The Jiang conjecture on the wavefront sets of local Arthur packets
Baiying Liu, Freydoon Shahidi
TL;DR
The paper studies Jiang's conjecture on the wave front sets of representations in local Arthur packets for classical $p$-adic groups, proposing that the maximal nilpotent orbits in a packet are governed by the $\ ext{Ghat}$-collapse of the Arthur parameter partition $\ul{p}(\psi)$ and related by Barbasch–Vogan duality. It extends the framework to a generalized setting with bitorsors and automorphisms, deriving dimension identities for nilpotent orbits and constructing explicit packet members whose wave front partitions realize the conjectured bounds in several cases. Under suitable assumptions, it also proves an enhanced Shahidi conjecture, linking temperedness to the existence of generic members in local Arthur packets. The combination of structural (collapse/expansion and BV duality) and constructive (explicit element in packets with prescribed wave front) methods advances understanding of how Arthur parameters determine wave front sets, with implications for endoscopic transfer and stability phenomena in the local Langlands program.
Abstract
This is a report on the progress made on a conjecture of Jiang on the upper bound nilpotent orbits in the wave front sets of representations in local Arthur packets of classical groups, which is a natural generalization of the Shahidi conjecture. We partially prove this conjecture, confirming the relation between the structure of wave front sets and the local Arthur parameters. Under certain assumptions, we also prove the enhanced Shahidi conjecture, which states that local Arthur packets are tempered if and only if they have generic members.