Section  \[subsec:CB\]. The following proposition is the analog of [@BW13 Theorem 2.18].

\[prop:mcT\] For any finite-dimensional $\U$-module $M$, the composition map $$\mc{T} := \Upsilon\circ \widetilde{\zeta} \circ T_{w_0}: M \longrightarrow M$$ is a $\bun$-module isomorphism.

Recall the bar involutions on $\U$ and its modules are denoted by $\Abar$, and the bar involution on $\bun$ is denoted by $\Bbar$. It is also understood that $\Abar(u) =\Abar(\imath(u))$ for $u\in \bun$.

We call a $\bun$-module $M$ equipped with an anti-linear involution $\Bbar$ [*involutive*]{} (or [*$\imath$-involutive*]{}) if $
\Bbar(u m) = \Bbar(u) \Bbar(m)$, $\forall u \in \bun, m \in M$. For any involutive $\U$-module $M$ with anti-linear involution $\Abar$, the anti-linear involution $$\Bbar := \Upsilon \circ \Abar : M \longrightarrow M$$ makes $M$ an $\imath$-involution ${\U^{\imath}}$-module (cf. [@BW13 Proposition 3.10]). In particular, since we know both $L(\la)$ and ${^{\omega}L}(\lambda)$ are involutive $\U$-modules, they are $\imath$-involutive ${\U^{\imath}}$-modules as well. The following theorem is the counterpart of [@BW13 Proposition 4.20].

\[thm:BCB\] Let $\la \in \La^+$. The $\bun$-module ${^{\omega}L}(\lambda)$ admits a unique basis $$\B^\imath(\lambda) := \{T^{\lambda}_{b} \mid b \in \B(\lambda)\}$$ which is $\Bbar$-invariant and of the form $$T^{\lambda}_{b} = b^+ \xi_{-\lambda} +\sum_{b' \prec b}
t^{\lambda}_{b;b'} b'^+  \xi_{-\lambda},
\quad \text{ for }\;  t^{\lambda}_{b;b'} \in q\Z[q].$$

\[def:CB\] $\B^\imath(\lambda)$ is called 