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The local Langlands conjecture

Olivier Taïbi

TL;DR

The notes formulate the local Langlands conjecture for connected reductive groups over local fields, introducing a conjectural map $\operatorname{LL}$ from irreducible representations to Langlands parameters with finite fibers and extending it through endoscopy to refine $L$-packets. They develop the dual-group framework ${}^L G$, parabolic embeddings, and the Weil-Deligne parametrization, and they analyze reductions to tempered/essentially discrete parameters, the unramified case via Satake, and Weil restriction. A refined theory for quasi-split and non-quasi-split groups is presented, incorporating Kaletha’s gerbe-based inner forms, endoscopic character relations, transfer factors, and the stabilized trace formula perspective. The text also connects the local correspondence to isocrystals and Tannakian formalisms, explaining how gerbes encode inner forms and how the Langlands program interfaces with Shimura varieties and Fargues–Scholze’s geometric perspective. Overall, it lays out a comprehensive, structurally unified program for the local Langlands correspondence across all local fields and inner forms, including the crucial endoscopic and geometric refinements.

Abstract

We formulate the local Langlands conjecture for connected reductive groups over local fields, including the internal parametrization of L-packets using endoscopy.

The local Langlands conjecture

TL;DR

The notes formulate the local Langlands conjecture for connected reductive groups over local fields, introducing a conjectural map from irreducible representations to Langlands parameters with finite fibers and extending it through endoscopy to refine -packets. They develop the dual-group framework , parabolic embeddings, and the Weil-Deligne parametrization, and they analyze reductions to tempered/essentially discrete parameters, the unramified case via Satake, and Weil restriction. A refined theory for quasi-split and non-quasi-split groups is presented, incorporating Kaletha’s gerbe-based inner forms, endoscopic character relations, transfer factors, and the stabilized trace formula perspective. The text also connects the local correspondence to isocrystals and Tannakian formalisms, explaining how gerbes encode inner forms and how the Langlands program interfaces with Shimura varieties and Fargues–Scholze’s geometric perspective. Overall, it lays out a comprehensive, structurally unified program for the local Langlands correspondence across all local fields and inner forms, including the crucial endoscopic and geometric refinements.

Abstract

We formulate the local Langlands conjecture for connected reductive groups over local fields, including the internal parametrization of L-packets using endoscopy.

Paper Structure

This paper contains 31 sections, 14 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Representations of reductive groups
  4. Setup
  5. Parabolic induction, the Jacquet functor and supercuspidal representations
  6. Asymptotic properties
  7. Harish-Chandra characters
  8. Langlands dual groups
  9. Based root data
  10. Langlands dual groups
  11. Parabolic subgroups and L-embeddings
  12. Langlands parameters
  13. Weil-Deligne groups
  14. Langlands parameters
  15. Reductions
  16. ...and 16 more sections

Key Result

Theorem 3.2

Let $\pi$ be an irreducible representation of $G(F)$.

Theorems & Definitions (38)

  • Remark 3.1
  • Theorem 3.2
  • proof
  • Definition 3.3
  • Proposition 3.4: WaldspurgerPlancherel
  • Proposition 3.5: WaldspurgerPlancherel
  • Remark 3.6
  • Theorem 3.7: Silberger_lang_class, BorelWallach, Dat_nutemp
  • Lemma 3.8
  • proof
  • ...and 28 more