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Stability of Elliptic Fargues-Scholze $L$-packets

Chenji Fu

TL;DR

The paper establishes stability for Harish-Chandra characters of FS L-packets attached to elliptic parameters by leveraging a geometric realization on Bun_G, the spectral action, and Hecke eigen-structures. It reduces stability to an equidistribution problem for weight multiplicities in high-weight representations, which is solved via Fourier analysis on finite abelian groups and precise growth estimates. The authors construct a derived packet \pi_0 = \mathcal{O}(S_{\varphi}) * \pi whose Harish-Chandra character is stable on elliptic regular semisimple elements, and show nonzero stability in characteristic zero; they also extend the stability to transfers across extended pure inner forms. The approach provides a self-contained, endoscopy-free route to stability, compatible with positive characteristic and deeply grounded in the geometric Langlands framework for p-adic groups.

Abstract

Let $F$ be a non-archimedean local field. Let $\overline{F}$ be an algebraic closure of $F$. Let $G$ be a connected reductive group over $F$. Let $\varphi$ be an elliptic $L$-parameter. For every irreducible representation $π$ of $G(F)$ with Fargues--Scholze $L$-parameter $\varphi$, we prove that there exists a finite set of irreducible representations $\{π_i\}_{i \in I}$ containing $π$, such that $π_i$ has Fargues--Scholze $L$-parameter $\varphi$ for all $i \in I$ and a certain non-zero $\mathbb{Z}$-linear combination $Θ_{π_0}$ of the Harish-Chandra characters of $\{π_i\}_{i \in I}$ is stable under $G(\overline{F})$ conjugation, as a function on the elliptic regular semisimple elements of $G(F)$. Moreover, if $F$ has characteristic zero, $Θ_{π_0}$ is a non-zero stable distribution on $G(F)$.

Stability of Elliptic Fargues-Scholze $L$-packets

TL;DR

The paper establishes stability for Harish-Chandra characters of FS L-packets attached to elliptic parameters by leveraging a geometric realization on Bun_G, the spectral action, and Hecke eigen-structures. It reduces stability to an equidistribution problem for weight multiplicities in high-weight representations, which is solved via Fourier analysis on finite abelian groups and precise growth estimates. The authors construct a derived packet \pi_0 = \mathcal{O}(S_{\varphi}) * \pi whose Harish-Chandra character is stable on elliptic regular semisimple elements, and show nonzero stability in characteristic zero; they also extend the stability to transfers across extended pure inner forms. The approach provides a self-contained, endoscopy-free route to stability, compatible with positive characteristic and deeply grounded in the geometric Langlands framework for p-adic groups.

Abstract

Let be a non-archimedean local field. Let be an algebraic closure of . Let be a connected reductive group over . Let be an elliptic -parameter. For every irreducible representation of with Fargues--Scholze -parameter , we prove that there exists a finite set of irreducible representations containing , such that has Fargues--Scholze -parameter for all and a certain non-zero -linear combination of the Harish-Chandra characters of is stable under conjugation, as a function on the elliptic regular semisimple elements of . Moreover, if has characteristic zero, is a non-zero stable distribution on .
Paper Structure (20 sections, 47 theorems, 139 equations)

This paper contains 20 sections, 47 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Sketch of the proof
  3. Comments on Theorem \ref{['Thm_main']}
  4. Acknowledgement
  5. Basic notions
  6. Algebraic groups and the regular representation
  7. Borovoi's algebraic fundamental group
  8. The Kottwitz set
  9. Recollections on Fargues--Scholze
  10. Harish-Chandra characters
  11. Recollections on Hansen--Kaletha--Weinstein
  12. Polynomial growth of $\chi(\operatorname{Char} V_{\mu_m})$
  13. Lemmas from Lie theory
  14. Lemmas about $H_g$
  15. Growth of $\chi(\operatorname{Char} V_{\mu_m})$
  16. ...and 5 more sections

Key Result

Theorem 1.0.3

(Theorem Thm_main_general and Thm_main_distribution) Let $\varphi$ be an elliptic $L$-parameter. For every $\pi \in \operatorname{Irr}_{{\overline{\mathbb{Q}}_{\ell}}}G(F)$ such that $\varphi_{\pi}^{\operatorname{FS}} = \varphi$, the Harish-Chandra character $\Theta_{\pi_0}$ of $\pi_0:={\mathcal{O}(

Theorems & Definitions (90)

  • Conjecture 1.0.2: kaletha2022representations, see kalethalocal for a more precise form
  • Theorem 1.0.3
  • Lemma 1.1.1
  • Theorem 1.1.2
  • Remark 1.1.3
  • Proposition 1.1.4
  • Definition 2.1.1: jantzen2003representation
  • Lemma 2.1.2
  • proof
  • Lemma 2.1.3
  • ...and 80 more