R(\alpha_{j})e(\nu)$ with $\nu=(i_{1},\dots,i_{t})$, and

(3.7.3) $R(\alpha_{j})=\bigoplus_{\nu^{\prime}\in
I^{\alpha_{j}}}R(\alpha_{j})e(\nu^{\prime})=\bigoplus_{\nu^{\prime}\in
I^{\alpha_{j}}}P_{j,\nu^{\prime}},$

where $P_{j,\nu^{\prime}}=\bigoplus_{w\in
S_{t}}\mathbf{k}[x_{1},\dots,x_{t}]\tau_{w}e(\nu^{\prime})$, which is
isomorphic to $P_{j}$ as $R(\alpha_{j})$-modules.

Since $\alpha_{j}$ is $\sigma$-stable, one can consider the category
$\mathscr{C}_{\alpha_{j}},\mathscr{L}_{\alpha_{j}}$ and
$\mathscr{P}_{\alpha_{j}}$. We define $\phi_{j}:\sigma^{*}L_{j}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,L_{j}$ by $\phi_{j}[w]=[\sigma(w)]$
(here $[w]$ is identified with the permutation
$(i_{1}^{\prime},\dots,i_{t}^{\prime})$ of $(i_{1},\dots,i_{t})$ on which
$\sigma$ acts). We define $L(j)=(L_{j},\phi_{j})$, which is a simple object in
$\mathscr{L}_{\alpha_{j}}$. We define $P(j)$ the projective cover of $L(j)$,
which lies in $\mathscr{P}_{\alpha_{j}}$. $P(j)$ is given as
$P(j)=(P_{j},\phi_{j})$, where $\phi_{j}:\sigma^{*}P_{j}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,P_{j}$ is the extension of
$\phi_{j}:\sigma^{*}L_{j}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,L_{j}$.

3.8. For $n\geq 1$, we define $L(j)^{(n)}\in\mathscr{L}_{n\alpha_{j}}$ by

(3.8.1) $L(j)^{(n)}=q_{j}^{\binom{n}{2}}L(j)^{\circ n}.$

By [M, Lemma 7.2], $L(j)^{(n)}$ is a self-dual simple object in
$\mathscr{L}_{n\alpha_{j}}$. Note that $L(j)^{(n)}=(L_{j}^{(n)},\phi_{nj})$,
where $L_{j}^{(n)}=q_{j}^{\binom{n}{2}}L_{j}^{\circ n}$, and
$\phi_{nj}:\sigma^{*}L_{j}^{(n)}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,L_{j}^{(n)}$ is induced from
$\phi_{j}:\sigma^{*}L_{j}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,L_{j}$. Let $P(j)^{(n)}$ be the
projective cover of $L(j)^{(n)}$. Then

(3.8.2) $P(j)^{(n)}=q_{j}^{\binom{n}{2}}P(j)^{\circ n}.$

We have $P(j)^{(n)}=(P_{j}^{(n)},\phi_{nj})$, where
$P_{j}^{(n)}=q_{j}^{\binom{n}{2}}P_{j}^{\circ n}$ is the projective cover of
$L_{j}^{(n)}$, and $\phi_{nj}:\sigma^{*}P_{j}^{(n)}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,P_{j}^{(n)}$ is the extension of
$\phi_{nj}:\sigma^{*}L_{j}^{(n)}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,L_{j}^{(n)}$.

3.9. Assume that $\beta\in Q_{+}^{\sigma}$ and $j\in J$. We define a functor
$F_{j}^{(n)}:\mathscr{C}_{\beta}\to\mathscr{C}_{\beta+n\alpha_{j}}$ by

(3.9.1) $F_{j}^{(n)}:P\mapsto P\circ P(j)^{(n)}.$

If we write $P=(P_{0},\phi)$, then the functor is given by

$F_{j}^{(n)}P=\bigl{(}R(\beta+n\alpha_{j})\otimes_{R(\beta)\otimes
R(n\alpha_{j})}(P_{0}\otimes P_{j}^{(n)}),\phi^{\prime}\bigr{)},$

where $\phi^{\prime}$ is obtained from
$\phi\otimes\phi_{nj}:\sigma^{*}(P_{0})\otimes\sigma^{*}(P_{j}^{(n)})\to
P_{0}\otimes P_{j}^{(n)}$. By 3.6, $F_{j}^{(n)}$ gives a functor
$\mathscr{P}_{\beta}\mapsto\mathscr{P}_{\beta+n\alpha_{j}}$. It induces an
operator $f_{j}^{(n)}:K(\mathscr{P}_{\beta})\to
K(\mathscr{P}_{\beta+n\alpha_{j}})$.

On the other hand, assume that $\beta\in Q_{+}^{\sigma}$ such that
$\beta-n\alpha_{j}\in Q_{+}^{\sigma}$ for $j\in J$. We define a functor
$E_{j}^{(n)}:\mathscr{L}_{\beta}\to\mathscr{L}_{\beta-n\alpha_{j}}$ as
follows; write $P(j)^{(n)}=(P_{j}^{(n)},\phi_{nj})$ as in 3.8, $P_{j}^{(n)}$
is the projective $R(n\alpha_{j})$-module. Let
$M=(M_{0},\phi)\in\mathscr{L}_{\beta}$, where $M_{0}\in
R(\beta)\operatorname{-{\bf gm}}$. Set

(3.9.3) $E_{j}^{(n)}:(M_{0},\phi)\mapsto\bigl{(}(R(\beta-n\alpha_{j})\circ
P_{j}^{(n)})^{\psi}\otimes_{R(\beta)}M_{0},\phi^{\prime}\bigr{)}=(M_{1},\phi^{\prime}),$

where $\phi^{\prime}$ is induced naturally from $\phi$ and $\phi_{nj}$. As in
2.6, $M_{0}\mapsto M_{1}$ is an exact functor. Hence $E_{j}^{(n)}$ is an exact
functor. It induces an operator
${e_{j}^{\prime}}^{(n)}:K(\mathscr{L}_{\beta})\to
K(\mathscr{L}_{\beta-n\alpha_{j}})$.

In the case where $n=1$, for $M=(M_{0},\phi_{0})\in\mathscr{C}_{\beta}$,
$E_{j}M=(\widetilde{E}_{j}M_{0},\phi)$, where $\widetilde{E}_{j}M_{0}$ is
given by

$\widetilde{E}_{j}M_{0}=(R(\beta-\alpha_{j})\otimes_{\mathbf{k}}P_{j}^{\psi})\otimes_{R(\beta-\alpha_{j})\otimes
R(\alpha_{j})}e(\beta-\alpha_{j},\alpha_{j})M_{0}.$

Note that $P_{j}=R(\alpha_{j})e(\nu)$ with $\nu=(i_{1},\dots,i_{t})$ by
(3.7.3). Thus $P_{j}^{\psi}=e(\nu)R(\alpha_{j})$, and so

(3.9.4)
$\widetilde{E}_{j}M_{0}=e(\beta-\alpha_{j},\alpha_{i_{1}},\dots,\alpha_{i_{t}})M_{0}=E_{i_{1}}\cdots
E_{i_{t}}M_{0},$

where $e(\beta-\alpha_{j},\alpha_{i_{1}},\dots,\alpha_{i_{t}})$ is defined
similarly to $e(\beta,\gamma)$ in 2.2, and $E_{i_{k}}$ is the functor defined
in (2.6.3)

The following result is a generalization of (2.6.5).

######  Lemma 3.10.

Assume that $j\in J$, $\beta\in Q_{+}^{\sigma}$. Then we have

$\displaystyle\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}f_{j}^{(n)}[P],[M]\operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}}$
$\displaystyle=\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}[P],e_{j}^{\prime(n)}[M]\operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}},\qquad(P\in\mathscr{P}_{\beta},M\in\mathscr{L}_{\beta+n\alpha_{j}}).$

###### Proof.

We define functors

$\displaystyle\widetilde{F}_{j}^{(n)}$
$\displaystyle:R(\beta)\operatorname{-{\bf gp}}\to
R(\beta+n\alpha_{j})\operatorname{-{\bf gp}},\quad P_{0}\mapsto P_{0}\circ
P_{j}^{(n)},$ $\displaystyle\widetilde{E}_{j}^{(n)}$
$\displaystyle:R(\beta+n\alpha_{j})\operatorname{-{\bf gm}}\to
R(\beta)\operatorname{-{\bf gm}},\quad M_{0}\mapsto(R(\beta)\circ
P_{j}^{(n)})^{\psi}\otimes_{R(\beta+n\alpha_{j})}M_{0}.$

Let $P=(P_{0},\phi)\in\mathscr{P}_{\beta}$,
$M=(M_{0},\phi^{\prime})\in\mathscr{L}_{\beta+n\alpha_{j}}$. We have

(3.10.1) $\displaystyle(\widetilde{F}_{j}^{(n)}$ $\displaystyle
P_{0})^{\psi}\otimes_{R(\beta+n\alpha_{j})}M_{0}$
$\displaystyle\simeq\biggl{(}P_{0}^{\psi}\otimes_{R(\beta)}(R(\beta)\otimes_{\mathbf{k}}(P_{j}^{(n)})^{\psi})_{R(\beta)\otimes
R(n\alpha_{j})}R(\beta+n\alpha_{j})\biggr{)}\otimes_{R(\beta+n\alpha_{j})}M_{0}$
$\displaystyle=P_{0}^{\psi}\otimes_{R(\beta)}\widetilde{E}_{j}^{(n)}M_{0}.$

Let $Z$ be the left hand side of (3.10.1). The isomorphisms
$\phi:\sigma^{*}P_{0}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,P_{0},\phi^{\prime}:\sigma^{*}M_{0}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,M_{0}$ induce the isomorphism
$\sigma^{*}Z\,\raise 2.0pt\hbox{$\underrightarrow{\sim}$}\,Z$, which coincides
with the isomorphism
$\sigma^{*}(P_{0}^{\psi}\otimes_{R(\beta)}\widetilde{E}_{j}^{(n)}M_{0})\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,P_{0}^{\psi}\otimes_{R(\beta)}\widetilde{E}_{j}^{(n)}M_{0}$
induced from $\phi$ and $\phi^{\prime}$, through the isomorphisms in (3.10.1).
Thus the lemma is proved. ∎

3.11. As discussed in 3.6, $K(\mathscr{P})$ is an associative algebra over
$\mathbf{Z}[\zeta_{\mathbf{n}},q,q^{-1}]$ with the basis
$\underline{\mathbf{B}}$. We define $\widetilde{\mathbf{A}}$ as the smallest
subring of $\mathbf{Z}[\zeta_{\mathbf{n}},q,q^{-1}]$ containing $\mathbf{A}$
such that all the structure constants of the algebra $K(\mathscr{P})$ with
respect to the basis $\underline{\mathbf{B}}$ lie in $\widetilde{\mathbf{A}}$.

