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A description of the depth-$r$ Bernstein center for rational depths

Sarbartha Bhattacharya, Tsao-Hsien Chen

TL;DR

This work develops a comprehensive description of the depth-$r$ Bernstein center $\mathcal{Z}^r(G)$ for split reductive groups over non-archimedean fields at rational depths, by realizing it as an inverse limit of refined parahoric Hecke algebras $A^r(G)$. It constructs maps from stable Moy-Prasad data to the center via $\xi^r$, attaches depth-$r$ Deligne–Lusztig parameters to irreducible representations through a well-behaved $\Theta_r$ map, and links these parameters to restricted depth-$r$ Langlands data. A central outcome is the decomposition of the smooth representation category $R(G)$ into a finite product of subcategories indexed by restricted DL parameters, paralleling a Langlands-informed partition of irreducibles. The depth-zero case is treated analogously, yielding depth-zero DL parameters and a corresponding center description. Collectively, these results illuminate how fractional-depth structure and stability principles govern the Bernstein center and MP-type parametrizations, with potential implications for LLC compatibility and packet decomposition in $p$-adic harmonic analysis.

Abstract

Let $G$ be a split connected reductive over a non-archimedean local field $k$. In this paper we give a description of the depth-$r$ Bernstein center of $G(k)$ for rational depths as a limit of depth-$r$ standard parahoric Hecke algebras, extending our previous work in the integral depths case (arXiv:2407.15128). Using this description, we construct maps from the space of stable functions on depth-$r$ Moy-Prasad quotients to the depth-$r$ center, and attach depth-$r$ Deligne-Lusztig parameters to smooth irreducible representations, with the assignment of parameters to irreducible representations shown to be consistent with restricted Langlands parameters for Moy-Prasad types described Chen-Debacker-Tsai (arXiv:2509.07780). As an application, we give a decomposition of the category of smooth representations into a product of full subcategories indexed by restricted depth-$r$ Langlands parameters.

A description of the depth-$r$ Bernstein center for rational depths

TL;DR

This work develops a comprehensive description of the depth- Bernstein center for split reductive groups over non-archimedean fields at rational depths, by realizing it as an inverse limit of refined parahoric Hecke algebras . It constructs maps from stable Moy-Prasad data to the center via , attaches depth- Deligne–Lusztig parameters to irreducible representations through a well-behaved map, and links these parameters to restricted depth- Langlands data. A central outcome is the decomposition of the smooth representation category into a finite product of subcategories indexed by restricted DL parameters, paralleling a Langlands-informed partition of irreducibles. The depth-zero case is treated analogously, yielding depth-zero DL parameters and a corresponding center description. Collectively, these results illuminate how fractional-depth structure and stability principles govern the Bernstein center and MP-type parametrizations, with potential implications for LLC compatibility and packet decomposition in -adic harmonic analysis.

Abstract

Let be a split connected reductive over a non-archimedean local field . In this paper we give a description of the depth- Bernstein center of for rational depths as a limit of depth- standard parahoric Hecke algebras, extending our previous work in the integral depths case (arXiv:2407.15128). Using this description, we construct maps from the space of stable functions on depth- Moy-Prasad quotients to the depth- center, and attach depth- Deligne-Lusztig parameters to smooth irreducible representations, with the assignment of parameters to irreducible representations shown to be consistent with restricted Langlands parameters for Moy-Prasad types described Chen-Debacker-Tsai (arXiv:2509.07780). As an application, we give a decomposition of the category of smooth representations into a product of full subcategories indexed by restricted depth- Langlands parameters.

Paper Structure

This paper contains 39 sections, 37 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. A description of the center
  3. Stable functions and Deligne-Lusztig parameters
  4. A conjecture on stability
  5. Organization and some conventions
  6. Some conventions:
  7. Acknowledgement
  8. Background and some notations
  9. Buildings- The split case
  10. Moy-Prasad Filtrations
  11. Filtrations for split Tori and its Lie algebra.
  12. Filtrations of $G(k)$.
  13. Filtrations of the Lie algebra and its dual
  14. The Bernstein center
  15. Stable Bernstein center.
  16. ...and 24 more sections

Key Result

Theorem 1.1

There is an explicit algebra isomorphism $[A^r]: A^r(G) \xrightarrow{\simeq} \mathcal{Z}^r(G)$ for a split reductive group $G$ and $r \in \mathbb Q_{\geq 0}$.

Theorems & Definitions (99)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.1
  • Remark 2.1
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 89 more