# Distinguished representations of SO(n+1,1) x SO(n,1), periods and branching laws

## TL;DR

This paper characterizes when certain irreducible representations of SO(n+1,1) and SO(n,1) have nonzero Hom spaces, confirming cases of the Gross--Prasad conjectures and constructing distinguished representations and periods.

## Contribution

It provides necessary and sufficient conditions for nonzero Hom spaces between representations of rank one orthogonal groups, advancing the understanding of branching laws and periods.

## Key findings

- Established criteria for nonzero Hom spaces in terms of $ heta$-stable parameters.
- Confirmed the Gross--Prasad conjectures for tempered representations of SO(n+1,1) and SO(n,1).
- Constructed nonzero periods and distinguished representations using these criteria.

## Abstract

Given irreducible representations $\Pi$ and $\pi$ of the rank one special orthogonal groups $G=SO(n+1,1)$ and $G'=SO(n,1)$ with nonsingular integral infinitesimal character, we state in terms of $\theta$-stable parameter necessary and sufficient conditions so that \[ \operatorname{Hom}_{G'}(\Pi|_{G'}, \pi )\not = \{0\}. \] In the special case that both $\Pi$ and $\pi$ are tempered, this implies the Gross--Prasad conjectures for tempered representations of $SO(n+1,1) \times SO(n,1)$ which are nontrivial on the center. We apply these results to construct nonzero periods and distinguished representations. If both $\Pi$ and $ \pi$ have the trivial infinitesimal character $\rho$ then we use a theorem that the periods are nonzero on the minimal $K$-type to obtain a nontrivial bilinear form on the $({\mathfrak g},K)$-cohomology of the representations.