Geometrization of Kottwitz-Shelstad transfer and Vogan's conjecture on local Arthur packets

Abstract

We give a geometric analogue of Kottwitz-Shelstad transfer for a general class of reductive groups over p-adic fields with cyclic component groups of trivial or prime order. We prove that this geometric transfer coincides with Kottwitz-Shelstad transfer, adapted to the language of disconnected groups, assuming the local Langlands correspondence satisfying certain desiderata, including the Kazhdan-Lusztig hypothesis for the ambient group and its endoscopic groups. As an application of this geometrization, we prove Vogan's conjecture on local Arthur packets over p-adic fields: for quasisplit classical groups and their pure inner forms, all local Arthur packets are ABV-packets, and the distributions attached to these ABV-packets of Arthur type satisfy Arthur's local conjectures on twisted endoscopic character relations. This demonstrates the following general principle: for any reductive group G over a local field, an Arthur packet should be defined as an ABV-packet of Arthur type, equipped with the endoscopic pairing calculated by traces of actions on vanishing cycles or microlocalizations.