Branching of unitary $\mathrm{O}(1,n+1)$-representations with non-trivial $(\mathfrak{g},K)$-cohomology

Let $G=\mathrm{O}(1,n+1)$ with maximal compact subgroup $K$ and let $\mathrm{\Pi }$ be a unitary irreducible representation of $G$ with non-trivial $(\mathfrak{g},K)$-cohomology. Then $\mathrm{\Pi }$ occurs inside a principal series representation of $G$, induced from the $\mathrm{O}(n)$-representation ${\bigwedge }^p({\mathbb{C}}^n)$ and characters of a minimal parabolic subgroup of $G$ at the limit of the complementary series. Considering the subgroup $G^{\mathrm{\prime }}=\mathrm{O}(1,n)$ of $G$ with maximal compact subgroup $K^{\mathrm{\prime }}$, we prove branching laws and explicit Plancherel formulas for the restrictions to $G^{\mathrm{\prime }}$ of all unitary representations occurring in such principal series, including the complementary series, all unitary $G$-representations with non-trivial $(\mathfrak{g},K)$-cohomology and further relative discrete series representations in the cases $p=0,n$. Discrete spectra are constructed explicitly as residues of $G^{\mathrm{\prime }}$-intertwining operators which resemble the Fourier transforms on vector bundles over the Riemannian symmetric space $G^{\mathrm{\prime }}\mathrm{/}{K}^{\mathrm{\prime }}$.