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

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

Doc. Math. 24, 373-420 (2019)
DOI: 10.25537/dm.2019v24.373-420
Communicated by Dan Ciubotaru


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 \(\mathrm{GL}_n(F)\) 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 \(\mathrm{SL}_n(F)\) 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.

Mathematics Subject Classification

20G25, 22E50


representation theory, Bernstein spectrum, Hecke algebras, stratified equivalence


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Aubert, Anne-Marie
CNRS, Sorbonne Université, Université Paris Diderot, Institut de Mathématiques de Jussieu, Paris Rive Gauche, IMJ-PRG F-75005 Paris, France
Baum, Paul
Mathematics Department, Pennsylvania State University, University Park, PA 16802, USA
Plymen, Roger
School of Mathematics, Manchester University, Manchester M13 9PL, England
Solleveld, Maarten
IMAPP, Radboud Universiteit Nijmegen, Heyendaalseweg 135, 6525AJ Nijmegen, The Netherlands