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

Abstract

For quantum symmetric pairs $(\textbf {U}, \textbf {U}^\imath )$ of Kac–Moody type, we construct $\imath$ -canonical bases for the highest weight integrable $\textbf U$ -modules and their tensor products regarded as $\textbf {U}^\imath$ -modules, as well as an $\imath$ -canonical basis for the modified form of the $\imath$ -quantum group $\textbf {U}^\imath$ . A key new ingredient is a family of explicit elements called $\imath$ -divided powers, which are shown to generate the integral form of $\dot {\textbf {U}}^\imath$ . 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.