Smooth Duals of Inner Forms of $\mathrm{GL}_n$ and $\mathrm{SL}_n$

Anne-Marie Aubert; Paul Baum; Roger Plymen; Maarten Solleveld

doi:10.4171/dm/684

JournalsdmVol. 24pp. 373–420

Smooth Duals of Inner Forms of $GL_{n}$ and $SL_{n}$

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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Abstract

Let $F$ 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 $GL_{n} (F)$ 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 $SL_{n} (F)$ 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 $GL_{n}$ and $SL_{n}$ . Doc. Math. 24 (2019), pp. 373–420

DOI 10.4171/DM/684