Let ${}_{\widetilde{\mathbf{A}}}K(\mathscr{P})$ be the
$\widetilde{\mathbf{A}}$-submodule of $K(\mathscr{P})$ spanned by
$\underline{\mathbf{B}}$. Then ${}_{\widetilde{\mathbf{A}}}K(\mathscr{P})$ is
the $\widetilde{\mathbf{A}}$-subalgebra of $K(\mathscr{P})$. Set
${}_{\widetilde{\mathbf{A}}}\underline{\mathbf{U}}_{q}^{-}=\widetilde{\mathbf{A}}\otimes_{\mathbf{A}}\underline{\mathbf{U}}_{q}^{-}$.
The following result was proved by McNamara [M].

######  Theorem 3.12 ([M, Thm. 6.1]).

There exists a unique $\widetilde{\mathbf{A}}$-algebra isomorphism

(3.12.1)
$\gamma_{0}:{}_{\widetilde{\mathbf{A}}}\underline{\mathbf{U}}_{q}^{-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\widetilde{\mathbf{A}}}K(\mathscr{P})$

such that $\gamma_{0}(f_{j}^{(n)})=[P(j)^{(n)}]$. Moreover, $\gamma_{0}$ is an
isometry.

The following result is a generalization of Proposition 2.9.

######  Proposition 3.13.

  1. (i)

Consider the actions of $f_{j}^{(n)}$ on $K(\mathscr{P})$ defined in (3.9.1),
and on $\underline{\mathbf{U}}_{q}^{-}$ by the left multiplication. Then there
exists an isomorphism of $\widetilde{\mathbf{A}}$-modules (actually anti-
algebra isomorphism)

(3.13.1)
$\gamma:{}_{\widetilde{\mathbf{A}}}\underline{\mathbf{U}}_{q}^{-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\widetilde{\mathbf{A}}}K(\mathscr{P})$

such that $\gamma$ maps
$1\in{}_{\widetilde{\mathbf{A}}}(\underline{\mathbf{U}}_{q}^{-})_{0}$ to
$[\mathbf{k},\operatorname{id}]\in{}_{\widetilde{\mathbf{A}}}K(\mathscr{P}_{0})$,
commuting with $f_{j}^{(n)}$. Moreover, $\gamma$ is an isometry.

  2. (ii)

Consider the actions of $e_{j}^{\prime(n)}$ on $K(\mathscr{L})$ defined in
(3.9.2), and on $(\underline{\mathbf{U}}_{q}^{-})^{*}$ as in 1.9. Then there
exists an isomorphism of $\widetilde{\mathbf{A}}$-modules,

(3.13.2) $\gamma^{*}:{}_{\widetilde{\mathbf{A}}}K(\mathscr{L})\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\widetilde{\mathbf{A}}}(\underline{\mathbf{U}}_{q}^{-})^{*}$

such that $\gamma^{*}$ maps
$[\mathbf{k},\operatorname{id}]\in{}_{\widetilde{\mathbf{A}}}K(\mathscr{L}_{0})$
to
$1^{*}\in{}_{\widetilde{\mathbf{A}}}(\underline{\mathbf{U}}_{q}^{-})^{*}_{0}$,
commuting with $e_{j}^{\prime(n)}$.

###### Proof.

The proof is similar to that of Proposition 2.9. We define
$\gamma:{}_{\widetilde{\mathbf{A}}}\underline{\mathbf{U}}_{q}^{-}\to{}_{\widetilde{\mathbf{A}}}K(\mathscr{P})$
by $\gamma=\gamma_{0}\circ*$, where
$*:{}_{\widetilde{\mathbf{A}}}\underline{\mathbf{U}}_{q}\to{}_{\widetilde{\mathbf{A}}}\underline{\mathbf{U}}_{q}^{-}$
is the anti-involution. Since the action of $f_{j}^{(n)}$ on $K(\mathscr{P})$
is defined by (3.9.1), (i) follows from Theorem 3.12. If we define
$\gamma^{*}$ as the transpose of $\gamma$, (ii) follows from (i) by Lemma
3.10. ∎

3.14.
$f_{j}^{(n)}\in{}_{\widetilde{\mathbf{A}}}\underline{\mathbf{U}}_{q}^{-}$
satisfies the relation $[n]^{!}_{j}f_{j}^{(n)}=f_{j}^{n}$. Hence by (3.13.1),
the operator $f_{j}^{(n)}$ on ${}_{\widetilde{\mathbf{A}}}K(\mathscr{P})$
satisfies a similar relation. Then by using Lemma 3.10, we obtain the
corresponding relation on the operator ${e^{\prime}}_{j}^{(n)}$ on
${}_{\widetilde{\mathbf{A}}}K(\mathscr{L})$. Thus we have

(3.14.1)
$[n]^{!}_{j}f_{j}^{(n)}=f_{j}^{n},\qquad[n]^{!}_{j}{e^{\prime}}_{j}^{(n)}={e^{\prime}}_{j}^{n}$

as operators on ${}_{\widetilde{\mathbf{A}}}K(\mathscr{P})$ and on
${}_{\widetilde{\mathbf{A}}}K(\mathscr{L})$.

3.15. In [M, Lemma 11.3], McNamara showed that
$\widetilde{\mathbf{A}}\subset\mathbf{Z}[\zeta_{\mathbf{n}}+\zeta_{\mathbf{n}}^{-1},q,q^{-1}]$,
and paused a question whether $\widetilde{\mathbf{A}}$ coincides with
$\mathbf{A}$ or not ([M, Question 11.2]). In this paper, we show that this
certainly holds, namely,

######  Theorem 3.16.

The structure constants in $K(\mathscr{P})$ with respect to the basis
$\underline{\mathbf{B}}$ are all contained in $\mathbf{A}$. Thus the
isomorphisms (3.12.1), (3.13.1) and (3.13.2) can be replaced by the
isomorphisms

$\gamma_{0},\gamma:{}_{\mathbf{A}}\underline{\mathbf{U}}_{q}^{-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}}K(\mathscr{P})\quad\text{
and}\quad\gamma^{*}:{}_{\mathbf{A}}K(\mathscr{L})\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}}(\underline{\mathbf{U}}_{q}^{-})^{*}.$

The theorem will be proved in the next section.

##  4\. The proof of Theorem 3.16

4.1. We keep the setup in Section 3. In particular, $\mathbf{n}$ is the order
of $\sigma:I\to I$, and $\zeta_{\mathbf{n}}$ is a primitive $\mathbf{n}$-th
root of unity in $\mathbf{C}$. If $\mathbf{n}=1$, Theorem 3.16 certainly holds
by Theorem 2.7. We prove the theorem by induction on $\mathbf{n}$. Hence we
assume that $\mathbf{n}>1$, and that the theorem holds for
$\mathbf{n}^{\prime}<\mathbf{n}$. We write
$\mathbf{n}=\mathbf{n}_{1}\mathbf{n}_{2}$, where $\mathbf{n}_{2}>1$ is a power
of a prime number $\ell$, and $\mathbf{n}_{1}$ is prime to $\mathbf{n}_{2}$.
Choose $a,b\in\mathbf{Z}$ such that $1=\mathbf{n}_{1}a+\mathbf{n}_{2}b$, and
set $\sigma_{1}=\sigma^{\mathbf{n}_{2}b},\sigma_{2}=\sigma^{\mathbf{n}_{1}a}$.
Thus $\sigma=\sigma_{1}\sigma_{2}$, and the order of $\sigma_{1}$ (resp.
$\sigma_{2}$) is $\mathbf{n}_{1}$ (resp. $\mathbf{n}_{2}$). We define
$\zeta_{\mathbf{n}_{1}}=(\zeta_{\mathbf{n}})^{\mathbf{n}_{2}b}$ and
$\zeta_{\mathbf{n}_{2}}=(\zeta_{\mathbf{n}})^{\mathbf{n}_{1}a}$. Then
$\zeta_{\mathbf{n}_{1}}$ (resp. $\zeta_{\mathbf{n}_{2}}$) is a primitive
$\mathbf{n}_{1}$-th root (resp. $\mathbf{n}_{2}$-th root) of unity in
$\mathbf{C}$. The ring homomorphism
$\mathbf{Z}[\zeta_{\mathbf{n}}]\to\mathbf{k}$ restricts to the ring
homomorphism $\mathbf{Z}[\zeta_{\mathbf{n}_{1}}]\to\mathbf{k}$, or to
$\mathbf{Z}[\zeta_{\mathbf{n}_{2}}]\to\mathbf{k}$.

4.2. Assume that $\beta\in Q_{+}^{\sigma}$. Let $\mathscr{C}_{\beta}$ be the
abelian category associated to $\sigma^{*}:R(\beta)\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,R(\beta)$ as in 3.3, namely, the objects
in $\mathscr{C}_{\beta}$ are $(M,\phi)$, where $M$ is a graded
$R(\beta)$-module, and $\phi:\sigma^{*}M\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,M$ satisfies the conditions in 3.1.
Since $\sigma_{1}$ is an admissible diagram automorphism on $X$, and $\beta$
is $\sigma_{1}$-invariant, one can define the abelian category
$\mathscr{C}^{\prime}_{\beta}$ by replacing $\sigma$ by $\sigma_{1}$.

