Trigonometric K-matrices for finite-dimensional representations of quantum affine algebras

Let \mathfrak{g} be a complex simple finite-dimensional Lie algebra and U_{q}(\widehat{\mathfrak{g}}) the corresponding quantum affine algebra. We prove that every irreducible finite-dimensional U_{q}(\widehat{\mathfrak{g}}) -module gives rise to a family of trigonometric solutions of Cherednik’s generalized reflection equation. These depend on the choice of a quantum affine symmetric pair U_{q}(\mathfrak{k})\subset U_{q}(\widehat{\mathfrak{g}}) . Our result relies on the construction of universal K-matrices for arbitrary quantum symmetric pairs we obtained in [Represent. Theory 26 , 764–824 (2022)] and the fact that every irreducible U_{q}(\widehat{\mathfrak{g}}) -module is generically irreducible under restriction to U_{q}(\mathfrak{k}) . In the case of small modules and Kirillov–Reshetikhin modules, we obtain new solutions of the standard and the transposed reflection equations.