this subsection, we fix $\lambda \in P^{+} \subset P_{{\mathrm{af}}}^{0}$ (see and ), and set $$\label{eq:J}
J:= \bigl\{ i \in I \mid {\langle \lambda,\,\alpha_i^{\vee} \rangle}=0 \bigr\} \subset I.$$

\[dfn:SBa\] For a rational number $0 < a < 1$, we define ${\mathrm{BG}^{{\frac{\infty}{2}}}_{\lambda,a}\bigl(({W^{J}})_{{\mathrm{af}}}\bigr)}$ to be the subgraph of ${\mathrm{BG}^{{\frac{\infty}{2}}}\bigl(({W^{J}})_{{\mathrm{af}}}\bigr)}$ with the same vertex set but having only the edges of the form $x {\xrightarrow{\,\beta\,}} y$ with $a{\langle x\lambda,\,\beta^{\vee} \rangle} \in {\mathbb{Z}}$.

\[dfn:SLS\] A semi-infinite Lakshmibai-Seshadri (LS for short) path of shape $\lambda $ is a pair $$\label{eq:SLS}
\pi = ({\mathbf{x}}\,;\, {\mathbf{a}}) 
     = (x_{1},\,\dots,\,x_{s} \,;\, a_{0},\,a_{1},\,\dots,\,a_{s}), \quad s \ge 1,$$ of a strictly decreasing sequence ${\mathbf{x}}: x_1 {\succ}\cdots {\succ}x_s$ of elements in $({W^{J}})_{{\mathrm{af}}}$ and an increasing sequence ${\mathbf{a}}: 0 = a_0 < a_1 < \cdots  < a_s =1$ of rational numbers satisfying the condition that there exists a directed path from $x_{u+1}$ to $x_{u}$ in ${\mathrm{BG}^{{\frac{\infty}{2}}}_{\lambda,a_{u}}\bigl(({W^{J}})_{{\mathrm{af}}}\bigr)}$ for each $u = 1,\,2,\,\dots,\,s-1$. We denote by ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ the set of all semi-infinite LS paths of shape $\lambda$.

Following [@INS Sect. 3.1] (see also [@NS16 Sect. 2.4]), we endow the set ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ with a crystal structure with weights in $P_{{\mathrm{af}}}$ by the map ${\mathop{\rm wt}\nolimits}:{\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \rightarrow P_{{\mathrm{af}}}$ and the root operators $e_{i}$, $f_{i}$, $i \in I_{{\mathrm{af}}}$; for 