# Stability of $\imath$canonical bases of irreducible finite type of real rank
one

Hideya Watanabe  (H. Watanabe) Osaka Central Advanced Mathematical Institute,
Osaka Metropolitan University, Osaka, 558-8585, Japan
<EMAIL_ADDRESS>
###### Abstract.

It has been known since their birth in Bao and Wang’s work that the
$\imath$canonical bases of $\imath$quantum groups are not stable in general.
In the author’s previous work, the stability of $\imath$canonical bases of
certain quasi-split types turned out to be closely related to the theory of
$\imath$crystals. In this paper, we prove the stability of $\imath$canonical
bases of irreducible finite type of real rank $1$, for which the theory of
$\imath$crystals has not been developed, by means of global and local crystal
bases.

###### 2020 Mathematics Subject Classification:

Primary 17B37; Secondary 17B10

##  1\. introduction

Let $A=(a_{i,j})_{i,j\in I}$ be a symmetrizable generalized Cartan matrix,
$\mathfrak{g}$ the associated Kac-Moody algebra, and
$\mathbf{U}=U_{q}(\mathfrak{g})$ the Drinfeld and Jimbo’s quantum group with
weight lattice $X$. Let $X^{+}$ denote the set of dominant weights. For each
$\lambda\in X^{+}$, let $V(\lambda)$ (resp., $V(-\lambda)$) denote the
irreducible integrable highest (resp., lowest) weight $\mathbf{U}$-module of
highest weight $\lambda$ (resp., lowest weight $-\lambda$) with highest weight
vector $v_{\lambda}$ (resp., lowest weight vector $v_{-\lambda}$). The
canonical bases (also known as global crystal bases) of the negative part
$\mathbf{U}^{-}$ and the positive part $\mathbf{U}^{+}$ of $\mathbf{U}$, and
of $V(\pm\lambda)$ for all $\lambda\in X^{+}$, were constructed for type ADE
in [16] and for general in [17] geometrically and in [11] algebraically.

In [18], Lusztig constructed the canonical basis of the tensor product
$V(-\lambda)\otimes V(\mu)$ for arbitrary $\lambda,\mu\in X^{+}$, from the
canonical bases of $V(-\lambda)$ and $V(\mu)$. A key ingredient of his
construction is the quasi-$R$-matrix, which intertwines the bar-involutions on
$\mathbf{U}\otimes\mathbf{U}$ and $\mathbf{U}$.

The canonical bases thus constructed are stable in the following sense. For
each $\lambda,\mu,\nu\in X^{+}$, there exists a unique $\mathbf{U}$-module
homomorphism

$\pi_{\lambda,\mu,\nu}:V(-\lambda-\nu)\otimes V(\mu+\nu)\rightarrow
V(-\lambda)\otimes V(\mu)$

which sends $v_{-\lambda-\nu}\otimes v_{\mu+\nu}$ to $v_{-\lambda}\otimes
v_{\mu}$. Then, each canonical basis element of $V(-\lambda-\nu)\otimes
V(\mu+\nu)$ is sent to either a canonical basis element of $V(-\lambda)\otimes
V(\mu)$ or $0$, and the kernel of $\pi_{\lambda,\mu,\nu}$ is spanned by a
subset of the canonical basis. In other words, the homomorphism
$\pi_{\lambda,\mu,\nu}$ is a based $\mathbf{U}$-module homomorphism.

From this stability property, we see that for each $\zeta\in X$, the family

(1.1) $\displaystyle\\{V(-\lambda)\otimes
V(\mu)\\}_{\begin{subarray}{c}\lambda,\mu\in X^{+}\\\
\mu-\lambda=\zeta\end{subarray}}$

of $\mathbf{U}$-modules with $\pi_{\lambda,\mu,\nu}$ for each
$\lambda,\mu,\nu\in X^{+}$ with $\mu-\lambda=\zeta$, forms a projective system
of based $\mathbf{U}$-modules. Then, the subspace $\dot{\mathbf{U}}1_{\zeta}$
of the modified quantum group $\dot{\mathbf{U}}=\bigoplus_{\zeta\in
X}\dot{\mathbf{U}}1_{\zeta}$ with canonical basis can be regarded as the
projective limit of the projective system above in a certain category of based
$\mathbf{U}$-modules. This construction led to an explicit description of the
crystal basis of modified quantum group in [13].

Our main interest in the present paper is the $\imath$quantum group
counterpart of the construction above. The $\imath$quantum group (also known
as the quantum symmetric pair coideal subalgebra) $\mathbf{U}^{\imath}$
associated with an admissible pair $(I_{\bullet},\tau)$ (in the sense of [14,
Definition 2.3]) and parameters $\varsigma_{i}\in\mathbb{Q}(q)^{\times}$ and
$\kappa_{i}\in\mathbb{Q}(q)$ for $i\in I\setminus I_{\bullet}$ is a certain
right coideal subalgebra of $\mathbf{U}$ which forms a quantum symmetric pair
$(\mathbf{U},\mathbf{U}^{\imath})$. For each Kac-Moody algebra $\mathfrak{g}$,
there exists a quantum symmetric pair
$(\mathbf{U},\mathbf{U}^{\imath})=(U_{q}(\mathfrak{g}\oplus\mathfrak{g}),U_{q}(\mathfrak{g}))$.
Such a quantum symmetric pair is said to be of diagonal type. Therefore, the
quantum group $\mathbf{U}$ itself is an instance of $\imath$quantum groups.

Let $w_{\bullet}$ denote the longest element of the Weyl group associated with
$I_{\bullet}$ (by the definition of admissible pairs, $I_{\bullet}$ is of
finite type). Bao and Wang constructed a $\mathbf{U}^{\imath}$-module
$V(w_{\bullet}\lambda,\mu)$ for each $\lambda,\mu\in X^{+}$ with a
distinguished basis, called the $\imath$canonical basis, and
$\mathbf{U}^{\imath}$-module homomorphisms

$\pi^{\imath}_{\lambda,\mu,\nu}:V(w_{\bullet}(\lambda+\tau\nu),\mu+\nu)\rightarrow
V(w_{\bullet}\lambda,\mu)$

for finite type in [5] and for the general case in [7]. To be more precise,
the $\mathbf{U}^{\imath}$-module $V(w_{\bullet}\lambda,\mu)$ is obtained by
restriction from the $\mathbf{U}$-submodule of $V(\lambda)\otimes V(\mu)$
generated by $v_{w_{\bullet}\lambda}\otimes v_{\mu}$, where
$v_{w_{\bullet}\lambda}\in V(\lambda)$ denotes the unique canonical basis
element of weight $w_{\bullet}\lambda$. And, the $\mathbf{U}^{\imath}$-module
homomorphism $\pi^{\imath}_{\lambda,\mu,\nu}$ is the unique one which sends
$v_{w_{\bullet}(\lambda+\tau\nu)}\otimes v_{\mu+\nu}$ to
$v_{w_{\bullet}\lambda}\otimes v_{\mu}$. A key ingredient of their
construction of the $\imath$canonical bases is the quasi-$K$-matrix, which
intertwines the bar-involutions on $\mathbf{U}$ and $\mathbf{U}^{\imath}$. The
existence of quasi-$K$-matrix and bar-involution on $\mathbf{U}^{\imath}$ in
general was formulated in [6] and proved in [15] after many partial results
(see Section $1$ of loc. cit.).

Set $X^{\imath}:=X/\\{\lambda+w_{\bullet}\tau\lambda\mid\lambda\in X\\}$, and
let $\overline{\cdot}:X\rightarrow X^{\imath}$ denote the quotient map. Then,
for each $\zeta\in X^{\imath}$, we obtain a projective system

(1.2)
$\displaystyle\\{V(w_{\bullet}\lambda,\mu)\\}_{\begin{subarray}{c}\lambda,\mu\in
X^{+}\\\ \overline{\mu+w_{\bullet}\lambda}=\zeta\end{subarray}}$

of $\mathbf{U}^{\imath}$-modules. This projective system can be seen as a
natural generalization of Lusztig’s one (1.1). In fact, they coincide with
each other when the quantum symmetric pair is of diagonal type since the
quasi-$K$-matrix and the $\mathbf{U}^{\imath}$-module
$V(w_{\bullet}\lambda,\mu)$ become the quasi-$R$-matrix and
$V(-\lambda)\otimes V(\mu)$, respectively. However, in contrast to the
projective system (1.1), the $\imath$canonical bases are not stable in the
projective system (1.2). They are merely asymptotically stable; nevertheless
this weaker stability property could still lead to the canonical basis (i.e.,
the $\imath$canonical basis) of the modified $\imath$quantum group in [5],
[7].

On the other hand, in [22], the author proved that the $\imath$canonical bases
are stable in the projective system (1.2) when $I_{\bullet}$ is empty,
$a_{i,\tau(i)}\in\\{2,0,-1\\}$ for all $i\in I$, and when the parameters
$\varsigma_{i},\kappa_{i}$ are chosen appropriately. As a result, he
interprets the subspace $\dot{\mathbf{U}}^{\imath}1_{\zeta}$ of the modified
$\imath$quantum group $\dot{\mathbf{U}}^{\imath}=\bigoplus_{\zeta\in
X^{\imath}}\dot{\mathbf{U}}^{\imath}1_{\zeta}$ with $\imath$canonical basis as
the projective limit of (1.2) in a certain category of based
$\mathbf{U}^{\imath}$-modules. In the proof, the theory of $\imath$crystals
developed in [21] plays a crucial role.

It is natural to expect that one can prove the stability of $\imath$canonical
bases for general quantum symmetric pairs by developing the theory of
$\imath$crystals. The theory of $\imath$crystals in [22] is based on many
explicit calculation involving the quantum symmetric pairs of real rank $1$,
just like the theory of crystals is based on calculation involving the quantum
group of rank $1$. Here, the real rank of a quantum symmetric pair refers the
number of $\tau$-orbits in $I\setminus I_{\bullet}$. The quantum group in a
quantum symmetric pair of real rank $1$ considered in [22] is either of type
$A_{1}$, $A_{1}\times A_{1}$, or $A_{2}$. Hence, its structure is relatively
simple. In general, the quantum group in a quantum symmetric pair of real rank
$1$ is not of finite type. Even if we restrict our attention to a quantum
symmetric pair of finite classical type, its rank can be arbitrarily high.
Hence, the same strategy as [22] is not applicable for general quantum
symmetric pairs.

In the present paper, we prove the stability of $\imath$canonical bases for
the quantum symmetric pair of irreducible finite type of real rank $1$ without
developing the theory of $\imath$crystals. Here, “irreducible” means that the
Dynkin diagram $I$, extended by adding edges between $i$ and $\tau(i)$ for all
$i\in I$, is connected as a graph. This is the first step toward the
generalization of the stability theorem of $\imath$canonical bases to general
quantum symmetric pairs. Since the stability of $\imath$canonical bases is
closely related to $\imath$crystals, the author expects that our new stability
theorem, in turn, stimulates an attempt to extend the theory of
$\imath$crystals to general quantum symmetric pairs.

Let us summarize our proof of the stability of $\imath$canonical bases. First,
we study the $\mathbf{U}$-module structure of $V(w_{\bullet}\lambda,\mu)$ by
investigating the crystal structure of its crystal basis. This enables us to
construct a $\mathbf{U}$-module homomorphism

(1.3) $\displaystyle V(w_{\bullet}(\lambda+\tau\nu),\mu+\nu)\rightarrow
V(\nu+w_{\bullet}\tau\nu)\otimes V(w_{\bullet}\lambda,\mu).$

Note that $\nu+w_{\bullet}\tau\nu$ is dominant. Next, verifying a sufficient
condition for a $\mathbf{U}$-module homomorphism to be based, which is given
in Proposition 2.4.6, we prove that the $\mathbf{U}$-module homomorphism above
is based. Finally, we prove that there exists a based
$\mathbf{U}^{\imath}$-module homomorphism
$V(\nu+w_{\bullet}\tau\nu)\rightarrow\mathbb{Q}(q)$ (here, $\mathbb{Q}(q)$ is
the trivial $\mathbf{U}^{\imath}$-module with $\imath$canonical basis
$\\{1\\}$) which sends the highest weight vector to $1$. Composing this based
$\mathbf{U}^{\imath}$-module homomorphism with the based $\mathbf{U}$-module
homomorphism in the second step, we obtain a based
$\mathbf{U}^{\imath}$-module homomorphism, which is identical to
$\pi^{\imath}_{\lambda,\mu,\nu}$. The first and second steps are applicable
for general quantum symmetric pairs. The final step is achieved by studying
the quantum symmetric pairs of irreducible finite type of real rank $1$ one-
by-one; there are only eight kinds of such quantum symmetric pairs.

The paper is organized as follows. In Section 2, we review well-known results
concerning canonical, or global crystal, and crystal bases, and based
$\mathbf{U}$-modules. Then, we give a sufficient condition for two based
$\mathbf{U}$-modules with a crystal morphism between their crystal bases
having a based $\mathbf{U}$-module homomorphism which lifts the crystal
morphism in Proposition 2.4.6. In Section 3, after recalling the definition of
$\imath$quantum groups and results of [7] about based
$\mathbf{U}^{\imath}$-modules, we construct the based $\mathbf{U}$-module
homomorphism (1.3). In Section 4, we finish our proof of the stability of
$\imath$canonical bases by studying certain $\mathbf{U}^{\imath}$-modules for
each quantum symmetric pair of irreducible finite type of real rank $1$ one-
by-one.

### Acknowledgement

This work was supported by JSPS KAKENHI Grant Numbers JP20K14286 and
JP21J00013.

##  2\. Quantum Group

The purpose of this section is to fix our notation about the quantum groups,
and then prove Proposition 2.4.6, which provides a sufficient condition for a
crystal morphism between the crystal bases of two based $\mathbf{U}$-modules
to be lifted to a based $\mathbf{U}$-module homomorphism.

###  2.1. Quantum group

Let $A=(a_{i,j})_{i,j\in I}$ be a symmetrizable generalized Cartan matrix with
a symmetrizing matrix $D=\operatorname{diag}(d_{i}\mid i\in I)$ with
$d_{i}\in\mathbb{Z}_{>0}$ being relatively prime. We often identify $I$ with
the Dynkin diagram of $A$. Let $Y$ and $X$ be finitely generated free abelian
groups with a perfect pairing $\langle,\rangle:Y\times
X\rightarrow\mathbb{Z}$. Let $\Pi^{\vee}=\\{h_{i}\mid i\in I\\}\subset Y$ and
$\Pi=\\{\alpha_{i}\mid i\in I\\}\subset X$ be linearly independent subsets
satisfying

$\langle h_{i},\alpha_{j}\rangle=a_{i,j}$

for all $i,j\in I$. Let $W$ denote the Weyl group associated with the
generalized Cartan matrix $A$. For each $i\in I$, let $s_{i}\in W$ denote the
simple reflection.

Let $q$ be an indeterminate. For each $i\in I$, $n\in\mathbb{Z}$, and
$m\in\mathbb{Z}_{\geq 0}$, set

$q_{i}:=q^{d_{i}},\ [n]_{i}:=\frac{q_{i}^{n}-q_{i}^{-n}}{q_{i}-q_{i}^{-1}},\
[m]_{i}!:=\prod_{n=1}^{m}[n]_{i}.$

When $d_{i}=1$, we sometimes omit the subscript “$i$” from notation above.

Let $\mathbf{U}$ denote the quantum group. Namely, $\mathbf{U}$ is the unital
associative $\mathbb{Q}(q)$-algebra with generators $\\{E_{i},F_{i},K_{h}\mid
i\in I,\ h\in Y\\}$ subject to the following relations: For each $i,j\in I$
and $h,h_{1},h_{2}\in Y$,

$\displaystyle K_{0}=1,\ K_{h_{1}}K_{h_{2}}=K_{h_{1}+h_{2}},$ $\displaystyle
K_{h}E_{i}=q^{\langle h,\alpha_{i}\rangle}E_{i}K_{h},\ K_{h}F_{i}=q^{\langle
h,-\alpha_{i}\rangle}F_{i}K_{h},$ $\displaystyle
E_{i}F_{j}-F_{j}E_{i}=\delta_{i,j}\frac{K_{i}-K_{i}^{-1}}{q_{i}-q_{i}^{-1}},$
$\displaystyle S_{i,j}(E_{i},E_{j})=S_{i,j}(F_{i},F_{j})=0\ \text{ if }i\neq
j,$

where

$K_{i}:=K_{d_{i}h_{i}},\
S_{i,j}(x,y):=\sum_{r+s=1-a_{i,j}}(-1)^{s}\frac{1}{[r]_{i}![s]_{i}!}x^{r}yx^{s}.$

The $\mathbf{U}$ is a Hopf algebra with comultiplication map $\Delta$ given by

$\displaystyle\Delta(E_{i})=E_{i}\otimes 1+K_{i}\otimes E_{i},$
$\displaystyle\Delta(F_{i})=1\otimes F_{i}+F_{i}\otimes K_{i}^{-1},$
$\displaystyle\Delta(K_{h})=K_{h}\otimes K_{h}$

for all $i\in I$, $h\in Y$.

