# A reciprocal branching problem for automorphic representations and global Vogan packets

## TL;DR

This paper explores the reciprocal branching problem for automorphic representations of special orthogonal groups, using the twisted automorphic descent method to find representations with prescribed restrictions, extending classical branching concepts.

## Contribution

It introduces a novel approach to the reciprocal branching problem for automorphic representations, applying the twisted automorphic descent method to special orthogonal groups.

## Key findings

- Established a framework for the reciprocal branching problem in automorphic representations.
- Demonstrated the applicability of the twisted automorphic descent method to classical groups.
- Extended the understanding of automorphic representation restrictions in the context of orthogonal groups.

## Abstract

Let $G$ be a group and $H$ be a subgroup of $G$. The classical branching rule (or symmetry breaking) asks: For an irreducible representation $\pi$ of $G$, determine the occurrence of an irreducible representation $\sigma$ of $H$ in the restriction of $\pi$ to $H$. The reciprocal branching problem of this classical branching problem is to ask: For an irreducible representation $\sigma$ of $H$, find an irreducible representation $\pi$ of $G$ such that $\sigma$ occurs in the restriction of $\pi$ to $H$. For automorphic representations of classical groups, the branching problem has been addressed by the well-known global Gan-Gross-Prasad conjecture. In this paper, we investigate the reciprocal branching problem for automorphic representations of special orthogonal groups using the twisted automorphic descent method as developed in [JZ15]. The method may be applied to other classical groups as well.