For each $(M,\phi)\in\mathscr{C}^{\prime}_{\beta}$, we define an object
$\sigma_{2}^{*}(M,\phi)=(M^{\prime},\phi^{\prime})\in\mathscr{C}^{\prime}_{\beta}$
as follows; set $M^{\prime}=\sigma_{2}^{*}M$, then
$\sigma_{2}^{*}(\phi):\sigma_{2}^{*}\sigma_{1}^{*}M\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,\sigma_{2}^{*}M$ induces an isomorphism
$\sigma_{1}^{*}M^{\prime}=\sigma_{1}^{*}\sigma_{2}^{*}M=\sigma_{2}^{*}\sigma_{1}^{*}M\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,\sigma_{2}^{*}M=M^{\prime}$, which we
denote by $\phi^{\prime}:\sigma_{1}^{*}M^{\prime}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,M^{\prime}$. Clearly
$\phi^{\prime}=\sigma_{2}^{*}(\phi)$ satisfies the relation (3.1.1), and we
obtain $(M^{\prime},\phi^{\prime})\in\mathscr{C}^{\prime}_{\beta}$. The
functor
$\sigma_{2}^{*}:\mathscr{C}^{\prime}_{\beta}\to\mathscr{C}^{\prime}_{\beta}$
is a periodic functor on the linear category $\mathscr{C}^{\prime}_{\beta}$.
Thus we define $\mathscr{C}^{\prime\prime}_{\beta}$ as the category associated
to the periodic functor $\sigma_{2}^{*}$ (see 3.1).

The category $\mathscr{C}_{\beta}^{\prime\prime}$ is described as follows;
objects are triples $(M,\phi_{1},\phi_{2})$ where
$(M,\phi_{1})\in\mathscr{C}^{\prime}_{\beta}$, and
$\phi_{2}:\sigma_{2}^{*}(M,\phi_{1})\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,(M,\phi_{1})$ in
$\mathscr{C}^{\prime}_{\beta}$ satisfies the condition that
$\phi_{2}\circ\sigma_{2}^{*}\phi_{2}\circ\cdots\circ(\sigma_{2}^{*})^{\mathbf{n}_{2}-1}\phi_{2}=\operatorname{id}_{\mathscr{C}^{\prime}_{\beta}}$.
A morphism from $(M,\phi_{1},\phi_{2})$ to
$(M^{\prime},\phi_{1}^{\prime},\phi^{\prime}_{2})$ is defined by using a
similar diagram as in (3.1.2) by replacing $M$ by $(M,\phi_{1})$ and $\sigma$
by $\sigma_{2}$.

By 3.3, $\mathscr{C}_{\beta}$ is equivalent to the category of graded
representations of the algebra
$R(\beta)\sharp(\mathbf{Z}/\mathbf{n}\mathbf{Z})$. Similarly,
$\mathscr{C}^{\prime}_{\beta}$ is equivalent to the category of graded
representations of the algebra
$R(\beta)\sharp(\mathbf{Z}/\mathbf{n}_{1}\mathbf{Z})$. By a similar argument,
$\mathscr{C}^{\prime\prime}_{\beta}$ is equivalent to the category of graded
representations of the algebra
$\bigl{(}R(\beta)\sharp(\mathbf{Z}/\mathbf{n}_{1}\mathbf{Z})\bigr{)}\sharp(\mathbf{Z}/\mathbf{n}_{2}\mathbf{Z})$.
Since
$\mathbf{k}[\mathbf{Z}/\mathbf{n}\mathbf{Z}]\simeq\mathbf{k}[\mathbf{Z}/\mathbf{n}_{1}\mathbf{Z}]\otimes_{\mathbf{k}}\mathbf{k}[\mathbf{Z}/\mathbf{n}_{2}\mathbf{Z}]$,
we see that

(4.2.1)
$\bigl{(}R(\beta)\sharp(\mathbf{Z}/\mathbf{n}_{1}\mathbf{Z})\bigr{)}\sharp(\mathbf{Z}/\mathbf{n}_{2}\mathbf{Z})\simeq
R(\beta)\sharp(\mathbf{Z}/\mathbf{n}\mathbf{Z}).$

Assume given $(M,\phi_{1},\phi_{2})\in\mathscr{C}^{\prime\prime}_{\beta}$.
$\phi_{2}:\sigma_{2}^{*}(M,\phi_{1})\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,(M,\phi_{1})$ induces an isomorphism
$\sigma_{2}^{*}M\,\raise 2.0pt\hbox{$\underrightarrow{\sim}$}\,M$, which we
also denote by $\phi_{2}$. Then
$\phi_{2}\circ\sigma_{2}^{*}(\phi_{1}):\sigma_{2}^{*}\sigma_{1}^{*}M\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,\sigma_{2}^{*}M\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,M$ gives an isomorphism
$\sigma^{*}M\,\raise 2.0pt\hbox{$\underrightarrow{\sim}$}\,M$, and the
correspondence
$(M,\phi_{1},\phi_{2})\mapsto(M,\phi_{2}\circ\sigma_{2}^{*}(\phi_{1}))$ gives
a functor $\mathscr{C}_{\beta}^{\prime\prime}\to\mathscr{C}_{\beta}$. By
(4.2.1), this gives an equivalence of the categories
$\mathscr{C}_{\beta}^{\prime\prime}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,\mathscr{C}_{\beta}$.

Recall that $\mathscr{P}_{\beta}$ is the full subcategory of finitely
generated graded projective objects in $\mathscr{C}_{\beta}$, which is
equivalent to the subcategory of finitely generated projective
$R(\beta)\sharp(\mathbf{Z}/\mathbf{n}\mathbf{Z})$-modules. Similarly, the full
subcategory $\mathscr{P}^{\prime}_{\beta}$ of $\mathscr{C}^{\prime}_{\beta}$,
and the full subcategory $\mathscr{P}_{\beta}^{\prime\prime}$ of
$\mathscr{C}^{\prime\prime}_{\beta}$ are defined. Let $\mathscr{L}_{\beta}$ be
the full subcategory of finite dimensional graded objects in
$\mathscr{C}_{\beta}$, namely, the full subcategory of finite dimensional
graded $R(\beta)\sharp(\mathbf{Z}/\mathbf{n}\mathbf{Z})$-modules. The full
subcategory $\mathscr{L}^{\prime}_{\beta}$ of $\mathscr{C}^{\prime}_{\beta}$,
and the full subcategory $\mathscr{L}_{\beta}^{\prime\prime}$ of
$\mathscr{C}^{\prime\prime}_{\beta}$ are defined similarly.

Summing up the above discussion, we obtain the following lemma.

######  Lemma 4.3.

The category $\mathscr{C}^{\prime\prime}_{\beta}$ (resp.
$\mathscr{P}^{\prime\prime}_{\beta}$, $\mathscr{L}^{\prime\prime}_{\beta}$) is
equivalent to the category $\mathscr{C}_{\beta}$ (resp. $\mathscr{P}_{\beta}$,
$\mathscr{L}_{\beta}$).

4.4. By the discussion in 3.2, if $(M,\phi)\in\mathscr{C}_{\beta}$, then
$(M,\zeta_{\mathbf{n}}\phi)\in\mathscr{C}_{\beta}$, which is denoted by
$\zeta_{\mathbf{n}}(M,\phi)$. Similarly, $\zeta_{\mathbf{n}_{1}}(M,\phi_{1})$
is defined for $(M,\phi_{1})\in\mathscr{C}^{\prime}_{\beta}$. For a given
$(M,\phi_{1},\phi_{2})\in\mathscr{C}_{\beta}^{\prime\prime}$, we have
$(M,\zeta_{\mathbf{n}_{1}}\phi_{1},\zeta_{\mathbf{n}_{2}}\phi_{2})\in\mathscr{C}_{\beta}^{\prime\prime}$,
which we denote by $\zeta_{\mathbf{n}}(M,\phi_{1},\phi_{2})$. Thus under the
category equivalence $\mathscr{C}_{\beta}^{\prime\prime}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,\mathscr{C}_{\beta}$, the action of
$\zeta_{\mathbf{n}}$ on $(M,\phi_{1},\phi_{2})$ corresponds to the action of
that on $(M,\phi_{2}\circ\sigma_{2}^{*}(\phi_{1}))$.

The notion of traceless elements in $\mathscr{C}_{\beta}$, and the
Grothendieck groups $K(\mathscr{P}_{\beta})$ and $K(\mathscr{L}_{\beta})$ were
defined in 3.2 and 3.3. This definition works also for
$\mathscr{C}^{\prime}_{\beta}$, and the Grothendieck group
$K(\mathscr{P}^{\prime}_{\beta}),K(\mathscr{L}^{\prime}_{\beta})$ are defined.

The notion of traceless elements for $\mathscr{C}^{\prime\prime}_{\beta}$ is
translated to our situation as follows. An object
$(A,\phi_{1},\phi_{2})\in\mathscr{C}^{\prime\prime}_{\beta}$ is said to be
traceless if there exists an object $(M,\phi)\in\mathscr{C}^{\prime}_{\beta}$,
an integer $t_{2}\geq 1$ which is a divisor of $\mathbf{n}_{2}$ such that
$(\sigma_{2}^{*})^{t_{2}}(M,\phi)\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,(M,\phi)$, and an isomorphism

(4.4.1)
$(A,\phi_{1})\simeq(M,\phi)\oplus\sigma_{2}^{*}(M,\phi)\oplus\cdots\oplus(\sigma_{2}^{*})^{t_{2}-1}(M,\phi),$

where $\phi_{2}:\sigma_{2}^{*}(A,\phi_{1})\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,(A,\phi_{1})$ is defined as in 3.2.
Furthermore, we assume that there exists an $R(\beta)$-module $M_{1}$, an
integer $t_{1}\geq 1$ which is a divisor of $\mathbf{n}_{1}$ such that
$(\sigma_{1}^{*})^{t_{1}}M_{1}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,M_{1}$, and an isomorphism

(4.4.2) $M\simeq
M_{1}\oplus(\sigma_{1}^{*})M_{1}\oplus\cdots\oplus(\sigma_{1}^{*})^{t_{1}-1}M_{1},$

where $\phi:\sigma_{1}^{*}M\,\raise 2.0pt\hbox{$\underrightarrow{\sim}$}\,M$
is defined as in 3.2. Finally, we assume that $t_{1}t_{2}\geq 2$.

