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Exceptional supercuspidal representations in small residue characteristic

In residue characteristics 2 and 3, this paper extends the Reeder–Yu construction of epipelagic representations to produce new supercuspidal representations of higher depth, including examples that do not arise from the original Reeder–Yu framework.

Original authors: Yiannis Fam

Published 2026-03-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yiannis Fam

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 you are trying to understand the hidden rhythms of a massive, complex machine. In the world of mathematics, this machine is a group (a set of symmetries), and the "rhythms" are representations (ways the group can act on things).

Mathematicians have spent decades trying to map out every possible rhythm this machine can make. For a long time, they had a master blueprint called Yu's Construction. It was like a universal recipe that worked perfectly when the machine's "fuel" (the residue characteristic $p$ ) was a large, clean number like 5, 7, or 11.

However, when the fuel is "dirty" or "small"—specifically the numbers 2 and 3—the master blueprint breaks down. It misses some very special, exotic rhythms. These missing rhythms are called exceptional supercuspidal representations. They are the "ghost notes" of the machine: rare, deep, and previously invisible to the standard recipe.

This paper, written by Yiannis Fam, is a guidebook for finding these ghost notes when the fuel is 2 or 3.

The Core Problem: The "Too-Simple" Recipe

Think of the standard recipe (Reeder–Yu construction) as a chef who only knows how to cook with fresh, whole ingredients.

The chef looks at the machine's structure and picks out the "top layer" of ingredients (mathematically, the quotient $G(k)_{x,+}/G(k)_{x,++}$ ).
If the ingredients are stable (they don't fall apart when shaken), the chef makes a dish (a representation).
The Problem: When the fuel is 2 or 3, the machine's "top layer" is too simple. It's like trying to bake a complex cake using only flour and water. The standard recipe says, "We can't make anything new here," because the ingredients look too basic.

But the author realizes: The machine is more complex than the top layer suggests.

The New Discovery: The "Hidden Basement"

The author, Yiannis Fam, decides to look deeper. Instead of just looking at the top layer, he digs into the basement of the machine's structure (mathematically, the abelianization of a slightly larger subgroup).

He finds that in the "small fuel" world (2 and 3), the basement is much bigger and more interesting than the top floor.

The Analogy: Imagine a building where the first floor looks like a plain, empty room. The standard recipe says, "Nothing interesting happens here." But the author climbs down a ladder to the basement and finds a secret garden full of exotic plants that the first floor was hiding.
The Innovation: He develops a new way to describe these hidden plants (characters of the group). He calls them $F_q$ -stable.
- Old Stability: "Is this plant strong enough to survive a hurricane?" (A very strict test).
- New $F_q$ -Stability: "Is this plant strong enough to survive a hurricane specifically in our local neighborhood?" (A slightly weaker, more practical test).
- Because the test is slightly easier, more plants pass! This allows the author to find rhythms that the old recipe missed.

The Three Special Cases (The Examples)

The author doesn't just talk about theory; he builds three specific examples to prove his method works:

The $G_2$ Machine with Fuel 2:
- He finds two new rhythms. Interestingly, these turn out to be the same as some rhythms found by the old recipe, but at a different location in the machine. It's like finding a hidden door that leads to a room you thought you already knew, but you entered it from a secret tunnel.
The $B_n$ Machine with Fuel 2:
- Here, he finds rhythms that are completely new. They are so deep and complex that the old recipe could never have found them. It's like discovering a new instrument in the orchestra that no one knew existed. These are "epipelagic" (a fancy word for "deep-sea") rhythms.
The $G_2$ Machine with Fuel 3:
- He finds a rhythm that is "halfway deep." The old recipe could only find rhythms that were "one-third deep" or "one-sixth deep." This new one is a unique depth that the old map didn't even have a coordinate for.

Why Does This Matter?

In the world of mathematics, specifically Number Theory and the Langlands Program (a grand theory connecting number puzzles to symmetry machines), these "ghost notes" are crucial.

Completing the Map: Just as explorers filled in the blank spots on old maps, this paper fills in the missing spots in our understanding of how these mathematical machines work when the numbers are small.
New Tools: The author provides a new algorithm (a set of instructions) to calculate these hidden structures. This is a toolkit that other mathematicians can now use to find even more hidden rhythms in other machines.
The "Formal Degree" Mystery: The paper also hints at a connection to the "formal degree" (a measure of how loud or significant a rhythm is). It suggests that these new rhythms have a specific "volume" that fits perfectly into a grand mathematical prediction called the Formal Degree Conjecture.

