{\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\lambda)$ for all $x \in W_{{\mathrm{af}}}$[;]{} in particular, we have $\Phi_{\lambda}({\mathcal{B}}_{0}(\lambda)) = {\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\lambda)$.

Let $x \in W_{{\mathrm{af}}}$. If $x$ is of the form $x = wt_{\xi}$ for some $w \in W$ and $\xi \in Q^{\vee}$, then $v_{x} \in V(\lambda)$ is a weight vector of weight $x\lambda = 
w\lambda-{\langle \lambda,\,\xi \rangle}\delta$; note that $w\lambda \in \lambda-Q^{+}$. Also, for $i \in I$ (resp., $i=0 \in I_{{\mathrm{af}}}$), the Chevalley generator $F_{i}$ (resp., $F_{0}$) of $U_{{\mathsf{q}}}({\mathfrak{g}}_{{\mathrm{af}}})$ acts on $V(\lambda)$ as a (linear) operator of weight $-\alpha_{i} \in Q$ (resp., $-\alpha_{0}=\theta-\delta \in Q + {\mathbb{Z}}_{< 0}\delta$). Therefore, the Demazure submodule $V_{x}^{-}(\lambda) = U_{{\mathsf{q}}}^{-}({\mathfrak{g}}_{{\mathrm{af}}})v_{x}$ has the weight space decomposition of the form: $$V_{x}^{-}(\lambda) = 
 \bigoplus_{k \in {\mathbb{Z}}}
\Biggl(
\underbrace{
   \bigoplus_{\gamma \in Q} 
   V_{x}^{-}(\lambda)_{\lambda+\gamma+k\delta}}_{=:V_{x}^{-}(\lambda)_{k}}
\Biggr),$$ where $V_{x}^{-}(\lambda)_{k} = \bigl\{ 0 \bigr\}$ for all $k > - {\langle \lambda,\,\xi \rangle}$; in addition, by Theorem \[thm:isom\], together with the definition of the map ${\mathop{\rm wt}\nolimits}:{\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \rightarrow P_{{\mathrm{af}}}$ (see ), we see that if $\gamma \notin -Q^{+}$, then $V_{x}^{-}(\lambda)_{\lambda+\gamma+k\delta} = \bigl\{0\bigr\}$ for all $k \in {\mathbb{Z}}$, since $W_{{\mathrm{af}}}\lambda \subset \lambda - Q^{+} +{\mathbb{Z}}\delta$ by the assumption that $\lambda \in P^{+}$. Here we claim that $V_{x}^{-}(\lambda)_{k}$ is finite-dimensional for all $k \in {\mathbb{Z}}$ with $k \le -{\langle \lambda,\,\xi \rangle}$; we show 