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Spectral mod p Satake isomorphism for GL_n

Heejong Lee

Abstract

Let $K/\mathbb{Q}_p$ be a finite extension with residue field $k$. By a work of Emerton--Gee, irreducible components inside the reduced special fiber of the moduli stack of rank $n$ étale $(\varphi,Γ)$-modules are labeled by Serre weights of $\mathrm{GL}_n(k)$. Let $σ$ be a non-Steinberg Serre weight and $\mathcal{C}_σ$ be the corresponding irreducible component. Motivated by the categorical $p$-adic local Langlands program, we construct a natural injective map $\mathcal{O}(\mathcal{C}_σ) \hookrightarrow \mathcal{H}(σ)$ from the ring of global functions on $\mathcal{C}_σ$ to the Hecke algebra of $σ$ compatible with the mod $p$ Satake isomorphism by Herzig and Henniart--Vignéras in a suitable sense. For sufficiently generic $σ$, we prove that it is an isomorphism. As an application, we obtain a natural stratification of the irreducible component whose strata are equipped with a parabolic structure. Our main input is a construction of a morphism from an integral Hecke algebra of a generic tame type to the ring of global functions on a tamely potentially crystalline Emerton--Gee stack.

Spectral mod p Satake isomorphism for GL_n

Abstract

Let be a finite extension with residue field . By a work of Emerton--Gee, irreducible components inside the reduced special fiber of the moduli stack of rank étale -modules are labeled by Serre weights of . Let be a non-Steinberg Serre weight and be the corresponding irreducible component. Motivated by the categorical -adic local Langlands program, we construct a natural injective map from the ring of global functions on to the Hecke algebra of compatible with the mod Satake isomorphism by Herzig and Henniart--Vignéras in a suitable sense. For sufficiently generic , we prove that it is an isomorphism. As an application, we obtain a natural stratification of the irreducible component whose strata are equipped with a parabolic structure. Our main input is a construction of a morphism from an integral Hecke algebra of a generic tame type to the ring of global functions on a tamely potentially crystalline Emerton--Gee stack.
Paper Structure (27 sections, 45 theorems, 102 equations)

This paper contains 27 sections, 45 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Galois theory
  4. Reductive groups
  5. Tame inertial types
  6. Inertial local Langlands
  7. Integral and mod $p$ Hecke algebras
  8. Hecke algebras: generalities
  9. Hecke algebra of Serre weights
  10. Integral Hecke algebras of tame types
  11. Locally algebraic types
  12. Mod $p$ reduction of integral Hecke algebras
  13. Moduli stack of Breuil--Kisin modules
  14. Breuil--Kisin modules
  15. Global functions
  16. ...and 12 more sections

Key Result

Theorem 1

If $\tau$ is tame and sufficiently generic, then there is a morphism which recovers psi after inverting $p$.

Theorems & Definitions (117)

  • Theorem 1: Corollary \ref{['cor:hecke-global-func']}
  • Theorem 2: Proposition \ref{['prop:upperbound']} and Theorem \ref{['thm:spec-satake']}
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 6: Theorem \ref{['thm:modp-parabolic']}
  • Remark 7
  • Remark 2.3.1
  • Definition 3.1.1
  • Lemma 3.1.2
  • ...and 107 more