Under the equivalence
$\mathscr{C}^{\prime\prime}_{\beta}\simeq\mathscr{C}_{\beta}$, the traceless
elements in $\mathscr{C}^{\prime\prime}_{\beta}$ correspond to the traceless
elements in $\mathscr{C}_{\beta}$. We define the Grothendieck group
$K(\mathscr{C}^{\prime\prime}_{\beta})$ in a similar way as in 3.2, by using
th action of $\zeta_{\mathbf{n}}$, and the traceless elements as above. Then
$K(\mathscr{C}^{\prime\prime}_{\beta})\simeq K(\mathscr{C}_{\beta})$. The
Grothendieck groups
$K(\mathscr{P}^{\prime\prime}_{\beta}),K(\mathscr{L}^{\prime\prime}_{\beta})$
are defined similarly, and we have

(4.4.3) $K(\mathscr{P}^{\prime\prime}_{\beta})\simeq
K(\mathscr{P}_{\beta}),\qquad K(\mathscr{L}^{\prime\prime}_{\beta})\simeq
K(\mathscr{L}_{\beta}).$

4.5. Recall that the functor
$\mathbf{D}:\mathscr{C}_{\beta}\to\mathscr{C}_{\beta}$ is given by
$\mathbf{D}(M,\phi)=(\mathbf{D}M,\mathbf{D}(\phi)^{-1})$ with respect to the
contravariant functor $\mathbf{D}:R(\beta)\operatorname{-Mod}\to
R(\beta)\operatorname{-Mod}$. Under the category equivalence
$\mathscr{C}^{\prime\prime}_{\beta}\simeq\mathscr{C}_{\beta}$, this is given
by

(4.5.1)
$\mathbf{D}:(M,\phi_{1},\phi_{2})\mapsto(\mathbf{D}M,\mathbf{D}(\phi_{1})^{-1},\mathbf{D}(\phi_{2})^{-1}),$

where $\mathbf{D}(\phi_{2})$ is the dual of
$\phi_{2}:\sigma_{2}^{*}(M,\phi_{1})\to(M,\phi_{1})$ in
$\mathscr{C}^{\prime}_{\beta}$, and it is also the dual of
$\phi_{2}:\sigma_{2}^{*}M\to M$ in $R(\beta)\operatorname{-Mod}$. Similar
results hold also for
$\mathscr{P}^{\prime\prime}_{\beta}\simeq\mathscr{P}_{\beta}$, and
$\mathscr{L}^{\prime\prime}_{\beta}\simeq\mathscr{L}_{\beta}$.

Here we note a lemma.

######  Lemma 4.6.

Assume that $(P,\phi)\in\mathscr{P}_{\beta}$ is a projective indecomposable
object, which is not traceless. Then $P$ is a projective indecomposable
$R(\beta)$-module.

###### Proof.

Let $(P,\phi)\in\mathscr{P}_{\beta}$ be a projective indecomposable object,
where $\phi:\sigma^{*}P\,\raise 2.0pt\hbox{$\underrightarrow{\sim}$}\,P$. We
assume that $(P,\phi)$ is not traceless. Then $P$ is a projective
$R(\beta)$-module. Let $P_{0}$ be an indecomposable direct summand of $P$. Let
$t$ be the smallest integer such that $(\sigma^{*})^{t}P_{0}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,P_{0}$. Then
$f\circ\sigma^{*}(\phi)\circ\cdots\circ(\sigma^{*})^{t-1}\phi:(\sigma^{*})^{t}P\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,P$ induces an isomorphism
$(\sigma^{*})^{t}P_{0}\,\raise 2.0pt\hbox{$\underrightarrow{\sim}$}\,P_{0}$,
which we denote by $\phi^{\prime}_{0}$. Set
$M=P_{0}\oplus\sigma^{*}P_{0}\oplus\cdots\oplus(\sigma^{*})^{t-1}P_{0}$. Then
$\sigma^{*}M\,\raise 2.0pt\hbox{$\underrightarrow{\sim}$}\,M$, and the
restriction of $\phi:\sigma^{*}P\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,P$ on $\sigma^{*}M$ gives an isomorphism
$\phi_{0}:\sigma^{*}M\,\raise 2.0pt\hbox{$\underrightarrow{\sim}$}\,M$, where
$\phi_{0}$ is defined by $\operatorname{id}:(\sigma^{*})^{k}P_{0}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,(\sigma^{*})^{k}P_{0}$ for
$k=1,\dots,t-1$, and $\phi_{0}^{\prime}:(\sigma^{*})^{t}P_{0}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,P_{0}$. Hence
$(M,\phi_{0})\in\mathscr{P}_{\beta}$ is a direct summand of $(P,\phi)$. Since
$(P,\phi)$ is indecomposable, we have $(P,\phi)=(M,\phi_{0})$. If $t\geq 2$,
then $(M,\phi_{0})$ is traceless. Hence by assumption, $t=1$, i.e., $P=P_{0}$
is indecomposable. The lemma holds. ∎

4.7. For a given $\beta\in Q_{+}^{\sigma}$, let
$\underline{\mathbf{B}}_{\beta}$ (resp. $\underline{\mathbf{B}}_{\beta}^{*}$)
be the $\mathbf{Z}[\zeta_{\mathbf{n}},q,q^{-1}]$-basis of
$K(\mathscr{P}_{\beta})$ (resp. $K(\mathscr{L}_{\beta})$) given in 3.4.
Similarly, for a given $\beta\in Q_{+}^{\sigma_{1}}$,
$\mathbf{Z}[\zeta_{\mathbf{n}_{1}},q,q^{-1}]$-basis of
$K(\mathscr{P}^{\prime}_{\beta})$ (resp. $K(\mathscr{L}^{\prime}_{\beta})$) is
defined, which we denote by $\mathbf{B}^{\prime}_{\beta}$ (resp.
$\mathbf{B}^{\prime*}_{\beta}$).

Let $(P,\phi,\phi^{\prime})\in\mathscr{P}^{\prime\prime}_{\beta}$ be a self-
dual projective indecomposable object corresponding to an element in
$\underline{\mathbf{B}}_{\beta}$ under the equivalence
$\mathscr{P}^{\prime\prime}_{\beta}\simeq\mathscr{P}_{\beta}$. Then $P\in
R(\beta)\operatorname{-{\bf gp}}$ is projective indecomposable by Lemma 4.6,
and $(P,\phi)\in\mathscr{P}^{\prime}_{\beta}$ is self-dual projective
indecomposable by (4.5.1). Thus we have

(4.7.1) The projection $(P,\phi_{1},\phi_{2})\mapsto(P,\phi_{1})$ induces a
map $\underline{\mathbf{B}}_{\beta}\to\pm\mathbf{B}^{\prime}_{\beta}$.

Let $\beta,\gamma\in Q_{+}^{\sigma}$. For
$(M,\phi)\in\mathscr{C}_{\beta},(N,\phi^{\prime})\in\mathscr{C}_{\gamma}$, the
convolution product is defined as $(M,\phi)\circ(N,\phi^{\prime})=(M\circ
N,\phi\circ\phi^{\prime})$ as in (3.6.1), by using the convolution product
$M\circ N\in R(\beta+\gamma)\operatorname{-Mod}$. Under the equivalence
$\mathscr{C}^{\prime\prime}_{\beta}\simeq\mathscr{C}_{\beta}$, this
convolution product is expressed as

(4.7.2)
$(M,\phi_{1},\phi_{2})\circ(N,\phi_{1}^{\prime},\phi_{2}^{\prime})=(M\circ
N,\phi_{1}\circ\phi_{1}^{\prime},\phi_{2}\circ\phi_{2}^{\prime}),$

where $(M\circ
N,\phi_{1}\circ\phi_{1}^{\prime})=(M,\phi_{1})\circ(N,\phi_{1}^{\prime})$
applied for the category $\mathscr{C}^{\prime}$, and
$\phi_{2}\circ\phi_{2}^{\prime}:\sigma_{2}^{*}(M\circ N)\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,M\circ N$ is defined similarly to
$\phi_{1}\circ\phi_{1}^{\prime}:\sigma_{1}^{*}(M\circ N)\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,(M\circ N)$.

Similarly to $\widetilde{\mathbf{A}}$ defined in 3.11, we define
$\widetilde{\mathbf{A}}_{1}$ as the smallest subring of
$\mathbf{Z}[\zeta_{\mathbf{n}_{1}},q,q^{-1}]$ containing $\mathbf{A}$ such
that the structure constants with respect to $\mathbf{B}^{\prime}$ are
contained in $\widetilde{\mathbf{A}}_{1}$. Since $\mathbf{n}_{1}<\mathbf{n}$,
by induction hypothesis, we have $\widetilde{\mathbf{A}}_{1}=\mathbf{A}$.

By (4.7.1) and (4.7.2), we have the following.

######  Lemma 4.8.

The structure constants of $\underline{\mathbf{B}}$ in
${}_{\widetilde{\mathbf{A}}}K(\mathscr{P})$ lie in
$\mathbf{Z}[\zeta_{\mathbf{n}_{2}},q,q^{-1}]$.

