Geodesic
Smooth Duals of Inner Forms of $\mathrm{GL}_n$ and $\mathrm{SL}_n$
Documenta mathematica, Tome 24 (2019), pp. 373-420
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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 GLn​(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 SLn​(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.
DOI : 10.4171/dm/684
Classification : 20G25, 22E50
Mots-clés : representation theory, Hecke algebras, Bernstein spectrum, stratified equivalence 
@article{10_4171_dm_684,
     author = {Paul Baum and Roger Plymen and Maarten Solleveld and Anne-Marie Aubert},
     title = {Smooth {Duals} of {Inner} {Forms} of $\mathrm{GL}_n$ and $\mathrm{SL}_n$},
     journal = {Documenta mathematica},
     pages = {373--420},
     year = {2019},
     volume = {24},
     doi = {10.4171/dm/684},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/684/}
}
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Paul Baum; Roger Plymen; Maarten Solleveld; Anne-Marie Aubert. Smooth Duals of Inner Forms of $\mathrm{GL}_n$ and $\mathrm{SL}_n$. Documenta mathematica, Tome 24 (2019), pp. 373-420. doi: 10.4171/dm/684

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