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Admissibility of Bernstein centers

This paper establishes a criterion for the admissibility of elements in the Bernstein center of totally disconnected locally compact groups as Harish-Chandra invariant distributions, thereby proving their local integrability for bounded depth or support in reductive $p$ -adic groups and generalizing prior complex-case results to both complex and mod- $\ell$ coefficients.

Original authors: Tsao-Hsien Chen, Cheng-Chiang Tsai

Published 2026-08-25

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CC BY 4.0

Original authors: Tsao-Hsien Chen, Cheng-Chiang Tsai

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

In the vast landscape of modern mathematics, there is a branch dedicated to understanding symmetry through the lens of groups, which are collections of objects that can be combined in specific ways. When these groups describe the symmetries of spaces that are built from numbers in a very specific, fragmented way—known as totally disconnected groups—they behave in a manner that is both rigid and incredibly rich. Mathematicians study these structures by looking at "representations," which are essentially ways of translating the abstract rules of the group into the language of linear algebra, where they can be calculated and visualized. A central tool in this field is the "Bernstein center," a sophisticated mathematical object that acts like a control panel, organizing all possible representations of a group into distinct, manageable categories. The question that has long intrigued researchers is whether the signals sent from this control panel can be translated into smooth, well-behaved functions that describe the group's behavior in a local, physical sense.

The paper by Chen and Tsai tackles this precise question for a specific class of these groups, which are defined over number systems that include p-adic numbers, a type of arithmetic that feels like a digital grid rather than a continuous line. The authors provide a clear, reliable test to determine when the abstract elements of the Bernstein center correspond to "admissible" distributions. In plain terms, an admissible distribution is a mathematical description of the group that is well-behaved enough to be studied locally; it does not explode or become chaotic when examined up close. The researchers prove that if a specific condition is met—namely, that the representations in a certain category are generated by vectors that remain unchanged under the action of a small, compact subgroup—then the corresponding distribution is guaranteed to be admissible. This finding is significant because it confirms that these abstract algebraic objects have a concrete, local existence, behaving like smooth functions that can be integrated and analyzed.

The authors demonstrate that this rule applies broadly, covering both the traditional setting where calculations are done with complex numbers and the more modern, challenging setting where calculations are performed using modular arithmetic. By establishing this criterion, they show that for any element in the Bernstein center that is restricted to a specific "depth" or a limited set of symmetry types, the resulting distribution is locally integrable. This means that mathematicians can now confidently treat these abstract signals as actual functions that vary smoothly across the group, at least in the regions where the group elements are regular and non-degenerate. The work unifies previous results that were known only for complex numbers and extends them to a wider, more difficult mathematical terrain, ensuring that the tools used to analyze these symmetries are robust and consistent.

Ultimately, the paper resolves a foundational issue by bridging the gap between the abstract algebraic classification of group representations and the concrete analytic properties of the distributions they generate. The researchers show that the structure of the group itself, specifically how its representations are built from fixed points of small subgroups, dictates the regularity of the associated distributions. This insight allows for the application of powerful analytical techniques to a wider range of mathematical problems, confirming that the local behavior of these complex systems is predictable and well-structured. The results hold true whether the underlying field of numbers is characteristic zero or a specific type of positive characteristic, offering a unified framework that strengthens the theoretical foundation for studying symmetries in p-adic environments.

Technical Summary: Admissibility of Bernstein Centers

Problem Statement
The paper addresses the question of when elements of the Bernstein center $Z(G)$ of a totally disconnected locally compact group $G$ correspond to admissible invariant distributions in the sense of Harish-Chandra. Specifically, the authors seek a criterion to determine if an element $z \in Z(G)$ , viewed as an essentially compact invariant distribution $f_z$ , satisfies the admissibility condition. This condition requires that for any point $x \in G$ , there exists a compact open pro- $p$ subgroup $U_x$ such that the convolution of the distribution with the character of any irreducible representation of a subgroup $U'_x \subset U_x$ is non-zero only if the representation has a specific non-vanishing fixed vector property relative to $G$ .

The authors aim to generalize existing results, particularly those of Moy and Tadić [MT02] in the complex case ( $k=\mathbb{C}$ ), to the setting of mod- $\ell$ coefficients (where $k$ is an algebraically closed field of characteristic $\ell \neq p$ ) and to establish local integrability results for distributions associated with bounded depth or bounded Bernstein support in reductive $p$ -adic groups.