4.9. Recall that $\underline{X}=(J,(\ ,\ ))$ is the Cartan datum induced from
$(X,\sigma)$, and $\underline{\mathbf{U}}_{q}^{-}$ is the quantum group
associated to $\underline{X}$. Here $\sigma=\sigma_{1}\sigma_{2}$, with
$\sigma_{2}$ : a power of $\ell$. Let $J_{1}$ be the set of
$\sigma_{1}$-orbits in $I$, and $X_{1}=(J_{1},(\ ,\ ))$ the Cartan datum
induced from $(X,\sigma_{1})$. $\sigma_{2}$ acts on $J_{1}$, and induces an
admissible diagram automorphism on $X_{1}$. The set of $\sigma_{2}$-orbits on
$J_{1}$ coincides with $J$, and the Cartan datum induced from
$(X_{1},\sigma_{2})$ is isomorphic to $\underline{X}$. We denote by
$\mathbf{U}_{q}^{\prime-}$ the quantum group associated to $X_{1}$.
$\sigma_{2}$ acts on $\mathbf{U}_{q}^{\prime-}$ as an automorphism.

Let ${}_{\mathbf{A}}K(\mathscr{P}^{\prime})$ be the $\mathbf{A}$-submodule of
$K(\mathscr{P}^{\prime})$ spanned by $\mathbf{B}^{\prime}$. Then by the
induction hypothesis, ${}_{\mathbf{A}}K(\mathscr{P}^{\prime})$ is the
$\mathbf{A}$-subalgebra of $K(\mathscr{P}^{\prime})$. By applying Theorem 3.16
to $K(\mathscr{P}^{\prime})$, we have an isomorphism of $\mathbf{A}$-modules

(4.9.1) $\gamma_{1}:{}_{\mathbf{A}}\mathbf{U}_{q}^{\prime-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}}K(\mathscr{P}^{\prime}),$

which maps $1\in{}_{\mathbf{A}}(\mathbf{U}_{q}^{\prime-})_{0}$ to
$[\mathbf{k},\operatorname{id}]\in{}_{\mathbf{A}}K(\mathscr{P}^{\prime}_{0})$,
and commutes with the actions of $f_{j_{1}}^{(n)}$ ($j_{1}\in
J_{1},n\in\mathbf{N}$).

Let ${}_{\mathbf{A}}K(\mathscr{L}^{\prime})$ be the $\mathbf{A}$-submodule of
$K(\mathscr{L}^{\prime})$ spanned by $\mathbf{B}^{\prime*}$. Then
${}_{\mathbf{A}}K(\mathscr{L}^{\prime})$ is $\mathbf{A}$-dual to
${}_{\mathbf{A}}K(\mathscr{P}^{\prime})$, and $\mathbf{B}^{\prime}$ and
$\mathbf{B}^{\prime*}$ are dual to each other. Let
$(\mathbf{U}_{q}^{\prime-})^{*}$ be the dual space of
$\mathbf{U}_{q}^{\prime-}$. By taking the transpose of $\gamma_{1}$, we have
an isomorphism of $\mathbf{A}$-modules,

(4.9.2) $\gamma_{1}^{*}:{}_{\mathbf{A}}K(\mathscr{L}^{\prime})\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}}(\mathbf{U}_{q}^{\prime-})^{*},$

which maps
$[\mathbf{k},\operatorname{id}]\in{}_{\mathbf{A}}K(\mathscr{L}^{\prime}_{0})$
to $1^{*}\in{}_{\mathbf{A}}(\mathbf{U}_{q}^{\prime-})^{*}_{0}$, and commutes
with the actions of $e_{j_{1}}^{\prime(n)}$ ($j\in J_{1},n\in\mathbf{N}$).

4.10. We consider the functor
$\tau=(\sigma_{2}^{-1})^{*}:\mathscr{C}^{\prime}_{\beta}\to\mathscr{C}^{\prime}_{\sigma_{2}(\beta)}$
for any $\beta\in Q_{+}^{\sigma_{1}}$. Note that the order of $\tau$ is equal
to $\mathbf{n}_{2}$, which is a power of $\ell$. Let
$\mathbf{A}^{\prime}=\mathbf{F}[q,q^{-1}]$ with
$\mathbf{F}=\mathbf{Z}/\ell\mathbf{Z}$. The discussion in Section 2 can be
applied to our situation, by replacing $K_{\operatorname{{\bf gp}}}$ (resp.
$K_{\operatorname{{\bf gm}}}$) by ${}_{\mathbf{A}}K(\mathscr{P}^{\prime})$
(resp. ${}_{\mathbf{A}}K(\mathscr{L}^{\prime})$) as follows. Set
${}_{\mathbf{A}^{\prime}}K(\mathscr{P}^{\prime})=\mathbf{A}^{\prime}\otimes_{\mathbf{A}}{}_{\mathbf{A}}K(\mathscr{P}^{\prime})$,
and define ${}_{\mathbf{A}^{\prime}}K(\mathscr{L}^{\prime})$ similarly.
$\mathbf{B}^{\prime}$ (resp. $\mathbf{B}^{\prime*}$) gives an
$\mathbf{A}^{\prime}$-basis of
${}_{\mathbf{A}^{\prime}}K(\mathscr{P}^{\prime})$ (resp.
${}_{\mathbf{A}^{\prime}}K(\mathscr{L}^{\prime})$). Let
${}_{\mathbf{A}^{\prime}}K(\mathscr{P}^{\prime})^{\tau}$ be the set of
$\tau$-fixed elements in ${}_{\mathbf{A}^{\prime}}K(\mathscr{P}^{\prime})$.
Then $\\{O(b)\mid b\in\mathbf{B}^{\prime}\\}$ gives an
$\mathbf{A}^{\prime}$-basis of
${}_{\mathbf{A}^{\prime}}K(\mathscr{P}^{\prime})^{\tau}$. Let
$\mathcal{J}^{\prime}$ be the subset of
${}_{\mathbf{A}^{\prime}}K(\mathscr{P}^{\prime})^{\tau}$ consisting of $O(x)$
for $x\in{}_{\mathbf{A}^{\prime}}K(\mathscr{P}^{\prime})$ such that $O(x)\neq
x$. Then $\mathcal{J}^{\prime}$ is an $\mathbf{A}^{\prime}$-submodule of
${}_{\mathbf{A}^{\prime}}K(\mathscr{P}^{\prime})^{\tau}$. Similarly,
$\mathbf{A}^{\prime}$-submodule $\mathcal{J}^{\prime*}$ of
${}_{\mathbf{A}^{\prime}}K(\mathscr{L}^{\prime})^{\tau}$ is defined. We define
the quotient modules $\mathcal{V}_{q}^{\prime}$ and
$\mathcal{V}^{\prime*}_{q}$ by

(4.10.1)
$\mathcal{V}_{q}^{\prime}={}_{\mathbf{A}^{\prime}}K(\mathscr{P}^{\prime})^{\tau}/\mathcal{J}^{\prime},\qquad\mathcal{V}^{\prime*}_{q}={}_{\mathbf{A}^{\prime}}K(\mathscr{L}^{\prime})^{\tau}/\mathcal{J}^{\prime*}.$

A $\sigma$-orbit $j$ in $I$ is regarded as a $\sigma_{2}$-orbit in $J_{1}$.
Thus for each $j\in J$, one can write as $j=\\{j_{1},\dots,j_{t}\\}$ with
$j_{k}\in J_{1}$. We define a functor
$\widetilde{F}_{j}^{(n)}:\mathscr{P}^{\prime}\to\mathscr{P}^{\prime}$ by
$\widetilde{F}_{j}^{(n)}=F_{j_{t}}^{(n)}\cdots F_{j_{1}}^{(n)}$. This induces
an $\mathbf{A}^{\prime}$-homomorphism
$\widetilde{f}_{j}^{(n)}=f_{j_{t}}^{(n)}\cdots f_{j_{1}}^{(n)}$ on
${}_{\mathbf{A}^{\prime}}K(\mathscr{P}^{\prime})$. As in 2.10,
$\widetilde{f}_{j}^{(n)}$ commutes with $\tau$, and acts on
$\mathcal{V}^{\prime}_{q}$.

Similarly, for each $j\in J$, we define a functor
$\widetilde{E}_{j}^{(n)}:\mathscr{L}^{\prime}\to\mathscr{L}^{\prime}$ by
$\widetilde{E}_{j}^{(n)}=E_{j_{1}}^{(n)}\cdots E_{j_{t}}^{(n)}$. This induces
an $\mathbf{A}^{\prime}$-homomorphism
$\widetilde{e}_{j}^{\prime(n)}=e_{j_{1}}^{(n)}\cdots e_{j_{t}}^{(n)}$
commuting with $\tau$. Thus we have an action of
$\widetilde{e}_{j}^{\prime(n)}$ on $\mathcal{V}_{q}^{\prime*}$. The pairing
$\operatorname{\langle}\ ,\
\operatorname{\rangle}:{}_{\mathbf{A}}K(\mathscr{P}^{\prime})\times{}_{\mathbf{A}}K(\mathscr{L}^{\prime})\to\mathbf{A}$
induces a perfect pairing $\operatorname{\langle}\ ,\
\operatorname{\rangle}:\mathcal{V}_{q}^{\prime}\times\mathcal{V}_{q}^{\prime*}\to\mathbf{A}^{\prime}$,
and $\widetilde{e}_{j}^{\prime(n)}$ coincides with the transpose of
$\widetilde{f}_{j}^{(n)}$.

