Stability of $$\imath $$ canonical Bases of Locally Finite Type

Abstract

We prove the stability conjecture of \(\imath \)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 \(\imath \)quantum groups of such type at \(q = \infty \). This result can be seen as a very restrictive version of the \(\imath \)crystal base theory for locally finite types.