Based modules over the $imath$quantum group of type AI

This paper studies classical weight modules over the $imath$quantum group $mathbf{U}^{imath}$ of type AI. We introduce the notion of based $mathbf{U}^{imath}$-modules by generalizing the notion of based modules over the quantum groups. We prove that each finite-dimensional irreducible classical weight $mathbf{U}^{imath}$-module with integer highest weight is a based $mathbf{U}^{imath}$-module. As a byproduct, a new combinatorial formula for the branching rule from $mathfrak{sl}_n$ to $mathfrak{so}_n$ is obtained.