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Gelfand-Kirillov bound for $p$ -adic Banach representations with infinitesimal character for $\text{GL}_2$ and quaternion units

The paper establishes an upper bound for the Gelfand-Kirillov dimension of admissible $p$ -adic Banach representations for $\text{GL}_2K$ and the unit group of quaternions, provided their locally analytic vectors possess an infinitesimal character.

Original authors: Reinier Sorgdrager

Published 2026-02-10

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Reinier Sorgdrager

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 a professional chef trying to organize a massive, high-end kitchen. This kitchen isn't just any kitchen; it’s a $p$ -adic kitchen, where the ingredients (numbers) follow strange, fractal-like rules. Instead of measuring things in grams or liters, everything is measured in powers of a prime number $p$ .

In this kitchen, the "recipes" are Banach representations. These are complex mathematical structures that describe how certain groups (like the symmetries of a shape) act on a space of functions.

The paper by Reinier Sorgdrager is essentially a study of "Kitchen Efficiency"—specifically, how much "space" or "complexity" these recipes actually take up.

1. The Concept: What is Gelfand-Kirillov (GK) Dimension?

Think of GK Dimension as the "Complexity Score" of a recipe.

A recipe for a single glass of water has a very low complexity score.
A recipe for a 10-course banquet with 50 different ingredients has a very high complexity score.

In mathematics, we want to know: "If I have a certain set of rules (an infinitesimal character), how complex can my recipe actually get?" If the complexity score is too high, the kitchen becomes unmanageable.

2. The Problem: The "Chaos" of $p$ -adic Symmetry

The author is looking at two specific types of "kitchens":

$GL_2$ : The symmetries of a 2D plane.
Quaternion Units: A more exotic, "twisted" version of symmetry.

Previously, mathematicians knew there was a limit to how complex these recipes could be, but the limit they had was a bit too loose—like saying, "A recipe for a meal will take somewhere between 1 and 100 hours." Sorgdrager wanted to tighten that bound, proving that the complexity is actually much lower: specifically, it's limited by the "size" of the field $K$ (the mathematical environment the kitchen sits in).

3. The Tool: The "Casimir Ideal" (The Master Organizer)

How do you prove a recipe isn't too complex? You look for constraints.

Imagine you are looking at a chaotic list of ingredients. Suddenly, you realize that every time a chef adds "Salt," they must also add "Pepper" in a specific ratio. This rule (a constraint) immediately reduces the number of possible combinations.

In this paper, the author uses something called a "Casimir Ideal." Think of this as a Master Rulebook. The "infinitesimal character" mentioned in the title is like a set of fundamental laws of physics for the kitchen. The author proves that these laws act like a "Master Rulebook" that forces the ingredients to behave. Because the ingredients are forced to follow these rules, they can't create an infinite variety of combinations. This "kills" the complexity, keeping the GK dimension low.

4. The Result: Tightening the Bounds

The main achievement (the Main Theorem) is proving that the complexity score (GK dimension) is $\leq [K : \mathbb{Q}_p]$ .

In our kitchen analogy, this is like proving: "No matter how fancy the banquet is, if it follows the laws of this kitchen, it will never require more than $X$ number of specialized tools."

Why does this matter? (The "Big Picture")

This paper is a piece of a much larger puzzle called the $p$ -adic Langlands Program. This program is like trying to find a "Universal Translator" between two different worlds:

World A: The world of Number Theory (the hidden patterns in prime numbers).
World B: The world of Harmonic Analysis (the study of waves and symmetries).

By proving that these representations have a controlled, predictable complexity, Sorgdrager is helping to ensure that the "Translator" actually works. He is proving that the objects we are trying to translate aren't so chaotic that they become impossible to understand.

Summary in a Nutshell

The Subject: Complex mathematical "recipes" (Banach representations).
The Goal: To find the maximum "complexity score" (GK dimension) of these recipes.
The Method: Using "Master Rules" (Casimir ideals) to show that the recipes are more organized than they look.
The Win: Proving the complexity is strictly limited, which helps mathematicians build a bridge between two massive fields of science.

This paper, titled "Gelfand-Kirillov Bound for $p$ -adic Banach Representations with Infinitesimal Character for $\text{GL}_2$ and Quaternion Units" by Reinier Sorgdrager, addresses a fundamental problem in the $p$ -adic Langlands program: bounding the complexity of certain $p$ -adic representations via the Gelfand-Kirillov (GK) dimension.

