The non-tempered [formula omitted] Arthur parameter and Gross–Prasad conjectures

Abstract

We provide a construction of local and automorphic non-tempered Arthur packets AΨ of the group SO(3,2) and its inner form SO(4,1) associated with Arthur's parameterΨ:LF×SL2(C)→O2(C)×SL2(C)→Sp4(C) and prove a multiplicity formula. We further study the restriction of the representations in AΨ to the subgroup SO(3,1). In particular, we discover that the local Gross–Prasad conjecture, formulated for generic L-packets, does not generalize naively to a non-generic A-packet. We also study the non-vanishing of the automorphic SO(3,1)-period on the group SO(4,1)×SO(3,1) and SO(3,2)×SO(3,1) for the representations above. The main tool is the local and global theta correspondence for unitary quaternionic similitude dual pairs.