Canonical bases arising from quantum symmetric pairs of Kac-Moody type

Abstract

For quantum symmetric pairs $(\mathbf{U}, \textbf{U}^\imath)$ of Kac-Moody type, we construct $\imath$canonical bases for the highest weight integrable $\mathbf{U}$-modules and their tensor products regarded as $\mathbf{U}^\imath$-modules, as well as an $\imath$canonical basis for the modified form of the $\imath$quantum group $\mathbf{U}^\imath$. A key new ingredient is a family of explicit elements called $\imath$divided powers, which are shown to generate the integral form of $\dot{\bf{U}}^\imath$. We prove a conjecture of Balagovic-Kolb, removing a major technical assumption in the theory of quantum symmetric pairs. Even for quantum symmetric pairs of finite type, our new approach simplifies and strengthens the integrality of quasi-K-matrix and the constructions of $\imath$canonical bases, by avoiding a case-by-case rank one analysis and removing the strong constraints on the parameters in a previous work.