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Tempered vs generic automorphic functions and the canonical filtration on automorphic functions

This paper introduces and studies a canonical filtration on the space of everywhere unramified automorphic functions over function fields, derived from the coherent singular support on the spectral side of the Langlands program, and proposes conjectures linking this cohomological filtration to one defined by the analytic spectrum of Hecke operators.

Original authors: Dennis Gaitsgory, Vincent Lafforgue, Sam Raskin

Published 2026-03-31

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CC BY 4.0

Original authors: Dennis Gaitsgory, Vincent Lafforgue, Sam Raskin

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 a massive, chaotic library filled with millions of books. These books represent automorphic functions—mathematical objects that encode deep secrets about numbers, shapes, and symmetries. For decades, mathematicians have been trying to organize this library.

This paper, written by three mathematical giants (Gaitsgory, Lafforgue, and Raskin), proposes a new, highly sophisticated way to sort these books. They don't just sort them by color or size; they sort them by the "shape of their shadows."

Here is the story of the paper, broken down into simple concepts and analogies.

1. The Two Sides of the Coin (The Langlands Connection)

To understand the library, you need to know about the Langlands Program. Think of this as a magical dictionary that translates between two different languages:

The Geometric Side (The Library): This is the world of shapes and bundles (called BunG). It's where the "automorphic functions" live.
The Spectral Side (The Blueprint): This is the world of symmetries and dual groups (called LS). It's where the "spectral data" lives.

The paper starts with a proven fact: There is a perfect one-to-one match between the books in the library and the blueprints. If you know the blueprint, you know the book, and vice versa.

2. The Problem: A Messy Library

The problem is that the library is messy. The books are piled up in a chaotic heap. Mathematicians want to find a specific type of book: the "Tempered" ones.

Tempered books are the "well-behaved," stable, and fundamental books. They are the ones that correspond to the most "pure" symmetries.
Generic books are the wild, chaotic ones that are harder to understand.

The big question is: How do we separate the Tempered books from the Generic ones?

3. The Solution: The "Shadow" Filtration

The authors introduce a new sorting method called a Filtration. Imagine you have a stack of objects, and you shine a light on them from different angles.

Some objects cast a simple, small shadow (like a dot).
Others cast a complex, jagged shadow (like a star or a cone).

In this paper, the "shadow" is called the Singular Support.

The Tempered books are the ones whose shadows are the simplest possible: a single point (the "zero" orbit).
The Generic books have more complex shadows.

The authors propose a system where you can peel away the library layer by layer. You start with the books that have the simplest shadows (the Tempered ones), then you remove them, and look at the next layer of complexity, and so on. This creates a Canonical Filtration—a perfect, natural order for the library.

4. The Big Discovery: The "Vacuum" Book

The paper's main result is about the very first layer of this filter: the Tempered layer.

They identify a specific, famous book in the library called the Vacuum Poincaré Function (let's call it the "Master Key").

The Discovery: They prove that if you take this "Master Key" and apply all the possible mathematical operations to it (using something called the "Excursion Algebra"), you generate exactly the entire collection of Tempered books.
The Analogy: Imagine you have a single seed (the Master Key). The authors prove that if you water it with the right nutrients (the Excursion Algebra), it grows into the entire forest of Tempered trees. You don't need to find the other trees individually; they are all hidden inside that one seed.

This is a huge deal because it gives a concrete way to find the "good" (Tempered) functions without having to guess.

5. The Rationality Conjecture: The "Rational" Blueprint

The authors also propose a guess (a conjecture) about the nature of these books.

They suspect that even though the math looks like it requires complex numbers (like $\sqrt{-1}$ ), the "Tempered" books are actually built from simple, rational numbers (like 1, 2, 3).
The Analogy: It's like discovering that a complex, swirling galaxy is actually made of the same simple Lego bricks that a child uses to build a house. They believe the "Tempered" layer is defined over the rational numbers, making it more fundamental and "real" than the chaotic layers above it.

6. The Ramanujan-Arthur Connection

Finally, the paper connects this new sorting method to two famous old problems in mathematics: the Ramanujan Conjecture and Arthur's Conjecture.

These conjectures are about predicting the "vibrations" (eigenvalues) of the books.
The authors suggest that their "Shadow Filtration" is the key to unlocking these old mysteries. If you sort the books by their shadows, the vibrations of the Tempered books turn out to be perfectly "pure" (related to Weil numbers), just as Ramanujan predicted.

Summary: What does this mean for the world?

This paper is like finding a new organizing system for the universe's code.

