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

### 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

## 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 $\mathrm{GL}_n(F)$ 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.

## Cite this article

Anne-Marie Aubert, Paul Baum, Roger Plymen, Maarten Solleveld, Smooth Duals of Inner Forms of $\mathrm{GL}_n$ and $\mathrm{SL}_n$. Doc. Math. 24 (2019), pp. 373–420

DOI 10.4171/DM/684