Smooth Duals of Inner Forms of and

  • Anne-Marie Aubert

    CNRS, Sorbonne Université, Université Paris Diderot, Institut de Mathématiques de Jussieu, Paris Rive Gauche, IMJ-PRG F-75005 Paris, France
  • Paul Baum

    Mathematics Department, Pennsylvania State University, University Park, PA 16802, USA
  • Roger Plymen

    School of Mathematics, Manchester University, Manchester M13 9PL, England
  • Maarten Solleveld

    IMAPP, Radboud Universiteit Nijmegen, Heyendaalseweg 135, 6525AJ Nijmegen, The Netherlands
Smooth Duals of Inner Forms of $\mathrm{GL}_n$ and $\mathrm{SL}_n$ cover
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Let be a non-archimedean local field. We prove that every Bernstein component in the smooth dual of each inner form of the general linear group is canonically in bijection with the extended quotient for the action, given by Bernstein, of a finite group on a complex torus. For inner forms of we prove that each Bernstein component is canonically in bijection with the associated twisted extended quotient. In both cases, the bijections satisfy naturality properties with respect to the tempered dual, parabolic induction, central character, and the local Langlands correspondence.

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Anne-Marie Aubert, Paul Baum, Roger Plymen, Maarten Solleveld, Smooth Duals of Inner Forms of and . Doc. Math. 24 (2019), pp. 373–420

DOI 10.4171/DM/684