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

Let $\mathfrak{g}$ be a complex simple Lie algebra and $U_q(\hat{\mathfrak{g}})$ the corresponding quantum affine algebra. We prove that every irreducible finite-dimensional $U_q(\hat{\mathfrak{g}})$-module gives rise to a family of trigonometric solutions of Cherednik's generalized reflection equation. These depend upon the choice of a quantum affine symmetric pair $U_q(\mathfrak{k})\subset U_q(\hat{\mathfrak{g}})$. Our result relies on the construction of universal K-matrices for arbitrary quantum symmetric pairs, obtained in our previous work, as well as the fact that every irreducible $U_q(\hat{\mathfrak{g}})$-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.