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

TL;DR

The paper develops a comprehensive framework for quasi-split affine quantum symmetric pairs of type $\mathsf{AIII}$, establishing a factorization formula for the Drinfeld–Cartan series $\boldsymbol\grave{\boldsymbol\Theta}_i(z)$ in terms of $\boldsymbol\phi_i^{\pm}$ and Cartan factors, and proving a group-like coproduct modulo the positive half. By reducing to rank-one cases and analyzing the relative braid group action, the authors handle all rank configurations, including the new rank-one cases where $a_{i,\tau(i)}$ equals $2,0,-1$. The results yield explicit eigenvalue descriptions on restricted representations and motivate a boundary $q$-character theory compatible with Frenkel–Reshetikhin theory, offering a robust algebraic route to boundary phenomena in quantum symmetric pairs. The work thus links Drinfeld-type loop realizations, coideal subalgebras, and boundary character theories, with promising geometric and categorification directions via Nakajima quiver varieties and orientifold KLR algebras.

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 $\boldsymbolΘ_i(z)$ 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.

Figures (1)

• Figure 1: (a) Satake diagram of type ${\mathsf{AIII}}_{2n-1}^{(\tau)}$ ($n \ge 2$); (b) Satake diagram of type ${\mathsf{AIII}}_{2n}^{(\tau)}$ ($n \ge 1$).

Theorems & Definitions (84)

• Theorem A: Theorem \ref{['thm: main overall']}
• Theorem B: Theorem \ref{['thm: coproduct main']}
• Theorem C: Theorem \ref{['cor: FR thm Oq']}
• Theorem D: Theorem \ref{['cor: comm diagram qchar actions']}
• Definition 2.1
• Theorem 2.2
• proof
• Lemma 2.3
• proof
• Definition 2.4