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Triality and adjoint lifting for GL(3)

This paper utilizes the stable twisted trace formula for the triality automorphism to establish the adjoint lifting of cuspidal representations of GL(3) with a discrete series local component to GL(8), while characterizing their isobaric decompositions and exploring implications for Ramanujan bounds and the strong Artin conjecture.

Original authors: Wee Teck Gan

Published 2026-05-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Wee Teck Gan

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

Imagine the universe of numbers and symmetries as a vast, intricate library. In this library, there are special "books" called automorphic representations. These books contain deep secrets about how numbers behave, but they are written in a very difficult, abstract language.

This paper, written by Wee Teck Gan, is like a master key that helps us translate one specific type of book (from the GL3 section of the library) into another, more complex type of book (from the GL8 section).

Here is a breakdown of the paper's journey, using simple analogies:

1. The Magic Key: "Triality"

The paper starts with a concept called Triality. Imagine a special 8-sided die (representing a mathematical object called $Spin_8$ ). This die has a magical property: if you rotate it in a specific way, it looks exactly the same, but the numbers on the faces have swapped places in a cycle of three. This "rotation" is called the triality automorphism.

The author uses this magical rotation to connect different parts of the library. Specifically, it connects the world of GL3 (which deals with 3x3 matrices) to the world of GL8 (which deals with 8x8 matrices).

2. The Main Mission: The "Adjoint Lifting"

The paper's primary goal is to prove that we can take a "book" (a representation) from the GL3 section and successfully translate it into the GL8 section. This process is called Adjoint Lifting.

The Analogy: Think of GL3 as a small, 3-dimensional sculpture. The "Adjoint Lifting" is a machine that takes this sculpture and builds a much larger, 8-dimensional version of it, preserving all its essential symmetries.
The Result: The author proves that this machine works, provided the original sculpture has at least one very specific, "rigid" feature (a discrete series component) at one location. This ensures the translation doesn't fall apart.

3. The Detective Work: What Does the Translated Book Look Like?

Once we have translated the book from GL3 to GL8, the paper asks: What does this new book actually contain?

The author acts like a detective, analyzing the "contents" of the new 8-dimensional book. He discovers that the new book isn't just one solid block; it's often a mixture (called an isobaric sum) of smaller, simpler books.

Depending on how the original GL3 book was created, the new GL8 book falls into one of four categories:

The "Cubic" Case: If the original book came from a specific type of 3-part number system (a cubic field extension), the new book breaks apart into tiny, single-number pieces (GL1) and medium-sized pieces (GL3).
The "Non-Cubic" Case: If the original book came from a non-symmetric 3-part system, the new book breaks into a 2-part piece and a 6-part piece.
The "Self-Dual" Case: If the original book is its own mirror image, the new book breaks into a 3-part piece and a 5-part piece.
The "Pure" Case: If the original book is very unique and doesn't fit the other categories, the new book might remain a single, solid 8-part piece, or it might split into two 4-part pieces.

This is crucial because it tells mathematicians exactly how complex the new object is.

4. Solving a Mystery: The "Artin Conjecture"

The paper applies this new translation machine to a famous unsolved puzzle called the Strong Artin Conjecture. This conjecture asks: "Can every 3-dimensional symmetry pattern found in number theory be found in our library of automorphic books?"

The author shows that for a specific type of pattern called "Tetrahedral" (which looks like a pyramid with a triangular base), the answer is YES. Because we now know how to translate GL3 to GL8, we can prove that these specific pyramid-shaped patterns exist as valid books in the library.

5. The "Ramanujan" Bound: How "Noisy" Can the Numbers Be?

Finally, the paper tackles a question about the "volume" or "noise" of the numbers in these books. The Ramanujan-Petersson Conjecture predicts that the numbers in these books should be perfectly quiet (having a value of exactly 1).

While we can't prove they are perfectly quiet yet, the author uses the new translation machine to prove they are very quiet.

