Compatibility of Drinfeld presentations for split affine Kac–Moody quantum symmetric pairs

Let \((\textbf{U}, \textbf{U}^\imath )\) be a split affine quantum symmetric pair of type \(\textsf{B}_n^{(1)}, \textsf{C}_n^{(1)}\) or \(\textsf{D}_n^{(1)}\). We prove factorization and coproduct formulae for the Drinfeld–Cartan operators \(\Theta _i(z)\) in the Lu–Wang Drinfeld-type presentation, generalizing the type \(\textsf{A}_n^{(1)}\) result from Przeździecki (arXiv:2311.13705). As an application, we show that a boundary analogue of the q-character map, defined via the spectra of these operators, is compatible with the usual q-character map. As an auxiliary result, we also produce explicit reduced expressions for the fundamental weights in the extended affine Weyl groups of classical types.