Summary in a Nutshell

Imagine a puzzle where the standard instructions only work for big, easy pieces. When the pieces are tiny and tricky (numbers 2 and 3), the instructions fail.

Yiannis Fam realized that the instructions were looking at the wrong part of the puzzle. By looking deeper into the "basement" of the structure and using a slightly more flexible rule for what counts as a "good piece," he found new, beautiful patterns that were previously invisible. He didn't just find them; he built a new set of instructions so anyone can find them too.

This is a breakthrough for understanding the fundamental symmetries of the universe when the math gets "small" and messy.

1. Problem Statement

The paper addresses the construction of supercuspidal representations of a connected reductive group $G(k)$ over a non-archimedean local field $k$ when the residue characteristic $p$ is small (specifically $p=2$ or $p=3$ ).

Context: For large $p$ , the construction of supercuspidal representations is largely exhausted by Yu's construction (extended to $p=2$ by Finzen and Stevens). However, in small residue characteristic, there exist irreducible supercuspidal representations that do not arise from Yu's construction.
Gap: Previous work by Reeder and Yu (2014) constructed a class of representations called epipelagic representations using "stable functionals" on the quotient $G(k)_{x,+}/G(k)_{x,++}$ (where $x$ is a point in the Bruhat-Tits building). This construction relies on Geometric Invariant Theory (GIT) stability.
Limitation: In small characteristic ( $p=2, 3$ ), the abelianization of the subgroup $G(k)_{x,+}$ is larger than the quotient $G(k)_{x,+}/G(k)_{x,++}$ . The Reeder-Yu construction, which only considers characters factoring through the latter quotient, fails to capture all possible epipelagic representations in these cases.
Goal: To extend the Reeder-Yu construction to higher depths by utilizing the full abelianization of $G(k)_{x,+}$ in small residue characteristic and to produce new examples of supercuspidal representations that are not covered by the original Reeder-Yu framework.

2. Methodology

The paper employs a combination of Bruhat-Tits theory, commutator relations in Lie algebras, and a refined stability criterion.

A. Abelianization of Compact Open Subgroups

The authors focus on the subgroup $G(k)_{x,+}$ , where $x$ is the barycentre of a facet in the Bruhat-Tits building.

Standard Case ( $p > 3$ ): The derived subgroup of $G(k)_{x,+}$ is contained in $G(k)_{x,++}$ , meaning the abelianization is simply $G(k)_{x,+}/G(k)_{x,++}$ .
Small Characteristic ( $p=2, 3$ ): The derived subgroup is strictly smaller. The authors explicitly compute the abelianization $V(x) = G(k)_{x,+}^{\text{ab}} / G(k)_{x,1}$ .
Algorithm: They develop an algorithm to determine the structure of $V(x)$ $V (x)$ as an $\mathbb{F}_q$ $F_{q}$ -vector space. This involves analyzing Chevalley commutator relations and identifying specific affine roots $\psi$ $ψ$ and tuples of roots that generate the abelianization.
- For $p=2$ and $G \neq G_2$ , the abelianization includes elements corresponding to sums of roots (e.g., $\psi+\theta$ ) and specific tuples $(\psi, \theta)$ that commute modulo higher filtration terms.
- This yields a vector space $V(x)$ that is strictly larger than the space used in Reeder-Yu.

B. $F_q$ -Stability

The Reeder-Yu construction requires a functional $\lambda$ to be stable in the sense of Geometric Invariant Theory (GIT) under the action of the reductive quotient $G_x$ .

New Definition: The author introduces $F_q$ -stability. A functional $\lambda \in V(x)^\vee$ is $F_q$ -stable if the Hilbert-Mumford criterion holds for all one-parameter subgroups defined over $\mathbb{F}_q$ (i.e., elements of $\text{Hom}_{\mathbb{F}_q}(\mathbb{G}_m, G_x)$ ), rather than requiring stability over the algebraic closure.
Significance: This is a strictly weaker condition than GIT stability. In small characteristic, there exist functionals that are $F_q$ -stable but not GIT-stable. This allows for the construction of representations from a broader set of characters.

