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

Abstract

Let $G=O(1,n+1)$ 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 $O(n)$-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 $G^{\prime }=O(1,n)$ of $G$ with maximal compact subgroup $K^{\prime }$, we prove branching laws and explicit Plancherel formulas for the restrictions to $G^{\prime }$ 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 $G^{\prime }$-intertwining operators which resemble the Fourier transforms on vector bundles over the Riemannian symmetric space $G^{\prime }/K^{\prime }$.

Final version, to appear in Annales de l'institut Fourier, 34 pages

References in corpus (2)

• Branching laws of unitary representations associated to minimal elliptic orbits for indefinite orthogonal group O(p,q)
• Symmetry breaking operators for real reductive groups of rank one