Drinfeld rational fractions for affine Kac-Moody quantum symmetric pairs

We formulate a precise connection between the new Drinfeld presentation of a quantum affine algebra $U_q\widehat{\mathfrak{g}}$ and the new Drinfeld presentation of affine coideal subalgebras of split type recently discovered by Lu and Wang. In particular, we establish a ``factorization formula'', expressing the commuting ``Drinfeld-Cartan''-type operators $\Theta_{i,k}$ in a coideal subalgebra in terms of the corresponding Drinfeld generators of $U_q\widehat{\mathfrak{g}}$, modulo the ``Drinfeld positive half" of $U_q\widehat{\mathfrak{g}}$. We study the spectra of these operators on finite dimensional representations, and describe them in terms of rational functions with an extra symmetry. These results can be seen as the starting point of a $q$-character theory for affine Kac--Moody quantum symmetric pairs. Additionally, we prove a compatibility result linking Lusztig's and the Lu-Wang-Zhang braid group actions.