C. Compact Induction

The main construction involves taking a character $\chi_\lambda$ of $G(k)_{x,+}$ (inflated from a functional $\lambda \in V(x)^\vee$ ) and forming the compactly induced representation:
$\pi_\lambda = \text{ind}_{G(k)_{x,+}}^{G(k)} \chi_\lambda$
The paper proves that if $\lambda$ is $F_q$ -stable, this induced representation decomposes into a finite direct sum of irreducible supercuspidal representations.

3. Key Contributions

Generalized Construction: The paper extends the Reeder-Yu epipelagic construction to higher depths in residue characteristics 2 and 3 by utilizing the full abelianization of $G(k)_{x,+}$ .
New Stability Notion: The introduction of $F_q$ -stability provides a rigorous framework to identify valid characters in small characteristic where standard GIT stability fails.
Explicit Abelianization: The paper provides an explicit algorithm and structural description of the abelianization $V(x)$ for $p=2$ (excluding $G_2$ ) and $p=3$ , identifying specific root combinations that survive in the abelianization.
Counter-Examples to Reeder-Yu Exhaustiveness: The author constructs explicit examples of epipelagic representations that do not arise from the original Reeder-Yu construction (which relies on GIT stability and the smaller quotient).

4. Main Results

The paper presents three specific families of examples demonstrating the power of the new method:

Case 1: $G = G_2$ with $q=2$

Setup: $x$ is the barycentre of an alcove.
Result: The authors construct two irreducible supercuspidal representations.
Comparison: These representations are shown to be isomorphic to epipelagic representations constructed by Reeder-Yu from a different point in the building. This demonstrates that the new method can recover known representations from different geometric data.

Case 2: $G$ of type $B_n$ with $q=2$

Setup: $x$ is the barycentre of an alcove.
Result: The authors produce irreducible supercuspidal representations of depth $\frac{1}{2(n-1)}$ .
Novelty: For $n \geq 3$ $n \geq 3$ , at least one of these representations does not arise from the Reeder-Yu construction.
- Reasoning: Reeder-Yu representations at this point have depths of the form $m/2^n$ (where $m|n$ ). The constructed depth $\frac{1}{2(n-1)}$ does not match this form for $n \geq 3$ .
- Mechanism: These representations are obtained using an $F_2$ -stable functional that is not GIT-stable, arising from the action of $SL_2(\mathbb{F}_2)$ on the abelianization.

Case 3: $G = G_2$ with $p=3$

Setup: $x$ is the barycentre of a specific facet (not the alcove barycentre).
Result: An irreducible supercuspidal representation of depth $1/2$ is constructed.
Novelty: This representation does not arise from the Reeder-Yu construction.
- Reasoning: Reeder-Yu representations for $G_2$ at depth $1/2$ correspond to specific stable points. The authors use formal degree calculations to show that the constructed representation has a formal degree incompatible with any representation arising from the Reeder-Yu stable points at that depth.
- Mechanism: The construction relies on the specific structure of the abelianization in characteristic 3 and the $F_3$ -stability of the functional.

5. Significance

Completeness of Classification: This work highlights that the classification of supercuspidal representations in small residue characteristic is more complex than in large characteristic. The "exceptional" nature of these representations requires tools beyond the standard Yu/Reeder-Yu frameworks.
Deepening Epipelagic Theory: By showing that epipelagic representations can exist at higher depths and be constructed from non-stable (in the GIT sense) functionals, the paper expands the scope of what constitutes an epipelagic representation.
Langlands Program Implications: The paper briefly discusses the Langlands parameters associated with these new representations. It suggests that these representations correspond to "wild" parameters with specific Swan conductors (e.g., $b(\phi)=4$ for the $G_2, p=3$ case), offering new test cases for the formal degree conjecture and the local Langlands correspondence in small characteristic.
Methodological Advancement: The explicit computation of abelianizations and the definition of $F_q$ -stability provide a blueprint for future investigations into supercuspidal representations for other exceptional groups and small characteristics.

In summary, Yiannis Fam successfully demonstrates that in residue characteristics 2 and 3, the landscape of supercuspidal representations is richer than previously understood, necessitating a generalized construction based on the full abelianization of Moy-Prasad subgroups and a relaxed stability condition.