this assertion by descending induction on $k$. Let $U_{{\mathsf{q}}}^{-}({\mathfrak{g}})$ denote the ${\mathbb{C}}({\mathsf{q}})$-subalgebra of $U_{{\mathsf{q}}}^{-}({\mathfrak{g}}_{{\mathrm{af}}})$ generated by $F_{i}$, $i \in I$. If $k=-{\langle \lambda,\,\xi \rangle}$, then the assertion is obvious since $V_{x}^{-}(\lambda)_{-{\langle \lambda,\,\xi \rangle}} = U_{{\mathsf{q}}}^{-}({\mathfrak{g}})v_{x}$ and $V(\lambda)$ is an integrable $U_{{\mathsf{q}}}({\mathfrak{g}}_{{\mathrm{af}}})$-module. Assume that $k < -{\langle \lambda,\,\xi \rangle}$. Observe that $V_{x}^{-}(\lambda)_{k}$ is a $U_{{\mathsf{q}}}^{-}({\mathfrak{g}})$-module generated by $F_{0}V_{x}^{-}(\lambda)_{k+1}$. Because $F_{0}V_{x}^{-}(\lambda)_{k+1}$ is finite-dimensional by our induction hypothesis, and $V(\lambda)$ is an integrable $U_{q}({\mathfrak{g}}_{{\mathrm{af}}})$-module, we deduce that $V_{x}^{-}(\lambda)_{k} = U_{{\mathsf{q}}}^{-}({\mathfrak{g}})(F_{0}V_{x}^{-}(\lambda)_{k+1})$ is also finite-dimensional, as desired.

Now, we define the graded character ${\mathop{\rm gch}\nolimits}V_{x}^{-}(\lambda)$ of $V_{x}^{-}(\lambda)$ to be $$\label{eq:gch}
{\mathop{\rm gch}\nolimits}V_{x}^{-}(\lambda) : = 
 \sum_{k \in {\mathbb{Z}}}
\Biggl(
 \sum_{\gamma \in Q} 
 \dim \bigl( V_{x}^{-}(\lambda)_{\lambda+\gamma+k\delta} \bigr) 
 e^{\lambda+\gamma}
\Biggr) q^{k};$$ observe that $$\label{eq:gch1}
{\mathop{\rm gch}\nolimits}V_{x}^{-}(\lambda) \in 
\bigl(\underbrace{{\mathbb{Z}}[e^{\nu} \mid \nu \in P]}_{={\mathbb{Z}[P]}}\bigr)
{[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]}q^{-{\langle \lambda,\,\xi \rangle}}.$$ For $\gamma \in Q$ and $k \in {\mathbb{Z}}$, we set ${\mathop{\rm fin}\nolimits}(\lambda+\gamma+k\delta):=\lambda+\gamma \in P$ and ${\mathop{\rm nul}\nolimits}(\lambda+\gamma+k\delta):=k \in {\mathbb{Z}}$. Then, by Theorem \[thm:isom\], we have $$\label{eq:gch2}
{\mathop{\rm gch}\nolimits}V_{x}^{-}(\lambda) = \sum_{\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\lambda)} 
e^{{\mathop{\rm fin}\nolimits}({\mathop{\rm wt}\nolimits}(\pi))} q^{{\mathop{\rm nul}\nolimits}({\mathop{\rm wt}\nolimits}(\pi))}.$$

Combinatorial standard monomial theory for semi-infinite LS paths. {#sec:SMT}
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In this section, we fix $\lambda,\,\mu \in P^{+} \subset P_{{\mathrm{af}}}^{0}$ (see and ), and set $$J:=\bigl\{i \in I \mid {\langle \lambda,\,\alpha_{i}^{\vee} \rangle}=0 \bigr\}, \qquad 
K:=\bigl\{i 