The quotient algebra
$\mathbf{V}_{q}^{\prime}={}_{\mathbf{A}^{\prime}}(\mathbf{U}^{\prime-}_{q})^{\sigma_{2}}/\mathbf{J}^{\prime}$
of ${}_{\mathbf{A}^{\prime}}(\mathbf{U}^{\prime-}_{q})^{\sigma_{2}}$ is
defined as in (1.6.1) by replacing
$\sigma:\mathbf{U}_{q}^{-}\to\mathbf{U}_{q}^{-}$ by
$\sigma_{2}:\mathbf{U}^{\prime-}_{q}\to\mathbf{U}^{\prime-}_{q}$. Also the
quotient module
$\mathbf{V}^{\prime*}_{q}={}_{\mathbf{A}^{\prime}}(\mathbf{U}^{\prime-}_{q})^{*,\sigma_{2}}/\mathbf{J}^{\prime*}$
is defined as in (1.11.1). ($\mathbf{J}^{\prime},\mathbf{J}^{\prime*}$ are the
objects corresponding to $\mathbf{J},\mathbf{J}^{*}$, respectively.) Note that
the Cartan datum induced from $(X_{1},\sigma_{2})$ coincides with
$\underline{X}$. Thus by Theorem 1.7 and Proposition 1.12, we see that there
exist isomorphisms

(4.10.2)
$\Phi_{1}:{}_{\mathbf{A}^{\prime}}\underline{\mathbf{U}}^{-}_{q}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,\mathbf{V}^{\prime}_{q},\qquad\Phi_{1}^{*}:\mathbf{V}^{\prime*}_{q}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}^{\prime}}(\underline{\mathbf{U}}^{-}_{q})^{*},$

where under these isomorphisms, the actions of $\widetilde{f}^{(n)}_{j}$
(resp. $\widetilde{e}_{j}^{\prime(n)}$) corresponds to the actions of
$f_{j}^{(n)}$ (resp. $e_{j}^{\prime(n)}$).

Since the isomorphism
$\gamma_{1}:{}_{\mathbf{A}}\mathbf{U}_{q}^{\prime-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,K(\mathscr{P}^{\prime})$ is compatible
with the action of $\sigma_{2}$ and $\tau$, it induces an isomorphism
${}_{\mathbf{A}^{\prime}}(\mathbf{U}_{q}^{\prime-})^{\sigma_{2}}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}^{\prime}}K(\mathscr{P}^{\prime})^{\tau}$,
which maps $\mathbf{J}^{\prime}$ onto $\mathcal{J}^{\prime}$. Similarly, the
isomorphism $\gamma_{1}^{*}:K(\mathscr{L}^{\prime})\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}}(\mathbf{U}_{q}^{\prime-})^{*}$
induces an isomorphism
${}_{\mathbf{A}^{\prime}}K(\mathscr{L}^{\prime})^{\tau}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}^{\prime}}(\mathbf{U}_{q}^{\prime-})^{*,\sigma_{2}}$,
which maps $\mathcal{J}^{\prime*}$ onto $\mathbf{J}^{\prime*}$. It follows
that $\gamma_{1}$ and $\gamma_{1}^{*}$ induce isomorphisms

(4.10.3) $\varTheta_{1}:\mathbf{V}_{q}^{\prime}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,\mathcal{V}^{\prime}_{q},\qquad\varTheta_{1}^{*}:\mathcal{V}_{q}^{\prime*}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,\mathbf{V}_{q}^{\prime*},$

where $\varTheta_{1}$ (resp. $\varTheta_{1}^{*}$ ) commutes with the actions
of $\widetilde{f}_{j}^{(n)}$ (resp. $\widetilde{e}_{j}^{\prime(n)}$).

4.11. Let $\widetilde{\mathbf{A}}$ be as in 3.11. By Lemma 4.8,
$\mathbf{A}\subset\widetilde{\mathbf{A}}\subset\mathbf{Z}[\zeta_{\mathbf{n}_{2}},q,q^{-1}]$.
We define

$\widetilde{\mathbf{A}}^{\prime}=\mathbf{A}^{\prime}\otimes_{\mathbf{A}}\widetilde{\mathbf{A}}\simeq\widetilde{\mathbf{A}}/\ell\widetilde{\mathbf{A}}.$

Then $\mathbf{A}^{\prime}$ is a subalgebra of
$\widetilde{\mathbf{A}}^{\prime}$.

We compare the structure of
${}_{\widetilde{\mathbf{A}}}K(\mathscr{L}^{\prime\prime})$ (resp.
${}_{\widetilde{\mathbf{A}}}K(\mathscr{P}^{\prime\prime})$) and
$\mathcal{V}^{\prime*}_{q}$ (resp. $\mathcal{V}^{\prime}_{q}$) by changing the
base ring to $\widetilde{\mathbf{A}}^{\prime}$. Set
${}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{P}^{\prime})=\widetilde{\mathbf{A}}^{\prime}\otimes_{\mathbf{A}^{\prime}}{}_{\mathbf{A}^{\prime}}K(\mathscr{P}^{\prime})$,
and
${}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{L}^{\prime})=\widetilde{\mathbf{A}}^{\prime}\otimes_{\mathbf{A}^{\prime}}{}_{\mathbf{A}^{\prime}}K(\mathscr{L}^{\prime})$.
The definition of $\mathcal{V}^{\prime}_{q}$ and $\mathcal{V}^{\prime*}_{q}$
in (4.8.1) can be applied by replacing $\mathbf{A}^{\prime}$ by
$\widetilde{\mathbf{A}}^{\prime}$, and one can define
${}_{\widetilde{\mathbf{A}}^{\prime}}\mathcal{V}^{\prime}_{q}$ and
${}_{\widetilde{\mathbf{A}}^{\prime}}\mathcal{V}_{q}^{\prime*}$ as the
quotients of
${}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{P}^{\prime})^{\tau}$ and
${}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{L}^{\prime})^{\tau}$,
respectively, where
$\mathcal{J}^{\prime}\subset{}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{P}^{\prime})^{\tau},\mathcal{J}^{\prime*}\subset{}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{L}^{\prime})^{\tau}$
are defined similarly.

Assume that $(M,\phi_{1},\phi_{2})\in\mathscr{L}^{\prime\prime}_{\beta}$ is a
traceless element. Then by 4.4, $(M,\phi_{1})\in\mathscr{L}^{\prime}_{\beta}$
is traceless, or there exists
$(M^{\prime},\phi^{\prime})\in\mathscr{L}^{\prime}_{\beta}$, and an integer
$t\geq 2$ which is a divisor of $\mathbf{n}_{2}$, such that
$(\sigma_{2}^{*})^{t}(M^{\prime},\phi^{\prime})\simeq(M^{\prime},\phi^{\prime})$
and that

$(M,\phi_{1})\simeq(M^{\prime},\phi^{\prime})\oplus\sigma_{2}^{*}(M^{\prime},\phi^{\prime})\oplus\cdots\oplus(\sigma_{2}^{*})^{t-1}(M^{\prime},\phi^{\prime}),$

where $\phi_{2}:\sigma_{2}^{*}(M,\phi_{1})\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,(M,\phi_{1})$ is defined as in (4.4.1).
It follows that $[M,\phi_{1}]=0$ in $K(\mathscr{L}^{\prime}_{\beta})$, or

$[M,\phi_{1}]=[M^{\prime},\phi^{\prime}]+\tau[M^{\prime},\phi^{\prime}]+\cdots+\tau^{t-1}[M^{\prime},\phi^{\prime}]\in\mathcal{J}^{\prime*}.$

In particular,

(4.11.1) If $(M,\phi_{1},\phi_{2})$ is traceless, then
$[M,\phi_{1}]\in\mathcal{J}^{\prime*}$.

Let $(L,\phi_{1},\phi_{2})\in\mathscr{L}^{\prime\prime}_{\beta}$ be a simple
object in $\mathscr{L}^{\prime\prime}_{\beta}$, which is not traceless, such
that
$[(L,\phi_{1},\phi_{2})]\in{}_{\widetilde{\mathbf{A}}}K(\mathscr{L}^{\prime\prime})$.
By [M, Thm. 3.6], $(L,\phi_{1})$ is a simple object in
$\mathscr{L}^{\prime}_{\beta}$ such that $\sigma_{2}^{*}(L,\phi_{1})\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,(L,\phi_{1})$. Here $L$ is a simple
$R(\beta)$-module, and is isomorphic to a self-dual simple module, up to
degree shift. By using the category equivalence
$\mathscr{L}^{\prime\prime}\simeq\mathscr{L}$, there exist self-dual simple
objects $(L,\phi_{1}^{\bullet},\phi_{2}^{\bullet})$ in
$\mathscr{L}_{\beta}^{\prime\prime}$. Then $(L,\phi_{1}^{\bullet})$ is self-
dual in $\mathscr{L}^{\prime}_{\beta}$, and there exists a power $\zeta_{1}$
of $\zeta_{\mathbf{n}_{1}}$, with degree shift $q^{a}$, such that
$(L,\phi_{1}^{\bullet})=\zeta_{1}q^{a}(L,\phi_{1})$. By our assumption in 4.1,
we may only consider the case where $\phi_{1}^{\bullet}=\pm\phi_{1}$, up to
degree shift. Now if $\mathbf{n}_{2}$ is odd,
$\phi^{\bullet}_{2}:\sigma_{2}^{*}(L,\phi^{\bullet}_{1})\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,(L,\phi^{\bullet}_{1})$ is unique, while
if $\mathbf{n}_{2}$ is even, there exist exactly two $\pm\phi^{\bullet}_{2}$
such that $(L,\phi_{1}^{\bullet},\phi_{2}^{\bullet})$ is self-dual. Thus for
$(L,\phi_{1},\phi_{2})$ as above, there exist $\zeta$ which is a power of
$\zeta_{\mathbf{n}_{2}}$, with some degree shift $q^{a}$, such that
$(L,\phi_{1},\phi_{2})=\zeta q^{a}(L,\phi_{1}^{\bullet},\phi_{2}^{\bullet})$.
We attach $\zeta
q^{a}[L,\phi_{1}]\in{}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{L}^{\prime}_{\beta})$
to $(L,\phi_{1},\phi_{2})\in\mathscr{L}^{\prime\prime}_{\beta}$. This gives a
well-defined map since we are working on
$\mathbf{F}=\mathbf{Z}/\ell\mathbf{Z}$.

