details, see Appendix \[sec:crystal\]. We denote by ${\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\lambda)$ the connected component of ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ containing $\pi_{\lambda}:=(e\,;\,0,1) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$.

If $\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ is of the form , then we set $$\label{eq:dir}
\iota(\pi):=x_{1} \in ({W^{J}})_{{\mathrm{af}}} \qquad 
 \text{(resp., $\kappa(\pi):=x_{s} \in ({W^{J}})_{{\mathrm{af}}}$)};$$ we call $\iota(\pi)$ (resp., $\kappa(\pi)$) the initial (resp., final) direction of $\pi$. For $x \in W_{{\mathrm{af}}}$, we set $$\label{eq:SLS-dem}
{\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\lambda) := 
 \bigl\{
   \pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \mid \kappa(\pi) {\succeq}{\Pi^{J}}(x)
 \bigr\}.$$

Extremal weight modules and their Demazure submodules. {#subsec:extremal}
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In this subsection, we fix $\lambda \in P^{+} \subset P_{{\mathrm{af}}}^{0}$ (see and ). Let $V(\lambda)$ denote the extremal weight module of extremal weight $\lambda$ over $U_{{\mathsf{q}}}({\mathfrak{g}}_{{\mathrm{af}}})$, which is an integrable $U_{{\mathsf{q}}}({\mathfrak{g}}_{{\mathrm{af}}})$-module generated by a single element $v_{\lambda}$ with the defining relation that $v_{\lambda}$ is an “extremal weight vector” of weight $\lambda$; recall from [@Kas02 Sect. 3.1] and [@Kas05 Sect. 2.6] that $v_{\lambda}$ is an extremal weight vector of weight $\lambda$ if and only if ($v_{\lambda}$ is a weight vector of weight $\lambda$ and) there exists a family $\{ v_{x} \}_{x \in W_{{\mathrm{af}}}}$ of weight vectors in $V(\lambda)$ such that $v_{e}=v_{\lambda}$, and such that for every $i \in I_{{\mathrm{af}}}$ and $x \in W_{{\mathrm{af}}}$ with $n:={\langle x\lambda,\,\alpha_{i}^{\vee} \rangle} \ge 0$ (resp., $\le 0$), the equalities $E_{i}v_{x}=0$ 