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

This paper establishes the triality twisted trace formula for PGSO(8) to derive a coarse classification of its automorphic representations, which subsequently yields a corresponding classification for the automorphic representations of G₂ through a comparative analysis of their respective trace formulas.

Original authors: Tuoping Du, Zhifeng Pen, Haoyang Wan

Published 2026-03-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tuoping Du, Zhifeng Pen, Haoyang Wan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Cosmic Detective Story

Imagine the universe of mathematics is filled with invisible, complex shapes called Groups. These groups describe symmetries—like how a snowflake looks the same when you rotate it, but on a much higher, abstract level.

Mathematicians want to understand the "music" these shapes play. This music is called Automorphic Representations. It's like trying to figure out every possible song a specific instrument can play.

This paper is about two specific instruments:

$G_2$ : A rare, exotic, and very rigid instrument (an "exceptional group").
$PGSOp8q$: A massive, complex machine with a special property called Triality.

The authors (Du, Peng, and Wang) are trying to figure out the songbook for the tiny instrument ( $G_2$ ) by using a giant, twisted mirror ($PGSOp8q$) to reflect the sound back at them.

1. The Magic of "Triality" (The Three-Way Mirror)

Most symmetries are simple. If you have a square, you can rotate it 90 degrees, and it looks the same. If you rotate it 180, it looks the same.

But the group $PGSOp8q$ is special. It has a property called Triality.

The Analogy: Imagine a three-sided prism. If you look at it from the front, you see a square. If you rotate it 120 degrees, you see a different square. Rotate it again, and you see a third square.
The Twist: In this mathematical world, these three "views" are actually the same object, just seen through different lenses. The paper uses a "Twisted Trace Formula," which is essentially a mathematical machine that takes a song played on one lens, twists it, and projects it onto the other two lenses.

The authors built a machine (the Triality Twisted Trace Formula) that connects these three views.

2. The Problem: The Exotic Instrument ( $G_2$ ) is Hard to Read

The group $G_2$ is the "stabilizer" of this triality.

The Analogy: Imagine the three-sided prism is spinning wildly. $G_2$ is the tiny, rigid core in the center that doesn't move while the prism spins around it. It's the anchor.
The Difficulty: Because $G_2$ is so small and rigid, it's incredibly hard to write down its songbook (classify its representations). The standard tools mathematicians use don't work well on it.

3. The Solution: The "Coarse Classification"

Since they can't read the songbook of $G_2$ directly, the authors decided to listen to the giant machine ($PGSOp8q$) instead.

The Strategy: They realized that the "songs" (representations) of the giant machine are related to the songs of the tiny core ( $G_2$ ).
The "Coarse" Classification: They didn't try to list every single note perfectly. Instead, they created a Coarse Classification.
- Analogy: Imagine you want to know what kind of music a specific jazz band plays. Instead of transcribing every single note of every solo, you just categorize the band by the types of instruments they use and the general "vibe" (the A-parameters).
- They found that the "vibes" of the giant machine ($PGSOp8q$) can be used to predict the "vibes" of the tiny core ( $G_2$ ).

4. The "Trace Formula": The Mathematical X-Ray

The Trace Formula is the main tool they use.

The Analogy: Think of it as an X-ray machine. You can't see the inside of the machine (the representations) directly. But if you send a sound wave (a function) through it, the way the sound bounces back (the "trace") tells you what's inside.
The Twist: Usually, you just bounce the sound off the object. Here, they "twist" the sound wave using the Triality symmetry before bouncing it. This allows them to see parts of the object that were previously hidden.

5. The "Endoscopic" Data: The Spy Network

To make the X-ray work, they needed to compare the giant machine to smaller, simpler machines called Endoscopic Groups (like $SL_3$ and $SO_4$ ).

The Analogy: Imagine trying to understand a complex city ($PGSOp8q$). You can't map the whole thing at once. So, you send spies to map the smaller neighborhoods ( $G_2$ , $SL_3$ , $SO_4$ ).
The authors mapped out exactly how these neighborhoods connect to the main city. They found that the "spies" from the neighborhood of $G_2$ are actually the ones that tell us the most about the center of the city.

6. The Result: A Map for the Future

The paper concludes with a major breakthrough:

They proved that the "songs" of the exotic $G_2$ group are actually just a subset of the "songs" of the giant $PGSOp8q$ group.
The Takeaway: They created a rough map (a coarse classification). It's not a perfect, high-definition map of every single street in the city of $G_2$ yet, but they have successfully identified which districts exist and how to get there.

In summary: The authors built a special, twisted mirror (Triality) to look at a giant, complex machine. By studying the reflections in that mirror, they were able to finally write down a rough guidebook for a tiny, mysterious instrument ( $G_2$ ) that had been too difficult to understand on its own. They didn't solve the whole puzzle, but they found the key to the door.

1. Problem Statement

The paper addresses the classification of automorphic representations for the exceptional Lie group $G_2$ and the adjoint group $PGSO(8)$. While Arthur's trace formula and endoscopic classification have been established for classical groups and $GL(n)$, the classification for exceptional groups remains challenging due to their complex root systems and lack of direct endoscopic relationships with general linear groups.

Specifically, the authors aim to:

Establish the triality twisted trace formula for $PGSO(8)$, including its discrete part.
Utilize the unique triality symmetry of the $D_4$ root system (which underlies $Spin(8)$ and $PGSO(8)$) to relate the automorphic spectrum of $PGSO(8)$ to that of $G_2$ .
Derive a coarse classification of automorphic representations for $G_2$ by comparing the standard trace formula of $G_2$ with the triality-twisted trace formula of $PGSO(8)$.

