A reciprocal branching problem for automorphic representations and global Vogan packets

Let G be a group and let H be a subgroup of G. The classical branching rule (or symmetry breaking) asks: For an irreducible representation πof G, determine the occurrence of an irreducible representation σ of H in the restriction of πto H. The reciprocal branching problem of this classical branching problem is to ask: For an irreducible representation σ of H, find an irreducible representation πof G such that σ occurs in the restriction of πto 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 [13]. The method may be applied to other classical groups as well.