Let $\overline{\cdot}$ denote the bar-involution on $\mathbf{U}$, i.e., the
$\mathbb{Q}$-algebra automorphism such that

$\overline{E_{i}}=E_{i},\ \overline{F_{i}}=F_{i},\ \overline{K_{h}}=K_{-h},\
\bar{q}=q^{-1}$

for all $i\in I$, $h\in Y$.

Let $\rho$ denote the anti-algebra automorphism on $\mathbf{U}$ such that

(2.1) $\displaystyle\rho(E_{i})=q_{i}^{-1}F_{i}K_{i},\
\rho(F_{i})=q_{i}K_{i}^{-1}E_{i},\ \rho(K_{h})=K_{h}$

for all $i\in I$, $h\in Y$.

For each $i\in I$, let $T_{i}$ denote both the algebra automorphism
$T^{\prime\prime}_{i,1}$ on $\mathbf{U}$ in [19, Proposition 37.1.2] and the
automorphism $T^{\prime\prime}_{i,1}$ on integrable $\mathbf{U}$-modules in
[19, 5.2.1]. For each $w\in W$ with a reduced expression $w=s_{i_{1}}\cdots
s_{i_{r}}$, set $T_{w}:=T_{i_{1}}\cdots T_{i_{r}}$.

Let $\mathbf{U}^{+}$ (resp., $\mathbf{U}^{-}$) denote the subalgebra of
$\mathbf{U}$ generated by $\\{E_{i}\mid i\in I\\}$ (resp., $\\{F_{i}\mid i\in
I\\}$). Let $(\mathcal{L}(\pm\infty),\mathcal{B}(\pm\infty))$ and
$\mathbf{B}(\pm\infty)$ denote the crystal base and global crystal basis of
$\mathbf{U}^{\mp}$ in [11, Theorems 4 and 6]. In this paper, crystal lattices
are considered over the subring $\mathbf{A}_{\infty}$ of $\mathbb{Q}(q)$
consisting of all rational functions regular at $q=\infty$. Let
$G_{\pm\infty}:\mathcal{B}(\pm\infty)\rightarrow\mathbf{B}(\pm\infty)$ denote
the bijection such that $G_{\pm\infty}(b)+q^{-1}\mathcal{L}(\pm\infty)=b$ for
all $b\in\mathcal{B}(\pm\infty)$. Set
$b_{\pm\infty}:=1+q^{-1}\mathcal{L}(\pm\infty)\in\mathcal{B}(\pm\infty)$.

###  2.2. Crystal

Let $\mathcal{B}$ be a crystal in the sense of [12, Section 1.2]. For each
$\lambda\in X$, set

$\mathcal{B}_{\lambda}:=\\{b\in\mathcal{B}\mid\operatorname{wt}(b)=\lambda\\}.$

We say that $b\in\mathcal{B}_{\lambda}$ is a highest weight element of weight
$\lambda$ if $\tilde{E}_{i}b=0$ for all $i\in I$. Let
$\mathcal{B}^{\text{hi}}\subset\mathcal{B}$ denote the set of highest weight
elements.

For each $b\in\mathcal{B}$, let $C(b)$ denote the connected component of
$\mathcal{B}$ containing $b$.

Given two crystals $\mathcal{B}_{1},\mathcal{B}_{2}$, their tensor product
$\mathcal{B}_{1}\otimes\mathcal{B}_{2}$ is the crystal with the following
structure:

(2.2) $\displaystyle\begin{split}&\operatorname{wt}(b_{1}\otimes
b_{2})=\operatorname{wt}(b_{1})+\operatorname{wt}(b_{2}),\\\
&\varepsilon_{i}(b_{1}\otimes b_{2})=\max(\varepsilon_{i}(b_{1})-\langle
h_{i},\operatorname{wt}(b_{2})\rangle,\varepsilon_{i}(b_{2})),\\\
&\varphi_{i}(b_{1}\otimes
b_{2})=\max(\varphi_{i}(b_{1}),\varphi_{i}(b_{2})+\langle
h_{i},\operatorname{wt}(b_{1})\rangle),\\\ &\tilde{E}_{i}(b_{1}\otimes
b_{2})=\begin{cases}\tilde{E}_{i}b_{1}\otimes b_{2}&\text{ if
}\varepsilon_{i}(b_{1})>\varphi_{i}(b_{2}),\\\
b_{1}\otimes\tilde{E}_{i}b_{2}&\text{ if
}\varepsilon_{i}(b_{1})\leq\varphi_{i}(b_{2}),\end{cases}\\\
&\tilde{F}_{i}(b_{1}\otimes
b_{2})=\begin{cases}b_{1}\otimes\tilde{F}_{i}b_{2}&\text{ if
}\varepsilon_{i}(b_{1})<\varphi_{i}(b_{2}),\\\ \tilde{F}_{i}b_{1}\otimes
b_{2}&\text{ if
}\varepsilon_{i}(b_{1})\geq\varphi_{i}(b_{2}).\end{cases}\end{split}$

Note that this structure is different from that in [12] due to difference of
convention.

The following lemma is easily deduced from the above.

######  Lemma 2.2.1.

Let $\mathcal{B}_{1},\mathcal{B}_{2}$ be crystals such that

$\varepsilon_{i}(b)=\max\\{k\in\mathbb{Z}_{\geq 0}\mid\tilde{E}_{i}^{k}b\neq
0\\},\ \varphi_{i}(b)=\max\\{k\in\mathbb{Z}_{\geq 0}\mid\tilde{F}_{i}^{k}b\neq
0\\}$

for all $i\in I$, $b\in\mathcal{B}_{1},\mathcal{B}_{2}$. Let
$(b_{1},b_{2})\in\mathcal{B}_{1}\times\mathcal{B}_{2}$. Then, we have
$b_{1}\otimes b_{2}\in(\mathcal{B}_{1}\otimes\mathcal{B}_{2})^{\text{hi}}$ if
and only if $b_{2}\in\mathcal{B}_{2}^{\text{hi}}$ and
$\varepsilon_{i}(b_{1})\leq\langle h_{i},\operatorname{wt}(b_{2})\rangle$ for
all $i\in I$.

###  2.3. Irreducible module $V(\pm\lambda)$

Let $X^{+}$ denote the set of dominant weights:

$X^{+}=\\{\lambda\in X\mid\langle h_{i},\lambda\rangle\geq 0\ \text{ for all
}i\in I\\}.$

For each $\lambda\in X^{+}$, let $V(\lambda)$ (resp., $V(-\lambda)$) denote
the irreducible integrable highest (resp., lowest) weight $\mathbf{U}$-module
of highest weight $\lambda$ (resp., lowest weight $-\lambda$). Let
$(\mathcal{L}(\pm\lambda),\mathcal{B}(\pm\lambda))$ and
$\mathbf{B}(\pm\lambda)$ denote the crystal base and global crystal basis of
$V(\pm\lambda)$ [11, Theorems 2 and 6]. For $w\in W$, let $b_{\pm
w\lambda}\in\mathcal{B}(\pm\lambda)$ and $v_{\pm
w\lambda}\in\mathbf{B}(\pm\lambda)$ denote the unique elements of weight $\pm
w\lambda$.

######  Remark 2.3.1.

Suppose that the Dynkin diagram $I$ is of finite type. Let $w_{0}\in W$ denote
the longest element. Then, for each $\lambda\in X^{+}$ and $w\in W$, the
symbol $v_{-w\lambda}$ represents both the vectors $v_{-w\lambda}\in
V(-\lambda)$ and $v_{ww_{0}(-w_{0}\lambda)}\in V(-w_{0}\lambda)$. In order to
make our notation consistent, we identify $V(-\lambda)$ with
$V(-w_{0}\lambda)$ under the $\mathbf{U}$-module isomorphism
$V(-\lambda)\rightarrow V(-w_{0}\lambda)$ which sends $v_{-\lambda}$ to
$v_{w_{0}(-w_{0}\lambda)}$.

Let $(,)$ denote the inner product on $V(\lambda)$ such that
$(v_{\lambda},v_{\lambda})=1$ and

$(xu,v)=(u,\rho(x)v)$

for all $x\in\mathbf{U}$, $u,v\in V(\lambda)$, where $\rho$ is the anti-
algebra automorphism on $\mathbf{U}$ given in (2.1). By [19, Lemma 19.1.4], we
have

$(G(b_{1}),G(b_{2}))\in\delta_{b_{1},b_{2}}+q^{-1}\mathbf{A}_{\infty}$

for all $b_{1},b_{2}\in\mathcal{B}(\lambda)$.

For each $\lambda\in X^{+}$, let

$\pi_{\pm\lambda}:\mathbf{U}^{\mp}\rightarrow V(\pm\lambda)$

denote the $\mathbf{U}^{\mp}$-module homomorphism given by
$\pi_{\pm\lambda}(x)=xv_{\pm\lambda}$. By [11, Theorem 5], it induces a map
$\pi_{\pm\lambda}:\mathcal{B}(\pm\infty)\rightarrow\mathcal{B}(\pm\lambda)\sqcup\\{0\\}$,
and the induced map restricts to a bijection

$\mathcal{B}(\pm\infty;\pm\lambda):=\\{b\in\mathcal{B}(\pm\infty)\mid\pi_{\pm\lambda}(b)\neq
0\\}\rightarrow\mathcal{B}(\pm\lambda).$

Let

(2.3)
$\displaystyle\iota_{\pm\lambda}:\mathcal{B}(\pm\lambda)\rightarrow\mathcal{B}(\pm\infty;\pm\lambda)$

denote its inverse. For each $b\in\mathcal{B}(\pm\lambda)$, we have the
following:

$\displaystyle G(b)=G_{\pm\infty}(\iota_{\pm\lambda}(b))v_{\pm\lambda},$
$\displaystyle\iota_{\pm\lambda}(b_{\pm\lambda})=b_{\pm\infty},$
$\displaystyle\operatorname{wt}(\iota_{\pm\lambda}(b))=\operatorname{wt}(b)\mp\lambda,$
$\displaystyle\varepsilon_{i}(\iota_{\lambda}(b))=\varepsilon_{i}(b),\
\varphi_{i}(\iota_{-\lambda}(b))=\varphi_{i}(b),$
$\displaystyle\tilde{F}_{i}(\iota_{\lambda}(b))=\iota_{\lambda}(\tilde{F}_{i}b)\text{
if }\tilde{F}_{i}b\neq 0,\
\tilde{E}_{i}\iota_{-\lambda}(b)=\iota_{-\lambda}(\tilde{E}_{i}b)\text{ if
}\tilde{E}_{i}b\neq 0$

for all $i\in I$, $b\in\mathcal{B}(\pm\lambda)$.

###  2.4. Based module

Set $\mathcal{A}:=\mathbb{Q}[q,q^{-1}]$. Let
$\dot{\mathbf{U}}=\bigoplus_{\lambda,\mu\in
X}1_{\lambda}\dot{\mathbf{U}}1_{\mu}$ denote the modified quantum group, and
${}_{\mathcal{A}}\dot{\mathbf{U}}$ its $\mathcal{A}$-form.

Following [4, Section 2.1], we define a based $\mathbf{U}$-module to be an
integrable $\mathbf{U}$-module $M$ equipped with a linear basis
$\mathbf{B}_{M}$ satisfying the following:

  * •

$\mathbf{B}_{M}\cap M_{\lambda}$ is a basis of $M_{\lambda}$ for all
$\lambda\in X$, where

$M_{\lambda}:=\\{m\in M\mid K_{h}m=q^{\langle h,\lambda\rangle}m\text{ for all
}h\in Y\\}.$

  * •

${}_{\mathcal{A}}M:=\mathcal{A}\mathbf{B}_{M}$ is a
${}_{\mathcal{A}}\dot{\mathbf{U}}$-submodule. We call it the
$\mathcal{A}$-form of $M$.

  * •

The $\mathbb{Q}$-linear map $\overline{\cdot}:M\rightarrow M$ sending $q^{n}v$
to $q^{-n}v$ for all $n\in\mathbb{Z}$ and $v\in\mathbf{B}_{M}$ satisfies
$\overline{xm}=\bar{x}\bar{m}$ for all $x\in\mathbf{U}$, $m\in M$. We call it
the bar-involution on $M$.

  * •

Setting $\mathcal{L}_{M}:=\mathbf{A}_{\infty}\mathbf{B}_{M}$ and
$\mathcal{B}_{M}:=\\{v+q^{-1}\mathcal{L}_{M}\mid v\in\mathbf{B}_{M}\\}$, the
pair $(\mathcal{L}_{M},\mathcal{B}_{M})$ forms a crystal base of $M$.

Let $(M,\mathbf{B}_{M})$ be a based $\mathbf{U}$-module. Then, the quotient
map
$\mathrm{ev}_{\infty}:\mathcal{L}_{M}\rightarrow\mathcal{L}_{M}/q^{-1}\mathcal{L}_{M}$
restricts to a $\mathbb{Q}$-linear isomorphism

$\mathcal{L}_{M}\cap{}_{\mathcal{A}}M\cap\overline{\mathcal{L}_{M}}\rightarrow\mathcal{L}_{M}/q^{-1}\mathcal{L}_{M}.$

Let $G_{M}$ denote its inverse. We sometimes omit the subscript $M$ of $G_{M}$
for simplicity.

A based submodule of a based module $(M,\mathbf{B}_{M})$ is a
$\mathbf{U}$-submodule $N\subset M$ spanned by a subset
$\mathbf{B}_{N}\subset\mathbf{B}_{M}$. Note that $(N,\mathbf{B}_{N})$ is a
based $\mathbf{U}$-module in its own right.

Let $(N,\mathbf{B}_{N})$ be a based submodule of a based $\mathbf{U}$-module
$(M,\mathbf{B}_{M})$. Then,

$(M/N,\\{v+N\mid v\in\mathbf{B}_{M}\setminus\mathbf{B}_{N}\\})$

is a based $\mathbf{U}$-module.

Let $(M,\mathbf{B}_{M}),(N,\mathbf{B}_{N})$ be based $\mathbf{U}$-modules, and
$f:M\rightarrow N$ a $\mathbf{U}$-module homomorphism. We say that $f$ is a
based $\mathbf{U}$-module homomorphism if
$f(\mathbf{B}_{M})\subset\mathbf{B}_{N}\sqcup\\{0\\}$ and
$\operatorname{Ker}f$ is a based submodule. This definition is reformulated as
follows.

######  Lemma 2.4.1.

Let $(M,\mathbf{B}_{M}),(N,\mathbf{B}_{N})$ be based $\mathbf{U}$-modules, and
$f:M\rightarrow N$ a $\mathbf{U}$-module homomorphism. Then, $f$ is a based
$\mathbf{U}$-module homomorphism if and only if it satisfies the following:

  * •

$f(\mathcal{L}_{M})\subset\mathcal{L}_{N}$; it induces a map

$\phi:\mathcal{B}_{M}\rightarrow\mathcal{B}_{N};\
b\mapsto\mathrm{ev}_{\infty}(f(G_{M}(b))).$

  * •

$f({}_{\mathcal{A}}M)\subset{}_{\mathcal{A}}N$.

  * •

$f\circ\psi_{M}=\psi_{N}\circ f$, where $\psi_{M},\psi_{N}$ denote the bar-
involution on $M,N$, respectively.

  * •

$\phi$ is injective on $\\{b\in\mathcal{B}_{M}\mid\phi(b)\neq 0\\}$.

