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Triality and adjoint lifting for GL(3)

Wee Teck Gan

TL;DR

The paper leverages the stable twisted trace formula for triality on PGSO8 to establish a weak adjoint lifting Ad from GL3 to GL8 for cuspidal GL3 representations with a discrete series local component. It identifies the elliptic endoscopic groups (G2, SL3, SO4) and analyzes both local and unramified transfer, obtaining a detailed description of possible isobaric decompositions of Ad(π) across several scenarios (automorphic induction from cubic fields, self-dual twists, and mixed cases). The work provides a concrete improvement toward the Ramanujan–Petersson bounds for GL3, proving that at unramified places the Satake eigenvalue growth satisfies |a| ≤ 21/65, and it outlines a program for endoscopic classification of discrete automorphic representations of G2. The results illuminate the interplay between triality, twisted endoscopy, and adjoint lifting, and set the stage for broader endoscopic classifications and sharper bounds in higher rank cases.

Abstract

Using the stable twisted trace formula for the triality automorphism, we show the adjoint lifting (to GL(8)) of cuspidal representations of GL(3) with a discrete series local component. We also describe the possible isobaric decompositions of the resulting automorphic representations on GL(8) and give an application towards Ramanujan bounds for GL(3).

Triality and adjoint lifting for GL(3)

TL;DR

The paper leverages the stable twisted trace formula for triality on PGSO8 to establish a weak adjoint lifting Ad from GL3 to GL8 for cuspidal GL3 representations with a discrete series local component. It identifies the elliptic endoscopic groups (G2, SL3, SO4) and analyzes both local and unramified transfer, obtaining a detailed description of possible isobaric decompositions of Ad(π) across several scenarios (automorphic induction from cubic fields, self-dual twists, and mixed cases). The work provides a concrete improvement toward the Ramanujan–Petersson bounds for GL3, proving that at unramified places the Satake eigenvalue growth satisfies |a| ≤ 21/65, and it outlines a program for endoscopic classification of discrete automorphic representations of G2. The results illuminate the interplay between triality, twisted endoscopy, and adjoint lifting, and set the stage for broader endoscopic classifications and sharper bounds in higher rank cases.

Abstract

Using the stable twisted trace formula for the triality automorphism, we show the adjoint lifting (to GL(8)) of cuspidal representations of GL(3) with a discrete series local component. We also describe the possible isobaric decompositions of the resulting automorphic representations on GL(8) and give an application towards Ramanujan bounds for GL(3).

Paper Structure

This paper contains 53 sections, 27 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Triality
  3. Symmetric composition algebras and ${\rm Spin}_8$
  4. Splitting of ${\rm Aut}$-sequence.
  5. Representations
  6. Chevalley-Steinberg épinglage.
  7. Twisted Endoscopic Groups
  8. Lie algebra computations.
  9. Elliptic endoscopic data.
  10. Twisted Levi subspaces
  11. Local Twisted Endoscopic Transfer
  12. Local K-forms
  13. Twisted orbital integrals
  14. Stable orbital integrals
  15. Local transfer
  16. ...and 38 more sections

Key Result

Theorem 1.2

Assume that $\pi$ is a cuspidal representation of ${\rm GL}_3$ over a number field, such that for some finite place $v_0$, the local component $\pi_{v_0}$ is a discrete series representation. Then the weak adjoint lifting ${\rm Ad}(\pi)$ exists as an automorphic representation of ${\rm GL}_8$.

Theorems & Definitions (46)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.2
  • Corollary 2.3
  • Proposition 3.2
  • proof
  • Proposition 4.6
  • proof
  • Proposition 4.7
  • ...and 36 more