Geodesic
Smooth Duals of Inner Forms of $\mathrm{GL}_n$ and $\mathrm{SL}_n$
Documenta mathematica, Tome 24 (2019), pp. 373-420.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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.
Classification : 20G25, 22E50
Keywords: representation theory, Bernstein spectrum, Hecke algebras, stratified equivalence 
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     author = {Aubert, Anne-Marie and Baum, Paul and Plymen, Roger and Solleveld, Maarten},
     title = {Smooth {Duals} of {Inner} {Forms} of {\(\mathrm{GL}_n\)} and {\(\mathrm{SL}_n\)}},
     journal = {Documenta mathematica},
     pages = {373--420},
     publisher = {mathdoc},
     volume = {24},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2019__24__a51/}
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Aubert, Anne-Marie; Baum, Paul; Plymen, Roger; Solleveld, Maarten. Smooth Duals of Inner Forms of \(\mathrm{GL}_n\) and \(\mathrm{SL}_n\). Documenta mathematica, Tome 24 (2019), pp. 373-420. http://geodesic.mathdoc.fr/item/DOCMA_2019__24__a51/