###### Proof.

The “only if” part is obvious. Let us prove the opposite direction. By the
assumption on $f$, we see that

$f(G_{M}(b))\in\mathcal{L}_{N}\cap{}_{\mathcal{A}}N\cap\overline{\mathcal{L}_{N}}$

for all $b\in\mathcal{B}_{M}$. Hence, we obtain

(2.4) $\displaystyle
f(G_{M}(b))=(G_{N}\circ\mathrm{ev}_{\infty})(f(G_{M}(b)))=G_{N}(\phi(b))$

for all $b\in\mathcal{B}_{M}$. This implies that
$f(\mathbf{B}_{M})\subset\mathbf{B}_{N}\sqcup\\{0\\}$. In order to complete
the proof, let us investigate $\operatorname{Ker}f$. Let
$v\in\operatorname{Ker}f$, and write

$v=\sum_{b\in\mathcal{B}_{M}}c_{b}G_{M}(b)$

for some $c_{b}\in\mathbb{Q}(q)$. Then, by equation (2.4), we have

$0=f(v)=\sum_{\begin{subarray}{c}b\in\mathcal{B}_{M}\\\ \phi(b)\neq
0\end{subarray}}c_{b}G_{N}(b).$

Since $\phi$ is injective on $\\{b\in\mathcal{B}_{M}\mid\phi(b)\neq 0\\}$, we
must have $c_{b}=0$ for all $b\in\mathcal{B}_{M}$ with $\phi(b)\neq 0$.
Therefore, we obtain

$v=\sum_{\begin{subarray}{c}b\in\mathcal{B}_{M}\\\
\phi(b)=0\end{subarray}}c_{b}G_{M}(b).$

Consequently,

$\operatorname{Ker}f=\mathbb{Q}(q)\\{G_{M}(b)\mid b\in\mathcal{B}_{M}\text{
and }\phi(b)=0\\}.$

This implies that $f$ is based. Thus, the proof completes. ∎

######  Example 2.4.2.

Let $\lambda,\mu,\lambda_{1},\ldots,\lambda_{r}\in X^{+}$.

  1. (1)

$(V(\pm\lambda),\mathbf{B}(\pm\lambda))$ is a based $\mathbf{U}$-module.

  2. (2)

Let $\mathbf{B}(\lambda_{1},\ldots,\lambda_{r})$ denote the canonical basis of
$V(\lambda_{1},\ldots,\lambda_{r}):=V(\lambda_{1})\otimes\cdots\otimes
V(\lambda_{r})$ constructed in [4, Theorem 2.9]. Then,
$(V(\lambda_{1},\ldots,\lambda_{r}),\mathbf{B}(\lambda_{1},\ldots,\lambda_{r}))$
is a based $\mathbf{U}$-module. Its crystal basis
$\mathcal{B}(\lambda_{1},\ldots,\lambda_{r})$ is the tensor product
$\mathcal{B}(\lambda_{1})\otimes\cdots\otimes\mathcal{B}(\lambda_{r})$.

  3. (3)

By [19, Proposition 25.1.2], for each $\lambda,\mu\in X^{+}$, there exists a
unique based $\mathbf{U}$-module homomorphism

$\chi_{\lambda,\mu}:V(\lambda+\mu)\rightarrow V(\lambda,\mu)$

such that $\chi_{\lambda,\mu}(v_{\lambda+\mu})=v_{\lambda}\otimes v_{\mu}$.

  4. (4)

By [19, Proposition 25.1.4], for each $\lambda\in X^{+}$, there exists a
unique based $\mathbf{U}$-module homomorphism

$\delta_{\lambda}:V(-\lambda)\otimes V(\lambda)\rightarrow\mathbb{Q}(q)$

such that $\delta_{\lambda}(v_{-\lambda}\otimes v_{\lambda})=1$. Here, we
identify $(\mathbb{Q}(q),\\{1\\})$ with the based $\mathbf{U}$-module
$(V(0),\mathbf{B}(0))$.

Let $\mathcal{C}^{+}$ denote the category of based $\mathbf{U}$-modules and
based homomorphisms consisting of $(M,\mathbf{B}_{M})$ with finite-dimensional
weight spaces satisfying the following: There exist finitely many
$\lambda_{1},\ldots,\lambda_{r}\in X$ such that for each $\lambda\in X$, we
have $M_{\lambda}\neq 0$ only if
$\lambda\in\bigcup_{k=1}^{r}\\{\lambda_{k}-\alpha\mid\alpha\in\sum_{i\in
I}\mathbb{Z}_{\geq 0}\alpha_{i}\\}$. In particular, $M$ is semisimple with
simple components of the form $V(\lambda)$, $\lambda\in X^{+}$.

Let $(M,\mathbf{B}_{M})\in\mathcal{C}^{+}$. For each $\lambda\in X^{+}$, let
$I_{\lambda}(M)$ denote the sum of all submodules of $M$ isomorphic to
$V(\lambda)$. Set

$\displaystyle M[>\lambda]:=\bigoplus_{\begin{subarray}{c}\mu\in X^{+}\\\
\mu>\lambda\end{subarray}}I_{\mu}(M),\quad M[\geq\lambda]:=M[>\lambda]\oplus
I_{\lambda}(M),$
$\displaystyle\mathcal{B}_{M}[>\lambda]:=\bigsqcup_{\begin{subarray}{c}b^{\prime}\in\mathcal{B}_{M}^{\text{hi}}\\\
\operatorname{wt}(b^{\prime})>\lambda\end{subarray}}C(b^{\prime}),\quad\mathcal{B}_{M}[\geq\lambda]:=\bigsqcup_{\begin{subarray}{c}b^{\prime}\in\mathcal{B}_{M}^{\text{hi}}\\\
\operatorname{wt}(b^{\prime})\geq\lambda\end{subarray}}C(b^{\prime}),$

where $<$ denotes the partial order on $X^{+}$ defined by saying
$\lambda\leq\mu$ if and only if $\mu-\lambda\in\sum_{i\in I}\mathbb{Z}_{\geq
0}\alpha_{i}$. Recall that $C(b^{\prime})$ denotes the connected component of
$\mathcal{B}_{M}$ containing $b^{\prime}$ (cf. Subsection 2.2). By the same
way as the proofs of [19, Propositions 27.1.7 and 27.1.8], we see that the
submodules $M[>\lambda]$ and $M[\geq\lambda]$ are based submodules with
crystal basis $\mathcal{B}_{M}[>\lambda]$ and $\mathcal{B}_{M}[\geq\lambda]$,
respectively, and that there exists a based $\mathbf{U}$-module isomorphism
$M[\geq\lambda]/M[>\lambda]\rightarrow
V(\lambda)^{|\mathcal{B}_{M,\lambda}^{\text{hi}}|}$ sending $G_{M}(b)$ to
$v_{\lambda}^{b}$ for all $b\in\mathcal{B}_{M,\lambda}^{\text{hi}}$, where
$v_{\lambda}^{b}$ denotes the highest weight vector of the $b$-th component.

For each $\lambda\in X^{+}$, let

$p_{\lambda}:M=\bigoplus_{\mu\in X^{+}}I_{\mu}(M)\rightarrow I_{\lambda}(M)$

denote the projection. For each $b\in\mathcal{B}_{M}^{\text{hi}}$, set

$v_{b}:=p_{\operatorname{wt}(b)}(G_{M}(b))\in I_{\operatorname{wt}(b)}(M).$

By the above, we see that $v_{b}$ is a highest weight vector of weight
$\operatorname{wt}(b)$ and that

$v_{b}\in\mathcal{L}_{M},\ \mathrm{ev}_{\infty}(v_{b})=b.$

For each $b\in\mathcal{B}_{M}^{\text{hi}}$, set

$M[\geq
b]:=M[>\operatorname{wt}(b)]\oplus\mathbf{U}v_{b},\quad\mathcal{B}_{M}[\geq
b]:=\mathcal{B}_{M}[>\operatorname{wt}(b)]\sqcup C(b).$

The connected component $C(b)$ is isomorphic to
$\mathcal{B}(\operatorname{wt}(b))$. Let

$\iota_{b}:C(b)\rightarrow\mathcal{B}(\operatorname{wt}(b))\xrightarrow{\iota_{\operatorname{wt}(b)}}\mathcal{B}(\infty;\operatorname{wt}(b))$

denote the composition of the isomorphism
$C(b)\rightarrow\mathcal{B}(\operatorname{wt}(b))$ and
$\iota_{\operatorname{wt}(b)}$ (cf. equation (2.3)).

######  Lemma 2.4.3.

Let $(M,\mathbf{B}_{M})\in\mathcal{C}^{+}$, and
$b\in\mathcal{B}_{M}^{\text{hi}}$. Then, $M[\geq b]$ is a based submodule of
$M$ with crystal basis $\mathcal{B}_{M}[\geq b]$. Moreover, there exists a
based $\mathbf{U}$-module isomorphism

$M[\geq b]/M[>\operatorname{wt}(b)]\rightarrow V(\operatorname{wt}(b))$

which sends $G_{M}(b)+M[>\operatorname{wt}(b)]$ to $v_{\operatorname{wt}(b)}$.

###### Proof.

The assertion is obvious from the above. ∎

######  Lemma 2.4.4.

Let $(M,\mathbf{B}_{M})\in\mathcal{C}^{+}$ and
$b\in\mathcal{B}_{M}^{\text{hi}}$. Then, we have $\overline{v_{b}}=v_{b}$.

###### Proof.

Set $\lambda:=\operatorname{wt}(b)$ and $u_{b}:=G_{M}(b)-v_{b}\in
M[>\lambda]$. Since the bar-involution on $M$ preserves $I_{\mu}(M)$ for all
$\mu\in X^{+}$, we see that $\overline{v_{b}}\in I_{\lambda}(M)$ and
$\overline{u_{b}}\in M[>\lambda]$. Hence, we obtain

$\overline{v_{b}}-v_{b}=u_{b}-\overline{u_{b}}\in I_{\lambda}(M)\cap
M[>\lambda]=0.$

This proves the assertion. ∎

######  Lemma 2.4.5.

Let $(M,\mathbf{B}_{M}),(N,\mathbf{B}_{N})\in\mathcal{C}^{+}$, and
$\phi:\mathcal{B}_{M}\rightarrow\mathcal{B}_{N}$ be a strict crystal morphism.
Then, there exists a unique $\mathbf{U}$-module homomorphism $f:M\rightarrow
N$ such that $f(v_{b})=v_{\phi(b)}$ for all $b\in\mathcal{B}_{M}^{\text{hi}}$.
Here, we set $v_{\phi(b)}=0$ if $\phi(b)=0$. Moreover, the following hold:

  * •

$f\circ\psi_{M}=\psi_{N}\circ f$, where $\psi_{M},\psi_{N}$ denote the bar-
involutions on $M,N$, respectively.

  * •

$f(\mathcal{L}_{M})\subset\mathcal{L}_{N}$.

  * •

$\phi\circ\mathrm{ev}_{\infty}=\mathrm{ev}_{\infty}\circ f$ on
$\mathcal{L}_{M}$.

###### Proof.

The existence of $f$ follows from easy observation that $v_{\phi(b)}$ is
either $0$ or a highest weight vector of weight $\operatorname{wt}(b)$. The
commutativity of $f$ and the bar-involutions follows from Lemma 2.4.4.

In order to show the remaining assertions, let
$b\in\mathcal{B}_{M}^{\text{hi}}$ and $b^{\prime}\in C(b)$. Then, we can write
$b^{\prime}=\tilde{F}_{i_{1}}\cdots\tilde{F}_{i_{r}}b$ for some
$i_{1},\ldots,i_{r}\in I$. Set
$v:=\tilde{F}_{i_{1}}\cdots\tilde{F}_{i_{r}}v_{b}\in\mathcal{L}_{M}$. Then, we
have

$\displaystyle\mathrm{ev}_{\infty}(v)=\tilde{F}_{i_{1}}\cdots\tilde{F}_{i_{r}}b=b^{\prime},$
$\displaystyle
f(v)=\tilde{F}_{i_{1}}\cdots\tilde{F}_{i_{r}}v_{\phi(b)}\in\mathcal{L}_{N},$
$\displaystyle\mathrm{ev}_{\infty}(f(v))=\tilde{F}_{i_{1}}\cdots\tilde{F}_{i_{r}}\phi(b)=\phi(b^{\prime}).$

These imply the remaining assertions. Thus, the proof completes. ∎

######  Proposition 2.4.6.

Let $(M,\mathbf{B}_{M}),(N,\mathbf{B}_{N})\in\mathcal{C}^{+}$,
$\phi:\mathcal{B}_{M}\rightarrow\mathcal{B}_{N}$ be a strict crystal morphism,
and $f:M\rightarrow N$ the $\mathbf{U}$-module homomorphism in Lemma 2.4.5.
Suppose that $f(E_{i}G_{M}(b))=E_{i}G_{N}(\phi(b))$ for all $i\in I$ and
$b\in\mathcal{B}_{M}^{\text{hi}}$. Then, we have $f(G_{M}(b))=G_{N}(\phi(b))$
for all $b\in\mathcal{B}_{M}$. Furthermore, if $\phi(b_{1})=\phi(b_{2})\neq 0$
implies $b_{1}=b_{2}$ for all $b_{1},b_{2}\in\mathcal{B}_{M}^{\text{hi}}$,
then $f$ is based.

###### Proof.

Let $b\in\mathcal{B}_{M}^{\text{hi}}$ and set $\lambda:=\operatorname{wt}(b)$.
Then, the number $D(b)$ of elements $b_{1}\in\mathcal{B}_{M}^{\text{hi}}$ such
that $\operatorname{wt}(b_{1})>\lambda$ is finite. We prove that
$f(G_{M}(b^{\prime}))=G_{N}(\phi(b^{\prime}))$ for all $b^{\prime}\in C(b)$ by
induction on $D(b)$.

Assume that our claim is true for all $b_{1}\in\mathcal{B}_{M}^{\text{hi}}$
with $D(b_{1})<D(b)$; note that we assume nothing when $D(b)=0$. Then, we have

(2.5) $\displaystyle f(G_{M}(b^{\prime\prime}))=G_{N}(\phi(b^{\prime\prime}))$

for all $b^{\prime\prime}\in\mathcal{B}[>\lambda]$.

Set

$u_{b}:=G_{M}(b)-v_{b}\in M[>\lambda],\quad
u_{\phi(b)}:=G_{N}(\phi(b))-v_{\phi(b)}\in N[>\lambda].$

We shall show that $f(u_{b})=u_{\phi(b)}$. By the assumption on $f$, we have

$E_{i}(f(u_{b})-u_{\phi(b)})=f(E_{i}G_{M}(b))-E_{i}G_{N}(\phi(b))=0$

for all $i\in I$. This implies that $f(u_{b})-u_{\phi(b)}$ is either $0$ or a
highest weight vector of weight $\lambda$. However, the submodule
$N[>\lambda]$, to which $f(u_{b})-u_{\phi(b)}$ belongs, has no highest weight
vector of weight $\lambda$. Hence, the equality $f(u_{b})=u_{\phi(b)}$
follows.

Now, we have

(2.6) $\displaystyle
f(G_{M}(b))=f(v_{b}+u_{b})=v_{\phi(b)}+u_{\phi(b)}=G_{N}(\phi(b)).$

Let $b^{\prime}\in C(b)$. We shall show that
$f(G_{M}(b^{\prime}))=G_{N}(\phi(b^{\prime}))$. By Lemma 2.4.3 (see also
Subsection 2.3), we have

$G_{M}(b^{\prime})=G_{\infty}(\iota_{b}(b^{\prime}))G_{M}(b)+\sum_{b^{\prime\prime}\in\mathcal{B}_{M}[>\lambda]}c_{b^{\prime\prime}}G_{M}(b^{\prime\prime})$

for some $b^{\prime\prime}\in\mathcal{A}$. By equations (2.5) and (2.6), we
obtain

$f(G_{M}(b^{\prime}))=G_{\infty}(\iota_{b}(b^{\prime}))G_{N}(\phi(b))+\sum_{b^{\prime\prime}\in\mathcal{B}_{M}[>\lambda]}c_{b^{\prime\prime}}G_{N}(\phi(b^{\prime\prime})).$

This implies that $f(G_{M}(b^{\prime}))\in{}_{\mathcal{A}}N$. By Lemma 2.4.5,
we see that

$f(G_{M}(b^{\prime}))\in\mathcal{L}_{N},\
\mathrm{ev}_{\infty}(f(G_{M}(b^{\prime})))=\phi(b^{\prime}),\
\overline{f(G_{M}(b^{\prime}))}=f(G_{M}(b^{\prime})).$

These imply that
$f(G_{M}(b^{\prime}))\in\mathcal{L}_{N}\cap{}_{\mathcal{A}}N\cap\overline{\mathcal{L}_{N}}$,
and hence,

$f(G_{M}(b^{\prime}))=(G_{N}\circ\mathrm{ev}_{\infty})(f(G_{M}(b^{\prime})))=G_{N}(\phi(b^{\prime})),$

as desired.