The Analogy: Imagine trying to guess the weight of a hidden object. Previous guesses said the object could be off by a certain amount. By using the new machine to look at the object from a different angle (the 8-dimensional view), the author tightens the guess.
The Result: The author proves that the "noise" in these numbers is much smaller than anyone previously knew. It's a significant improvement in our understanding of how close these numbers are to being perfect.

Summary

In short, this paper:

Proves a new way to translate mathematical objects from a 3-dimensional world to an 8-dimensional world using a magical "triality" rotation.
Classifies exactly what these new 8-dimensional objects look like (are they solid blocks or mixtures?).
Solves a specific case of a century-old puzzle about number patterns (the Artin Conjecture).
Improves the precision of our estimates for how "noisy" these mathematical numbers can be.

It is a work of pure mathematical architecture, building a bridge between two distant islands of knowledge and showing us exactly what lies on the other side.

Technical Summary: Triality and Adjoint Lifting for GL3

Problem Statement

This paper addresses the existence and structural properties of the adjoint lifting for cuspidal automorphic representations of $GL_3$ to $GL_8$ . Specifically, it seeks to establish the functorial lifting associated with the adjoint representation $\text{Ad}: GL_3(\mathbb{C}) \to GL_8(\mathbb{C})$ . While the lifting for $GL_2$ (Gelbart-Jacquet) is classical, the $GL_3$ case presents significant challenges due to the complexity of the target group and the lack of a direct trace formula application for the specific map.

The paper also investigates the isobaric decomposition of the resulting lifted representation $\text{Ad}(\pi)$ . It aims to classify the possible shapes of this decomposition (e.g., whether it remains cuspidal or splits into smaller cuspidal components) and to determine the conditions under which $\pi$ is an automorphic induction or a twist of a self-dual representation.

Methodology

The core methodology relies on the principle of triality applied to the group $PGSO_8$ and the stable twisted trace formula.

Triality Automorphism: The authors utilize the order-3 outer automorphism $\theta$ of the split group $PGSO_8$ (and its simply-connected cover $Spin_8$ ). This automorphism permutes the three 8-dimensional representations of $Spin_8$ .
Twisted Endoscopy: The paper identifies the twisted endoscopic groups associated with the pair $(PGSO_8, \theta)$ $(P GS O_{8}, θ)$ . Through Lie algebra computations involving the fixed points of $s \cdot \theta^\vee$ $s \cdot θ^{\lor}$ in the dual Lie algebra $\mathfrak{spin}_8$ $spin_{8}$ , the authors determine that the elliptic twisted endoscopic groups are:
- The exceptional group $G_2$
- $SL_3$
- $SO_4$
Functorial Composition: The adjoint lifting is constructed by composing known functorial liftings with the triality automorphism. The map $\tilde{\text{Ad}}: PGL_3(\mathbb{C}) \to Spin_8(\mathbb{C})$ $\tilde{Ad} : P G L_{3} (C) \to S p i n_{8} (C)$ is identified as the unique lift of the adjoint representation. The strategy involves:
- Establishing a twisted endoscopic lifting from $SL_3$ to $PGSO_8$ .
- Combining this with the pullback from $GL_3$ to $SL_3$ and the standard lifting from $PGSO_8$ to $SO_8$ (and subsequently to $GL_8$ ).
Stable Twisted Trace Formula: The existence of the lifting is proven using the stable twisted trace formula for $(PGSO_8, \theta)$ , as developed by Moeglin and Waldspurger. The authors isolate specific Hecke-Satake families to show that the spectral side of the formula detects the desired lifting.
Cuspidality Criteria: To analyze the decomposition of $\text{Ad}(\pi)$ , the paper employs L-function identities (specifically involving $\wedge^3$ and $\text{Sym}^2$ L-functions) and known cuspidality criteria for Rankin-Selberg and exterior square liftings (e.g., results by Kim, Shahidi, Ramakrishnan, Wang, Asgari, and Raghuram).

Key Contributions and Results

1. Existence of the Adjoint Lifting

Theorem 1.2: The paper proves the existence of the weak adjoint lifting $\text{Ad}(\pi)$ for a cuspidal representation $\pi$ of $GL_3$ over a number field, provided $\pi$ has a discrete series local component at some finite place $v_0$ .

