0}:=\bigl\{wt_{\xi} \mid w \in W,\, \xi \in Q^{\vee,+} \bigr\} \subset W_{{\mathrm{af}}}.$$

Denote by ${\Delta_{{\mathrm{af}}}}$ the set of real roots of ${\mathfrak{g}}_{{\mathrm{af}}}$, and by ${\Delta_{{\mathrm{af}}}^{+}}\subset {\Delta_{{\mathrm{af}}}}$ the set of positive real roots; we know from [@Kac Proposition 6.3] that ${\Delta_{{\mathrm{af}}}}= 
\bigl\{ \alpha + n \delta \mid \alpha \in \Delta,\, n \in {\mathbb{Z}}\bigr\}$, and ${\Delta_{{\mathrm{af}}}^{+}}= 
\Delta^{+} \sqcup 
\bigl\{ \alpha + n \delta \mid \alpha \in \Delta,\, n \in {\mathbb{Z}}_{> 0}\bigr\}$. For $\beta \in {\Delta_{{\mathrm{af}}}}$, we denote by $\beta^{\vee} \in {\mathfrak{h}}_{{\mathrm{af}}}$ its dual root, and $s_{\beta} \in W_{{\mathrm{af}}}$ the corresponding reflection; if $\beta \in {\Delta_{{\mathrm{af}}}}$ is of the form $\beta = \alpha + n \delta$ with $\alpha \in \Delta$ and $n \in {\mathbb{Z}}$, then $s_{\beta} =s_{\alpha} t_{n\alpha^{\vee}} \in W \ltimes Q^{\vee}$.

Finally, let $U_{{\mathsf{q}}}({\mathfrak{g}}_{{\mathrm{af}}})$ denote the quantized universal enveloping algebra over ${\mathbb{C}}({\mathsf{q}})$ associated to ${\mathfrak{g}}_{{\mathrm{af}}}$, with $E_{i}$ and $F_{i}$, $i \in I_{{\mathrm{af}}}$, the Chevalley generators corresponding to $\alpha_{i}$ and $-\alpha_{i}$, respectively. We denote by $U_{{\mathsf{q}}}^{-}({\mathfrak{g}}_{{\mathrm{af}}})$ the negative part of $U_{{\mathsf{q}}}({\mathfrak{g}}_{{\mathrm{af}}})$, that is, the ${\mathbb{C}}({\mathsf{q}})$-subalgebra of $U_{{\mathsf{q}}}({\mathfrak{g}}_{{\mathrm{af}}})$ generated by $F_{i}$, $i \in I_{{\mathrm{af}}}$.

Parabolic semi-infinite Bruhat graph. {#subsec:SiBG}
-------------------------------------

In this subsection, we fix a subset $J \subset I$. We set ${Q_{J}}:= \bigoplus_{i \in {J}} {\mathbb{Z}}\alpha_i$, ${Q_{J}^{\vee}}:= \bigoplus_{i \in {J}} {\mathbb{Z}}\alpha_i^{\vee}$, ${Q_{J}^{\vee,+}}:= \sum_{i \in 