The remaining assertion follows from Lemma 2.4.1. Thus, the proof completes. ∎

###  2.5. Based submodule $V(w\lambda,\mu)$

For each $\lambda,\mu\in X^{+}$ and $w\in W$, let $V(w\lambda,\mu)\subset
V(\lambda)\otimes V(\mu)$ denote the $\mathbf{U}$-submodule generated by
$v_{w\lambda}\otimes v_{\mu}$. By [7, Theorem 2.2], the $V(w\lambda,\mu)$ is a
based submodule of $V(\lambda,\mu)$. Set

$\displaystyle\mathcal{L}(w\lambda,\mu):=\mathcal{L}(\lambda,\mu)\cap
V(w\lambda,\mu),$
$\displaystyle\mathcal{B}(w\lambda,\mu):=\\{v+q^{-1}\mathcal{L}(w\lambda,\mu)\mid
v\in\mathbf{B}(\lambda,\mu)\cap V(w\lambda,\mu)\\}.$

Then, $(\mathcal{L}(w\lambda,\mu),\mathcal{B}(w\lambda,\mu))$ forms a crystal
base of $V(w\lambda,\mu)$.

######  Lemma 2.5.1.

Let $\lambda,\mu,\nu\in X^{+}$ and $w\in W$. Then, $V(\nu)\otimes
V(w\lambda,\mu)$ is a based submodule of $V(\nu,\lambda,\mu)$ $($see Example
2.4.2 (2) for the based module structure of $V(\nu,\lambda,\mu)$$)$.

###### Proof.

It suffices to show that $G(b_{1}\otimes b_{2}\otimes b_{3})\in V(\nu)\otimes
V(w\lambda,\mu)$ for all $b_{1}\otimes b_{2}\otimes
b_{3}\in\mathcal{B}(\nu)\otimes\mathcal{B}(w\lambda,\mu)\subset\mathcal{B}(\nu,\lambda,\mu)$.
By the construction of $G(b_{1}\otimes b_{2}\otimes b_{3})$ in [4, Theorem
2.9], we have

$G(b_{1}\otimes b_{2}\otimes b_{3})=\sum_{b^{\prime}_{1}\otimes
b^{\prime}_{2}\otimes
b^{\prime}_{3}\in\mathcal{B}(\nu,\lambda,\mu)}c_{b^{\prime}_{1},b^{\prime}_{2},b^{\prime}_{3}}G(b^{\prime}_{1})\otimes
G(b^{\prime}_{2}\otimes b^{\prime}_{3})$

for some $c_{b^{\prime}_{1},b^{\prime}_{2},b^{\prime}_{3}}\in\mathcal{A}$ such
that $c_{b^{\prime}_{1},b^{\prime}_{2},b^{\prime}_{3}}=0$ unless
$G(b^{\prime}_{2}\otimes b^{\prime}_{3})$ belongs to the smallest based
$\mathbf{U}$-submodule of $V(\lambda,\mu)$ containing $G(b_{2}\otimes b_{3})$.
Since $V(w\lambda,\mu)$ is based submodule of $V(\lambda,\mu)$ containing
$G(b_{2}\otimes b_{3})$, we see that
$c_{b^{\prime}_{1},b^{\prime}_{2},b^{\prime}_{3}}=0$ unless
$G(b^{\prime}_{2}\otimes b^{\prime}_{3})\in V(w\lambda,\mu)$. This proves the
assertion. ∎

##  3\. $\imath$Quantum group

In this section, after recalling the notion of $\imath$quantum groups and
based $\mathbf{U}^{\imath}$-modules, we prove the existence of certain based
$\mathbf{U}$-module homomorphisms in Proposition 3.4.7. This reduces the
problem of stability of $\imath$canonical bases to that of existence of based
$\mathbf{U}^{\imath}$-module homomorphism
$\delta^{\imath}_{\nu}:V(\nu+w_{\bullet}\tau\nu)\rightarrow\mathbb{Q}(q)$
sending the highest weight vector to $1$ for each $\nu\in X^{+}$.

###  3.1. Admissible pair

An admissible pair [14, Definition 2.3] is a pair $(I_{\bullet},\tau)$
consisting of a subset $I_{\bullet}\subset I$ of finite type and a Dynkin
diagram automorphism $\tau$ on $I$ satisfying the following:

  * •

$\tau^{2}=\mathrm{id}$.

  * •

$w_{\bullet}(\alpha_{j})=-\alpha_{\tau(j)}$ for all $j\in I_{\bullet}$, where
$w_{\bullet}$ denotes the longest element of the Weyl group $W_{I_{\bullet}}$
for $I_{\bullet}$.

  * •

$\langle\rho^{\vee}_{\bullet},\alpha_{i}\rangle\in\mathbb{Z}$ for all $i\in
I_{\circ}:=I\setminus I_{\bullet}$ with $\tau(i)=i$, where
$\rho^{\vee}_{\bullet}$ denotes half the sum of positive coroots for
$I_{\bullet}$.

In order to clarify which Dynkin diagram $I$ is considered, we sometimes
denote an admissible pair by the triple $(I,I_{\bullet},\tau)$.

An admissible pair $(I_{\bullet},\tau)$ is said to be _irreducible_ if for
each $i,j\in I$, there exists a sequence $i=i_{1},\ldots,i_{r}=j\in I$ such
that for each $k=1,\ldots,r-1$, we have either $a_{i_{k},i_{k+1}}\neq 0$ or
$i_{k+1}=\tau(i_{k})$.

An admissible pair is said to be of _finite type_ if the Dynkin diagram $I$ is
of finite type. The admissible pairs of finite type with $I_{\bullet}\neq I$
are identical to the Satake diagrams in [1] (cf. [14, after Definition 2.3]).

The number of $\tau$-orbits in $I_{\circ}$ is called the _real rank_ of
$(I_{\bullet},\tau)$.

Later, we will restrict our attention to the irreducible admissible pairs of
finite type of real rank $1$. Here is the list of such admissible pairs; for
the labels of Dynkin diagrams (except type $A_{1}\times A_{1}$), we follow [8,
Figs. 2.5 and 2.6]:

  * •

Type AI.

    * –

$I=\\{1\\}$: type $A_{1}$.

    * –

$I_{\bullet}=\emptyset$.

    * –

$\tau=\mathrm{id}$.

  * •

Type AII.

    * –

$I=\\{1,2,3\\}$: type $A_{3}$.

    * –

$I_{\bullet}=\\{1,3\\}$.

    * –

$\tau=\mathrm{id}$.

  * •

Type AIII.

    * –

$I=\\{1,2\\}$: type $A_{1}\times A_{1}$.

    * –

$I_{\bullet}=\emptyset$.

    * –

$\tau(1)=2,\ \tau(2)=1$.

  * •

Type AIV.

    * –

$I=\\{1,\ldots,n\\}$: type $A_{n}$ with $n\geq 2$.

    * –

$I_{\bullet}=\\{2,\ldots,n-1\\}$.

    * –

$\tau(i)=n-i+1$.

  * •

Type BII.

    * –

$I=\\{1,\ldots,n\\}$: type $B_{n}$ with $n\geq 2$.

    * –

$I_{\bullet}=\\{2,\ldots,n\\}$.

    * –

$\tau=\mathrm{id}$.

  * •

Type CII.

    * –

$I=\\{1,\ldots,n\\}$: type $C_{n}$ with $n\geq 3$.

    * –

$I_{\bullet}=\\{1,3,\ldots,n\\}$.

    * –

$\tau=\mathrm{id}$.

  * •

Type DII.

    * –

$I=\\{1,\ldots,n\\}$: type $D_{n}$ with $n\geq 4$.

    * –

$I_{\bullet}=\\{2,\ldots,n\\}$.

    * –

$\tau(i)=\begin{cases}n&\text{ if }i=n-1\text{ and }n\in 2\mathbb{Z},\\\
n-1&\text{ if }i=n\text{ and }n\in 2\mathbb{Z},\\\ i&\text{
otherwise}.\end{cases}$

  * •

Type FII.

    * –

$I=\\{1,2,3,4\\}$: type $F_{4}$.

    * –

$I_{\bullet}=\\{1,2,3\\}$.

    * –

$\tau=\mathrm{id}$.

###  3.2. $\imath$Quantum group

Assume that $\tau$ induces involutive automorphisms on $Y$ and $X$ such that

$\tau h_{i}=h_{\tau(i)},\ \tau\alpha_{i}=\alpha_{\tau(i)},\ \langle\tau
h,\tau\lambda\rangle=\langle h,\lambda\rangle$

for all $i\in I$, $h\in Y$, $\lambda\in X$. Note that this assumption is
always satisfied when $I$ is of finite type and $Y=\sum_{i\in
I}\mathbb{Z}h_{i}$.

Set

$Y^{\imath}:=\\{h\in Y\mid h+w_{\bullet}\tau h=0\\},\quad
X^{\imath}:=X/\\{\lambda+w_{\bullet}\tau\lambda\mid\lambda\in X\\}.$

For each $\lambda\in X$, let $\bar{\lambda}\in X^{\imath}$ denote the image of
$\lambda$. The perfect pairing $\langle,\rangle:Y\times
X\rightarrow\mathbb{Z}$ induces a bilinear map
$\langle,\rangle:Y^{\imath}\times X^{\imath}\rightarrow\mathbb{Z}$ such that

$\langle h,\bar{\lambda}\rangle=\langle h,\lambda\rangle$

for all $h\in Y^{\imath}$, $\lambda\in X$.

For each $i\in I_{\circ}$, chose
$\varsigma_{i}\in\mathbb{Z}[q,q^{-1}]^{\times}$ and
$\kappa_{i}\in\mathbb{Z}[q,q^{-1}]$ in a way such that

  * •

$\varsigma_{i}=\varsigma_{\tau(i)}$ if $\langle
h_{i},w_{\bullet}\alpha_{\tau(i)}\rangle=0$.

  * •

$\kappa_{i}=0$ unless $\tau(i)=i$, $\langle h_{i},\alpha_{j}\rangle=0$ for all
$j\in I_{\bullet}$, and $\langle h_{k},\alpha_{i}\rangle\in 2\mathbb{Z}$ for
all $k\in I_{\circ}$ with $\tau(k)=k$ and $\langle h_{k},\alpha_{j}\rangle=0$
for all $j\in I_{\bullet}$.

  * •

$\varsigma_{\tau(i)}=(-1)^{\langle
2\rho_{\bullet}^{\vee},\alpha_{i}\rangle}q_{i}^{-\langle
h_{i},2\rho_{\bullet}+w_{\bullet}\alpha_{\tau(i)}\rangle}\overline{\varsigma_{i}}$,
where $\rho_{\bullet}$ denotes half the sum of positive roots for
$I_{\bullet}$.

  * •

$\overline{\kappa_{i}}=\kappa_{i}$.

The first two conditions are needed to make the $\imath$quantum group below to
be a reasonable size (cf. [14, Theorem 10.8], see also [2, Remark 3.3]). The
third one guarantees the existence of the bar-involution on
$\mathbf{U}^{\imath}$ (cf. [15, Corollary 4.2]). The fourth one was used to
construct the quasi-$K$-matrix in [3, Theorem 6.10]. The constraint
$\varsigma_{i},\kappa_{i}\in\mathbb{Z}[q,q^{-1}]$ is necessary for the theory
of $\imath$canonical bases in [7].

The $\imath$quantum group is the subalgebra
$\mathbf{U}^{\imath}\subset\mathbf{U}$ generated by

$\\{E_{j},B_{i},K_{h}\mid j\in I_{\bullet},\ i\in I,\ h\in Y^{\imath}\\},$

where

$B_{i}:=\begin{cases}F_{i}&\text{ if }i\in I_{\bullet},\\\
F_{i}+\varsigma_{i}T_{w_{\bullet}}(E_{\tau(i)})K_{i}^{-1}+\kappa_{i}K_{i}^{-1}&\text{
if }i\in I_{\circ}.\end{cases}$

By [15, Corollary 4.2], there exists a unique $\mathbb{Q}$-algebra
automorphism $\overline{\cdot}$ on $\mathbf{U}^{\imath}$ such that

$\overline{E_{j}}=E_{j},\ \overline{B_{i}}=B_{i},\ \overline{K_{h}}=K_{-h},\
\bar{q}=q^{-1}$

for all $j\in I_{\bullet}$, $i\in I$, $h\in Y^{\imath}$. We call it the
$\imath$bar-involution on $\mathbf{U}^{\imath}$.

Let $\mathbf{U}_{I_{\bullet}}\subset\mathbf{U}^{\imath}$ denote the subalgebra
generated by $\\{E_{j},F_{j},K_{j}^{\pm 1}\mid j\in I_{\bullet}\\}$. It is the
quantum group associated with $I_{\bullet}$. Similarly, let
$\mathbf{U}_{I_{\bullet}}^{+}$, $\mathcal{B}_{I_{\bullet}}(-\infty)$,
$\mathcal{B}_{I_{\bullet}}(-\infty;-\lambda)$,
$\mathcal{B}_{I_{\bullet}}(-\lambda)$, and so on denote the same things
without subscripts, but associated with $I_{\bullet}$.

###  3.3. Based module

Let $\dot{\mathbf{U}}^{\imath}=\bigoplus_{\zeta,\eta\in
X^{\imath}}1_{\zeta}\dot{\mathbf{U}}^{\imath}1_{\eta}$ denote the modified
$\imath$quantum group (cf. [7, Section 3.5], [21, Section 3.3]), and
${}_{\mathcal{A}}\dot{\mathbf{U}}^{\imath}$ its $\mathcal{A}$-form.

Following [7, Definition 6.11], we define a based $\mathbf{U}^{\imath}$-module
to be a pair $(M,\mathbf{B}^{\imath}_{M})$ consisting of a weight
$\mathbf{U}^{\imath}$-module $M$ and its linear basis
$\mathbf{B}^{\imath}_{M}$ satisfying the following:

  * •

$\mathbf{B}^{\imath}_{M}\cap M_{\zeta}$ forms a basis of $M_{\zeta}$, where

$M_{\zeta}:=\\{m\in M\mid K_{h}m=q^{\langle h,\zeta\rangle}\text{ for all
}h\in Y^{\imath}\\}.$

  * •

${}_{\mathcal{A}}M:=\mathcal{A}\mathbf{B}^{\imath}_{M}$ is a
${}_{\mathcal{A}}\dot{\mathbf{U}}^{\imath}$-submodule. We call it the
$\mathcal{A}$-form of $M$.

  * •

The $\mathbb{Q}$-linear map $\overline{\cdot}:M\rightarrow M$ sending $q^{n}v$
to $q^{-n}v$ for all $n\in\mathbb{Z}$ and $v\in\mathbf{B}^{\imath}_{M}$
satisfies $\overline{xm}=\bar{x}\bar{m}$ for all $x\in\mathbf{U}^{\imath}$ and
$m\in M$. We call it the $\imath$bar-involution on $M$.

  * •

Setting $\mathcal{L}_{M}:=\mathbf{A}_{\infty}\mathbf{B}^{\imath}_{M}$, the
quotient map
$\mathrm{ev}_{\infty}:\mathcal{L}_{M}\rightarrow\mathcal{L}_{M}/q^{-1}\mathcal{L}_{M}$
restricts to a $\mathbb{Q}$-linear isomorphism
$\mathcal{L}_{M}\cap{}_{\mathcal{A}}M\cap\overline{\mathcal{L}_{M}}\rightarrow\mathcal{L}_{M}/q^{-1}\mathcal{L}_{M}$;
let $G^{\imath}_{M}$ denote its inverse.

