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The B(G)-parametrization of the local Langlands correspondence

Alexander Bertoloni Meli, Masao Oi

TL;DR

The paper develops a canonical extension of the local Langlands correspondence from the basic B(G)_{bas}-twist (basic inner forms) to the full Kottwitz set B(G), for local fields and both quasi-split and inner-twisted forms. It shows that a B(L)_{bas}-LLC for Levi subgroups L suffices to induce a B(G)-LLC for G by composing L-parameters with the L-embedding L ↪ G and by using the representation theory of the disconnected group S_{φ}. The construction yields a natural, choice-independent bijection between Irr(S_{φ}) and the union of L-packets Π_{φ}(G_b) over all b ∈ B(G), compatible with duality and endoscopy, and compatible with the broader Fargues–Scholze program linking G-bundles and L-parameters. A regular-part endoscopic identity is established to relate transfers of stable distributions on endoscopic groups to the regular parts of representations in Π_{φ}(G_b), illustrating the interplay between Langlands data and endoscopic transfer in the non-basic setting.

Abstract

This article is on the parametrization of the local Langlands correspondence over local fields for non-quasi-split groups according to the philosophy of Vogan. We show that a parametrization indexed by the basic part of the Kottwitz set (which is an extension of the set of pure inner twists) implies a parametrization indexed by the full Kottwitz set. On the Galois side, we consider irreducible algebraic representations of the full centralizer group of the $L$-parameter (i.e not a component group). When $F$ is a $p$-adic field, we discuss a generalization of the endoscopic character identity.

The B(G)-parametrization of the local Langlands correspondence

TL;DR

The paper develops a canonical extension of the local Langlands correspondence from the basic B(G)_{bas}-twist (basic inner forms) to the full Kottwitz set B(G), for local fields and both quasi-split and inner-twisted forms. It shows that a B(L)_{bas}-LLC for Levi subgroups L suffices to induce a B(G)-LLC for G by composing L-parameters with the L-embedding L ↪ G and by using the representation theory of the disconnected group S_{φ}. The construction yields a natural, choice-independent bijection between Irr(S_{φ}) and the union of L-packets Π_{φ}(G_b) over all b ∈ B(G), compatible with duality and endoscopy, and compatible with the broader Fargues–Scholze program linking G-bundles and L-parameters. A regular-part endoscopic identity is established to relate transfers of stable distributions on endoscopic groups to the regular parts of representations in Π_{φ}(G_b), illustrating the interplay between Langlands data and endoscopic transfer in the non-basic setting.

Abstract

This article is on the parametrization of the local Langlands correspondence over local fields for non-quasi-split groups according to the philosophy of Vogan. We show that a parametrization indexed by the basic part of the Kottwitz set (which is an extension of the set of pure inner twists) implies a parametrization indexed by the full Kottwitz set. On the Galois side, we consider irreducible algebraic representations of the full centralizer group of the -parameter (i.e not a component group). When is a -adic field, we discuss a generalization of the endoscopic character identity.
Paper Structure (31 sections, 42 theorems, 100 equations)

This paper contains 31 sections, 42 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Review of the Kottwitz set
  4. Representation theory of disconnected reductive groups
  5. S-groups of L-parameters as disconnected reductive groups
  6. Weyl group constructions
  7. Review of the B(G)bas form of the conjectural correspondence.
  8. The enhanced Archimedean basic correspondence
  9. Statement of main theorem
  10. The construction
  11. The easy map
  12. The map in the other direction
  13. Independence of choices
  14. Properties of the correspondence
  15. Endoscopic character identity
  16. ...and 16 more sections

Key Result

Theorem 1.2

Let $G$ be a quasi-split connected reductive group with a fixed Whittaker datum $\mathfrak{w}$. Suppose that there is an LLC for $G$ and its $B(G)_{\mathrm{bas}}$-inner twists as well as an LLC for each proper Levi subgroup $L \subset G$ and its $B(L)_{\mathrm{bas}}$-inner twists. Then there is a na

Theorems & Definitions (97)

  • Conjecture 1.1: fargues--scholze
  • Theorem 1.2: See §\ref{['s: basiccorrespondence']}, §\ref{['s: theconstruction']}
  • Theorem 1.3: Theorem \ref{['thm: ECI']}
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3: AcharHardestyRicherepthrydisconnected
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 87 more