by $\ell (\cdot)$ the length function of $W$, and let $w_0$ be the longest element of $W$. The following lemma is well-known (cf. [@BW13 Lemma 1.5]).

\[lem:Tw0\] The following identities hold: $$T_{w_0}(K_{\alpha_i}) =K^{-1}_{\alpha_{-i}}, \quad T_{w_0}(E_{\alpha_i}) = -F_{\alpha_{-i}}K_{\alpha_{-i}}, \quad 
T_{w_0}(F_{\alpha_{-i}}) = - K^{-1}_{\alpha_i}E_{\alpha_i}, \quad \text{ for } i \in \I.$$

Let $\Lambda^+ = \{\lambda \in \Lambda \mid 2(\alpha_i , \lambda)/(\alpha_i,\alpha_i) \in {\N}, \forall i \in \I\}$ be the set of dominant weights. Note that $\mu \in \Lambda^+$ if and only if $\mu ^{\inv} \in \Lambda^+$, since the bilinear pairing $(\cdot,\cdot)$ on $ \Lambda$ is invariant under $\inv : \Lambda \rightarrow \Lambda$.

Let $M(\lambda)$ be the Verma module of $\U$ with highest weight $\lambda\in \Lambda$ and with a highest weight vector denoted by $\eta$ or $\eta_{\lambda}$. We define a $\U$-module $^\omega M(\lambda)$, which has the same underlying vector space as $M(\lambda)$ but with the action twisted by the involution $\omega$ given in Proposition \[prop:invol\]. When considering $\eta$ as a vector in $^\omega M(\lambda)$, we shall denote it by $\xi$ or $\xi_{-\lambda}$. The Verma module $M(\lambda)$ associated to dominant $\la \in \La^+$ has a unique finite-dimensional simple quotient $\U$-module, denoted by $L(\lambda)$. Similarly we define the $\U$-module $^\omega L(\lambda)$. For $\la \in \Lambda^+$, 