It organizes chaos: It provides a clear, step-by-step way to separate the stable, fundamental mathematical objects from the chaotic ones.
It finds the seed: It proves that the most important, stable objects can all be generated from a single, simple starting point (the Vacuum Poincaré function).
It bridges gaps: It connects the abstract world of geometry with the concrete world of number theory, suggesting that the "good" numbers are rational and well-behaved.

In short, the authors have handed us a new map to navigate the most complex library in mathematics, showing us exactly where the "treasure" (the Tempered functions) is hidden and how to dig it up.

This paper, authored by Dennis Gaitsgory, Vincent Lafforgue, and Sam Raskin, establishes a deep connection between the geometric Langlands program and the classical theory of automorphic forms over function fields. It introduces a canonical filtration on the space of automorphic functions derived from the "coherent singular support" on the spectral side and investigates its relationship with analytic properties (temperedness) and rationality.

Here is a detailed technical summary of the paper's problem, methodology, key contributions, results, and significance.

1. Problem Statement and Context

The paper operates within the framework of the Geometric Langlands Conjecture for a smooth complete curve $X$ over a finite field $\mathbb{F}_q$ and a reductive group $G$ .

The Geometric Equivalence: The authors rely on the established equivalence (from [GR]) between the category of sheaves on the moduli stack of $G$ -bundles with nilpotent singular support, $\text{Shv}_{\text{Nilp}}(\text{Bun}_G)$ , and the category of ind-coherent sheaves on the stack of restricted $\check{G}$ -local systems, $\text{IndCoh}_{\text{Nilp}}(\text{LS}^{\text{restr}}_{\check{G}})$ .
The Classical Limit: By taking the categorical trace of the Frobenius endomorphism on both sides, one obtains an isomorphism between the space of $\mathbb{Q}_\ell$ -valued automorphic functions, $\text{Funct}_c(\text{Bun}_G(\mathbb{F}_q), \mathbb{Q}_\ell)$ , and the global sections of the dualizing sheaf on the spectral stack:
$\text{Funct}_c(\text{Bun}_G(\mathbb{F}_q), \mathbb{Q}_\ell) \xrightarrow{\sim} \Gamma^{\text{IndCoh}}(\text{LS}^{\text{arithm}}_{\check{G}}, \omega_{\text{LS}^{\text{arithm}}_{\check{G}}})$
The Core Question: The spectral side possesses a natural filtration indexed by closed $\text{Ad}$ $Ad$ -invariant subsets of the nilpotent cone of the Lie algebra $\check{\mathfrak{g}}$ $\overset{ˇ}{g}$ , induced by the coherent singular support of sheaves. The paper asks:
1. What does this filtration look like on the automorphic side?
2. How does the "zero" term of this filtration (corresponding to the zero orbit) relate to the space of tempered automorphic functions?
3. Does this filtration respect rational structures ( $\mathbb{Q}$ vs. $\mathbb{Q}_\ell$ )?
4. How does this relate to the Arthur and Ramanujan-Arthur conjectures regarding the spectral decomposition of $L^2$ spaces?

2. Methodology

The authors employ a blend of derived algebraic geometry, categorical trace theory, and spectral analysis.

Categorical Traces: They utilize the theory of traces of endofunctors on DG-categories (specifically $\text{Shv}$ and $\text{IndCoh}$ ) to translate geometric equivalences into isomorphisms of vector spaces of functions.
Singular Support Filtration: On the spectral side, they define subcategories $\text{IndCoh}_Y(\text{LS}^{\text{restr}}_{\check{G}})$ for closed $\text{Ad}$ -invariant subsets $Y \subset \check{\mathfrak{g}}$ . The filtration on the trace space is induced by the inclusion of these subcategories.
Beilinson Projectors: A crucial technical tool is the use of Beilinson projectors. Since the "vacuum Poincaré function" (a fundamental automorphic function) does not naturally lie in the subcategory $\text{Shv}_{\text{Nilp}}$ , the authors construct a specific projector $P_a$ associated with algebraic substacks $Z_a \subset \text{LS}^{\text{restr}}_{\check{G}}$ to "force" the object into the nilpotent support category while preserving the trace.
Excursion Algebra: They utilize the excursion algebra $\text{Exc}(X, \check{G}) = \Gamma(\text{LS}^{\text{arithm}}_{\check{G}}, \mathcal{O})$ , which acts on the space of automorphic functions. This algebra plays the role of the Hecke algebra in the spectral picture.
Rationality Analysis: The authors investigate the $\mathbb{Q}$ -structure of these spaces, leveraging the fact that the excursion algebra has a known rational structure (from [GaLiRe]) to conjecture compatibility with the filtration.

