Based morphisms for characters of quantum symmetric pairs

TL;DR

The paper classifies one-dimensional $\mathbf{B}$-modules that appear inside finite-dimensional based $\mathbf{U}$-modules for Hermitian quantum symmetric pairs and shows that the natural projections are based-$\mathbf{B}$-morphisms. It develops an $\imath$crystal framework at $q=\infty$ to analyze spherical vectors and proves compatibility with integral forms of both canonical and dual canonical bases. A rank-one reduction strategy is employed to transfer results from the explicit AI and AIV rank-one cases to general Hermitian types, yielding an explicit description of branching rules and integrable characters over $\mathbb{Q}(q)$. The work extends prior results from the algebraic closure to the base field $\mathbb{Q}(q)$ and provides a detailed compatibility theory with integral structures, enabling control over spherical vectors and their $q$-limits in a unified framework.

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.

Theorems & Definitions (67)

• Theorem 1
• Theorem 2: Proposition \ref{['prop:fieldextend']}
• Theorem 3: Theorem \ref{['thm:mainthm']}
• Theorem 4: Theorem \ref{['lem:integral1']}
• Definition 2.1: Drinfel'd Jimbo quantum group
• Definition 2.2: Based module
• Example 2.3
• Definition 2.4: Based submodules and based morphisms
• Example 2.5
• Lemma 2.8