Automorphic Representations of Low Rank Groups

The following sections are included:

• PREFACE

• CONTENTS

Let F be a global field, F_v the completion at a place v, 𝔸 the ring of adèles of F. Let H, or H₀, be the F-group SL(2), and G the F-group PGL(3). This part studies the lifting (or correspondence) of automorphic forms of H(𝔸) = SL(2, 𝔸) to those of G(𝔸) = PGL(3, 𝔸). It provides an intrinsic definition of packets of admissible and automorphic representations of SL(2, F_v) and SL(2, 𝔸). This definition is not based on relations to representations of GL(2, F_v) and GL(2, 𝔸), but rather on character relations and the lifting. This approach applies to groups other than SL(n). The work establishes multiplicity one theorem for cuspidal representations of SL(2, 𝔸), proves rigidity theorem for packets of these, computes the multiplicity of a cuspidal representation in a packet of a cuspidal representation, and determines the self-contragredient admissible representations of PGL(3, F_v) and the self-contragredient automorphic representations of PGL(3, 𝔸). The lifting is compatible with the symmetric square (or adjoint) three-dimensional representation of the dual group of H in . It is defined by means of twisted character relations. It is studied here by means of comparison of orbital integrals and of twisted trace formulae.

The following sections are included:

• Summary

• Introduction

• Hecke algebra

• Norms

• Local lifting

• Orthogonality

The following sections are included:

• Fundamental lemma
  • SL(2) to tori

• Differential forms

• Matching orbital integrals

• Germ expansion

The following sections are included:

• Introduction

• Geometric side

• Analytic side

• Trace formulae

The following sections are included:

• Introduction

• The comparison

• Appendix: Mathematica program

The following sections are included:

• Approximation

• Main theorems

• Characters and genericity
  • Germs of twisted characters

The following sections are included:

• Introduction

• Proof of theorem, anisotropic case

• Proof of theorem, isotropic case

The following sections are included:

• Functorial overview

• Statement of results

The following sections are included:

• Introduction

• Conjugacy classes

• Orbital integrals

• Fundamental lemma

• Admissible representations

• Representations of U(2,1;ℂ/ℝ)

• Fundamental lemma again

The following sections are included:

• Stable trace formula

• Twisted trace formula

• Restricted comparison

• Trace identity

• The σ-endo-lifting e′

• The quasi-endo-lifting e

• Unitary symmetric square

The following sections are included:

• Local identity

• Separation

• Specific lifts

• Whittaker models and twisted characters

• Global lifting

• Concluding remarks

The following sections are included:

• Statement of results

• The Zeta function

The following sections are included:

• The Shimura variety

• Decomposition of cohomology

• Galois representations

The following sections are included:

• Stabilization and the test function

• Functorial overview of basechange for U(3)

• Local results on basechange for U(3)

• Global results on basechange for U(3)

• Spectral side of the stable trace formula

• Proper endoscopic group

The following sections are included:

• The reflex field

• The representation of the dual group

• Local terms at p

• The eigenvalues at p

• Terms at p for the endoscopic group

The following sections are included:

• Representation of the real GL(2)

• Representations of U(2,1)

• Finite-dimensional representations

The following sections are included:

• Stable case

• Unstable case

• Nontempered case

The following sections are included:

• REFERENCES

• INDEX

• AUTOMORPHIC REPRESENTATIONS OF LOW RANK GROUPS