Arthur packets for metaplectic groups

TL;DR

This paper defines Arthur packets for metaplectic groups over local and number fields, establishing character relations and a multiplicity formula for automorphic spectra, confirming conjectures and extending prior work.

Contribution

It introduces a new definition of Arthur packets for metaplectic groups and derives a multiplicity formula for automorphic spectra, extending Gan-Ichino's results and confirming a conjecture of Gan.

Findings

Arthur packets characterized by endoscopic relations

Multiplicity formula for automorphic spectrum established

Compatibility with known results in low rank cases

Abstract

For metaplectic groups over a local field of characteristic zero, we define the Arthur packet attached to any Arthur parameter $\psi$ as a multi-set of unitary genuine irreducible representations, characterized by endoscopic character relations. Over number fields, we obtain a multiplicity formula for the genuine discrete $L^2$-automorphic spectrum in terms of global Arthur parameters and $\epsilon$-factors, by leveraging the trace formula for metaplectic groups. This confirms a conjecture of Gan, and extends earlier results of Gan-Ichino on the Shimura-Waldspurger correspondences, whereas their works play a critical role in our proof. Furthermore, all these are shown to be compatible with existing results in rank one (Waldspurger) and two (Gan-Ichino).