An analogue of irreducible cuspidal representations for the group $PGL(2)$ over a two-dimensional local field

This paper constructs an analogue of irreducible cuspidal representations for $PGL(2)$ over a two-dimensional local field $K=F((t))$ using quadratic extensions and non-Galois-invariant characters, demonstrating that while the construction parallels the classical case, the resulting representations exhibit distinct behavior upon restriction to the Borel subgroup and motivating a generalized definition of cuspidality for split reductive groups.

The Gist

The Big Picture: A New Kind of Music

Imagine you are a music theorist studying a specific type of instrument, the PGL(2) group. For a long time, mathematicians have studied how this instrument sounds when played in a "standard" setting (a field called $F$, like the p-adic numbers). They know exactly what the "purest" notes are. In this world, these pure notes are called irreducible cuspidal representations.

Think of these "pure notes" as a specific, unique melody that cannot be broken down into smaller, simpler tunes. If you try to play this melody on a smaller, simpler instrument (a subgroup called $P$), it still sounds like one single, unbreakable note.

The Problem:
The authors of this paper ask: "What happens if we change the instrument?" Instead of playing in the standard setting ($F$), they want to play in a much more complex, layered setting called a Two-Dimensional Local Field ($K = F((t))$).

Imagine the standard field $F$ is a flat, 2D sheet of paper. The new field $K$ is like a stack of infinite sheets of paper, or a 3D block of time and space. It's a much more complicated environment.

The authors want to find the "pure notes" (cuspidal representations) in this new, complex 3D world.

The Main Discovery: Similar but Different

The paper is essentially a detective story. The authors try to build these "pure notes" in the new 3D world using a recipe they know works in the 2D world.

1. The Recipe (The Construction)
In the old world, you make a pure note by taking a "quadratic extension" (a specific type of mathematical doubling, like adding a square root) and a "character" (a specific pattern or rhythm).

• The Analogy: Imagine you have a secret code (the character) and a special key (the quadratic extension). If you lock them together correctly, you get a unique, unbreakable melody.

The authors show that you can do the exact same thing in the new 3D world. You take a quadratic extension of the new field and a character, and you build a representation.

2. The Surprise (The Twist)
Here is where the story gets interesting. In the old 2D world, if you take your pure note and play it on the smaller instrument ($P$), it sounds exactly like the only possible pure note that instrument can make. It's unique. Everyone agrees on what it sounds like.

But in the new 3D world, things are weird.

• The Analogy: Imagine you have a unique melody. In the old world, if you play it on a small flute, everyone agrees it's "The Flute Song."
• In the new world, the authors find that while their new "pure notes" are still unbreakable when played on the small flute, they don't all sound the same.
• Some sound like "The Flute Song," but others sound like a slightly different, yet still unbreakable, melody.
• The Result: The restriction of these new representations to the smaller group is irreducible (it's still one solid note), but it is not isomorphic (it's not identical) to the "standard" note everyone expected.

This is a major difference. In the old world, there was only one way to be "pure" on the small instrument. In the new world, there are many different ways to be pure, and they depend on the "depth" or complexity of the original note.

The "Depth" Concept

The authors introduce a concept called Depth.

• Analogy: Think of the field $K$ as a deep ocean.
  • Depth 0: You are on the surface.
  • Depth 1: You are one meter down.
  • Depth 2: You are two meters down.
• The "pure notes" they construct have different depths.
• In the old world, depth didn't really matter for the final sound on the small instrument.
• In the new world, the depth determines the sound. A note from depth 1 sounds different from a note from depth 2 when played on the small instrument.

The "Associated Graded" Trick

The authors realize that while the notes sound different, they are all related.

• Analogy: Imagine you have a complex, multi-layered cake. If you look at the whole cake, it looks different from a simple sponge cake. But if you take a slice and look at the layers one by one (the "associated graded"), you see that every layer is actually made of the same basic sponge.
• The authors prove that if you "peel back" the layers of their new, complex 3D representations, the underlying structure (the "sponge") is actually the standard, expected melody.
• So, the new notes are extensions of the old notes. They are more complex, but they are built on the same foundation.

Why Does This Matter?

This paper is important because it maps out a new territory in mathematics.

