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Stabilization of the trace formula for metaplectic groups

Wen-Wei Li

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.

Stabilization of the trace formula 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 -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.

Paper Structure

This paper contains 163 sections, 256 theorems, 1194 equations.

Table of Contents

  1. Introduction
  2. Invariant trace formula for metaplectic groups
  3. Stabilization: desiderata
  4. Refined matching
  5. Strategy of the proof
  6. Metaplectic features
  7. Structure of this work
  8. Acknowledgements
  9. Preliminaries
  10. General notations
  11. Basics
  12. Local and global fields
  13. Varieties and groups
  14. Linear algebra
  15. Combinatorics
  16. ...and 148 more sections

Key Result

Theorem 1.2.2

For all infinitesimal character $\nu$, we have as genuine invariant distributions on $\tilde{G}_V$. Consequently, for all $t \geq 0$.

Theorems & Definitions (671)

  • Definition 1.2.1
  • Theorem 1.2.2: infra. Theorem \ref{['prop:spectral-stabilization']}
  • Theorem 1.2.3: infra. Theorem \ref{['prop:spectral-stabilization-disc']}
  • Remark 1.3.1
  • Theorem 1.4.1: infra. Theorems \ref{['prop:stabilization-geom-LTF']}, \ref{['prop:stabilization-disc-LTF']}
  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Definition 2.2.1
  • Remark 2.2.2
  • ...and 661 more