Stabilization of the trace formula for metaplectic groups

TL;DR

This paper extends the stabilization of the trace formula to metaplectic groups, enabling a deeper understanding of their automorphic spectrum through endoscopic transfer and stable formulas.

Contribution

It provides the first stabilization of the full trace formula for metaplectic groups, linking it to stable trace formulas of orthogonal groups and adapting existing strategies.

Findings

Decomposition of the invariant trace formula for metaplectic groups.

Stabilization of the local trace formula over local fields.

Establishment of endoscopic transfer for metaplectic groups.

Abstract

We stabilize the full Arthur-Selberg trace formula for the metaplectic covering of symplectic groups over a number field. This provides a decomposition of the invariant trace formula for metaplectic groups, which encodes information about the genuine $L^2$-automorphic spectrum, into a linear combination of stable trace formulas of products of split odd orthogonal groups via endoscopic transfer. By adapting the strategies of Arthur and Moeglin-Waldspurger from the linear case, the proof is built on a long induction process that mixes up local and global, geometric and spectral data. As a by-product, we also stabilize the local trace formula for metaplectic groups over any local field of characteristic zero.