The notions of based $\mathbf{U}^{\imath}$-submodules and based
$\mathbf{U}^{\imath}$-module homomorphisms are defined in the same way as
based $\mathbf{U}$-modules. The following can be proved by the same way as
Lemma 2.4.1.

######  Lemma 3.3.1.

Let $(M,\mathbf{B}_{M}),(N,\mathbf{B}_{N})$ be based
$\mathbf{U}^{\imath}$-modules, and $f:M\rightarrow N$ a
$\mathbf{U}^{\imath}$-module homomorphism. Then, $f$ is a based
$\mathbf{U}^{\imath}$-module homomorphism if and only if it satisfies the
following:

  * •

$f(\mathcal{L}_{M})\subset\mathcal{L}_{N}$; it induces a map

$\phi:\mathcal{B}_{M}\rightarrow\mathcal{B}_{N};\
b\mapsto\mathrm{ev}_{\infty}(f(G_{M}(b))).$

  * •

$f({}_{\mathcal{A}}M)\subset{}_{\mathcal{A}}N$.

  * •

$f\circ\psi^{\imath}_{M}=\psi^{\imath}_{N}\circ f$, where
$\psi^{\imath}_{M},\psi^{\imath}_{N}$ denote the $\imath$bar-involution on
$M,N$, respectively.

  * •

$\phi$ is injective on $\\{b\in\mathcal{B}_{M}\mid\phi(b)\neq 0\\}$.

######  Example 3.3.2.

Let $\lambda,\mu,\lambda_{1},\ldots,\lambda_{r}\in X^{+}$ and $w\in W$.

  1. (1)

Let $\mathbf{B}^{\imath}(\lambda_{1},\ldots,\lambda_{r})$ denote the
$\imath$canonical basis of the based $\mathbf{U}$-module
$V(\lambda_{1},\ldots,\lambda_{r})$ in the sense of [7, Theorem 6.12]. Then,
$(V(\lambda_{1},\ldots,\lambda_{r}),\mathbf{B}^{\imath}(\lambda_{1},\ldots,\lambda_{r}))$
forms a based $\mathbf{U}^{\imath}$-module.

  2. (2)

By [7, Theorem 6.13 (2)], $V(w\lambda,\mu)$ is a based
$\mathbf{U}^{\imath}$-submodule of $V(\lambda,\mu)$; set
$\mathbf{B}^{\imath}(w\lambda,\mu):=\mathbf{B}^{\imath}(\lambda,\mu)\cap
V(w\lambda,\mu)$.

  3. (3)

Let $\dot{\mathbf{B}}^{\imath}$ denote the $\imath$canonical basis of
$\dot{\mathbf{U}}^{\imath}$ in the sense of [7, Theorem 7.2]. Then,
$(\dot{\mathbf{U}}^{\imath},\dot{\mathbf{B}}^{\imath})$ is a based
$\mathbf{U}^{\imath}$-module.

######  Lemma 3.3.3.

Let $\lambda,\mu,\nu\in X^{+}$ and $w\in W$. Then, $V(\nu)\otimes
V(w\lambda,\mu)$ is a based $\mathbf{U}^{\imath}$-submodule of
$V(\nu,\lambda,\mu)$.

###### Proof.

It suffices to show that $G^{\imath}(b_{1}\otimes b_{2}\otimes b_{3})\in
V(\nu)\otimes V(w\lambda,\mu)$ for all $b_{1}\otimes b_{2}\otimes
b_{3}\in\mathcal{B}(\nu)\otimes\mathcal{B}(w\lambda,\mu)\subset\mathcal{B}(\nu,\lambda,\mu)$.
By the construction of $G^{\imath}(b_{1}\otimes b_{2}\otimes b_{3})$ in [7,
Theorem 6.12], we have

$G^{\imath}(b_{1}\otimes b_{2}\otimes b_{3})=\sum_{b^{\prime}_{1}\otimes
b^{\prime}_{2}\otimes
b^{\prime}_{3}\in\mathcal{B}(\nu,\lambda,\mu)}c_{b^{\prime}_{1},b^{\prime}_{2},b^{\prime}_{3}}G(b^{\prime}_{1}\otimes
b^{\prime}_{2}\otimes b^{\prime}_{3})$

for some $c_{b^{\prime}_{1},b^{\prime}_{2},b^{\prime}_{3}}\in\mathcal{A}$ such
that $c_{b^{\prime}_{1},b^{\prime}_{2},b^{\prime}_{3}}=0$ unless
$G(b^{\prime}_{1}\otimes b^{\prime}_{2}\otimes b^{\prime}_{3})$ belongs to the
smallest based $\mathbf{U}$-submodule of $V(\nu,\lambda,\mu)$ containing
$G(b_{1}\otimes b_{2}\otimes b_{3})$. Since $V(\nu)\otimes V(w\lambda,\mu)$ is
a based $\mathbf{U}$-submodule containing $G(b_{1}\otimes b_{2}\otimes
b_{3})$, we see that $c_{b^{\prime}_{1},b^{\prime}_{2},b^{\prime}_{3}}=0$
unless $G(b^{\prime}_{1}\otimes b^{\prime}_{2}\otimes b^{\prime}_{3})\in
V(\nu)\otimes V(w\lambda,\mu)$. This proves the assertion. ∎

###  3.4. Based submodule $V(w_{\bullet}\lambda,\mu)$

Let $\lambda,\mu\in X^{+}$. Let
$C_{I_{\bullet}}(b_{\lambda})\subset\mathcal{B}(\lambda)$ denote the connected
component of $\mathcal{B}(\lambda)$ containing $b_{\lambda}$ as the crystal of
type $I_{\bullet}$. Namely,

$C_{I_{\bullet}}(b_{\lambda})=\\{\tilde{F}_{j_{1}}\cdots\tilde{F}_{j_{r}}b_{\lambda}\mid
j_{1},\ldots,j_{r}\in I_{\bullet}\\}\setminus\\{0\\}.$

Note that it is isomorphic to $\mathcal{B}_{I_{\bullet}}(w_{\bullet}\lambda)$
as a crystal of type $I_{\bullet}$; we regard $-w_{\bullet}\lambda$ as a
dominant weight for $I_{\bullet}$.

######  Lemma 3.4.1.

We have

$C_{I_{\bullet}}(b_{\lambda})=\\{b\in\mathcal{B}(\lambda)\mid\operatorname{wt}(b)\geq
w_{\bullet}\lambda\\}.$

###### Proof.

Since
$C_{I_{\bullet}}(b_{\lambda})\simeq\mathcal{B}_{I_{\bullet}}(w_{\bullet}\lambda)$,
we see that $\operatorname{wt}(b)\geq w_{\bullet}\lambda$ for all $b\in
C_{I_{\bullet}}(b_{\lambda})$. This proves the containment “$\subset$”.

We shall prove the opposite direction. Let $b\in\mathcal{B}(\lambda)$ be such
that $\operatorname{wt}(b)\geq w_{\bullet}\lambda$. Since
$\operatorname{wt}(b)\leq\lambda$, we can write

(3.1) $\displaystyle\lambda-\operatorname{wt}(b)=\sum_{j\in
I_{\bullet}}m_{j}\alpha_{j}$

for some $m_{j}\geq 0$. On the other hand, we have

$b=\tilde{F}_{i_{1}}\cdots\tilde{F}_{i_{r}}b_{\lambda}$

for some $i_{1},\ldots,i_{r}\in I$. Taking equation (3.1) into account, we see
that $i_{1},\ldots,i_{r}\in I_{\bullet}$. This implies that $b\in
C_{I_{\bullet}}(b_{\lambda})$. Thus, the proof completes. ∎

By Subsection 2.3, there exist maps

$\pi=\pi_{I_{\bullet};w_{\bullet}\lambda}:\mathcal{B}_{I_{\bullet}}(-\infty)\rightarrow
C_{I_{\bullet}}(b_{\lambda})\sqcup\\{0\\},\
\iota=\iota_{I_{\bullet};w_{\bullet}\lambda}:C_{I_{\bullet}}(b_{\lambda})\hookrightarrow\mathcal{B}_{I_{\bullet}}(-\infty;w_{\bullet}\lambda)$

such that

(3.2) $\displaystyle\begin{split}&\pi\circ\iota=\mathrm{id},\\\
&\iota(b_{w_{\bullet}\lambda})=b_{-\infty},\\\
&\operatorname{wt}(\iota(b))=\operatorname{wt}(b)-w_{\bullet}\lambda,\\\
&\varphi_{j}(\iota(b))=\varphi_{j}(b),\\\
&\tilde{E}_{j}\iota(b)=\iota(\tilde{E}_{j}b)\text{ if }\tilde{E}_{j}b\neq
0\end{split}$

for all $j\in I_{\bullet}$ and $b\in C_{I_{\bullet}}(b_{\lambda})$.

######  Lemma 3.4.2.

Let $b\in\mathcal{B}_{I_{\bullet}}(-\infty)$ and set
$\pi:=\pi_{I_{\bullet};w_{\bullet}\lambda}$. Then, we have

$G(b)(v_{w_{\bullet}\lambda}\otimes v_{\mu})=G(\pi(b))\otimes
v_{\mu}=G(\pi(b)\otimes b_{\mu}).$

In particular, $G(b^{\prime}\otimes b_{\mu})=G(b^{\prime})\otimes v_{\mu}\in
V(w_{\bullet}\lambda,\mu)$ for all $b^{\prime}\in
C_{I_{\bullet}}(b_{\lambda})$.

###### Proof.

We have

$G(b)(v_{w_{\bullet}\lambda}\otimes v_{\mu})=G(b)v_{w_{\bullet}\lambda}\otimes
v_{\mu}=G(\pi(b))\otimes v_{\mu}.$

The left-hand side belongs to ${}_{\mathcal{A}}V(w_{\bullet}\lambda,\mu)$, and
is bar-invariant. On the other hand, the right-hand side belongs to
$\mathcal{L}(\lambda,\mu)$, and its image under $\mathrm{ev}_{\infty}$ is
$\pi(b)\otimes b_{\mu}$. Therefore, we have $G(\pi(b))\otimes
v_{\mu}\in\mathcal{L}(w_{\bullet}\lambda,\mu)\cap{}_{\mathcal{A}}V(w_{\bullet}\lambda,\mu)\cap\overline{\mathcal{L}(w_{\bullet}\lambda,\mu)}$,
and consequently,

$G(\pi(b))\otimes v_{\mu}=(G\circ\mathrm{ev}_{\infty})(G(\pi(b))\otimes
v_{\mu})=G(\pi(b)\otimes b_{\mu}),$

as desired. ∎

######  Proposition 3.4.3.

Let $\lambda,\mu\in X^{+}$ and $b\in\mathcal{B}(w_{\bullet}\lambda,\mu)$.
Then, $b$ is a highest weight element if and only if $b=b_{1}\otimes b_{\mu}$
for some $b_{1}\in C_{I_{\bullet}}(b_{\lambda})$ such that
$\varepsilon_{i}(b_{1})\leq\langle h_{i},\mu\rangle$ for all $i\in I$.

###### Proof.

Let $b\in\mathcal{B}(w_{\bullet}\lambda,\mu)^{\text{hi}}$. Since the
$\mathbf{U}$-module $V(w_{\bullet}\lambda,\mu)$ is generated by a global
crystal basis element $v_{w_{\bullet}\lambda}\otimes v_{\mu}$ of weight
$w_{\bullet}\lambda+\mu$, we have
$I_{\lambda^{\prime}}(V(w_{\bullet}\lambda,\mu))=0$ unless
$\lambda^{\prime}\geq w_{\bullet}\lambda+\mu$. Hence, we obtain
$\operatorname{wt}(b)\geq w_{\bullet}\lambda+\mu$. Now, the “only if” part
follows from Lemmas 2.2.1 and 3.4.1.

Let us prove the opposite direction. Let $b_{1}\in
C_{I_{\bullet}}(b_{\lambda})$ be such that $\varepsilon_{i}(b)\leq\langle
h_{i},\mu\rangle$ for all $i\in I$. By Lemma 2.2.1, we have $b:=b_{1}\otimes
b_{\mu}\in\mathcal{B}(\lambda,\mu)^{\text{hi}}$. By Lemma 3.4.2, we see that
$G(b)\in V(w_{\bullet}\lambda,\mu)$. Hence, we obtain

$b=\mathrm{ev}_{\infty}(G(b))\in\mathcal{B}(w_{\bullet}\lambda,\mu).$

This completes the proof. ∎

For each $\lambda,\mu\in X^{+}$, set

$\displaystyle C_{I_{\bullet};\mu}(b_{\lambda}):=\\{b\in
C_{I_{\bullet}}(b_{\lambda})\mid\varepsilon_{i}(b)\leq\langle
h_{i},\mu\rangle\ \text{ for all }i\in I\\}.$

Then, by Proposition 3.4.3, we obtain

(3.3)
$\displaystyle\mathcal{B}(w_{\bullet}\lambda,\mu)^{\text{hi}}=\\{b\otimes
b_{\mu}\mid b\in C_{I_{\bullet};\mu}(b_{\lambda})\\}.$

For each $\nu\in X^{+}$, consider the compositions

$\displaystyle\pi=\pi_{I_{\bullet};w_{\bullet}\lambda,\nu}:C_{I_{\bullet}}(b_{\lambda+\tau\nu})\xrightarrow{\iota_{I_{\bullet};w_{\bullet}(\lambda+\tau\nu)}}\mathcal{B}_{I_{\bullet}}(-\infty;w_{\bullet}(\lambda+\tau\nu))\xrightarrow{\pi_{I_{\bullet};w_{\bullet}\lambda}}C_{I_{\bullet}}(b_{\lambda})\sqcup\\{0\\},$
$\displaystyle\iota=\iota_{I_{\bullet};w_{\bullet}\lambda,\nu}:C_{I_{\bullet}}(b_{\lambda})\xrightarrow{\iota_{I_{\bullet};w_{\bullet}\lambda}}\mathcal{B}_{I_{\bullet}}(-\infty;w_{\bullet}\lambda)\xrightarrow{\pi_{I_{\bullet};w_{\bullet}(\lambda+\tau\nu)}}C_{I_{\bullet}}(b_{\lambda+\tau\nu}).$

They satisfy the following (cf. equation (3.2)):

(3.4) $\displaystyle\begin{split}&\pi\circ\iota=\mathrm{id},\\\
&\iota(b_{w_{\bullet}\lambda})=b_{w_{\bullet}(\lambda+\tau\nu)},\\\
&\varphi_{j}(\iota(b))=\varphi_{j}(b),\\\
&\operatorname{wt}(\iota(b))=\operatorname{wt}(b)+w_{\bullet}\tau\nu,\\\
&\tilde{E}_{j}\iota(b)=\iota(\tilde{E}_{j}b)\text{ if }\tilde{E}_{j}b\neq
0\end{split}$

for all $j\in I_{\bullet}$ and $b\in C_{I_{\bullet}}(b_{\lambda})$.

######  Lemma 3.4.4.

Let $\lambda,\mu,\nu\in X^{+}$ and $b\in C_{I_{\bullet}}(b_{\lambda})$. Set
$\iota:=\iota_{I_{\bullet};w_{\bullet}\lambda,\nu}$. Then, we have
$\iota(b)\in C_{I_{\bullet};\mu+\nu}(b_{\lambda+\tau\nu})$ if and only if
$b\in C_{I_{\bullet};\mu}(b_{\lambda})$.

###### Proof.

By equation (3.4), we have

$\varepsilon_{j}(\iota(b))=\varphi_{j}(\iota(b))-\langle
h_{j},\operatorname{wt}(\iota(b))\rangle=\varphi_{j}(b)-\langle
h_{j},\operatorname{wt}(b)+w_{\bullet}\tau\nu\rangle=\varepsilon_{j}(b)+\langle
h_{j},\nu\rangle$

for all $j\in I_{\bullet}$, and

$\varepsilon_{i}(\iota(b))=0=\varepsilon_{i}(b)$

for all $i\in I_{\circ}$. Now, the assertion follows from the definition of
$C_{I_{\bullet};\mu}(b_{\lambda})$. ∎

######  Lemma 3.4.5.

Let $\nu\in X^{+}$. Then, we have $\nu+w_{\bullet}\tau\nu\in X^{+}$.

