its quotient $\La_\inv$ which will be used in quantum symmetric pairs.

The involution $\inv$ and the lattice $\Lambda_\inv$ {#subsec:theta}
----------------------------------------------------

Let $q$ be an indeterminate. For $r \in \N$, we define the following index sets: $$\begin{aligned}
  \label{eq:I}
\begin{split}
\I_{2r+1} &= \{i \in \Z \mid -r \le i \le r\},  
 \\
\I_{2r}  &= \big\{i \in \Z+\hf \mid -r < i < r \big\}.
\end{split}\end{aligned}$$

Set $k =2r+1$ or $2r$, and we use the shorthand notation $\I =\I_k$ in the remainder of Section \[sec:prelim\], and very often through out this paper. Let $$\Pi (= \Pi_k)= \big \{\alpha_i=\varepsilon_{i-\hf}-\varepsilon_{i+\hf} \mid i \in \I \big \}$$ be the simple system of type $A_{k}$, and let $\Phi$ be the associated root system. Denote by $$\Lambda (= \Lambda_k) = \sum_{{i \in \I} } \big( \Z \varepsilon_{i - \hf} + \Z \varepsilon_{i + \hf} \big)$$ the integral weight lattice, and denote by $(\cdot, \cdot)$ the standard bilinear pairing on $\Lambda$ such that $(\varepsilon_a, 
\varepsilon_b) = \delta_{ab}$ for all $a,b$. For any $\mu = \sum_{i}c_i\alpha_i \in \N {\Pi}$, set $\hgt(\mu) = \sum_{i} c_i$. Let $\inv$ be the involution of the weight lattice $\Lambda$ such that $$\inv(\varepsilon_{i-\hf}) = - \varepsilon_{-i+\hf}, \quad \text{ for all } i \in \I.$$ We shall 