Shifted twisted Yangians of quasi-split ADE types

Associated to all quasi-split Satake diagrams of type ADE and even spherical coweights $\mu$, we introduce the shifted iYangians ${}^{\mathrm{ı}}{Y}_{\mu }$ and establish their PBW bases. We construct the iGKLO representations of ${}^{\mathrm{ı}}{Y}_{\mu }$, which factor through quotients called truncated shifted iYangians ${}^{\mathrm{ı}}{Y}_{\mu }^{\lambda }$. In type AI with $\mu$ dominant, a variant of ${}^{\mathrm{ı}}{Y}_{\mu }^{N{\varpi }_1^{\vee }}$ is identified with the truncated shifted iYangians in another definition, which are isomorphic to finite W-algebras of type BCD. These new family of algebras has connections and applications to fixed point loci of affine Grassmannian slices which will be developed in a sequel.