Pro-$p$ Iwahori-Hecke modules in semisimple rank one and singularity categories

This paper establishes an equivalence between the homotopy category of Hovey's Gorenstein projective model structure on pro-$p$ Iwahori-Hecke modules for semisimple rank one groups and the singularity category of an explicit scheme, thereby recovering Grosse-Klönne's mod-$p$ Langlands correspondence for $\mathrm{GL}_2$ and providing a complete explicit description for $\mathrm{SL}_2$ and $\mathrm{PGL}_2$.

The Gist

Imagine you are trying to understand the hidden structure of a massive, complex machine (the group $G$) by listening to the sounds it makes when you tap it with different hammers. In the world of mathematics, this machine is a group of symmetries related to numbers, and the "sounds" are mathematical objects called modules.

This paper, written by Nicolas Dupré, is about figuring out exactly what these sounds look like when the machine is relatively simple (specifically, for groups like $GL_2$, $SL_2$, and $PGL_2$) and when the "tapping" happens in a very specific, strange mathematical universe called characteristic $p$ (think of it as a world where counting wraps around after a prime number, like 2, 3, or 5).

Here is the story of the paper, broken down into simple concepts:

1. The Two Worlds: The Machine and the Map

The paper connects two very different ways of looking at the same problem:

• The Algebra World (The Machine): This is the "Hecke Algebra." Imagine a giant library of instructions (functions) that tell you how to move parts of the machine. Mathematicians study the "modules" (the things these instructions act upon). Some of these modules are "supersingular"—they are the most exotic, weird, and interesting ones.
• The Geometry World (The Map): This is the "Singularity Category." Imagine a landscape made of hills and valleys. Some parts of this landscape are smooth, but others have sharp, jagged points called singularities (like the tip of a star or the crossing point of two roads).

The Big Discovery: Dupré proves that the "weird sounds" from the Algebra World are actually identical to the "jagged points" on the Geometric Map.

• Analogy: It's like realizing that the static noise on an old radio is actually a perfect, encoded map of a mountain range. If you know the shape of the mountains, you know the static, and vice versa.

2. The "Gorenstein" Filter

To make this connection, the author uses a special filter called a Gorenstein model structure.

• Analogy: Imagine you have a bucket of sand mixed with pebbles. You want to study the pebbles, but the sand is getting in the way. The "Gorenstein" filter is a magical sieve that removes all the "easy" pebbles (those with finite complexity) and only lets the "infinite" or "eternal" pebbles through.
• The paper studies what happens when you only look at these eternal pebbles. This filtered view is called the Homotopy Category ($Ho(HG)$).

3. The Three Cases: $GL_2$, $SL_2$, and $PGL_2$

The author solves the puzzle for three specific versions of the machine.

• Case A: $GL_2$ (The General Linear Group)

  • This is the most "standard" version.
  • The Result: The filtered algebra world is perfectly equivalent to the singularity category of a specific scheme (a geometric object) built by Dotto, Emerton, and Gee.
  • Why it matters: This recovers a famous result called the Mod-p Langlands Correspondence.
  • Analogy: Think of this as finding a perfect key that unlocks a door. The key (the algebra) fits the lock (the Galois representations) exactly. The paper shows that the "static" on the radio is actually a perfect map of the terrain on the other side of the door.

• Case B: $PGL_2$ (The Projective Linear Group)

  • This is a slightly simpler version where some symmetries are ignored.
  • The Result: It behaves very similarly to $GL_2$. The map is still a perfect match.

• Case C: $SL_2$ (The Special Linear Group)

  • This is the tricky one. It has a "twist" (a sign character) that the others don't.
  • The Result: The map isn't a perfect 1-to-1 match anymore. It's like a many-to-one relationship. Several different "sounds" from the algebra map to the same "jagged point" on the geometric map.
  • Analogy: Imagine a group of twins (the algebra modules) walking into a room. In the $GL_2$ case, every twin has a unique face. In the $SL_2$ case, some twins look identical to the person watching from the window (the geometric map). The paper explains exactly how these "twins" are grouped together (these are called L-packets).

4. The "Chain of Projective Lines"

The geometric maps the author builds look like chains of projective lines (think of them as infinite loops or circles) glued together at specific points.

