Based morphisms for characters of quantum symmetric pairs

We study based one-dimensional modules of quantum symmetric pairs over the field $\mathbb{Q}(q)$. We provide a complete classification of one-dimensional $\mathbf{B}$-modules that appear as submodules of simple finite-dimensional based $\mathbf{U}$-modules and determine the corresponding branching rules. The main result of this paper shows that the corresponding projections are morphisms of based $\mathbf{B}$-modules. To this end we characterize one-dimensional modules at $q=\infty$, thus developing a $\imath$crystal basis theory for these modules. This is then applied to show compatibility with the integral forms of the (dual-)canonical basis.