path from $x$ to $y$ in ${\mathrm{BG}^{{\frac{\infty}{2}}}\bigl(({W^{J}})_{{\mathrm{af}}}\bigr)}$; we write $x {\prec}y$ if $x {\preceq}y$ and $x \ne y$.

In the case $J = \emptyset$, the semi-infinite Bruhat order on $W_{{\mathrm{af}}}$ is just the generic Bruhat order introduced in [@Lu80]; see [@INS Appendix A.3] for details. Also, for a general $J$, the parabolic semi-infinite Bruhat order on $({W^{J}})_{{\mathrm{af}}}$ is nothing but the partial order on $J$-alcoves introduced in [@Lu97] when we take a special point to be the origin.

In Appendix \[sec:basic\], we recall some of the basic properties of the semi-infinite Bruhat order.

For $x \in ({W^{J}})_{{\mathrm{af}}}$, let ${\mathrm{Lift}}(x)$ denote the set of lifts of $x$ in $W_{{\mathrm{af}}}$ with respect to the map ${\Pi^{J}}:W_{{\mathrm{af}}} \twoheadrightarrow ({W^{J}})_{{\mathrm{af}}}$, that is, $$\label{eq:lift}
{\mathrm{Lift}}(x):=
\bigl\{ x' \in W_{{\mathrm{af}}} \mid {\Pi^{J}}(x') = x \bigr\};$$ for an explicit description of ${\mathrm{Lift}}(x)$, see Lemma \[lem:lift\]. The following proposition will be proved in Appendix \[sec:prf-Deo\].

\[prop:Deo\] If $x \in W_{{\mathrm{af}}}$ and $y \in ({W^{J}})_{{\mathrm{af}}}$ satisfy the condition that $y {\succeq}{\Pi^{J}}(x)$, then the set $${\mathrm{Lift}_{{\succeq}x}(y)}:=
\bigl\{ y' \in {\mathrm{Lift}}(y) \mid y' {\succeq}x \bigr\}$$ has the minimum element with respect to the semi-infinite Bruhat order on $W_{{\mathrm{af}}}$[;]{} we denote this element by $\min {\mathrm{Lift}_{{\succeq}x}(y)}$.

Semi-infinite Lakshmibai-Seshadri paths. {#subsec:SLS}
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In 