3. Key Contributions and Results

A. The Main Theorem: Image of the Poincaré Function

The central technical result (Theorem 1.3.10) identifies the image of the vacuum Poincaré function ( $poincVac^!$ ) under the Langlands isomorphism $L_G$ .

Result: Under the isomorphism $L_G$ , the function $poincVac^!$ corresponds to the image of the unit element $1 \in \Gamma(\text{LS}^{\text{arithm}}_{\check{G}}, \mathcal{O})$ under the canonical map:
$\Gamma(\text{LS}^{\text{arithm}}_{\check{G}}, \mathcal{O}) \to \Gamma^{\text{IndCoh}}(\text{LS}^{\text{arithm}}_{\check{G}}, \omega)$
Significance: This bridges the gap between the "geometric" construction of the Poincaré function (via sheaves) and the "spectral" construction (via the structure sheaf of the moduli stack). It proves that the Poincaré function generates the "zero" part of the filtration.

B. The Filtration and Tempered Functions

The authors define a filtration on $\text{Funct}_c(\text{Bun}_G(\mathbb{F}_q), \mathbb{Q}_\ell)$ indexed by nilpotent orbits.

The Zero Term: They prove (Corollary 2.4.5) that the term corresponding to the zero orbit, denoted $\text{Funct}_c(\text{Bun}_G(\mathbb{F}_q), \mathbb{Q}_\ell)^{\text{temp}}$ , is exactly the subspace generated by the Poincaré function under the action of the excursion algebra.
Non-degenerate Functions: This subspace is identified with the space of non-degenerate automorphic functions.
Irreducible Support: They show that the cuspidal non-degenerate functions correspond precisely to the locus of irreducible local systems on the spectral side (Theorem 2.5.8).

C. Rationality Conjectures

The paper proposes a series of conjectures (joint with D. Kazhdan) regarding the rationality of these structures:

Conjecture 2.7.3: The filtration defined by nilpotent orbits is defined over $\mathbb{Q}$ .
Conjecture 2.7.10: The action of the excursion algebra on automorphic functions is compatible with the rational structures.
Implication: These conjectures imply that the space of tempered functions is defined over $\mathbb{Q}$ , not just $\mathbb{Q}_\ell$ .

D. Connection to Arthur and Ramanujan Conjectures

The authors propose a web of conjectures linking the cohomological filtration to the analytic decomposition of $L^2$ spaces (Arthur's conjecture).

Spectral Decomposition: They conjecture that the closure of the filtration term $Y$ in the $L^2$ space corresponds to the direct sum of Arthur components associated with nilpotent orbits contained in $Y$ .
Ramanujan-Arthur: They formulate an $\ell$ -adic version of the Ramanujan-Arthur conjecture (Conjecture 3.4.6), stating that cuspidal automorphic functions decompose according to nilpotent orbits, with eigenvalues of Hecke operators being "Weil numbers" of specific weights.
Hierarchy: They establish logical implications showing that a "theorem-in-progress" (Conjecture 3.6.5 regarding the support of discrete spectra on elliptic Arthur parameters) would imply the Ramanujan-Arthur conjecture.

4. Significance

Unification of Geometric and Classical Langlands: The paper provides a rigorous mechanism to transfer the sophisticated "singular support" filtration from the geometric (categorical) side to the classical (function-theoretic) side. This gives a precise geometric meaning to the decomposition of automorphic forms.
Characterization of Temperedness: It offers a new, geometric characterization of tempered automorphic forms. Traditionally defined analytically (via unitary representations), tempered forms are here identified as the "zero-orbit" component of a filtration derived from the geometry of the spectral stack.
Rationality of Automorphic Forms: The conjectures regarding the $\mathbb{Q}$ -structure of the filtration and the excursion algebra action are significant for arithmetic geometry, suggesting that the deep spectral properties of automorphic forms are intrinsic to the rational structure of the underlying moduli spaces.
Path to Ramanujan-Arthur: By framing the Ramanujan-Arthur conjecture in terms of the support of discrete spectra on the coarse moduli space of local systems, the paper provides a concrete geometric strategy (via the work in progress [GaLaRa]) to prove these long-standing conjectures.
Technical Innovation: The use of Beilinson projectors to handle objects outside the nilpotent support category and the explicit calculation of the trace of the Poincaré function represent significant technical advances in the application of derived algebraic geometry to number theory.

In summary, this paper acts as a bridge, translating the abstract categorical machinery of the geometric Langlands program into concrete statements about the structure, rationality, and spectral decomposition of classical automorphic functions, while proposing a roadmap for proving the Ramanujan-Arthur conjecture.