Smooth Duals of Inner Forms of and
Anne-Marie AubertCNRS, Sorbonne Université, Université Paris Diderot, Institut de Mathématiques de Jussieu, Paris Rive Gauche, IMJ-PRG F-75005 Paris, France
Paul BaumMathematics Department, Pennsylvania State University, University Park, PA 16802, USA
Roger PlymenSchool of Mathematics, Manchester University, Manchester M13 9PL, England
Maarten SolleveldIMAPP, Radboud Universiteit Nijmegen, Heyendaalseweg 135, 6525AJ Nijmegen, The Netherlands
Let 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 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 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 and . Doc. Math. 24 (2019), pp. 373–420DOI 10.4171/DM/684