###### Proof.

For each $i\in I$, we have

$\langle h_{i},\nu+w_{\bullet}\tau\nu\rangle=\langle h_{i},\nu\rangle+\langle
w_{\bullet}h_{\tau(i)},\nu\rangle.$

Now, the assertion follows from easy observation that
$w_{\bullet}h_{\tau(i)}=-h_{i}$ if $i\in I_{\bullet}$ and that
$w_{\bullet}h_{\tau(i)}$ is a positive coroot if $i\in I_{\circ}$. ∎

By Lemma 3.4.5, we can apply Lemma 2.5.1 to see that
$V(\nu+w_{\bullet}\tau\nu)\otimes V(w_{\bullet}\lambda,\mu)$ is a based
$\mathbf{U}$-submodule of $V(\nu+w_{\bullet}\tau\nu,\lambda,\mu)$.

######  Lemma 3.4.6.

Let $b\in\mathcal{B}_{I_{\bullet}}(-\infty)$, and set
$\pi:=\pi_{I_{\bullet};w_{\bullet}\lambda}$. Then, we have

$G(b)(v_{\nu+w_{\bullet}\tau\nu}\otimes v_{w_{\bullet}\lambda}\otimes
v_{\mu})=v_{\nu+w_{\bullet}\tau\nu}\otimes G(\pi(b))\otimes
v_{\mu}=G(b_{\nu+w_{\bullet}\tau\nu}\otimes\pi(b)\otimes b_{\mu}).$

In particular, $G(b_{\nu+w_{\bullet}\tau\nu}\otimes b^{\prime}\otimes
b_{\mu})=v_{\nu+w_{\bullet}\tau\nu}\otimes G(b^{\prime})\otimes v_{\nu}$ for
all $b^{\prime}\in C_{I_{\bullet}}(b_{\lambda})$.

###### Proof.

Noting that

$E_{j}v_{\nu+w_{\bullet}\tau\nu}=0,\quad\langle
h_{j},\nu+w_{\bullet}\tau\nu\rangle=0$

for all $j\in I_{\bullet}$, we see that the same argument as the proof of
Lemma 3.4.2 proves the assertion. ∎

######  Proposition 3.4.7.

Let $\lambda,\mu,\nu\in X^{+}$ and set
$\iota:=\iota_{I_{\bullet};w_{\bullet}\lambda,\nu}$ and
$\pi:=\pi_{I_{\bullet};w_{\bullet}\lambda,\nu}$. Then, there exists a unique
based $\mathbf{U}$-module homomorphism

$f:V(w_{\bullet}(\lambda+\tau\nu),\mu+\nu)\rightarrow
V(\nu+w_{\bullet}\tau\nu)\otimes V(w_{\bullet}\lambda,\mu)$

such that

$f(G(b\otimes b_{\mu+\nu}))=G(b_{\nu+w_{\bullet}\tau\nu}\otimes\pi(b)\otimes
b_{\mu})$

for all $b\in C_{I_{\bullet};\mu+\nu}(b_{\lambda+\tau\nu})$.

###### Proof.

First, we observe that there exists a strict crystal morphism

$\phi:\mathcal{B}(w_{\bullet}(\lambda+\tau\nu),\mu+\nu)\rightarrow\mathcal{B}(\nu+w_{\bullet}\tau\nu)\otimes\mathcal{B}(w_{\bullet}\lambda,\mu)$

such that

$\phi(b\otimes b_{\mu+\nu})=b_{\nu+w_{\bullet}\tau\nu}\otimes\pi(b)\otimes
b_{\mu}$

for all $b\in C_{I_{\bullet};\mu+\nu}(b_{\lambda+\tau\nu})$. This follows from
equation (3.3), and that $b\otimes b_{\mu+\nu}$ is a highest weight element of
weight $\operatorname{wt}(b)+\mu+\nu$ and
$b_{\nu+w_{\bullet}\tau\nu}\otimes\pi(b)\otimes b_{\mu}$ is either $0$ or a
highest weight element of weight $\operatorname{wt}(b)+\mu+\nu$ (see Lemma
3.4.4) for all $b\in C_{I_{\bullet};\mu+\nu}(b_{\lambda})$.

Let $f:V(w_{\bullet}(\lambda+\tau\nu),\mu+\nu)\rightarrow
V(\nu+w_{\bullet}\tau\nu)\otimes V(w_{\bullet}\lambda,\mu)$ denote the
$\mathbf{U}$-module homomorphism in Lemma 2.4.5. By Proposition 2.4.6, in
order to prove the assertion, it suffices to show that

$f(E_{i}G(b))=E_{i}G(\phi(b))$

for all $i\in I$ and
$b\in\mathcal{B}(w_{\bullet}(\lambda+\tau\nu),\mu+\nu)^{\text{hi}}$ by
induction on $D(b)$ (see the proof of Proposition 2.4.6 for the definition of
$D(b)$). By weight consideration, we see that $E_{i}G(b)=0=E_{i}G(\phi(b))$ if
$i\in I_{\circ}$. Hence, we only need to consider the case when $i\in
I_{\bullet}$.

Assume that our claim is true for all $b^{\prime}$ with $D(b^{\prime})<D(b)$;
note that we assume nothing when $D(b)=0$. Set $\zeta:=\operatorname{wt}(b)$.
Then, by Proposition 2.4.6, $f$ restricts to a based $\mathbf{U}$-module
homomorphism

$V(w_{\bullet}(\lambda+\tau\nu),\mu+\nu)[>\zeta]\rightarrow
V(\nu+w_{\bullet}\tau\nu)\otimes V(w_{\bullet}\lambda,\mu).$

Let $b_{1}\in C_{I_{\bullet};\mu+\nu}(b_{\lambda+\tau\nu})$ be such that
$b=b_{1}\otimes b_{\mu+\nu}$. Set
$\tilde{b}_{1}:=\iota_{I_{\bullet};w_{\bullet}(\lambda+\tau\nu)}(b_{1})$. Let
us write

$E_{i}G(\tilde{b}_{1})=\sum_{b^{\prime}\in\mathcal{B}_{I_{\bullet}}(-\infty)}c_{b^{\prime}}G(b^{\prime})$

for some $c_{b^{\prime}}\in\mathcal{A}$. Then, using Lemma 3.4.2, we compute
as

$\displaystyle E_{i}G(b)$ $\displaystyle=E_{i}G(b_{1}\otimes b_{\mu+\nu})$
$\displaystyle=E_{i}G(\tilde{b}_{1})(v_{w_{\bullet}(\lambda+\tau\nu)}\otimes
v_{\mu+\nu})$
$\displaystyle=\sum_{b^{\prime}}c_{b^{\prime}}G(b^{\prime})(v_{w_{\bullet}(\lambda+\tau\nu)}\otimes
v_{\mu+\nu})$
$\displaystyle=\sum_{b^{\prime}}c_{b^{\prime}}G(\pi^{\prime}(b^{\prime})\otimes
b_{\mu+\nu}),$

where $\pi^{\prime}:=\pi_{I_{\bullet};w_{\bullet}(\lambda+\tau\nu)}$. Since
$E_{i}G(b)\in V(w_{\bullet}(\lambda+\tau\nu),\mu+\nu)[>\zeta]$, we have
$c_{b^{\prime}}=0$ unless $\pi^{\prime}(b^{\prime})\otimes
b_{\mu+\nu}\in\mathcal{B}(w_{\bullet}(\lambda+\tau\nu),\mu+\nu)[>\zeta]$. By
our induction hypothesis, this implies that

$f(E_{i}G(b))=\sum_{b^{\prime}}c_{b^{\prime}}G(\phi(\pi^{\prime}(b^{\prime})\otimes
b_{\mu+\nu})).$

On the other hand, using Lemma 3.4.6, we compute as

$\displaystyle E_{i}G(\phi(b))$
$\displaystyle=E_{i}G(b_{\nu+w_{\bullet}\tau\nu}\otimes\pi(b_{1})\otimes
b_{\mu})$
$\displaystyle=E_{i}G(\tilde{b}_{1})(v_{\nu+w_{\bullet}\tau\nu}\otimes
v_{w_{\bullet}\lambda}\otimes v_{\mu})$
$\displaystyle=\sum_{b^{\prime}}c_{b^{\prime}}G(b^{\prime})(v_{\nu+w_{\bullet}\tau\nu}\otimes
v_{w_{\bullet}\lambda}\otimes v_{\mu})$
$\displaystyle=\sum_{b^{\prime}}c_{b^{\prime}}G(b_{\nu+w_{\bullet}\tau\nu}\otimes\pi^{\prime\prime}(b^{\prime})\otimes
b_{\mu}),$

where $\pi^{\prime\prime}:=\pi_{I_{\bullet};w_{\bullet}\lambda}$. Hence, in
order to complete the proof, it suffices to show that
$\phi(\pi^{\prime}(b^{\prime})\otimes
b_{\mu+\nu})=b_{\nu+w_{\bullet}\tau\nu}\otimes\pi^{\prime\prime}(b^{\prime})\otimes
b_{\mu}$ for all $b^{\prime}$ with $c_{b^{\prime}}\neq 0$.

Let $b^{\prime}\in\mathcal{B}_{I_{\bullet}}(-\infty)$ be such that
$c_{b^{\prime}}\neq 0$. Suppose first that $\pi^{\prime}(b^{\prime})=0$. Then,
we have $\pi^{\prime\prime}(b^{\prime})=\pi(\pi^{\prime}(b^{\prime}))=0$, and
hence,

$\phi(\pi^{\prime}(b^{\prime})\otimes
b_{\mu+\nu})=0=b_{\nu+w_{\bullet}\tau\nu}\otimes\pi^{\prime\prime}(b^{\prime})\otimes
b_{\mu},$

as desired. Next, suppose that $\pi^{\prime}(b^{\prime})\neq 0$. Then, there
exists $b_{2}\in C_{I_{\bullet};\mu+\nu}(b_{\lambda+\tau\nu})$ such that
$\pi^{\prime}(b^{\prime})\otimes b_{\mu}\in C(b_{2}\otimes b_{\mu+\nu})$. This
implies that there exist $j_{1},\ldots,j_{r}\in I_{\bullet}$ such that
$\tilde{F}_{j_{r}}\cdots\tilde{F}_{j_{1}}(b_{2}\otimes
b_{\mu+\nu})=\pi^{\prime}(b^{\prime})\otimes b_{\mu+\nu}$. Moreover, we must
have

(3.5) $\displaystyle\varepsilon_{j_{k+1}}(b_{2}^{k})\geq\langle
h_{j_{k+1}},\mu+\nu\rangle,\ \varphi_{j_{k+1}}(b_{2}^{k})>0$

for all $k=0,\ldots,r-1$, where

$b_{2}^{0}:=b_{2},\ b_{2}^{k+1}:=\tilde{F}_{j_{k+1}}b_{2}^{k}.$

Then, we compute as

$\displaystyle\phi(\pi^{\prime}(b^{\prime})\otimes b_{\mu+\nu})$
$\displaystyle=\tilde{F}_{j_{r}}\cdots\tilde{F}_{j_{1}}\phi(b_{2}\otimes
b_{\mu+\nu})$
$\displaystyle=\tilde{F}_{j_{r}}\cdots\tilde{F}_{j_{1}}(b_{\nu+w_{\bullet}\tau\nu}\otimes\pi(b_{2})\otimes
b_{\mu})$
$\displaystyle=b_{\nu+w_{\bullet}\tau\nu}\otimes\pi(b_{2}^{r})\otimes
b_{\mu}.$

The last equality follows from the tensor product rule (2.2) and equations
(3.4) and (3.5). Since

$\pi(b_{2}^{r})=\pi(\pi^{\prime}(b^{\prime}))=\pi^{\prime\prime}(b^{\prime}),$

our claim follows. Thus, the proof completes. ∎

######  Corollary 3.4.8.

Let $\lambda,\mu,\nu\in X^{+}$. Assume that there exists a based
$\mathbf{U}^{\imath}$-module homomorphism
$\delta^{\imath}_{\nu}:V(\nu+w_{\bullet}\tau\nu)\rightarrow\mathbb{Q}(q)$ such
that $\delta^{\imath}_{\nu}(v_{\nu+w_{\bullet}\tau\nu})=1$. Then, there exists
a based $\mathbf{U}^{\imath}$-module homomorphism
$\pi^{\imath}_{\lambda,\mu,\nu}:V(w_{\bullet}(\lambda+\tau\nu),\mu+\nu)\rightarrow
V(w_{\bullet}\lambda,\mu)$ such that
$\pi^{\imath}_{\lambda,\mu,\nu}(v_{w_{\bullet}(\lambda+\tau\nu)}\otimes
v_{\mu+\nu})=v_{w_{\bullet}\lambda}\otimes v_{\mu}$.

###### Proof.

Let $f:V(w_{\bullet}(\lambda+\tau\nu),\mu+\nu)\rightarrow
V(\nu+w_{\bullet}\tau\nu)\otimes V(w_{\bullet}\lambda,\mu)$ denote the based
$\mathbf{U}$-module homomorphism in Proposition 3.4.7. Then, we have

$f(v_{w_{\bullet}(\lambda+\tau\nu)}\otimes
v_{\mu+\nu})=f(G(b_{w_{\bullet}(\lambda+\tau\nu)}\otimes
b_{\mu+\nu}))=G(b_{\nu+w_{\bullet}\tau\nu}\otimes
b_{w_{\bullet}\lambda}\otimes b_{\mu})=v_{\nu+w_{\bullet}\tau\nu}\otimes
v_{w_{\bullet}\lambda}\otimes v_{\mu}.$

Set
$\pi^{\imath}=\pi^{\imath}_{\lambda,\mu,\nu}:=(\delta^{\imath}_{\nu}\otimes\mathrm{id})\circ
f$. Noting that
$G^{\imath}(b_{\nu+w_{\bullet}\tau\nu})=v_{\nu+w_{\bullet}\tau\nu}$, we obtain

$\pi^{\imath}(v_{w_{\bullet}(\lambda+\tau\nu)}\otimes
v_{\mu+\nu})=(\delta^{\imath}_{\nu}\otimes\mathrm{id})(v_{\nu+w_{\bullet}\tau\nu}\otimes
v_{w_{\bullet}\lambda}\otimes v_{\mu})=v_{w_{\bullet}\lambda}\otimes v_{\mu}.$

This implies that $\pi^{\imath}$ commutes with the $\imath$bar-involutions. On
the other hand, both $f$ and $\delta^{\imath}_{\nu}\otimes\mathrm{id}$
preserve the crystal lattices and $\mathcal{A}$-forms. Let
$\phi:\mathcal{B}(w_{\bullet}(\lambda+\tau\nu),\mu+\nu)\rightarrow\mathcal{B}(\nu+w_{\bullet}\tau\nu)\otimes\mathcal{B}(w_{\bullet}\lambda,\mu)\sqcup\\{0\\}$
and
$\delta^{\imath}_{\nu}:\mathcal{B}(\nu+w_{\bullet}\tau\nu)\rightarrow\\{1\\}\sqcup\\{0\\}$
denote the induced maps. Then, $\pi^{\imath}$ preserves the crystal lattices,
and the induced map

$\pi^{\imath}=\pi^{\imath}_{\lambda,\mu,\nu}:=(\delta^{\imath}_{\nu}\otimes\mathrm{id})\circ\phi:\mathcal{B}(w_{\bullet}(\lambda+\tau\nu),\mu+\nu)\rightarrow\mathcal{B}(w_{\bullet}\lambda,\mu)\sqcup\\{0\\}$

is injective on
$\\{b\in\mathcal{B}(w_{\bullet}(\lambda+\tau\nu),\mu+\nu)\mid\pi^{\imath}(b)\neq
0\\}$. By Lemma 3.3.1, we see that $\pi^{\imath}$ is based. Thus, the proof
completes. ∎

##  4\. Irreducible finite type of real rank $1$

In this section, we assume the following:

  * •

The admissible pair $(I,I_{\bullet},\tau)$ is of irreducible finite type of
real rank $1$ (cf. Subsection 3.1).

  * •

$Y=\sum_{i\in I}\mathbb{Z}h_{i}$.

  * •

$X=\operatorname{Hom}_{\mathbb{Z}}(Y,\mathbb{Z})$.

  * •

$\kappa_{i}=0$ for all $i\in I_{\circ}$.

