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Geometric structure in smooth dual and local Langlands conjecture

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Japanese Journal of Mathematics Aims and scope

Abstract

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.

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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu – Paris Rive Gauche, U.M.R. 7586 du C.N.R.S., U.P.M.C., 4 place Jussieu, 75005, Paris, France

    Anne-Marie Aubert

  2. Mathematics Department, Pennsylvania State University, University Park, PA, 16802, USA

    Paul Baum

  3. School of Mathematics, Southampton University, Southampton, SO17 1BJ, England

    Roger Plymen

  4. School of Mathematics, Manchester University, Manchester, M13 9PL, England

    Roger Plymen

  5. Radboud Universiteit Nijmegen, Heyendaalseweg 135, 6525AJ, Nijmegen, the Netherlands

    Maarten Solleveld

Authors
  1. Anne-Marie Aubert
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  2. Paul Baum
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  3. Roger Plymen
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  4. Maarten Solleveld
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Corresponding author

Correspondence to Paul Baum.

Additional information

Communicated by: Takeshi Saito

This article is based on the 11th Takagi Lectures that the second author delivered at the University of Tokyo on November 17 and 18, 2012.

The second author was partially supported by NSF grant DMS-0701184.

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Aubert, AM., Baum, P., Plymen, R. et al. Geometric structure in smooth dual and local Langlands conjecture. Jpn. J. Math. 9, 99–136 (2014). https://doi.org/10.1007/s11537-014-1267-x

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  • DOI: https://doi.org/10.1007/s11537-014-1267-x

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