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

Abstract

Let $G=\operatorname{O}(1,n+1)$ with maximal compact subgroup $K$ and let $\Pi$ be a unitary irreducible representation of $G$ with non-trivial $(\mathfrak{g},K)$-cohomology. Then $\Pi$ occurs inside a principal series representation of $G$, induced from the $\operatorname{O}(n)$-representation $\bigwedge\nolimits^p(\mathbb{C}^n)$ and characters of a minimal parabolic subgroup of $G$ at the limit of the complementary series. Considering the subgroup $G'=\operatorname{O}(1,n)$ of $G$ with maximal compact subgroup $K'$, we prove branching laws and explicit Plancherel formulas for the restrictions to $G'$ 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 $G'$-intertwining operators which resemble the Fourier transforms on vector bundles over the Riemannian symmetric space $G'/K'$.