A reciprocal branching problem for automorphic representations and global Vogan packets

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 π of G, determine the occurrence of an irreducible representation σ of H in the restriction of π to H. The reciprocal branching problem of this classical branching problem is to ask: For an irreducible representation σ of H, find an irreducible representation π of G such that σ occurs in the restriction of π 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 [17]. The method may be applied to other classical groups as well.

Bessel Descents and Branching Problems

“…Hence it is a desirable question to find the precise information about the global Arthur parameter of π that gives an answer the Cuspidal Reciprocal Branching Problem! In a recent work of Jiang-Liu-Xu ( [13]), a special situation when H m is SO(3) is explicitly studied. In this case, we are able to prove in some situation that the twisted automorphic descent method may still construct a cuspidal π with generic global Arthur parameter (Theorem 6.3 of [13]).…”

“…In a recent work of Jiang-Liu-Xu ( [13]), a special situation when H m is SO(3) is explicitly studied. In this case, we are able to prove in some situation that the twisted automorphic descent method may still construct a cuspidal π with generic global Arthur parameter (Theorem 6.3 of [13]). We will leave further details of discussion about this mysterious question to our future work.…”

“…(4) In [19], the idea and method of the twisted automorphic descents was used to prove the non-vanishing of certain tensor product L-functions for classical groups. (5) In [13], the twisted automorphic descents were used to test the Branching Problem and the Reciprocal Branching Problem for cuspidal automorphic representations of classical groups.…”