Take $(M,\phi_{1},\phi_{2})\in\mathscr{L}_{\beta}^{\prime\prime}$, and
consider the composition series of $(M,\phi_{1},\phi_{2})$, where each
composition factor is isomorphic to a simple object
$(L,\phi^{\prime}_{1},\phi^{\prime}_{2})\in\mathscr{L}^{\prime\prime}_{\beta}$
such that $\sigma_{2}^{*}(L,\phi^{\prime}_{1})\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,(L,\phi^{\prime}_{1})$, or a traceless
element. Accordingly, $(M,\phi_{1})\in\mathscr{C}^{\prime}_{\beta}$ has a
composition series, where each composition factor is isomorphic to a simple
object $(L,\phi_{1}^{\prime})$ such that $\sigma_{2}^{*}L\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,L$, or a traceless element in
$\mathscr{L}^{\prime}_{\beta}$.

In view of (4.11.1), we obtain a well-defined
$\widetilde{\mathbf{A}}^{\prime}$-module homomorphism

(4.11.2)
$\Psi^{*}:{}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{L}^{\prime\prime})\to{}_{\widetilde{\mathbf{A}}^{\prime}}\mathcal{V}_{q}^{\prime*},$

where $[(L,\phi_{1},\phi_{2})]$ goes to $[\zeta q^{a}(L,\phi_{1})]$ for a
simple module $L\in R(\beta)\operatorname{-{\bf gm}}$ such that
$\sigma^{*}L\,\raise 2.0pt\hbox{$\underrightarrow{\sim}$}\,L$. By considering
the transpose of (4.11.2), one can define an
$\widetilde{\mathbf{A}}^{\prime}$-module homomorphism

(4.11.3)
$\Psi:{}_{\widetilde{\mathbf{A}}^{\prime}}\mathcal{V}_{q}^{\prime}\to{}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{P}^{\prime\prime}).$

Under the isomorphisms $K(\mathscr{L})\simeq K(\mathscr{L}^{\prime\prime})$,
$K(\mathscr{P})\simeq L(\mathscr{P}^{\prime\prime})$, $e_{j}^{\prime(n)}$
(resp, $f_{j}^{(n)}$ ) acts on
${}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{L}^{\prime\prime})$ (resp. on
${}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{P}^{\prime\prime})$).

######  Lemma 4.12.

  1. (i)

The map $\Psi^{*}$ commutes with the action of $e_{j}^{\prime(n)}$ on
${}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{L}^{\prime\prime})$, and that
of $\widetilde{e}_{j}^{\prime(n)}$ on
${}_{\widetilde{\mathbf{A}}^{\prime}}\mathcal{V}_{q}^{\prime*}$.

  2. (ii)

The map $\Psi$ commutes with the action of $f_{j}^{(n)}$ on
${}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{P}^{\prime\prime})$, and that
of $\widetilde{f}_{j}^{(n)}$ on
${}_{\widetilde{\mathbf{A}}^{\prime}}\mathcal{V}_{q}^{\prime}$.

###### Proof.

(ii) follows from (i) by the discussion in 4.10, and by Lemma 3.10.

We prove (i). First consider the case where $n=1$. Assume that $\beta\in
Q_{+}^{\sigma}$ such that $\beta-\alpha_{j}\in Q_{+}^{\sigma}$. Take
$(M,\phi)\in\mathscr{L}_{\beta}$, which corresponds to
$(M,\phi_{1},\phi_{2})\in\mathscr{L}^{\prime\prime}_{\beta}$ under the
equivalence $\mathscr{L}^{\prime\prime}_{\beta}\simeq\mathscr{L}_{\beta}$. By
the discussion in 3.9, $E_{j}(M,\phi)$ is written as
$E_{j}(M,\phi)=(\widetilde{E}_{j}M,\phi^{\prime})$, where $\widetilde{E}_{j}M$
is given as in (3.9.4), and $\phi^{\prime}$ is determined uniquely from
$\phi$. Thus $E_{j}(M,\phi_{1},\phi_{2})$ is written as
$E_{j}(M,\phi_{1},\phi_{2})=(\widetilde{E}_{j}(M,\phi_{1}),\phi_{2}^{\prime})$,
where
$\widetilde{E}_{j}(M,\phi_{1})\in\mathscr{L}^{\prime}_{\beta-\alpha_{j}}$ is
defined as in 4.10, and $\phi_{2}^{\prime}$ is determined uniquely from
$\phi_{1},\phi_{2}$. The functor
$(M,\phi_{1})\mapsto\widetilde{E}_{j}(M,\phi_{1})$ induces the operation
$\widetilde{e}^{\prime}_{j}:K(\mathscr{L}^{\prime}_{\beta})\to
K(\mathscr{L}^{\prime}_{\beta-\alpha_{j}})$, and
$\widetilde{e}^{\prime}_{j}([M,\phi_{1}])$ is $\tau$-invariant. Hence it
induces the operation $\widetilde{e}^{\prime}_{j}$ on
${}_{\widetilde{\mathbf{A}}^{\prime}}\mathcal{V}_{q}^{\prime*}$. This shows
that the map $\Psi^{*}$ commutes with the action of $e^{\prime}_{j}$ on
${}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{L}^{\prime\prime})$ and that
of $\widetilde{e}^{\prime}_{j}$ on
${}_{\widetilde{\mathbf{A}}^{\prime}}\mathcal{V}_{q}^{\prime*}$.

We have $[n]^{!}_{j}{e^{\prime}}^{(n)}_{j}={e^{\prime}}^{n}_{j}$ on
${}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{L}^{\prime\prime})$ by
(3.14.1). On the other hand, we have
$\widetilde{e^{\prime}}^{(n)}_{j}={e^{\prime}}^{(n)}_{j_{t}}\cdots{e^{\prime}}^{(n)}_{j_{1}}$
on ${}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{L}^{\prime})$. Note that
$[n]^{!}_{j_{k}}{e^{\prime}}^{(n)}_{j_{k}}={e^{\prime}}^{n}_{j_{k}}$, where
$q_{j}=q^{(\alpha_{j},\alpha_{j})/2}=q^{t(\alpha_{i},\alpha_{i})/2}=q_{i}^{t}$
for $i=j_{k}$. Hence we have

$([n]^{!}_{i})^{t}\widetilde{e^{\prime}}^{(n)}_{j}=\widetilde{e^{\prime}}_{j}^{n}.$

Since $([n]^{!}_{i})^{t}=[n]^{!}_{j}$ in $\widetilde{\mathbf{A}}^{\prime}$,
and ${}_{\widetilde{\mathbf{A}}^{\prime}}\mathcal{V}_{q}^{\prime*}$ is a free
$\widetilde{\mathbf{A}}^{\prime}$-module, the assertion holds for any $n$.
Thus (i) is proved. ∎

######  Proposition 4.13.

  1. (i)

The map
$\Psi:{}_{\widetilde{\mathbf{A}}^{\prime}}\mathcal{V}_{q}^{\prime}\to{}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{P}^{\prime\prime})$
gives an isomorphism of $\widetilde{\mathbf{A}}^{\prime}$-modules. Furthermore
the isomorphism
$\gamma:{}_{\widetilde{\mathbf{A}}^{\prime}}\underline{\mathbf{U}}_{q}^{-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{P})$
factors through isomorphisms

(4.13.1)
$\textstyle{\gamma:{}_{\widetilde{\mathbf{A}}^{\prime}}\underline{\mathbf{U}}_{q}^{-}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{1}}$$\textstyle{{}_{\widetilde{\mathbf{A}}^{\prime}}\mathbf{V}^{\prime}_{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varTheta_{1}}$$\textstyle{{}_{\widetilde{\mathbf{A}}^{\prime}}\mathcal{V}^{\prime}_{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{{}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{P}),}$

where $\Phi_{1},\varTheta_{1},\Psi$ commute with the action of $f_{j}^{(n)}$
or $\widetilde{f}_{j}^{(n)}$.

  2. (ii)

The map
$\Psi^{*}:{}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{L}^{\prime\prime})\to{}_{\widetilde{\mathbf{A}}^{\prime}}\mathcal{V}_{q}^{\prime*}$
gives an isomorphism of $\widetilde{\mathbf{A}}^{\prime}$-modules. Furthermore
the isomorphism
$\gamma^{*}:{}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{L})\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\widetilde{\mathbf{A}}^{\prime}}(\underline{\mathbf{U}}_{q}^{-})^{*}$
factors through isomorphisms

(4.13.2)
$\textstyle{\gamma^{*}:{}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{L}^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi^{*}}$$\textstyle{{}_{\widetilde{\mathbf{A}}^{\prime}}\mathcal{V}^{\prime*}_{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varTheta^{*}_{1}}$$\textstyle{{}_{\widetilde{\mathbf{A}}^{\prime}}\mathbf{V}^{\prime*}_{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{1}^{*}}$$\textstyle{{}_{\widetilde{\mathbf{A}}^{\prime}}(\underline{\mathbf{U}}_{q}^{-})^{*},}$

where $\Psi^{*},\varTheta_{1}^{*},\Phi_{1}^{*}$ commute with the action of
$e_{j}^{\prime(n)}$ or $\widetilde{e}_{j}^{\prime(n)}$.

