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

TL;DR

This paper analyzes how certain unitary representations of the orthogonal group O(1,n+1) with non-trivial cohomology decompose when restricted to a subgroup O(1,n), providing explicit branching laws and Plancherel formulas.

Contribution

It provides explicit branching laws and Plancherel formulas for the restriction of unitary representations with non-trivial cohomology from O(1,n+1) to O(1,n), including complementary and discrete series.

Findings

Derived explicit decomposition formulas for restrictions of unitary representations.

Constructed discrete spectra as residues of intertwining operators.

Extended results to include complementary and relative discrete series representations.

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 $G'/K'$.