Geometric structure and the local Langlands conjecture

2012/11/01 by Anne‐Marie Aubert, Anne-Marie Aubert, Paul Baum +6 · 5 citations
Mathematics · #20G05 #22E50 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT) #Representation Theory (math.RT) #math.NT #math.RT #msc:20G05 #msc:22E50

paper · pdf · doi:10.48550/arxiv.1211.0180

75 pages. Some minor changes and corrections have been made

openalex publication_date 2012/11/01 · arxiv created 2013/05/20 · arxiv updated 2013/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that a strengthened form of the local Langlands conjecture is valid throughout the principal series of any connected split reductive p-adic group. The method of proof is to establish the presence of a very simple geometric structure, in both the smooth dual and the Langlands parameters. We prove that this geometric structure is present, in the same way, for the general linear group, including all of its inner forms. With these results as evidence, we give a detailed formulation of a general geometric structure conjecture.

Citations

• Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique

Cited by

• L-packets and depth for SL₂(K) with K a local function field of characteristic 2
• On Satake parameters for representations with parahoric fixed vectors
• Morita equivalence for k-algebras
• Geometric structure for the principal series of a split reductive p-adic group with connected centre
• The principal series of p-adic groups with disconnected centre

Related

• Extended quotients in the principal series of reductive p-adic groups
• The local Langlands conjecture for the p-adic inner form of Sp(4)
• The Hiraga-Ichino-Ikeda Conjecture for Principal Series of Split p-adic Groups
• Local Langlands in families: The banal case
• Geometric structure in the principal series of the p-adic group G₂