• Visual: Imagine a string of beads. Some beads are smooth circles, but where they touch, they form a sharp "X" shape (a singularity).
• The paper shows that the "eternal" algebra modules live exactly at these sharp "X" intersections. If you move away from the intersection, the module becomes "boring" (finite complexity) and disappears from our filtered view.

5. The "Spherical Module" (The Hero of the Story)

To prove all this, the author uses a special tool called the Spherical Module.

• Analogy: Think of this as a "universal translator" or a "master key." It's a specific mathematical object that knows how to speak both the language of the Algebra and the language of the Geometry. By using this key, the author translates the complex algebraic problems into geometric ones, solves them, and translates them back.

6. Why Should You Care?

This paper is a piece of the Langlands Program, which is often called the "Grand Unified Theory of Mathematics." It tries to connect:

1. Number Theory (how numbers behave, specifically Galois representations).
2. Harmonic Analysis (how symmetries and waves work, specifically Hecke algebras).

The Takeaway:
Dupré has shown that for these specific groups, the "weird, infinite" behavior of the number-theoretic symmetries is exactly the same as the "jagged, singular" geometry of a specific shape.

• If you understand the shape of the jagged mountain, you understand the behavior of the numbers.
• If you understand the numbers, you can draw the map of the mountain.

The paper also clarifies that while this connection is perfect for some groups ($GL_2$), for others ($SL_2$), the connection is a bit "fuzzy," grouping multiple number-theoretic objects into single geometric points. This "fuzziness" is not a mistake; it's a fundamental feature of how these symmetries work, and the paper maps it out with precision.

In a nutshell: The paper builds a bridge between a noisy radio (algebra) and a jagged mountain range (geometry), proving they are two sides of the same coin, and explaining exactly how the static noise encodes the shape of the peaks.

Technical Summary

1. Problem Statement and Context

The paper investigates the structure of the category of modules over the pro-$p$ Iwahori-Hecke algebra ($H_G$) associated with split reductive groups $G$ of semisimple rank one over a non-archimedean local field $F$ (specifically $G = GL_2(F), SL_2(F), PGL_2(F)$).

• The Object of Study: The focus is on the homotopy category $Ho(H_G)$, derived from Hovey's Gorenstein projective model structure on the category of $H_G$-modules. This category is obtained by localizing the module category $Mod(H_G)$ by trivializing all modules of finite projective dimension.
• The Goal: To provide an explicit description of $Ho(H_G)$ and relate it to singularity categories of specific schemes parametrizing semisimple Galois representations. This aims to lift the known mod-$p$ Langlands correspondences (specifically the bijection between supersingular Hecke modules and Galois representations established by Große-Klönne) to an equivalence of triangulated categories.
• The Challenge: While the center $Z(H_G)$ relates to Galois representations, the homotopy category $Ho(H_G)$ contains more information than just the center. For groups like $SL_2$, the center fails to capture the full homotopy structure due to specific singularities in the algebra that are invisible to the center.

2. Methodology

The author employs a combination of representation theory, homological algebra, and algebraic geometry:

1. Model Category Theory: Utilizes Hovey's Gorenstein projective model structure. The homotopy category $Ho(H_G)$ is identified with the singularity category of the ring $H_G$ (via the equivalence between Gorenstein projective modules and acyclic complexes of injectives).
2. Bruhat-Tits Theory and Resolutions: Uses the Ollivier-Schneider resolution to reduce the study of $Ho(H_G)$ to the homotopy categories of finite Hecke algebras associated with maximal compact subgroups ($H_{x_0}$). This reduction relies on the fact that the model structure on $Mod(H_G)$ is a "right transfer" of the product model structures on the faces of the Bruhat-Tits building.
3. Block Decomposition: Decomposes the Hecke algebra $H_G$ using central idempotents $e_\gamma$ corresponding to orbits of characters of the finite torus $T(\mathbb{F}_q)$. This separates the algebra into regular and non-regular components.
4. Explicit Algebraic Computation:
   • Analyzes the structure of the algebras $e_\gamma H_{x_0}$ and $e_\gamma H_G$.
   • For $GL_2$ and $PGL_2$, shows that regular components are matrix algebras over their centers.
   • For $SL_2$, identifies a unique non-regular component (the "sign" character) where the algebra is not a matrix algebra over its center, requiring a larger central subalgebra $\tilde{Z}$ to capture the structure.
5. Geometric Construction: Constructs explicit schemes $X_{q,G}$ (chains of projective lines) whose singular loci correspond to the singularities of the Hecke algebras.
6. DGAs (Differential Graded Algebras): For the $GL_2$ case, constructs a generator for the homotopy category to realize it as the derived category of an explicit DGA.

