On the intersection of local Arthur packets for classical groups

Abstract

In this paper, for symplectic and split odd special orthogonal groups, following Atobe's refinement on Moeglin's construction of local Arthur packets, we give new algorithms, comparing to those given by Atobe ([Ato22]), to determine whether a given representation is of Arthur type, and determine all local Arthur packets containing a representation of Arthur type. Our algorithms depend on certain operators on construction data, each of which preserves representations, forming a proper subset of those used by Atobe. In particular, for any representation π of Arthur type, we give a precise formula to compute the set {local Arthur parameter ψ | the local Arthur packet Π ψ contains π}.Our main results have four applications as follows. (1). We give a classification of all the local Arthur packets which contain tempered representations, and give the precise number of the tempered representations in any local Arthur packet and describe their L-data. We also give a classification of all the local Arthur packets which have non-trivial intersection with a given tempered local Arthur packet. (2). We prove the enhanced Shahidi's conjecture, which says that local Arthur packets are tempered if and only if they have generic members. In particular, this implies that any irreducible tempered generic representation lives in exactly one local Arthur packet. (3). We give a complete description of the construction data for representations in the L-packet of any given local Arthur packet. (4). For each representation of Arthur type, we specify a unique construction data satisfying the property that if this representation lies in the L-packet of a local Arthur packet then this construction data provides exactly the corresponding local Arthur parameter. This result suggests an answer to the mysterious question of determining which local Arthur parameter is "the" parameter for a representation if it lies in several local Arthur packets.