the counterpart of [@BW13 §2.3, §2.4 and §4.4].

There is a unique family of elements $\Upsilon_\mu \in {}_\mA\U_{-\mu}^-$ for $\mu \in {\N}{\Pi}$ such that $\Upsilon_0 = 1$, and $\Upsilon = \sum_{\mu}\Upsilon_\mu \in \widehat{\U}^-$ intertwines the bar involution $\psi_{\imath}$ on $\bun$ and the bar involution $\psi$ on $\U$ via the embedding $\imath$; that is, $\Upsilon$ satisfies the following identity (in $\widehat{\U}$): $$\label{eq:star}
 \imath(\psi_{\imath}{u}) \Upsilon = \Upsilon\  \psi({\imath(u)}), \quad \text{ for all } u \in \bun.$$ Moreover, $\Upsilon_\mu = 0$ unless $\mu^{\inv} = \mu$. We also have $\Upsilon \cdot \ov{\Upsilon} =1.$

Consider a $\Qq$-valued function $\zeta$ on $\Lambda$ such that $$\begin{aligned}
\zeta (\mu+\alpha_0)&=-q \zeta (\mu) \quad (\text{only for the pair $(\U_q(\mathfrak{sl}_{2r+1}), {\U^{\imath}}_q(\mathfrak{sl}_{2r+1})$}), 
 \notag \\ 
\zeta (\mu+\alpha_i) &= -q^{(\alpha_i, \mu+\alpha_i)-(\alpha_{-i},\mu)} \zeta (\mu), 
  \label{eq:zeta0} \\
\zeta (\mu+\alpha_{-i}) &= -q^{(\alpha_{-i}, \mu+\alpha_{-i}) - (\alpha_{i}, \mu)-1} \zeta (\mu),
\quad \forall \mu \in \Lambda, \; i \in \Ihf. 
\notag\end{aligned}$$ Such $\zeta$ clearly exists (but not unique). For any weight $\U$-module $M$, define a $\Qq$-linear map on $ M$ as follows: $$\begin{aligned}
  \label{eq:zeta}
\begin{split}
\widetilde{\zeta}&: M \longrightarrow M, 
 \\
\widetilde{\zeta}  (m &)  =  \zeta (\mu)m, \quad \forall   m \in M_{\mu}.
\end{split}\end{aligned}$$

Recall that $w_0$ is the longest element of $W$ and $T_{w_0}$ is the associated braid group element from 