Methodology
The paper employs a structural approach based on the decomposition of the category of smooth representations $R(G)$ and the properties of the Bernstein center acting on these categories.

Categorical Decomposition: The authors utilize the decomposition of the category of smooth representations $R(G) = R(G)^\heartsuit \oplus R(G)^\spadesuit$ . This induces a corresponding decomposition of the Bernstein center $Z(G) = Z(G)^\heartsuit \oplus Z(G)^\spadesuit$ .
Fixed Vector Generation: The core of their method relies on identifying subcategories $R(G)^\heartsuit$ where every representation $\pi$ is generated by its fixed vectors under a specific compact open pro- $p$ subgroup $U$ (i.e., $\pi$ is generated by $(V_\pi)^U$ ).
Admissibility Criterion (Theorem 2.3): The authors prove that if a decomposition $R(G) = R(G)^\heartsuit \oplus R(G)^\spadesuit$ $R (G) = R (G)^{♡} \oplus R (G)^{♠}$ exists such that all $\pi \in R(G)^\heartsuit$ $π \in R (G)^{♡}$ are generated by $U$ $U$ -fixed vectors, then any $z \in Z(G)^\heartsuit$ $z \in Z (G)^{♡}$ yields a $U$ $U$ -admissible distribution $f_z$ $f_{z}$ .
- Mechanism: The proof involves showing that if $f_z * \Theta_\rho \neq 0$ for an irreducible representation $\rho$ of a subgroup $U' \subset U$ , there must exist a representation $\pi \in R(G)^\heartsuit$ such that $z(\pi) \neq 0$ and $\text{Hom}_{U'}(\check{\rho}, \pi) \neq 0$ . By expressing a non-zero vector in the image of the embedding of $\check{\rho}$ into $\pi$ as a sum of $U$ -fixed vectors, the authors demonstrate that the fixed vector space $(V_\rho)^{U' \cap gUg^{-1}}$ must be non-zero for some $g \in G$ , satisfying the admissibility definition.
Application to Reductive Groups: The authors apply this general criterion to connected reductive groups over $p$ -adic fields. They leverage known results regarding depth decompositions (Moy-Prasad) and Bernstein supports to identify specific subcategories $R(G)^\heartsuit$ that satisfy the fixed-vector generation property.

Key Contributions and Results

General Criterion for Admissibility: The paper establishes a sufficient condition (Theorem 2.3) for an element of the Bernstein center to define an admissible invariant distribution. This condition is purely categorical, relying on the generation of representations by fixed vectors of a compact open subgroup.
Local Integrability for Bounded Depth: For a reductive group $G$ over a $p$ -adic field, the authors show that any element $z$ in the Bernstein center with bounded depth (i.e., $z \in Z(G)_{\leq r}$ ) corresponds to an admissible distribution. Consequently, these distributions satisfy Harish-Chandra's local character expansion and are represented by locally constant functions on the regular semisimple locus (Corollary 3.4).
Local Integrability for Bounded Bernstein Support: Similarly, for elements with bounded Bernstein support (restriction to all but finitely many Bernstein blocks is zero), the corresponding distributions are admissible and satisfy the same local integrability properties (Corollary 3.5).
Mod- $\ell$ Generalization: The results hold uniformly for both complex coefficients ( $k=\mathbb{C}$ ) and mod- $\ell$ coefficients ( $k$ of characteristic $\ell \neq p$ ). In the mod- $\ell$ case, the results regarding local integrability (specifically representation by a function via iterated geometric series) rely on recent generalizations of Harish-Chandra's theory found in [T26].

Significance and Claims
The paper claims to provide a unified framework for understanding the admissibility of Bernstein centers, generalizing the complex case results of Moy and Tadić [MT02].

Novelty in Mod- $\ell$ : The authors explicitly state that the results for $\ell > 0$ appear to be new and do not follow directly from the methods used in the complex case literature.
Alternative Proofs: In the case $\ell=0$ , the paper offers an alternative argument for the main results of [MT02] via the depth decomposition and fixed-vector generation properties.
Scope: The methods are presented as applicable to both complex and mod- $\ell$ coefficients, extending the reach of admissibility criteria to a broader arithmetic setting. The authors also note an expectation that these results may extend to local function fields of characteristic $p$ (provided $p$ is sufficiently large) and to coefficients in $\mathbb{Z}[1/p]$ -algebras with modifications, though these are presented as expectations rather than proven theorems within the text.

Technical Summary: Admissibility of Bernstein Centers

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