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Arthur's multiplicity formula for even orthogonal and unitary groups

Rui Chen, Jialiang Zou

Abstract

Let G be an even orthogonal or unitary group over a number field. Based on the same observation used in arXiv:1705.10106, we prove the Arthur's multiplicity formula for the generic part of the automorphic discrete spectrum of G by using the theta lift. We also consider a class of non-generic A-parameters and obtain a multiplicity formula in this case. In particular, we obtain a description of the full automorphic discrete spectrum of even orthogonal or unitary groups with Witt index less or equal to one.

Arthur's multiplicity formula for even orthogonal and unitary groups

Abstract

Let G be an even orthogonal or unitary group over a number field. Based on the same observation used in arXiv:1705.10106, we prove the Arthur's multiplicity formula for the generic part of the automorphic discrete spectrum of G by using the theta lift. We also consider a class of non-generic A-parameters and obtain a multiplicity formula in this case. In particular, we obtain a description of the full automorphic discrete spectrum of even orthogonal or unitary groups with Witt index less or equal to one.

Paper Structure

This paper contains 32 sections, 46 theorems, 339 equations.

Table of Contents

  1. Introduction
  2. Statement of results in generic case
  3. Near equivalence classes and Arthur parameters
  4. Local Langlands correspondence
  5. Multiplicity formula
  6. A special case
  7. Preliminaries
  8. Theta lifts
  9. Unitary representations of low rank
  10. Some inequalities
  11. Review of results for quasi-split groups
  12. Remarks on Whittaker data
  13. Weak transfer to the general linear group
  14. Attaching Arthur parameters
  15. A key equality: Generic case
  16. ...and 17 more sections

Key Result

Theorem 2.1

There exists a decomposition where $L^2_\psi(G)$ is a full near equivalence class of irreducible representations $\pi$ in $L^2_{\mathop{\mathrm{disc}}\nolimits}(G)$ such that the $L$-parameter of $\pi_v$ is $\phi_{\psi_v}$ for almost all places $v$ of $F$.

Theorems & Definitions (95)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Proposition 2.8
  • Theorem 3.1
  • Remark 3.2
  • ...and 85 more