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Local Jacquet-Langlands correspondence and parametric degrees

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Abstract.

Let F be a non-Archimedean local field with finite residue field. Let n be a positive integer, let G = GL n (F), and let D be a central F-division algebra of dimension n 2. The Jacquet-Langlands correspondence gives a canonical bijection π↦πD from the set of equivalence classes of irreducible, smooth, essentially square-integrable representations π of G to the set of equivalence classes of irreducible smooth representations of D ![![times;. We give a necessary and sufficient condition, in terms of dimτ, for an irreducible smooth representation τ of D × to be of the form πD, for an irreducible supercuspidal representation π of G, thereby solving an old problem. This relies on the explicit classification of the irreducible smooth representations of G and the parallel classification of the irreducible representations of D ×.

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Authors and Affiliations

  1. King’s College, Department of Mathematics, Strand, WC2R 2LS, London, UK

    Colin J. Bushnell

  2. Université de Paris-Sud, Département de Mathématiques &, UMR 8628 du CNRS, Bâtiment, 425, 91405 cedex, France

    Guy Henniart

Authors
  1. Colin J. Bushnell
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  2. Guy Henniart
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Correspondence to Colin J. Bushnell.

Additional information

This paper was written while the first-named author was visiting, and partly supported by, Université de Paris-Sud. At that time, the second-named author was enjoying the hospitality of the IHES, during a stay at the CNRS granted by Université de Paris-Sud; he would like to thank all those institutions. The work was also partially supported by EU network ‘‘Arithmetical Algebraic Geometry’’.

Mathematics Subject Classification (2000): 22E50

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Bushnell, C., Henniart, G. Local Jacquet-Langlands correspondence and parametric degrees. manuscripta math. 114, 1–7 (2004). https://doi.org/10.1007/s00229-004-0452-2

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  • DOI: https://doi.org/10.1007/s00229-004-0452-2

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