###### Proof.

We define a map
$\widetilde{\gamma}:{}_{\widetilde{\mathbf{A}}^{\prime}}\underline{\mathbf{U}}_{q}^{-}\to{}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{P}^{\prime\prime})$
as the composite of $\Phi_{1},\varTheta_{1}$ and $\Psi$,

$\textstyle{\widetilde{\gamma}:{}_{\widetilde{\mathbf{A}}^{\prime}}\underline{\mathbf{U}}_{q}^{-}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{1}}$$\textstyle{{}_{\widetilde{\mathbf{A}}^{\prime}}\mathbf{V}^{\prime}_{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varTheta_{1}}$$\textstyle{{}_{\widetilde{\mathbf{A}}^{\prime}}\mathcal{V}^{\prime}_{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{{}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{P}^{\prime\prime}).}$

By (4.10.2), the action of $f_{j}^{(n)}$ on
${}_{\widetilde{\mathbf{A}}^{\prime}}\underline{\mathbf{U}}_{q}^{-}$ and that
of $\widetilde{f}_{j}^{(n)}$ on
${}_{\widetilde{\mathbf{A}}^{\prime}}\mathbf{V}_{q}^{\prime}$ are compatible
with $\Phi_{1}$. By (4.10.3), the action of $\widetilde{f}_{j}^{(n)}$ is
compatible with $\varTheta_{1}$. By Lemma 4.12, the action of
$\widetilde{f}_{j}^{(n)}$ and that of $f_{j}^{(n)}$ are compatible with
$\Psi$. It follows that $\widetilde{\gamma}$ is compatible with the action of
$f_{j}^{(n)}$. Moreover, $\widetilde{\gamma}$ maps
$1\in{}_{\widetilde{\mathbf{A}}^{\prime}}\underline{\mathbf{U}}_{q}^{-}$ to
$[(\mathbf{k},\operatorname{id},\operatorname{id})]\in{}_{\widetilde{\mathbf{A}}}K(\mathscr{P}^{\prime\prime}_{0})$.
On the other hand, by Proposition 3.13, we have an anti-algebra isomorphism
$\gamma:{}_{\widetilde{\mathbf{A}}^{\prime}}\underline{\mathbf{U}}_{q}^{-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{P})$,
which commutes with the action of $f_{j}^{(n)}$, under the identification
$K(\mathscr{P})\simeq K(\mathscr{P}^{\prime\prime})$. Since $\gamma$ is
determined uniquely by the action of $f_{j}^{(n)}$, we conclude that
$\gamma=\widetilde{\gamma}$. In particular, $\Psi$ is an isomoprhism, and (i)
is proved. (ii) follows from (i). The proposition is proved. ∎

4.14. We are now ready to prove Theorem 3.16. Let $\underline{\mathbf{B}}$ be
the basis of ${}_{\widetilde{\mathbf{A}}}K(\mathscr{P})$ given in 3.4, and
$\underline{\mathbf{B}}^{*}$ be the dual basis of
${}_{\widetilde{\mathbf{A}}}K(\mathscr{L})$. Similarly, we obtain the basis
$\mathbf{B}^{\prime}$ of $K(\mathscr{P}^{\prime})$, and its dual basis
$\mathbf{B}^{\prime*}$ of $K(\mathscr{L}^{\prime})$ over a suitable extension
of $\mathbf{A}$. But by our assumption in 4.1, we may assume that
$\mathbf{B}^{\prime}$ is a basis of ${}_{\mathbf{A}}K(\mathscr{P}^{\prime})$,
and $\mathbf{B}^{\prime*}$ is the dual basis of
${}_{\mathbf{A}}K(\mathscr{L}^{\prime})$. Let $(B^{\prime})^{\tau}$ (resp.
$(\mathbf{B}^{\prime*})^{\tau}$ be the set of $\tau$-fixed elements in
$\mathbf{B}^{\prime}$ (resp. in $\mathbf{B}^{\prime*}$). Then the image
$\underline{\mathbf{B}}^{\prime}$ of $(\mathbf{B}^{\prime})^{\tau}$ in
$\mathcal{V}_{q}$ gives an $\mathbf{A}^{\prime}$-basis of
$\mathcal{V}_{q}^{\prime}$, and the image $\underline{\mathbf{B}}^{\prime*}$
of $(\mathbf{B}^{\prime*})^{\tau}$ gives an $\mathbf{A}^{\prime}$-basis of
$\mathcal{V}_{q}^{\prime*}$. Now by the construction of the map
$\Psi^{*}:{}_{\widetilde{\mathbf{A}}^{\prime}}K(\mathscr{L}^{\prime\prime})\to{}_{\widetilde{\mathbf{A}}^{\prime}}\mathcal{V}_{q}^{\prime*}$,
and by Proposition 4.13, $\Psi^{*}(\underline{\mathbf{B}}^{*})$ coincides with
$\underline{\mathbf{B}}^{\prime*}$. Hence $\Psi^{-1}(\underline{\mathbf{B}})$
coincides with $\underline{\mathbf{B}}^{\prime}$. In the diagram (4.13.1), the
maps $\Phi_{1},\varTheta_{1}$ are defined over $\mathbf{A}^{\prime}$. Thus
$\gamma^{-1}(\underline{\mathbf{B}})$ gives an $\mathbf{A}^{\prime}$-basis of
${}_{\mathbf{A}^{\prime}}\underline{\mathbf{U}}_{q}^{-}$.

On the other hand, since originally $\gamma$ is an isomorphism
${}_{\widetilde{\mathbf{A}}}\underline{\mathbf{U}}_{q}^{-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\widetilde{\mathbf{A}}}K(\mathscr{P})$,
we see that $\gamma^{-1}(\underline{\mathbf{B}})$ is an
$\widetilde{\mathbf{A}}$-basis of
${}_{\widetilde{\mathbf{A}}}\underline{\mathbf{U}}_{q}^{-}$. Fix $\beta\in
Q_{-}$, and let $Y$ (resp. $Y^{\prime}$) be the
$\widetilde{\mathbf{A}}$-submodule (resp. $\mathbf{A}$-submodule) of
$\underline{\mathbf{U}}_{q}^{-}$ spanned by
$\gamma^{-1}(\underline{\mathbf{B}}_{\beta})$. Then $Y/Y^{\prime}$ is a
finitely generated $\mathbf{Z}$-module, and by the above discussion,
$\ell(Y/Y^{\prime})=0$. Hence by Nakayama’s lemma, $Y$ coincides with
$Y^{\prime}$ if the base ring is extended from
$\mathbf{A}=\mathbf{Z}[q,q^{-1}]$ to $\mathbf{Z}_{\ell}[q,q^{-1}]$, where
$\mathbf{Z}_{\ell}$ is the ring of $\ell$-adic integers. It follows that, for
$b,b^{\prime}\in\gamma^{-1}(\underline{\mathbf{B}})$, we have
$bb^{\prime}=\sum_{b^{\prime\prime}\in\gamma^{-1}(\underline{\mathbf{B}})}a_{bb^{\prime}}^{b^{\prime\prime}}b^{\prime\prime}$,
where

$a_{bb^{\prime}}^{b^{\prime\prime}}\in\mathbf{Z}[\zeta_{\mathbf{n}_{2}},q,q^{-1}]\cap\mathbf{Z}_{\ell}[q,q^{-1}]=\mathbf{Z}[q,q^{-1}].$

Thus $\gamma^{-1}(\underline{\mathbf{B}})$ is an $\mathbf{A}$-basis of
${}_{\mathbf{A}}\mathbf{U}_{q}^{-}$. Since $\gamma$ is an anti-algebra
isomorphism, we see that the structure constants of the basis
$\underline{\mathbf{B}}$ all lie in $\mathbf{A}$, namely we have
$\widetilde{\mathbf{A}}=\mathbf{A}$. Theorem 3.16 is proved.

As a corollary to Theorem 3.16, a refinement of Proposition 4.13 is obtained.

######  Corollary 4.15.

  1. (i)

There exist isomorphisms of $\mathbf{A}$-modules,

$\gamma:{}_{\mathbf{A}}\underline{\mathbf{U}}_{q}^{-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}}K(\mathscr{P}),\qquad\gamma^{*}:{}_{\mathbf{A}}K(\mathscr{L})\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}}(\underline{\mathbf{U}}_{q}^{-})^{*}.$

  2. (ii)

The isomorphisms $\gamma,\gamma^{*}$ can be factored, as
$\mathbf{A}^{\prime}$-modules, through

##  5\.  KLR algebras and global bases

5.1. In this section, we assume that $R=\bigoplus_{\beta\in Q_{+}}R(\beta)$ is
symmetric type, namely, the Cartan matrix $A=(a_{ij})$ is symmetric, and
$Q_{i,j}(u,v)$ is a polynomial in $u-v$. Furthermore, we assume that
$\mathbf{k}$ is an algebraically closed field of characteristic 0. We consider
the isomorphisms $\gamma:{}_{\mathbf{A}}\underline{\mathbf{U}}_{q}^{-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}}K(\mathscr{P})$ and
$\gamma^{*}:{}_{\mathbf{A}}K(\mathscr{L})\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}}\underline{\mathbf{U}}_{q}^{-}$
as in Corollary 4.15. In the last paragraph of [M], he states a result that
$\gamma^{-1}(\underline{\mathbf{B}})$ coincides with the canonical basis, and
also coincides with the lower global basis of
$\underline{\mathbf{U}}_{q}^{-}$, with a brief indication for the proof, which
relies on the geometric argument. In this section we give a simple proof of
this fact, without using the geometry.

Let $\widetilde{\varTheta}_{0}:{}_{\mathbf{A}}\mathbf{U}_{q}^{-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,K_{\operatorname{{\bf gp}}}$ be the
isomorphism of $\mathbf{A}$-algebras as given in Theorem 2.7. Recall that
$\mathbf{B}$ is the basis of $K_{\operatorname{{\bf gp}}}$, and
$\mathbf{B}^{*}$ is the basis of $K_{\operatorname{{\bf gm}}}$ given in 2.4.
Let $\mathcal{B}^{*}$ be the upper global basis of
${}_{\mathbf{A}}(\mathbf{U}_{q}^{-})^{*}$ as given in Theorem 1.10, and
$\mathcal{B}$ the lower global basis of ${}_{\mathbf{A}}\mathbf{U}_{q}^{-}$.
Also let $\mathcal{B}^{\natural}$ be the canonical basis of
${}_{\mathbf{A}}\mathbf{U}_{q}^{-}$. In the case where $X$ is symmetric, it is