Based morphisms for characters of quantum symmetric pairs

TL;DR

This paper classifies one-dimensional modules of quantum symmetric pairs, develops a crystal basis theory for them, and demonstrates their compatibility with canonical bases, advancing understanding of their structure and representations.

Contribution

It provides a complete classification of one-dimensional modules for quantum symmetric pairs and introduces a new crystal basis theory for these modules.

Findings

Classified all one-dimensional modules as submodules of finite-dimensional based modules.

Established morphisms of based modules for these one-dimensional modules.

Proved compatibility with integral forms of canonical bases.

Abstract

We study based one-dimensional modules of quantum symmetric pairs over the field $\mathbb{Q}(q)$. We provide a complete classification of one-dimensional $\mathbf{B}$-modules that appear as submodules of simple finite-dimensional based $\mathbf{U}$-modules and determine the corresponding branching rules. The main result of this paper shows that the corresponding projections are morphisms of based $\mathbf{B}$-modules. To this end we characterize one-dimensional modules at $q=\infty$, thus developing a $\imath$crystal basis theory for these modules. This is then applied to show compatibility with the integral forms of the (dual-)canonical basis.