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The triality of the twisted discrete trace formula for PGSO(8)

Tuoping Du, Zhifeng Pen, Haoyang Wan

Abstract

In this paper, we establish the triality twisted trace formula for PGSO(8), including its discrete part, and obtain a coarse classification of its automorphic representations by combining the properties of triality. By comparing the standard trace formula for G_2 with the triality twisted trace formula for PGSO(8), we derive a corresponding coarse classification for automorphic representations of G_2. Specifically, we construct the triality-twisted elliptic endoscopic data for PGSO(8), and the elliptic endoscopic data for G_2. Based on these constructions and the general framework of trace formulas, we establish the relevant trace formulas. Utilizing the triality property of PGSO(8), we obtain a coarse classification of its automorphic representations, which in turn yields a coarse classification for those of G_2.

The triality of the twisted discrete trace formula for PGSO(8)

Abstract

In this paper, we establish the triality twisted trace formula for PGSO(8), including its discrete part, and obtain a coarse classification of its automorphic representations by combining the properties of triality. By comparing the standard trace formula for G_2 with the triality twisted trace formula for PGSO(8), we derive a corresponding coarse classification for automorphic representations of G_2. Specifically, we construct the triality-twisted elliptic endoscopic data for PGSO(8), and the elliptic endoscopic data for G_2. Based on these constructions and the general framework of trace formulas, we establish the relevant trace formulas. Utilizing the triality property of PGSO(8), we obtain a coarse classification of its automorphic representations, which in turn yields a coarse classification for those of G_2.
Paper Structure (18 sections, 29 theorems, 285 equations, 1 figure)

This paper contains 18 sections, 29 theorems, 285 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Triality
  3. Octonion algebra.
  4. Root system
  5. Clifford algebra,
  6. Spin groups
  7. Triality
  8. Triality and Arthur Parameters
  9. Triality for Octonion Algebras
  10. Elliptic Endoscopic Data
  11. Triality twisted elliptic endoscopic data
  12. Trace formula of $\operatorname{G}_{2}$
  13. The Triality Twisted Trace Formula for $\operatorname{PGSO}(8)$
  14. Expansion of discrete invariant distribution
  15. A-parameters
  16. ...and 3 more sections

Key Result

Theorem 1.0.1

If $F$ is a local field, and $G=\operatorname{G}_{2}$, then the elliptic endoscopic data of $G$ are and where $s_{3}=(-1,-1,1,-1,-1,1,1)$ and $s_{4}=(1,1,1,-1,-1,-1,-1)$.

Figures (1)

  • Figure 1: To define the Langlands correspondence

Theorems & Definitions (52)

  • Theorem 1.0.1
  • Theorem 1.0.2
  • Theorem 1.0.3
  • Theorem 1.0.4
  • Definition 2.3.1
  • Proposition 2.3.2
  • Definition 2.4.1
  • Definition 2.5.1
  • Remark 2.5.2
  • Theorem 2.5.3: Hurwitz
  • ...and 42 more