  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.
