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The stack of spherical Langlands parameters

Thibaud van den Hove

TL;DR

This work defines the stack of spherical Langlands parameters Loc$_{{}^cG}^{\mathrm{sph}}$ using inertia invariants $\widehat{G}^I$ and embeds it as a closed substack of the full parameter stack Loc$_{{}^cG}$, enabling ramified groups to be treated within the Langlands framework. It develops integral transfer maps between parahoric Hecke algebras and connects them to the Fargues–Scholze spectral action, establishing compatibility with inner forms, central isogenies, and parabolic induction. Leveraging these tools, it proves Eichler–Shimura congruence relations for Hodge type Shimura varieties with ramification at $p$ by relating Frobenius actions to explicit Hecke polynomials via the inertia-invariant dual group. The results unify geometric and spectral approaches to Langlands parameters and Shimura variety cohomology in the ramified setting, extending prior ramified cases and providing a coherent framework for future categorical Langlands developments.

Abstract

For a reductive group over a nonarchimedean local field, we define the stack of spherical Langlands parameters, using the inertia-invariants of the Langlands dual group. This generalizes the stack of unramified Langlands parameters in case the group is unramified. We then use this stack to deduce the Eichler--Shimura congruence relations for Hodge type Shimura varieties, without restrictions on the ramification.

The stack of spherical Langlands parameters

TL;DR

This work defines the stack of spherical Langlands parameters Loc using inertia invariants and embeds it as a closed substack of the full parameter stack Loc, enabling ramified groups to be treated within the Langlands framework. It develops integral transfer maps between parahoric Hecke algebras and connects them to the Fargues–Scholze spectral action, establishing compatibility with inner forms, central isogenies, and parabolic induction. Leveraging these tools, it proves Eichler–Shimura congruence relations for Hodge type Shimura varieties with ramification at by relating Frobenius actions to explicit Hecke polynomials via the inertia-invariant dual group. The results unify geometric and spectral approaches to Langlands parameters and Shimura variety cohomology in the ramified setting, extending prior ramified cases and providing a coherent framework for future categorical Langlands developments.

Abstract

For a reductive group over a nonarchimedean local field, we define the stack of spherical Langlands parameters, using the inertia-invariants of the Langlands dual group. This generalizes the stack of unramified Langlands parameters in case the group is unramified. We then use this stack to deduce the Eichler--Shimura congruence relations for Hodge type Shimura varieties, without restrictions on the ramification.
Paper Structure (10 sections, 18 theorems, 22 equations)

This paper contains 10 sections, 18 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Eichler--Shimura relations in the ramified case
  3. The stack of spherical Langlands parameters
  4. Outline
  5. Acknowledgements
  6. The stack of spherical Langlands parameters
  7. Integral transfer maps
  8. On the Fargues--Scholze correspondence for representations with parahoric fixed vectors
  9. Congruence relations
  10. The orbit of a pinning-preserving automorphism, by Sean Cotner

Key Result

Theorem 1.1

Let $\mathop{\mathrm{Sh}}\nolimits_{K',\overline{E}}$ be a Hodge type Shimura variety with Iwahori level at $p$, and $\Lambda$ a $\mathbb{Z}_\ell[\sqrt{p}]$-algebra, under a mild assumption on $\ell\neq p$. Then, the inertia subgroup of $\mathop{\mathrm{Gal}}\nolimits(\overline{E}/E)$ acts unipotent

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Theorem 2.5
  • proof
  • ...and 33 more