we let ${}_\mA L(\lambda) ={}_\mA\U^- \eta$ and $^\omega _\mA L(\lambda) ={}_\mA \U^+ \xi$ be the $\mA$-submodules of $L(\lambda)$ and $^\omega L(\lambda) $, respectively.

We call a $\U$-module $M$ equipped with an anti-linear involution $\Abar$ is called [*involutive*]{} if $
\Abar(u m) = \Abar(u) \Abar(m)$, $\forall u \in \U, m \in M$. The $\U$-modules ${^{\omega}L}(\lambda)$ and $L(\lambda)$ are both involutive. Given any two involutive $\U$-modules $M$ and $M'$, Lusztig showed that their tensor product $M \otimes M'$ is also involutive ([@Lu94 §27.31]).

In [@Lu90; @Lu94] and [@Ka], the canonical basis $\bold{B}$ of ${_\mA \U^+} \cong {_\mA \U^-}$ has been constructed. For any element $b \in \B$, when considered as an element in $\U^-$ or $\U^+$, we shall denote it by $b^-$ or $b^+$, respectively. In [@Lu94], subsets $\B(\lambda)$ of $\B$ is also constructed for each $\lambda \in \Lambda^+$, such that $\{b^-\eta_{\lambda} \mid b \in \B(\lambda)\}$ gives the canonical basis of ${}_\mA L(\lambda)$. Similarly $\{b^+ \xi_{-\lambda} \mid b \in \B(\lambda)\}$ gives the canonical basis of ${^{\omega}_\mA L(\lambda)}$.

Quantum Symmetric pairs {#sec:QSP}
=======================

In this section we shall develop the theory of $\imath$-canonical bases for the quantum symmetric pairs : $$(\U_q(\mathfrak{sl}_{2r+1}), {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}))\quad \text{ and } \quad (\U_q(\mathfrak{sl}_{2r+2}), {\U^{\imath}}_q(\mathfrak{sl}_{2r+2})).$$

The definitions of the quantum 