1. The Problem

The central objective is to provide an upper bound for the Gelfand-Kirillov dimension of admissible $p$ -adic Banach representations that possess an infinitesimal character.

In the context of the $p$ -adic Langlands program, specifically regarding "patched modules" $M_\infty$ (which are candidates for the correspondence), it is crucial to know if these representations are non-zero. A key way to prove non-zeroness is through "Miracle Flatness," which requires knowing the GK dimension of the special fiber. While previous work (e.g., [BHH+23b]) established bounds for smooth mod $p$ representations, the Banach setting remained less explored. Specifically, the author seeks to improve upon existing bounds for connected reductive groups $G$ , moving from the general bound of $< 2d_G$ (where $d_G$ is the dimension of the flag variety) to the sharper bound of $\le d_G$ .

2. Methodology

The author employs a sophisticated approach involving non-commutative algebra, Lie theory, and the theory of $p$ -adic distribution algebras. The methodology can be broken down into several technical stages:

Filtration and Associated Graded Rings: The author uses the theory of $p$ -valued groups and their associated filtered rings. By studying the distribution algebras $D^r(H, L)$ and their associated graded rings $\text{gr}^r M$ , the author translates questions about the dimension of a representation into questions about the Krull dimension of a graded module over a Noetherian ring.
The "Casimir Ideal" Approach: The core strategy is to show that the associated graded module $\text{gr } M$ is annihilated by a specific ideal, which the author terms the "Casimir ideal." This ideal is generated by the principal symbols of elements in the center of the universal enveloping algebra (the Casimir operators) and the central elements of the Lie algebra.
Lazard’s Isomorphism: The author utilizes Lazard’s isomorphism to relate the graded ring of the completed group ring to the universal enveloping algebra of the graded Lie algebra. This allows the infinitesimal action of the Lie algebra to be "transported" to the associated graded module.
$p$ -power Subgroups and $\epsilon$ -torsion Freeness: To handle the technicalities of the filtrations, the author studies the groups of $p$ -powers $H^{p^N}$ . A crucial technical lemma (Proposition 5.1) proves that for sufficiently large $N$ , the associated graded module is $\epsilon$ -torsion free, which ensures that the information from the infinitesimal action is preserved in the reduction modulo $\epsilon$ .

3. Key Contributions

Refinement of Bounds: The paper provides the first affirmative answer to a question posed by [DPS23], proving that for $\text{GL}_2$ and quaternion units, the GK dimension is bounded by $d_G$ (specifically $[K : \mathbb{Q}_p]$ ) rather than $2d_G$ .
Conceptualization of the Casimir Ideal: The author provides a more conceptual explanation of the ideals used in previous mod $p$ studies, showing they arise naturally from the Lie-theoretic structure of the group.
Generalization to Ramified Fields: Unlike previous works that often assumed $K/\mathbb{Q}_p$ was unramified, this paper explicitly handles the complexities of ramification by carefully constructing $p$ -valuations and analyzing the non-integrality of idempotents.
New Tools for $p$ -adic Lie Groups: The paper develops several lemmas regarding the $\epsilon$ -torsion freeness and the behavior of principal symbols under $p$ -power maps, which are useful for the broader $p$ -adic representation theory community.

4. Main Results

Theorem (Main Theorem): Let $p > 2$ and let $K$ and $L$ be $p$ -adic fields. Let $\Pi$ be an admissible $L$ -Banach space representation of either $\text{GL}_2(K)$ or the group of units of the quaternion algebra $D^\times$ over $K$ . If the locally analytic vectors of $\Pi$ have an infinitesimal character, then:
$\text{GKdim}(\Pi) \le [K : \mathbb{Q}_p]$

Additional Results:

Optimality: For $\text{GL}_2(K)$ , the bound is shown to be optimal, as principal series representations achieve exactly $[K : \mathbb{Q}_p]$ .
Application to Langlands: The result is applied to the patched modules $M_\infty$ appearing in the $p$ -adic Langlands program, concluding that the associated Banach representations $\Pi_y$ have $\text{GKdim} \le [K : \mathbb{Q}_p]$ .

5. Significance

This paper is significant because it bridges the gap between the algebraic properties of Lie algebras (infinitesimal characters) and the geometric/topological complexity of Banach representations (GK dimension). By providing a sharp upper bound, it offers a tool for verifying the non-vanishing of representations in the $p$ -adic Langlands program. Furthermore, the "Casimir ideal" framework provides a roadmap for extending these results to higher-rank groups like $\text{GL}_n$ , which is a major frontier in current research.