For each $i\in I$, let $\varpi_{i}\in X$ denote the $i$-th fundamental weight;

$\langle h_{j},\varpi_{i}\rangle=\delta_{i,j}$

for all $j\in I$.

In Subsections 4.1–4.5, we study certain $\mathbf{U}^{\imath}$-modules in
order to prove Proposition 4.6.1. Then, we prove the stability of
$\imath$canonical bases in Theorem 4.6.2.

###  4.1. Type A

Let $n\geq 1$ and consider the Dynkin diagram $I=\\{1,\ldots,n\\}$ of type
$A_{n}$. For each $r\in I$ and $n+1\geq i_{1}>\cdots i_{r}\geq 1$, let
$b_{i_{1},\ldots,i_{r}}\in\mathcal{B}(\varpi_{r})$ denote the unique element
of weight $\sum_{k=1}^{r}(\varpi_{i_{k}}-\varpi_{i_{k}-1})$; here, we set
$\varpi_{0}:=0$. Set $v_{i_{1},\ldots,i_{r}}:=G(b_{i_{1},\ldots,i_{r}})\in
V(\varpi_{r})$.

#### Type AI

Set

(4.1) $\displaystyle\begin{split}&\varsigma_{1}:=q^{-1},\\\
&\varpi:=2\varpi_{1}.\end{split}$

Since

$\varpi_{1}+w_{\bullet}\tau\varpi_{1}=2\varpi_{1},$

we have

$\nu+w_{\bullet}\tau\nu\in\mathbb{Z}_{\geq 0}\varpi$

for all $\nu\in X^{+}$.

######  Lemma 4.1.1.

Let $K:V(\varpi_{1})\rightarrow V(\varpi_{1})$ denote the linear isomorphism
given by

$K(v_{1})=v_{2},\quad K(v_{2})=v_{1}.$

Then, $K$ is a based $\mathbf{U}^{\imath}$-module isomorphism.

###### Proof.

By Theorem [5, Theorem 4.18], we see that there exists a
$\mathbf{U}^{\imath}$-module isomorphism $K^{\prime}$ such that
$K^{\prime}(v_{1})=v_{2}$. Then, we have

$K^{\prime}(v_{2})=K^{\prime}(B_{1}v_{1})=B_{1}v_{2}=v_{1}.$

Hence, we obtain $K^{\prime}=K$.

It remains to show that $K$ is based. We have

$v_{2}=B_{1}v_{1}\in\mathcal{L}(\varpi_{1})\cap{}_{\mathcal{A}}V(\varpi_{1})\cap\psi^{\imath}(\mathcal{L}(\varpi_{1})),$

where $\psi^{\imath}$ denote the $\imath$bar-involution on $V(\varpi_{1})$.
This implies that $v_{2}=G^{\imath}(b_{2})$. Therefore, the $K$ is a based
$\mathbf{U}^{\imath}$-module isomorphism. Thus, the proof completes. ∎

######  Lemma 4.1.2.

There exists $w_{0}\in\mathcal{L}(\varpi)$ such that
$\mathbf{U}^{\imath}w_{0}\simeq\mathbb{Q}(q)$ and
$\mathrm{ev}_{\infty}(w_{0})=b_{\varpi}$.

###### Proof.

Noting that $\varpi=2\varpi_{1}$ and $V(\varpi_{1})=V(-\varpi_{1})$ (see
Remark 2.3.1), consider the composition

$g:V(\varpi)\xrightarrow{\chi_{\varpi_{1},\varpi_{1}}}V(\varpi_{1})\otimes
V(\varpi_{1})\xrightarrow{K\otimes\mathrm{id}}V(\varpi_{1})\otimes
V(\varpi_{1})\xrightarrow{\delta_{\varpi_{1}}}\mathbb{Q}(q).$

For the definitions of based $\mathbf{U}$-module homomorphisms
$\chi_{\varpi_{1},\varpi_{1}}$ and $\delta_{\varpi_{1}}$, see Example 2.4.2
(3) and (4), respectively.

By the definition, $g$ is a $\mathbf{U}^{\imath}$-module homomorphism
preserving the crystal lattices. For each $b\in\mathcal{B}(\varpi)$,
considering at $q=\infty$, we obtain

$\mathrm{ev}_{\infty}(g(G(b)))=\delta_{b,b_{\varpi}}.$

Hence, $\operatorname{Ker}g$ has a basis $\\{w_{b}\mid
b\in\mathcal{B}(\varpi)\setminus\\{b_{\varpi}\\}\\}$ such that
$w_{b}\in\mathcal{L}(\varpi)$ and $\mathrm{ev}_{\infty}(w_{b})=b$. Therefore,
the complement $W_{0}\subset V(\varpi)$ of $\operatorname{Ker}g$ with respect
to the inner product on $V(\varpi)$ (cf. Subsection 2.3) is spanned by a
vector $w_{0}\in\mathcal{L}(\varpi)$ satisfying
$\mathrm{ev}_{\infty}(w_{0})=b_{\varpi}$. Now, the assertion follows from the
following:

$W_{0}\simeq
V(\varpi)/\operatorname{Ker}g\simeq\operatorname{Im}g=\mathbb{Q}(q).$

Thus, the proof completes. ∎

#### Type AII

Set

(4.2) $\displaystyle\begin{split}&\varsigma_{2}:=q,\\\
&\varpi:=\varpi_{2}.\end{split}$

Since

$\varpi_{i}+w_{\bullet}\tau\varpi_{i}=\begin{cases}\varpi_{2}&\text{ if
}i=1,3,\\\ 2\varpi_{2}&\text{ if }i=2\end{cases}$

for all $i\in I$, we have

$\nu+w_{\bullet}\tau\nu\in\mathbb{Z}_{\geq 0}\varpi$

for all $\nu\in X^{+}$.

######  Lemma 4.1.3.

We have $B_{2}v_{4,3}=q^{2}v_{3,1}$.

###### Proof.

Using [19, Proposition 5.2.2], we compute as

$\displaystyle
B_{2}v_{4,3}=(F_{2}+qT_{w_{\bullet}}(E_{2})K_{2}^{-1})(v_{4,3})$
$\displaystyle=q^{2}T_{w_{\bullet}}(E_{2}T_{w_{\bullet}}^{-1}(v_{4,3}))$
$\displaystyle=q^{2}T_{w_{\bullet}}(E_{2}v_{4,3})$
$\displaystyle=q^{2}T_{w_{\bullet}}(v_{4,2})$ $\displaystyle=q^{2}v_{3,1}.$

This proves the assertion. ∎

######  Lemma 4.1.4.

There exists $w_{0}\in\mathcal{L}(\varpi)$ such that
$\mathbf{U}^{\imath}w_{0}\simeq\mathbb{Q}(q)$ and
$\mathrm{ev}_{\infty}(w_{0})=b_{\varpi}$.

###### Proof.

By direct calculation and Lemma 4.1.3, we see that the vector

$w_{0}:=v_{2,1}-q^{-2}v_{4,3}\in V(\varpi)$

spans the $\mathbf{U}^{\imath}$-submodule isomorphic to $\mathbb{Q}(q)$.
Clearly, we have $w_{0}\in\mathcal{L}(\varpi)$ and
$\mathrm{ev}_{\infty}(w_{0})=b_{2,1}=b_{\varpi}$. Thus, the proof completes. ∎

#### Type AIII

Set

(4.3) $\displaystyle\begin{split}&\varsigma_{1}=\varsigma_{2}:=1,\\\
&\varpi:=\varpi_{1}+\varpi_{2}.\end{split}$

Since

$\varpi_{i}+w_{\bullet}\tau\varpi_{i}=\varpi_{1}+\varpi_{2}$

for all $i\in I$, we have

$\nu+w_{\bullet}\tau\nu\in\mathbb{Z}_{\geq 0}\varpi$

for all $\nu\in X^{+}$.

For each $i=1,2$, the $V(\varpi_{i})$ is two-dimensional. Let $v_{1}^{i}$
(resp., $v_{2}^{i}$) denote the highest (resp., lowest) weight vector.

######  Lemma 4.1.5.

Let $K:V(\varpi_{1})\rightarrow V(\varpi_{2})$ denote the linear isomorphism
givne by

$K(v_{1}^{1})=v_{2}^{2},\ K(v_{2}^{1})=v_{1}^{2}.$

Then, $K$ is a based $\mathbf{U}^{\imath}$-module isomorphism.

###### Proof.

Noting that

$B_{1}v^{1}_{1}=v^{1}_{2},\quad B_{2}v^{2}_{1}=v^{2}_{2},$

the assertion can be proved by the same way as Lemma 4.1.1 ∎

######  Lemma 4.1.6.

There exists $w_{0}\in\mathcal{L}(\varpi)$ such that
$\mathbf{U}^{\imath}w_{0}\simeq\mathbb{Q}(q)$ and
$\mathrm{ev}_{\infty}(w_{0})=b_{\varpi}$.

###### Proof.

Using Lemma 4.1.5, the same argument as in the proof of Lemma 4.1.2 proves the
assertion. ∎

#### Type AIV

Set

(4.4) $\displaystyle\begin{split}&\varsigma_{1}:=1,\
\varsigma_{n}:=(-1)^{n}q^{n-1}\\\ &\varpi:=\varpi_{1}+\varpi_{n}.\end{split}$

Since

$\varpi_{i}+w_{\bullet}\tau\varpi_{i}=\varpi_{1}+\varpi_{n}$

for all $i\in I$, we have

$\nu+w_{\bullet}\tau\nu\in\mathbb{Z}_{\geq 0}\varpi$

for all $\nu\in X^{+}$.

######  Lemma 4.1.7.

We have $B_{n}v_{n}=v_{n+1}+v_{1}$.

###### Proof.

Using [19, Proposition 5.2.2], we compute as

$\displaystyle
B_{n}v_{n}=(F_{n}+(-1)^{n}q^{n-1}T_{w_{\bullet}}(E_{1})K_{n}^{-1})v_{n}$
$\displaystyle=v_{n+1}+(-1)^{n}q^{n-2}T_{w_{\bullet}}(E_{1}T_{w_{\bullet}}^{-1}(v_{n}))$
$\displaystyle=v_{n+1}+(-1)^{2}T_{w_{\bullet}}(E_{1}v_{2})$
$\displaystyle=v_{n+1}+v_{1}.$

This proves the assertion. ∎

######  Lemma 4.1.8.

Let $K:V(\varpi_{1})\rightarrow V(\varpi_{1})$ denote the linear isomorphism
givne by

$K(v_{i})=\begin{cases}v_{n+1}&\text{ if }i=1,\\\ v_{i}&\text{ if
}i=2,\ldots,n,\\\ v_{1}&\text{ if }i=n+1.\end{cases}$

Then, it is a based $\mathbf{U}^{\imath}$-module isomorphism.

###### Proof.

Noting that

$\displaystyle v_{2}=B_{1}v_{1},$ $\displaystyle v_{3}=B_{2}v_{2},$
$\displaystyle\cdots,$ $\displaystyle v_{n}=B_{n-1}v_{n-1}$

and that

$v_{n+1}=B_{n}v_{n}-v_{1}$

by Lemma 4.1.7, the assertion can be proved by the same way as Lemma 4.1.1 ∎

######  Lemma 4.1.9.

There exists $w_{0}\in\mathcal{L}(\varpi)$ such that
$\mathbf{U}^{\imath}w_{0}\simeq\mathbb{Q}(q)$ and
$\mathrm{ev}_{\infty}(w_{0})=b_{\varpi}$.

###### Proof.

Using Lemma 4.1.8, the same argument as in the proof of Lemma 4.1.2 proves the
assertion. ∎

###  4.2. Type BII

Let $n\geq 2$ and consider the Dynkin diagram $I=\\{1,\ldots,n\\}$ of type
$B_{n}$. Set $L:=\\{1,\ldots,n,0,\bar{n},\ldots,\bar{1}\\}$, and equip the set
$\mathcal{B}:=\\{b_{i}\mid i\in L\\}$ with the crystal structure as in [8,
Example 2.22]. Then, $\mathcal{B}$ is identical to $\mathcal{B}(\varpi_{1})$.
For each $i\in L$, set $v_{i}:=G(b_{i})\in V(\varpi_{1})$.

Set

(4.5) $\displaystyle\begin{split}&\varsigma_{1}:=q^{2n-3},\\\
&\varpi:=\varpi_{1}.\end{split}$

Since

$\varpi_{i}+w_{\bullet}\tau\varpi_{i}=\begin{cases}2\varpi_{1}&\text{ if
}i\neq n,\\\ \varpi_{1}&\text{ if }i=n\end{cases}$

for all $i\in I$, we have

$\nu+w_{\bullet}\tau\nu\in\mathbb{Z}_{\geq 0}\varpi$

for all $\nu\in X^{+}$.

######  Lemma 4.2.1.

We have $B_{1}v_{\bar{1}}=q^{2n-1}v_{2}$.

###### Proof.

Using [19, Proposition 5.2.2], we compute as

$\displaystyle
B_{1}v_{\bar{1}}=(F_{1}+q^{2n-3}T_{w_{\bullet}}(E_{1})K_{1}^{-1})v_{\bar{1}}$
$\displaystyle=q^{2n-1}T_{w_{\bullet}}(E_{1}T_{w_{\bullet}}^{-1}(v_{\bar{1}}))$
$\displaystyle=q^{2n-1}T_{w_{\bullet}}(E_{1}v_{\bar{1}})$
$\displaystyle=q^{2n-1}T_{w_{\bullet}}(v_{\bar{2}})$
$\displaystyle=q^{2n-1}v_{2}.$

This proves the assertion. ∎

######  Lemma 4.2.2.

There exists $w_{0}\in\mathcal{L}(\varpi)$ such that
$\mathbf{U}^{\imath}w_{0}\simeq\mathbb{Q}(q)$ and
$\mathrm{ev}_{\infty}(w_{0})=b_{\varpi}$.

###### Proof.

By direct calculation and Lemma 4.2.1, we see that the vector

$w_{0}:=v_{1}-q^{-2n+1}v_{\bar{1}}\in V(\varpi)$

spans the $\mathbf{U}^{\imath}$-submodule isomorphic to $\mathbb{Q}(q)$.
Clearly, we have $w_{0}\in\mathcal{L}(\varpi)$ and
$\mathrm{ev}_{\infty}(w_{0})=b_{1}=b_{\varpi}$. Thus, the proof completes. ∎

###  4.3. Type CII

Let $n\geq 3$ and consider the Dynkin diagram $I=\\{1,\ldots,n\\}$ of type
$C_{n}$. Set $L:=\\{1,\ldots,n,\bar{n},\ldots,\bar{1}\\}$, and equip the set
$\mathcal{B}:=\\{b_{i}\mid i\in L\\}$ with the crystal structure as in [8,
Example 2.23]. Then, $\mathcal{B}$ is identical to $\mathcal{B}(\varpi_{1})$.

The $\mathbf{U}$-module $V(\varpi_{2})$ can be embedded into
$V(\varpi_{1})\otimes V(\varpi_{1})$ in a way such that

$v_{\varpi_{2}}\mapsto v_{2}\otimes v_{1}-q^{-1}v_{1}\otimes v_{2}.$

Then, the crystal basis $\mathcal{B}(\varpi_{2})$ is identified with the
connected component $C(b_{2}\otimes b_{1})$ of
$\mathcal{B}\otimes\mathcal{B}$. For each $(i,j)\in L^{2}$ such that
$b_{i}\otimes b_{j}\in\mathcal{B}(\varpi_{2})$, set

$v_{i,j}=\begin{cases}v_{\bar{k}}\otimes v_{k}+q^{-1}v_{\overline{k-1}}\otimes
v_{k-1}\\\ -q^{-1}v_{k-1}\otimes v_{\overline{k-1}}-q^{-2}v_{k}\otimes
v_{\bar{k}}&\text{ if }(i,j)=(\overline{k},k)\text{ for some
}k\in\\{2,\ldots,n\\},\\\ v_{i}\otimes v_{j}-q^{-1}v_{j}\otimes v_{i}&\text{
otherwise}.\end{cases}$

Then, $\\{v_{i,j}\mid b_{i}\otimes b_{j}\in\mathcal{B}(\varpi_{2})\\}$ forms a
free basis of $\mathcal{L}(\varpi_{2})$.