3. Key Contributions and Results

A. Explicit Description of $Ho(H_G)$ (Theorem A)

The paper provides a complete decomposition of the homotopy category for $G = SL_2, PGL_2, GL_2$:

• For $PGL_2$ ($p>2$): $Ho(H_{PGL_2})$ is equivalent to a product of copies of $Mod(k^2)$ (the category of vector spaces over $k^2$), indexed by regular Weyl group orbits.
• For $SL_2$ ($p>2$): $Ho(H_{SL_2})$ is equivalent to $C \times \prod_{\gamma \text{ reg}} (Mod(k^2) \times Mod(k^2))$.
  • Here, $C = Mod(k) \times Mod(k)$ corresponds to the "sign" character component, which is non-trivial only when $p>2$.
  • The factor $Mod(k^2) \times Mod(k^2)$ arises because the regular components of $H_{SL_2}$ are not matrix algebras over their centers, unlike the $GL_2$ case.
• Significance: This reveals that for $SL_2$, the homotopy category is strictly larger than what the center $Z(H_{SL_2})$ suggests, as the center is a polynomial ring (regular) for the sign component, while the algebra itself has infinite global dimension.

B. Equivalence with Singularity Categories (Theorem B)

The author establishes a functor $L_{G,*}: Ho(H_G) \to Sing(X_{q,G})$ relating the Hecke modules to the singularity category of a scheme $X_{q,G}$:

• $G = GL_2$ and $PGL_2$: The functor is an equivalence of triangulated categories.
  • $X_{q, GL_2}$ is the scheme introduced by Dotto-Emerton-Gee and Pépin-Schmidt (a chain of projective lines).
  • This equivalence lifts the Große-Klönne bijection: simple supersingular modules of infinite projective dimension correspond bijectively to singular points of $X_{q,G}$.
• $G = SL_2$: The functor is not an equivalence but a well-defined map.
  • The map on objects is surjective onto the singular points of $X_{q, SL_2}$.
  • The fibers of this map correspond to L-packets of simple supersingular modules (pairs of modules mapping to the same geometric point).
  • This geometrically explains why the mod-$p$ Langlands correspondence for $SL_2$ is not a bijection but involves L-packets.

C. DGA Description for $GL_2$ (Section 5)

For $G = GL_2$, the author identifies a specific generator (the "Gorenstein projective spherical module") and proves that $Ho(H_{GL_2})$ is triangle equivalent to the derived category $D(\tilde{A})$ of an explicit Differential Graded Algebra $\tilde{A}$.

• The endomorphism ring of a simple supersingular module in $Ho(H_{GL_2})$ is shown to be isomorphic to the finite Hecke algebra $H_{x_0}$ (specifically the regular component).

4. Significance and Impact

1. Geometric Interpretation of Homotopy Categories: The paper successfully translates abstract homological properties of Hecke algebras into geometric language (singularity categories of schemes). This provides a new framework for understanding the "shape" of the mod-$p$ Langlands program.
2. Resolution of the $SL_2$ Case: It clarifies the discrepancy between the center of the Hecke algebra and the full module category for $SL_2$. The construction of the scheme $X_{q, SL_2}$ (which includes an extra $\mathbb{P}^1$ compared to previous constructions by Ardakov-Schneider) is necessary to fully capture the homotopy category, specifically the "sign" component.
3. Lifting the Langlands Correspondence: The results show that the classical bijection between supersingular modules and Galois representations is the "shadow" of a deeper equivalence between triangulated categories. The failure of this equivalence for $SL_2$ (resulting in L-packets) is now seen as a geometric phenomenon related to the specific singularities of the scheme.
4. Computational Tools: The explicit descriptions of $Ho(H_G)$ as products of simple categories ($Mod(k^2)$) and the DGA description for $GL_2$ provide powerful tools for future computations in the mod-$p$ Langlands program, particularly for calculating extensions and derived structures.

In summary, Dupré's work bridges the gap between the algebraic structure of pro-$p$ Iwahori-Hecke algebras and the geometry of Galois representation moduli spaces, offering a unified categorical perspective on the mod-$p$ Langlands correspondence for rank one groups.