Stability of \(\imath \)canonical Bases of Locally Finite Type

We prove the stability conjecture of \(\imath \) $ı$ canonical bases, which was raised by Huanchen Bao and Weiqiang Wang in 2016, for all locally finite types. To this end, we characterize the trivial module over the \(\imath \) $ı$ quantum groups of such type at \(q = \infty \) $q=\infty$ . This result can be seen as a very restrictive version of the \(\imath \) $ı$ crystal base theory for locally finite types.