and $F_{i}^{(n)}v_{x}=v_{s_{i}x}$ (resp., $F_{i}v_{x}=0$ and $E_{i}^{(-n)}v_{x}=v_{s_{i}x}$) hold, where for $i \in I_{{\mathrm{af}}}$ and $k \in {\mathbb{Z}}_{\ge 0}$, the $E_{i}^{(k)}$ and $F_{i}^{(k)}$ are the $k$-th divided powers of $E_{i}$ and $F_{i}$, respectively; note that the weight of $v_{x}$ is $x\lambda$. Also, for each $x \in W_{{\mathrm{af}}}$, we define the Demazure submodule $V_{x}^{-}(\lambda)$ of $V(\lambda)$ by $$\label{eq:dem}
V_{x}^{-}(\lambda):=U_{{\mathsf{q}}}^{-}({\mathfrak{g}}_{{\mathrm{af}}})v_{x}.$$

We know from [@Kas94 Proposition 8.2.2] that $V(\lambda)$ has a crystal basis ${\mathcal{B}}(\lambda)$ and the corresponding global basis $\bigl\{G(b) \mid b \in {\mathcal{B}}(\lambda)\bigr\}$; we denote by $u_{\lambda}$ the element of ${\mathcal{B}}(\lambda)$ such that $G(u_{\lambda})=v_{\lambda}$, and by ${\mathcal{B}}_{0}(\lambda)$ the connected component of ${\mathcal{B}}(\lambda)$ containing $u_{\lambda}$. Also, we know from [@Kas05 Sect. 2.8] (see also [@NS16 Sect. 4.1]) that $V_{x}^{-}(\lambda) \subset V(\lambda)$ is compatible with the global basis of $V(\lambda)$, that is, there exists a subset ${\mathcal{B}}_{x}^{-}(\lambda)$ of the crystal basis ${\mathcal{B}}(\lambda)$ such that $$\label{eq:deme}
V_{x}^{-}(\lambda) = 
\bigoplus_{b \in {\mathcal{B}}_{x}^{-}(\lambda)} {\mathbb{C}}({\mathsf{q}}) G(b) 
\subset
V(\lambda) = 
\bigoplus_{b \in {\mathcal{B}}(\lambda)} {\mathbb{C}}({\mathsf{q}}) G(b).$$

For every $x \in W_{{\mathrm{af}}}$, we have $V_{x}^{-}(\lambda) = V_{{\Pi^{J}}(x)}^{-}(\lambda)$ and ${\mathcal{B}}_{x}^{-}(\lambda) = {\mathcal{B}}_{{\Pi^{J}}(x)}^{-}(\lambda)$.

We know the following from [@INS Theorem 3.2.1] and [@NS16 Theorem 4.2.1].

\[thm:isom\] There exists an isomorphism $\Phi_{\lambda}:{\mathcal{B}}(\lambda) \stackrel{\sim}{\rightarrow} {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ of crystals such that $\Phi(u_{\lambda}) = \pi_{\lambda}$ and such that $\Phi_{\lambda}({\mathcal{B}}_{x}^{-}(\lambda)) = 