\text{$x\beta \in {\Delta_{{\mathrm{af}}}^{+}}$ for all $\beta \in ({\Delta_{J}})_{{\mathrm{af}}}^{+}$} \bigr\};\end{aligned}$$ note that if ${J}= \emptyset$, then $(W^{\emptyset})_{{\mathrm{af}}}=W_{{\mathrm{af}}}$ and $(W_{\emptyset})_{{\mathrm{af}}}=\bigl\{e\bigr\}$. We know from [@Pet97] (see also [@LS10 Lemma 10.6]) that for each $x \in W_{{\mathrm{af}}}$, there exist a unique $x_1 \in ({W^{J}})_{{\mathrm{af}}}$ and a unique $x_2 \in ({W_{J}})_{{\mathrm{af}}}$ such that $x = x_1 x_2$; we define a (surjective) map $$\label{eq:PiJ}
{\Pi^{J}}: W_{{\mathrm{af}}} \twoheadrightarrow ({W^{J}})_{{\mathrm{af}}}, \quad x \mapsto x_{1},$$ where $x= x_1 x_2$ with $x_1 \in ({W^{J}})_{{\mathrm{af}}}$ and $x_2 \in ({W_{J}})_{{\mathrm{af}}}$.

\[dfn:sell\] Let $x \in W_{{\mathrm{af}}}$, and write it as $x = w t_{\xi}$ for $w \in W$ and $\xi \in Q^{\vee}$. We define the semi-infinite length ${\ell^{{\frac{\infty}{2}}}}(x)$ of $x$ by: ${\ell^{{\frac{\infty}{2}}}}(x) = \ell (w) + 2 {\langle \rho,\,\xi \rangle}$.

\[dfn:SiB\]

The (parabolic) semi-infinite Bruhat graph ${\mathrm{BG}^{{\frac{\infty}{2}}}\bigl(({W^{J}})_{{\mathrm{af}}}\bigr)}$ is the ${\Delta_{{\mathrm{af}}}^{+}}$-labeled, directed graph with vertex set $({W^{J}})_{{\mathrm{af}}}$ whose directed edges are of the following form: $x {\xrightarrow{\,\beta\,}} s_{\beta} x$ for $x \in ({W^{J}})_{{\mathrm{af}}}$ and $\beta \in {\Delta_{{\mathrm{af}}}^{+}}$, where $s_{\beta } x \in ({W^{J}})_{{\mathrm{af}}}$ and ${\ell^{{\frac{\infty}{2}}}}(s_{\beta} x) = {\ell^{{\frac{\infty}{2}}}}(x) + 1$. When $J=\emptyset$, we write ${\mathrm{BG}^{{\frac{\infty}{2}}}(W_{{\mathrm{af}}})}$ for $\mathrm{BG}^{{\frac{\infty}{2}}}\bigl((W^{\emptyset})_{{\mathrm{af}}}\bigr)$.

The semi-infinite Bruhat order is a partial order ${\preceq}$ on $({W^{J}})_{{\mathrm{af}}}$ defined as follows: for $x,\,y \in ({W^{J}})_{{\mathrm{af}}}$, we write $x {\preceq}y$ if there exists a directed 