1. It confirms the map: It shows that the "pure notes" (cuspidal representations) exist in this complex 3D world, just like they do in the 2D world.
2. It reveals new terrain: It shows that the rules are slightly different. The "uniqueness" we relied on in the old world is gone. We have to be more careful.
3. It opens the door: By showing how to build these notes, the authors provide a toolkit for other mathematicians to explore even larger, more complex groups (like $PGL(n)$) in these strange 2D fields.

Summary in One Sentence

The authors successfully built "pure, unbreakable musical notes" for a complex, 3D mathematical world, discovering that while these notes work beautifully, they come in many different "flavors" depending on their depth, unlike the single, unique flavor found in the simpler, 2D world.

Technical Summary

1. Problem Statement and Context

The Core Problem:
The paper addresses the representation theory of the group $G(K) = \text{PGL}(2, K)$, where $K = F((t))$ is a two-dimensional local field (a local field of Laurent series over a local non-archimedean field $F$). The authors seek to construct and classify irreducible cuspidal representations of $G(K)$, which serve as the two-dimensional analogue of the well-understood cuspidal representations of $G(F)$.

The Challenge:
In the classical case ($G(F)$), irreducible cuspidal representations are characterized by the property that their restriction to the Borel subgroup $P(F)$ is irreducible. Furthermore, all such restrictions are isomorphic to a unique irreducible cuspidal representation of $P(F)$.
However, for $G(K)$, the situation is more complex:

1. The category of smooth representations involves pro-vector spaces rather than standard vector spaces.
2. The restriction of a cuspidal representation of $G(K)$ to the Borel subgroup $P(K)$ is irreducible, but not unique (unlike the classical case).
3. There is no direct analogue of the Jacquet functor (coinvariants with respect to the unipotent radical) that works in the same way for $K$.

Goal:
To construct a class of "special" representations of $G(K)$ (analogous to cuspidal representations) starting from characters of quadratic extensions of $K$, and to analyze their properties, specifically their restriction to $P(K)$.

---

2. Methodology and Framework

The authors employ a multi-layered approach combining algebraic geometry, the theory of pro-algebraic groups, and Mackey theory adapted to infinite-dimensional settings.

A. Mathematical Setting

• Fields: $F$ is a local non-archimedean field of odd residue characteristic. $K = F((t))$ and $O = F[[t]]$.
• Groups:
  • $G = \text{PGL}(2)$.
  • $P$: The Borel subgroup (upper triangular matrices).
  • $G(O)$ and $G'(O)$: Pro-algebraic groups defined as stabilizers of specific lattices in the Lie algebra. $G'(O)$ is the normalizer of the Iwahori subgroup $I$.
  • $G(K)$: The group of points over the 2D local field, treated as a group object in the category of ind-schemes.
• Categories: The authors work within the category of smooth representations on pro-vector spaces ($\text{Vect}$), as defined in their previous work [7]. This is necessary because $G(K)$ acts on infinite-dimensional spaces that are projective limits of finite-dimensional ones.

B. Key Definitions

1. Special Representations:
   • For $G(O)$ or $G'(O)$: An irreducible representation $\pi$ (dim $>1$) is special if its restriction to $P(O)$ is irreducible.
   • For $G(K)$: An irreducible representation $\Pi$ is special if its restriction to $P(K)$ is irreducible.
2. Elliptic Representations:
   • A representation is elliptic if its "support" (in the sense of the restriction to a congruence subgroup) corresponds to an elliptic conjugacy class in the Lie algebra $\mathfrak{g}(F)$.
   • The authors prove that for $G(O)$ and $G'(O)$, Special $\iff$ Elliptic.
3. Cuspidality for $G(K)$:
   • Defined via a filtration condition on the restriction to the unipotent radical of $P(K)$. A representation is cuspidal if a specific filtration (related to the depth of the representation) is dense.

C. Construction Strategy

The construction proceeds in two main stages:

1. Classification at the "Compact" Level ($G(O)$ and $G'(O)$):
   • Classify special/elliptic representations of the pro-algebraic groups $G(O)$ and $G'(O)$.
   • These are constructed from quadratic extensions $L/K$ and characters $\theta: L^*/K^* \to \mathbb{C}^*$ that are not Galois invariant ($\theta \neq \bar{\theta}$).
   • The classification distinguishes between unramified and ramified extensions of $K$.
2. Induction to $G(K)$:
   • Use a compact induction functor (adapted for ind-schemes and pro-vector spaces) to induce these special representations from $G(O)$ (or $G'(O)$) to $G(K)$.
   • Analyze the resulting representation $\Pi(\pi)$ on $G(K)$.