For each $i\in I$ and $k\in\\{2,\ldots,n\\}$, one can straightforwardly verify
that

(4.6) $\displaystyle
F_{i}v_{\bar{k},k}=\begin{cases}[2]v_{\bar{\imath},i+1},&\text{ if }k=i+1,\\\
v_{\bar{\imath},i+1}&\text{ if }k=i+2\text{ or }k=i<n,\\\ 0&\text{
otherwise},\end{cases}$

and

(4.7) $\displaystyle
E_{i}v_{\bar{k},k}=\begin{cases}[2]v_{\overline{i+1},i},&\text{ if }k=i+1,\\\
v_{\overline{i+1},i}&\text{ if }k=i+2\text{ or }k=i<n,\\\ 0&\text{
otherwise}.\end{cases}$

######  Lemma 4.3.1.

Set

$w^{\prime}_{0}:=-\frac{1}{[2]}v_{\bar{2},2}+\sum_{k=3}^{n}(-1)^{k-3}\frac{[n-k+1]}{[n-2]}v_{\bar{k},k}.$

Then, we have $E_{j}w^{\prime}_{0}=F_{j}w^{\prime}_{0}=0$ for all $j\in
I_{\bullet}$.

###### Proof.

Since $w^{\prime}_{0}$ is a weight vector of weight $0$, the identity
$E_{j}w^{\prime}_{0}=0$ follows from $F_{j}w^{\prime}_{0}=0$.

By equation (4.6), we have

$F_{i}w^{\prime}_{0}=\begin{cases}[2]\cdot(-\frac{1}{[2]})+1&\text{ if
}i=1,\\\ \frac{(-1)^{i-3}}{[n-2]}([n-i+1]-[2][n-i]+[n-i-1])&\text{ if
}i=3,\ldots,n-1,\\\ 0&\text{ if }i=n.\end{cases}$

Now, the assertion follows from a well-known identity

$[2][m]=[m+1]+[m-1]$

valid for all $m\in\mathbb{Z}$. Thus, the proof completes. ∎

Set

(4.8) $\displaystyle\begin{split}&\varsigma_{2}:=q^{n-1},\\\
&\varpi:=\varpi_{2}.\end{split}$

Since

$\varpi_{i}+w_{\bullet}\tau\varpi_{i}=\begin{cases}\varpi_{2}&\text{ if
}i=1,\\\ 2\varpi_{2}&\text{ if }i\neq 1\end{cases}$

for all $i\in I$, we have

$\nu+w_{\bullet}\tau\nu\in\mathbb{Z}_{\geq 0}\varpi$

for all $\nu\in X^{+}$.

######  Lemma 4.3.2.

There exists $w_{0}\in\mathcal{L}(\varpi)$ such that
$\mathbf{U}^{\imath}w_{0}\simeq\mathbb{Q}(q)$ and
$\mathrm{ev}_{\infty}(w_{0})=b_{\varpi}$.

###### Proof.

By direct calculation, equations (4.6) and (4.7), and Lemma 4.3.1, we see that
the vector

$w_{0}:=v_{2,1}-\frac{q^{-n+1}[2][n-2]}{[n]}w^{\prime}_{0}+q^{-2n+1}v_{\bar{1},\bar{2}}$

spans the $\mathbf{U}^{\imath}$-submodule isomorphic to $\mathbb{Q}(q)$.
Clearly, we have $w_{0}\in\mathcal{L}(\varpi)$ and
$\mathrm{ev}_{\infty}(w_{0})=b_{2,1}=b_{\varpi}$. Thus, the proof completes. ∎

###  4.4. Type DII

Let $n\geq 4$ and consider the Dynkin diagram $I=\\{1,\ldots,n\\}$ of type
$D_{n}$. Set $L:=\\{1,\ldots,n,\bar{n},\ldots,\bar{1}\\}$, and equip the set
$\mathcal{B}:=\\{b_{i}\mid i\in L\\}$ with the crystal structure as in [8,
Example 2.24]. Then, $\mathcal{B}$ is identical to $\mathcal{B}(\varpi_{1})$.
For each $i\in L$, set $v_{i}:=G(b_{i})\in V(\varpi_{1})$.

Set

(4.9) $\displaystyle\begin{split}&\varsigma_{1}:=q^{n-2},\\\
&\varpi:=\varpi_{1}.\end{split}$

Since

$\varpi_{i}+w_{\bullet}\tau\varpi_{i}=\begin{cases}2\varpi_{1}&\text{ if
}i\neq n-1,n,\\\ \varpi_{1}&\text{ if }i=n-1,n\end{cases}$

for all $i\in I$, we have

$\nu+w_{\bullet}\tau\nu\in\mathbb{Z}_{\geq 0}\varpi$

for all $\nu\in X^{+}$.

######  Lemma 4.4.1.

We have $B_{1}v_{\bar{1}}=q^{n-1}v_{2}$.

###### Proof.

Using [19, Proposition 5.2.2], we compute as

$\displaystyle
B_{1}v_{\bar{1}}=(F_{1}+q^{n-2}T_{w_{\bullet}}(E_{1})K_{1}^{-1})v_{\bar{1}}$
$\displaystyle=q^{n-1}T_{w_{\bullet}}(E_{1}T_{w_{\bullet}}^{-1}(v_{\bar{1}}))$
$\displaystyle=q^{n-1}T_{w_{\bullet}}(E_{1}v_{\bar{1}})$
$\displaystyle=q^{n-1}T_{w_{\bullet}}(v_{\bar{2}})$
$\displaystyle=q^{n-1}v_{2}.$

This proves the assertion. ∎

######  Lemma 4.4.2.

There exists $w_{0}\in\mathcal{L}(\varpi)$ such that
$\mathbf{U}^{\imath}w_{0}\simeq\mathbb{Q}(q)$ and
$\mathrm{ev}_{\infty}(w_{0})=b_{\varpi}$.

###### Proof.

By direct calculation and Lemma 4.4.1, we see that the vector

$w_{0}:=v_{1}-q^{-n+1}v_{\bar{1}}\in V(\varpi)$

spans the $\mathbf{U}^{\imath}$-submodule isomorphic to $\mathbb{Q}(q)$.
Clearly, we have $w_{0}\in\mathcal{L}(\varpi)$ and
$\mathrm{ev}_{\infty}(w_{0})=b_{1}=b_{\varpi}$. Thus, the proof completes. ∎

###  4.5. Type FII

Consider the Dynkin diagram $I=\\{1,2,3,4\\}$ of type $F_{4}$, and the crystal
$\mathcal{B}(\varpi_{4})$ (see [8, Fig. 5.8]). For each $\lambda\in
X\setminus\\{0\\}$ with $\mathcal{B}(\varpi_{4})_{\lambda}\neq\emptyset$, let
$b_{\lambda}\in\mathcal{B}(\varpi_{4})$ denote the unique element of weight
$\lambda$. Set

$b_{0}^{1}:=\tilde{F}_{4}b_{-\varpi_{3}+2\varpi_{4}},\quad
b_{0}^{2}:=\tilde{F}_{3}b_{-\varpi_{2}+2\varpi_{3}-\varpi_{4}}.$

Then, we have $\mathcal{B}(\varpi_{4})_{0}=\\{b_{0}^{1},b_{0}^{2}\\}$.

Set

(4.10) $\displaystyle\begin{split}&\varsigma_{1}:=q^{5},\\\
&\varpi:=\varpi_{4}.\end{split}$

Since

$\varpi_{i}+w_{\bullet}\tau\varpi_{i}=\begin{cases}2\varpi_{4}&\text{ if
}i=1,4,\\\ 4\varpi_{4}&\text{ if }i=2,\\\ 3\varpi_{4}&\text{ if
}i=3\end{cases}$

for all $i\in I$, we have

$\nu+w_{\bullet}\tau\nu\in\mathbb{Z}_{\geq 0}\varpi$

for all $\nu\in X^{+}$.

######  Lemma 4.5.1.

There exists $w_{0}\in\mathcal{L}(\varpi)$ such that
$\mathbf{U}^{\imath}w_{0}\simeq\mathbb{Q}(q)$ and
$\mathrm{ev}_{\infty}(w_{0})=b_{\varpi}$.

###### Proof.

Using a computer program GAP [9] with a package QuaGroup [10], we see that

$w_{0}:=G(b_{\varpi_{4}})-\frac{q^{-5}[2]}{[3]}\left(G(b_{0}^{2})-\frac{1}{[2]}G(b_{0}^{1})\right)+q^{-11}G(b_{-\varpi_{4}})$

spans a $\mathbf{U}^{\imath}$-submodule of $V(\varpi)$ isomorphic to
$\mathbb{Q}(q)$. Clearly, we have $w_{0}\in\mathcal{L}(\varpi)$ and
$\mathrm{ev}_{\infty}(w_{0})=b_{\varpi_{4}}=b_{\varpi}$. Thus, the proof
completes. ∎

###  4.6. Stability of $\imath$canonical bases

In the sequel, we set $\varsigma_{i}$ for $i\in I_{\circ}$ and $\varpi\in
X^{+}$ as in (4.1)–(4.5), (4.8)–(4.10).

######  Proposition 4.6.1.

For each $m\in\mathbb{Z}_{\geq 0}$, there exists a unique based
$\mathbf{U}^{\imath}$-module homomorphism
$g_{m}:V(m\varpi)\rightarrow\mathbb{Q}(q)$ such that $g_{m}(v_{m\varpi})=1$.

###### Proof.

We prove the assertion by induction on $m$. The case when $m=0$ is trivial.
Let us prove for $m=1$.

Let $w_{0}\in V(\varpi)$ be as in Lemmas 4.1.2, 4.1.4, 4.1.6, 4.1.9,, 4.2.2,
4.3.2, 4.4.2, and 4.5.1. Set $W_{0}:=\mathbb{Q}(q)w_{0}$ and $W_{1}\subset
V(\varpi)$ the complement of $W_{0}$ with respect to the inner product in
Subsection 2.3. Then, we can write

$v_{\varpi}=cw_{0}+w_{1}$

for some $c\in 1+\mathbf{A}_{\infty}$ and $w_{1}\in W_{1}$. Define a
$\mathbf{U}^{\imath}$-module homomorphism

$g_{1}:V(\varpi)=W_{0}\oplus W_{1}\rightarrow\mathbb{Q}(q)$

by

$g_{1}(w_{0})=c^{-1},\ g_{1}(W_{1})=0.$

Then, it preserves the crystal lattices, and we have

$g_{1}(v_{\varpi})=1.$

This identity implies that $g_{1}$ preserves the $\mathcal{A}$-forms and
commutes with the $\imath$bar-involutions. Moreover, the induced map
$\gamma_{1}:\mathcal{B}(\varpi)\rightarrow\\{1\\}\sqcup\\{0\\}$ satisfies

$\gamma_{1}(b)=\delta_{b,b_{\varpi}}$

for all $b\in\mathcal{B}(\varpi)$. Therefore, by Lemma 3.3.1, we see that
$g_{1}$ is a based $\mathbf{U}^{\imath}$-module homomorphism.

Now, assume that $m>1$ and the assertion is true for $m-1$. Consider the
composition

$g:V(m\varpi)\xrightarrow{\chi_{(m-1)\varpi,\varpi}}V((m-1)\varpi)\otimes
V(\varpi)\xrightarrow{g_{m-1}\otimes\mathrm{id}}V(\varpi)\xrightarrow{g_{1}}\rightarrow\mathbb{Q}(q).$

Then, it preserves the crystal lattices and $\mathcal{A}$-forms, and we have

$g(v_{m\varpi})=g_{1}(g_{m-1}(v_{(m-1)\varpi})\otimes
v_{\varpi})=g_{1}(v_{\varpi})=1.$

This identity implies that $g=g_{m}$ and that $g_{m}$ commutes with the
$\imath$bar-involutions. Moreover, the induced map
$\gamma_{m}:\mathcal{B}(m\varpi)\rightarrow\\{1\\}\sqcup\\{0\\}$ satisfies

$\gamma(b)=\delta_{b,b_{m\varpi}}$

for all $b\in\mathcal{B}(m\varpi)$. Therefore, by Lemma 3.3.1, we see that
$g_{m}$ is a based $\mathbf{U}^{\imath}$-module homomorphism. Thus, the proof
completes. ∎

######  Theorem 4.6.2.

Let $\lambda,\mu,\nu\in X^{+}$. Then, there exists a unique based
$\mathbf{U}^{\imath}$-module homomorphism

$\pi^{\imath}_{\lambda,\mu,\nu}:V(w_{\bullet}(\lambda+\tau\nu),\mu+\nu)\rightarrow
V(w_{\bullet}\lambda,\mu)$

such that

$\pi^{\imath}_{\lambda,\mu,\nu}(v_{w_{\bullet}(\lambda+\tau\nu)}\otimes
v_{\mu+\nu})=v_{w_{\bullet}\lambda}\otimes v_{\mu}.$

###### Proof.

The assertion follows from Corollary 3.4.8 and Proposition 4.6.1 since we have
$\nu+w_{\bullet}\tau\nu\in\mathbb{Z}_{\geq 0}\varpi$. ∎

######  Corollary 4.6.3.

For each $\lambda,\mu\in X^{+}$, there exists a unique based
$\mathbf{U}^{\imath}$-module homomorphism

$\pi^{\imath}_{\lambda,\mu}:\dot{\mathbf{U}}^{\imath}1_{\overline{\mu+w_{\bullet}\lambda}}\rightarrow
V(w_{\bullet}\lambda,\mu)$

such that

$\pi^{\imath}_{\lambda,\mu}(1_{\overline{\mu+w_{\bullet}\lambda}})=v_{w_{\bullet}\lambda}\otimes
v_{\mu}.$

###### Proof.

Define a $\mathbf{U}^{\imath}$-module homomorphism
$\pi^{\imath}_{\lambda,\mu}$ by

$\pi^{\imath}_{\lambda,\mu}(x):=x\cdot(v_{w_{\bullet}\lambda}\otimes
v_{\mu}).$

We shall show that it is based.

Let $x\in\dot{\mathbf{B}}^{\imath}$ (see Example 3.3.2 (3)). By [7, Theorem
7.2], there exists $\nu\in X^{+}$ such that

$\pi^{\imath}_{\lambda+\tau\nu,\mu+\nu}(x)\in\mathbf{B}^{\imath}(w_{\bullet}(\lambda+\tau\nu),\mu+\nu).$

Then, by Theorem 4.6.2, we see that

$\pi^{\imath}_{\lambda,\mu}(x)=\pi^{\imath}_{\lambda,\mu,\nu}(\pi^{\imath}_{\lambda+\tau\nu,\mu+\nu}(x))\in\mathbf{B}^{\imath}(w_{\bullet}\lambda,\mu)\sqcup\\{0\\}.$

It remains to show that $\operatorname{Ker}\pi^{\imath}_{\lambda,\mu}$ is
spanned by a subset of $\dot{\mathbf{B}}^{\imath}$. Let
$v\in\operatorname{Ker}\pi^{\imath}_{\lambda,\mu}$, and write

$v=\sum_{k=1}^{r}c_{k}x_{k}$

for some $c_{1},\ldots,c_{r}\in\mathbb{Q}(q)^{\times}$ and
$x_{1},\ldots,x_{r}\in\dot{\mathbf{B}}^{\imath}$. Again by [7, Theorem 7.2],
there exists $\nu\in X^{+}$ such that the vectors
$\pi^{\imath}_{\lambda+\tau\nu,\mu+\nu}(x_{k})$ are distinct $\imath$canonical
basis elements. Since the homomorphism $\pi^{\imath}_{\lambda,\mu,\nu}$ is
based, the nonzero vectors of the form
$\pi^{\imath}_{\lambda,\mu}(x_{k})=\pi^{\imath}_{\lambda,\mu,\nu}(\pi^{\imath}_{\lambda+\tau\nu,\mu+\nu}(x_{k}))$
are distinct. Hence, we must have $\pi^{\imath}_{\lambda,\mu}(x_{k})=0$ for
all $k=1,\ldots,r$. This implies that

$\operatorname{Ker}\pi^{\imath}_{\lambda,\mu}=\mathbb{Q}(q)\\{x\in\dot{\mathbf{B}}^{\imath}\mid\pi^{\imath}_{\lambda,\mu}(x)=0\\},$

as desired. Thus, the proof completes. ∎

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