Note: The authors note that the local condition can likely be removed via a more extensive analysis of the trace formula involving all twisted endoscopic groups, a task deferred to future joint work.

2. Classification of Isobaric Decompositions

The paper provides a comprehensive "cuspidality criterion" for $\text{Ad}(\pi)$ , categorizing the decomposition based on the origin of $\pi$ :

Automorphic Induction from Cubic Extensions:
- Galois Case: If $\pi$ is induced from a cyclic cubic extension, $\text{Ad}(\pi)$ contains $GL_1$ -summands (specifically, the cubic characters associated with the extension). The decomposition is of type $1+1+3+3$ or $1+8$ (sum of 8 cubic characters).
- Non-Galois Case: If $\pi$ is induced from a non-cyclic cubic extension, $\text{Ad}(\pi)$ contains a cuspidal $GL_2$ -summand. The decomposition is of type $2+6$ , $2+3+3$ , or $2+4$ .
Twist of Self-Dual Representations:
- If $\pi$ is a twist of a self-dual cuspidal representation (which corresponds to the Gelbart-Jacquet lift of a $GL_2$ form), $\text{Ad}(\pi)$ contains a cuspidal $GL_3$ -summand. The decomposition is of type $3+5$ .
General Case (Non-Induction, Non-Self-Dual):
- If $\pi$ $π$ is not an automorphic induction from a cubic extension and not a twist of a self-dual representation, $\text{Ad}(\pi)$ $Ad (π)$ is either:
  - Cuspidal: Occurring when the base change of $\pi$ to any quadratic extension is not an automorphic induction.
  - Sum of two $GL_4$ s: Occurring when the base change to a quadratic extension $K$ is an automorphic induction from four distinct non-cyclic cubic extensions of $K$ . In this case, $\text{Ad}(\pi)$ is the sum of two self-dual cuspidal $GL_4$ representations with non-trivial central characters.

3. L-Function Properties

The paper establishes that for the general case (where $\text{Ad}(\pi)$ is a discrete orthogonal L-parameter), the order of the pole at $s=1$ of the exterior cube L-function $L_S(s, \text{Ad}(\pi), \wedge^3)$ equals the number of summands in the isobaric decomposition of $\text{Ad}(\pi)$ .

4. Applications

Strong Artin Conjecture: The existence of the lifting implies the Strong Artin Conjecture for 3-dimensional "tetrahedral" Galois representations (Theorem 1.4). These are representations analogous to the tetrahedral 2-dimensional representations proven by Langlands.
Ramanujan-Petersson Bound: The paper derives a new bound for the Hecke eigenvalues of $GL_3$ cuspidal representations.
Theorem 1.5: For a cuspidal representation $\pi$ of $GL_3$ with a unitary central character and a discrete series local component, if $|\lambda| = q_v^a$ is an eigenvalue of the Satake parameter, then:
$|a| \leq \frac{1}{4} - \frac{1}{130}$
This improves upon previous bounds (e.g., Blomer-Brumley's $5/14$ ) by utilizing the properties of the adjoint lift to $GL_8$ and applying Luo-Rudnick-Sarnak bounds to the lifted representation.

Significance and Claims

The paper claims to be the first to explicitly construct the adjoint lifting for $GL_3$ using the principle of triality and the stable twisted trace formula. It highlights that the triality automorphism of $PGSO_8$ provides a powerful mechanism to relate $GL_3$ to $GL_8$ via the exceptional group $G_2$ and $SL_3$ .

The authors emphasize that the results provide a complete description of the fibers and image of the adjoint lifting in terms of automorphic induction and self-duality. This contributes to the understanding of the multiplicity and rigidity of cuspidal L-packets for $SL_3$ , a topic where multiplicity-one properties fail for $n \geq 3$ .

The paper modestly acknowledges that the removal of the "discrete series" local condition in the existence theorem relies on further technical work regarding the cancellation of terms in the trace formula, which is the subject of ongoing collaboration. However, the structural analysis of the lifting (Theorem 1.3) is presented as a complete classification assuming the lifting exists.