Japanese Journal of Mathematics

, Volume 9, Issue 2, pp 99–136

Geometric structure in smooth dual and local Langlands conjecture

  • Anne-Marie Aubert
  • Paul Baum
  • Roger Plymen
  • Maarten Solleveld
Special Feature: The Takagi Lectures

DOI: 10.1007/s11537-014-1267-x

Cite this article as:
Aubert, AM., Baum, P., Plymen, R. et al. Jpn. J. Math. (2014) 9: 99. doi:10.1007/s11537-014-1267-x


This expository paper first reviews some basic facts about p-adic fields, reductive p-adic groups, and the local Langlands conjecture. If G is a reductive p-adic group, then the smooth dual of G is the set of equivalence classes of smooth irreducible representations of G. The representations are on vector spaces over the complex numbers. In a canonical way, the smooth dual is the disjoint union of subsets known as the Bernstein components. According to a conjecture due to ABPS (Aubert–Baum–Plymen–Solleveld), each Bernstein component has a geometric structure given by an appropriate extended quotient. The paper states this ABPS conjecture and then indicates evidence for the conjecture, and its connection to the local Langlands conjecture.

Keywords and phrases

reductive p-adic group local Langlands conjecture Bernstein components 

Mathematics Subject Classification (2010)

11F85 22E50 11R39 20G05 

Copyright information

© The Mathematical Society of Japan and Springer Japan 2014

Authors and Affiliations

  • Anne-Marie Aubert
    • 1
  • Paul Baum
    • 2
  • Roger Plymen
    • 3
    • 4
  • Maarten Solleveld
    • 5
  1. 1.Institut de Mathématiques de Jussieu – Paris Rive Gauche, U.M.R. 7586 du C.N.R.S., U.P.M.C.ParisFrance
  2. 2.Mathematics DepartmentPennsylvania State UniversityUniversity ParkUSA
  3. 3.School of MathematicsSouthampton UniversitySouthamptonEngland
  4. 4.School of MathematicsManchester UniversityManchesterEngland
  5. 5.Radboud Universiteit NijmegenNijmegenthe Netherlands