# Branching laws for discrete series of some affine symmetric spaces

## TL;DR

This paper investigates the restriction of discrete series representations of affine symmetric spaces, using period integrals and symmetry-breaking operators, and explores conjectures related to Arthur packets and the Gross-Prasad conjectures.

## Contribution

It introduces non-vanishing symmetry-breaking operators for discrete spectrum representations and discusses conjectures on restrictions within Arthur packets.

## Key findings

- Constructed explicit symmetry-breaking operators for certain representations.
- Established non-vanishing conditions for these operators.
- Proposed conjectures relating to Arthur and Vogan packets.

## Abstract

In this paper we study branching laws for certain unitary representations. This is done on the smooth vectors via a version of the {\it period integrals}, studied in number theory, and also closely connected to the {\it symmetry-breaking operators}, introduced by T.~Kobayashi. We exhibit non-vanishing symmetry breaking operators for the restriction of a representation $\Pi$ in the discrete spectrum for real hyperboloids to representations of smaller orthogonal groups. In the last part we discuss some conjectures for the restriction of representations in Arthur packets containing the representation $\Pi$ and the corresponding Arthur-Vogan packets to smaller orthogonal groups; these are inspired by the Gross-Prasad conjectures.