Orthogonal Howe duality and dynamical (split) symmetric pairs

Inspired by Etingof--Varchenko's dynamical fusion, dynamical $R$-matrix, and dynamical Weyl group for Lie algebras, we introduce, for split symmetric pairs, versions of dynamical fusion, dynamical $K$-matrix, and dynamical Weyl group. We then turn to the study of $(\mathfrak{so}_{2n},O_m)$-duality and prove that the standard Knizhnik-Zamolodchikov and dynamical operators (both differential and difference) on the $\mathfrak{so}_{2n}$-side are exchanged with the symmetric pair analogs, for $O_m\subset GL_m$, on the $O_m$-side.