a semi-infinite LS path $\eta$ with respect to $x$ (based on the existence of the semi-infinite analog of the so-called Deodhar lift), and of the description of tensor product decomposition of crystals in terms of ${\iota(\bullet,\,x)}$ (Theorem \[thm:SMT\] and Theorem \[thm:Dem\]). We remark that our ${\iota(\eta,\,x)}$ is an analogue of the one in [@LiSe03; @LeSh14] in the setting of level-zero extremal weight modules over $U_{{\mathsf{q}}}({\mathfrak{g}}_{{\mathrm{af}}})$. Using them, we obtain our Pieri-Chevalley formula:

\[fPCf\] For $\lambda \in P^{+}$ and $x \in W_{{\mathrm{af}}}^{\ge 0}:=W \times Q^{\vee,+}$, we have $$\label{fPC}
[{\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \lambda )] \cdot [{\mathcal{O}}_{{\mathbf{Q}_{G}}( x )}] = 
 \sum_{\eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x} (- w_{\circ} \lambda )} 
 e^{{\mathop{\rm fin}\nolimits}(\eta)}q^{{\mathop{\rm nul}\nolimits}(\eta)} \cdot [ {\mathcal{O}}_{{\mathbf{Q}_{G}}( {\iota(\eta,\,x)} )}] \quad \in K^{\prime}_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}}),$$ where ${\mathop{\rm fin}\nolimits}(\eta) \in P$ and ${\mathop{\rm nul}\nolimits}(\eta) \in {\mathbb{Z}}$ for $\eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}(- w_{\circ} \lambda )$ are defined by: $${\mathop{\rm wt}\nolimits}(\eta) = {\mathop{\rm fin}\nolimits}(\eta) + {\mathop{\rm nul}\nolimits}(\eta) \delta.$$

Once generalities on $K^{\prime}_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}})$ and the semi-infinite SMT are given, our strategy for the proof of Theorem \[fPCf\] is along the line of [@LiSe03]. Namely, we compare the functionals $$P \ni \lambda \mapsto \sum_{i \ge 0}(-1)^{i} 
  {\mathop{\rm gch}\nolimits}H^{i}( {\mathbf{Q}_{G}}, {\mathcal{E}}\otimes_{{\mathcal{O}}_{{\mathbf{Q}_{G}}}} {\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \lambda ) ) \in {\mathbb{C}}[P]{(\hspace{-1pt}(q^{-1})\hspace{-1pt})},$$ where $[{\mathcal{E}}]$ is taken from the both sides of .