---

3. Key Contributions and Results

A. Classification of Special Representations of $G(O)$ and $G'(O)$

The paper establishes a bijection between special representations and specific characters of quadratic extensions:

• For $G'(O)$: Special representations correspond bijectively to pairs $(L, \theta)$ where $L/K$ is a ramified quadratic extension and $\theta$ is a character of $L^*/K^*$ with $\theta \neq \bar{\theta}$.
• For $G(O)$: Special representations correspond to pairs $(L, \theta)$ where $L/K$ is an unramified quadratic extension. The set of such representations forms a $\mathbb{Z}/2\mathbb{Z}$-torsor over the set of such pairs (due to subtleties involving quadratic characters).
• Depth: The depth of the representation is determined by the conductor of the character $\theta$.

B. Induction to $G(K)$ and Cuspidality

• Theorem 1.5: If $\pi$ is a special representation of $G(O)$ or $G'(O)$, then the induced representation $\Pi(\pi) = \text{ind}_{G(O)}^{G(K)} \pi$ is irreducible and cuspidal.
• Restriction to $P(K)$: The restriction $\Pi(\pi)|_{P(K)}$ is irreducible.
• Non-Uniqueness of Restriction: Unlike the classical case ($G(F)$), the restriction $\Pi(\pi)|_{P(K)}$ is not isomorphic to a single "standard" representation for all $\pi$. Instead, the restriction depends on the depth of $\pi$.
  • However, the authors show that all such restrictions admit a $\mathbb{Z}$-filtration.
  • The associated graded representation of this filtration is isomorphic to a specific "standard" irreducible cuspidal representation $\Upsilon$ of $P(K)$.
  • This graded structure is the primary difference between the representation theory of $G(K)$ and $G(F)$.

C. Non-Existence of Direct Extension

• Corollary 11.2: There is no irreducible representation of $G(K)$ whose restriction to $P(K)$ is isomorphic to the "standard" cuspidal representation $\Upsilon$ (the direct analogue of the unique $P(F)$ representation).
• This highlights a fundamental structural difference: the "standard" representation of $P(K)$ cannot be extended to $G(K)$, whereas in the 1D case, it can.

D. Appendix: Generalized Cuspidality

The authors propose a definition of cuspidality for smooth representations of $H(K)$ for any split reductive group $H$. A representation is cuspidal if the filtrations defined by the kernels of projections to coinvariants of unipotent radicals of all proper standard parabolic subgroups are dense.

---

4. Significance and Implications

1. Extension of Local Langlands: This work is a crucial step in extending the Local Langlands Correspondence to higher-dimensional local fields. It provides the first concrete construction of the "cuspidal" side of the correspondence for $G(K)$.
2. New Phenomena in 2D Fields: The paper reveals that the representation theory of groups over 2D local fields is not merely a trivial generalization of the 1D case. The failure of the uniqueness of the restriction to the Borel subgroup and the necessity of working with filtrations and associated graded objects represent a fundamental shift in the theory.
3. Pro-Vector Space Framework: The rigorous treatment of representations on pro-vector spaces and ind-schemes provides a necessary technical foundation for future work in geometric representation theory over higher-dimensional fields.
4. Connection to Double Affine Hecke Algebras: The authors note a connection to the double affine Hecke algebra ($H_{q,t}$), suggesting that these constructed representations may provide a geometric realization of modules over these algebras, linking number theory with algebraic geometry.

Summary of the "Story"

The paper constructs a family of irreducible representations for $\text{PGL}(2)$ over a 2D local field by inducing "special" representations from compact subgroups. These special representations are classified by characters of quadratic extensions. While the induced representations are irreducible and cuspidal, their behavior upon restriction to the Borel subgroup differs significantly from the classical 1D case: they do not all restrict to the same representation, but rather form a family whose "associated graded" pieces are all isomorphic to a standard model. This work establishes the existence and basic properties of these representations, laying the groundwork for a full classification and a potential higher-dimensional Langlands program.