Compatibility of Drinfeld presentations and $q$-characters for affine Kac-Moody quantum symmetric pairs: quasi-split case

TL;DR

This paper proves new factorization and coproduct formulas for affine quantum symmetric pairs of type AIII, and introduces a boundary $q$-character map compatible with existing $q$-character theory.

Contribution

It establishes generalized formulas for Drinfeld--Cartan series and constructs a boundary $q$-character map compatible with the classical one.

Findings

Factorization and coproduct formulas for $oldsymbol heta_i(z)$ are proven.

A boundary $q$-character map is constructed and shown to be compatible with the original.

Results extend previous split type findings to quasi-split affine quantum symmetric pairs.

Abstract

Let $(\mathbf{U}, \mathbf{U}^\imath)$ be a quasi-split affine quantum symmetric pair of type $\mathsf{AIII}$. This case is of particular interest thanks to the existence of geometric realizations and Schur--Weyl dualities. We establish factorization and coproduct formulae for the Drinfeld--Cartan series $\boldsymbol\Theta_i(z)$ in the Lu--Pan--Wang--Zhang `new Drinfeld'-style presentation, generalizing the split type results from [Prz23, LP25a]. As an application, we construct a boundary analogue of the $q$-character map, and show that it is compatible with Frenkel and Reshetikhin's original $q$-character homomorphism.