2. Methodology

The authors employ a sophisticated combination of Lie theory, representation theory, and the Langlands program framework:

Triality Symmetry: The core methodology relies on the outer automorphism group of the $D_4$ root system, which is isomorphic to the symmetric group $S_3$ . This "triality" permutes the three 8-dimensional representations of $Spin(8)$: the standard representation and the two half-spin representations. This symmetry descends to an automorphism $\theta$ of order 3 on $PGSO(8)$.
Twisted Trace Formula: The authors construct the trace formula for the twisted group $\tilde{G} = PGSO(8) \rtimes \langle \theta \rangle$ $\tilde{G} = P GS O (8) ⋊ ⟨ θ ⟩$ . This involves:
- Defining $\theta$ -twisted elliptic endoscopic data for $PGSO(8)$.
- Constructing the Langlands-Kottwitz-Shelstad transfers of test functions.
- Stabilizing the trace formula using the work of Moeglin, Waldspurger, and Arthur.
Endoscopic Comparison:
- The twisted endoscopic groups for $PGSO(8)$ are identified as $G_2$ , $SL(3)$, and $SO(4)$.
- The authors compare the twisted trace formula of $PGSO(8)$ with the standard trace formula of $G_2$ .
- They utilize the fact that $G_2$ is the stabilizer of the triality action on $PGSO(8)$.
Arthur Parameters: The classification is framed in terms of A-parameters (Arthur parameters). The authors analyze the set of $\theta$ -stable A-parameters for $PGSO(8)$ and map them to automorphic representations.

3. Key Contributions and Results

A. Construction of Twisted Endoscopic Data

The paper explicitly constructs the triality twisted elliptic endoscopic data for $PGSO(8)$.

Theorem 4.1.1: Identifies the twisted endoscopic groups for $PGSO(8) \rtimes \theta$ $P GS O (8) ⋊ θ$ as:
1. $G_2$ (associated with the identity element).
2. $SL(3)$ (associated with a specific semisimple element $s_3$ ).
3. $SO(4)$ (associated with a specific semisimple element $s_4$ ).
The paper also constructs the standard elliptic endoscopic data for $G_2$ itself, identifying $G_2$ , $PGL(3)$, and $SO(4)$ as its endoscopic groups (Theorem 4.1.2).

B. The Triality Twisted Trace Formula

The authors establish the full trace formula and its discrete part for $PGSO(8)$ twisted by triality.

Theorem 6.0.1: The twisted trace formula $I(f)$ is expressed as a sum of stable distributions on the endoscopic groups:
$I(f) = S^{G_2}(f_1) + \frac{1}{4}S^{SO(4)}(f_4) + \frac{1}{3}S^{SL(3)}(f_3)$
where $f_1, f_4, f_3$ are the transfers of the test function $f$ .
The discrete part $I_{disc, t}(f)$ is similarly expanded, providing a spectral decomposition involving intertwining operators and traces over the endoscopic groups.

C. Coarse Classification of Automorphic Representations

By comparing the trace formulas, the authors derive a classification for $G_2$ based on the parameters of $PGSO(8)$.

Theorem 7.2.7: Establishes a correspondence between the set of $O$ -stable A-parameters for $PGSO(8)$ (where $O = Out(PGSO(8)) \cong S_3$ ) and the set of $O$ -orbits of automorphic representations of $PGSO(8)$.
Theorem 9.0.3 (Main Result for $G_2$ ): The discrete automorphic spectrum of $G_2$ is non-zero only for parameters that arise from the triality-stable A-parameters of $PGSO(8)$. Specifically:
$L^2_{disc}(G_2(F) \backslash G_2(\mathbb{A})) = \bigoplus_{\psi \in \Psi_O(PGSO(8))} L^2_{disc, \psi}(G_2(F) \backslash G_2(\mathbb{A}))$
This implies that the automorphic representations of $G_2$ can be parametrized (coarsely) by the triality-stable A-parameters of $PGSO(8)$.

D. Technical Nuances

The paper distinguishes between stable and semi-stable A-parameters. A parameter is stable if all its irreducible components are $\theta$ -stable; otherwise, it is semi-stable (arising from lifts of $SO(4)$ parameters).
The authors note that while the correspondence is established, the spaces $L^2_{disc, \psi}$ may vanish for certain parameters $\psi$ , making this a "coarse" or "weak" classification pending further study of the stable multiplicity formula.

4. Significance

Bridging Exceptional Groups: This work provides a crucial link between the exceptional group $G_2$ and the classical-type group $PGSO(8)$ via triality. It demonstrates how the rich symmetry of $D_4$ can be exploited to solve classification problems for $G_2$ , which is otherwise difficult to handle directly.
Advancement of Trace Formula Theory: The paper successfully applies the stabilization of the twisted trace formula (a major achievement by Moeglin and Waldspurger) to the specific case of triality, a non-standard twist.
Foundation for Future Work: While the classification is currently "coarse" (some parameters may yield zero contribution), the paper lays the necessary groundwork for an explicit endoscopic classification of $G_2$ . It defines the correct set of parameters and the structural framework for future papers to determine the exact multiplicity formulas and non-vanishing conditions.
Methodological Innovation: The construction of explicit twisted endoscopic data for $PGSO(8)$ and the detailed analysis of the triality automorphism on the level of A-parameters and representations serve as a template for studying other groups with outer automorphisms.

In summary, the paper successfully establishes the triality twisted trace formula for $PGSO(8)$ and uses it to derive a parameterization of the discrete spectrum of $G_2$ , marking a significant step toward the full endoscopic classification of exceptional groups.