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On the categorical local Langlands conjectures for depth-zero regular supercuspidal representations

Chenji Fu

TL;DR

This thesis develops the depth-zero aspect of the categorical local Langlands program for split, simply connected G, by computing the L-parameter side and the depth-zero representation block side and verifying the conjecture for GL_n. It describes the L-parameter component X_φ/Ĝ containing a tame regular semisimple elliptic parameter φ, showing X_φ/Ĝ ≅ [*/S_ψ] × μ with μ a product of μ_{ℓ^{k_i}}, and frames the representation side via a Broué-type equivalence between blocks of depth-zero parahoric representations and blocks of finite groups of Lie type, with compact induction providing the key equivalence. In the GL_n case, explicit matrices and tori computations yield a concrete equivalence between Rep_Λ(GL_n(F))_{[π]} and QCoh on the corresponding L-parameter component, realized through the spectral action on Bun_G and LL in families. The work thus integrates moduli of L-parameters, finite-type block theory, and spectral actions to realize the categorical local Langlands correspondence for depth-zero regular supercuspidal blocks and points toward a generalizable framework for all reductive groups.

Abstract

Let F be a non-archimedean local field with residue characteristic p. Let l be a prime number different from p. Let G be a connected reductive group which is split, semi-simple, and simply connected. On the one hand, we describe the category of quasi-coherent sheaves on the connected component of the stack of L-parameters over Z_l-bar containing a tame, regular semisimple, elliptic L-parameter over F_l-bar. On the other hand, we describe the block of Rep_{Z_l-bar}G(F) containing a depth-zero regular supercuspidal irreducible representation πover F_l-bar. For G=GL_n, we compute both sides explicitly and verify the categorical local Langlands conjecture for depth-zero supercuspidal blocks.

On the categorical local Langlands conjectures for depth-zero regular supercuspidal representations

TL;DR

This thesis develops the depth-zero aspect of the categorical local Langlands program for split, simply connected G, by computing the L-parameter side and the depth-zero representation block side and verifying the conjecture for GL_n. It describes the L-parameter component X_φ/Ĝ containing a tame regular semisimple elliptic parameter φ, showing X_φ/Ĝ ≅ [*/S_ψ] × μ with μ a product of μ_{ℓ^{k_i}}, and frames the representation side via a Broué-type equivalence between blocks of depth-zero parahoric representations and blocks of finite groups of Lie type, with compact induction providing the key equivalence. In the GL_n case, explicit matrices and tori computations yield a concrete equivalence between Rep_Λ(GL_n(F))_{[π]} and QCoh on the corresponding L-parameter component, realized through the spectral action on Bun_G and LL in families. The work thus integrates moduli of L-parameters, finite-type block theory, and spectral actions to realize the categorical local Langlands correspondence for depth-zero regular supercuspidal blocks and points toward a generalizable framework for all reductive groups.

Abstract

Let F be a non-archimedean local field with residue characteristic p. Let l be a prime number different from p. Let G be a connected reductive group which is split, semi-simple, and simply connected. On the one hand, we describe the category of quasi-coherent sheaves on the connected component of the stack of L-parameters over Z_l-bar containing a tame, regular semisimple, elliptic L-parameter over F_l-bar. On the other hand, we describe the block of Rep_{Z_l-bar}G(F) containing a depth-zero regular supercuspidal irreducible representation πover F_l-bar. For G=GL_n, we compute both sides explicitly and verify the categorical local Langlands conjecture for depth-zero supercuspidal blocks.

Paper Structure

This paper contains 46 sections, 36 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. $L$-parameter side
  3. Heuristics on the component $X_{\varphi}/\hat{G}$
  4. Ingredients of the computation
  5. Representation side
  6. Equivalence to the block of a finite group of Lie type
  7. Proof of the main theorem for the representation side
  8. The example of $\mathop{\mathrm{\operatorname{GL}}}\nolimits_n$
  9. The categorical local Langlands conjecture for $\mathop{\mathrm{\operatorname{GL}}}\nolimits_n$
  10. Preliminaries and notations
  11. Logical dependence and suggestions for reading
  12. Acknowledgements
  13. TRSELP components of the stack of $L$-parameters
  14. The connected component $X_{\varphi}$ containing a TRSELP $\varphi$
  15. Recollections on the moduli space of Langlands parameters
  16. ...and 31 more sections

Key Result

Theorem 1.1.1

Assume that $Z(\hat{G})$ is finite. Let $\varphi \in Z^1(W_F, \hat{G}(\overline{\mathbb{F}}_{\ell}))$ be a TRSELP. Let $\psi \in Z^1(W_F, \hat{G}(\overline{\mathbb{Z}}_{\ell}))$ be a lift of $\varphi$. Let $X_{\varphi}$ be the connected component of $Z^1(W_F, \hat{G})_{\overline{\mathbb{Z}}_{\ell}}$ where $S_{\psi}:=C_{\hat{G}}(\psi)$ is the schematic centralizer of $\psi$ in $\hat{G}$.

Theorems & Definitions (83)

  • Theorem 1.1.1
  • Theorem 1.2.1
  • Example 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3: Schematic centralizer
  • Remark 2.1.4
  • Definition 2.1.5
  • Example 2.1.6
  • Remark 2.1.7
  • Lemma 2.1.8
  • ...and 73 more