Smooth Duals of Inner Forms of $GL_n$ and $SL_n$

TL;DR

This paper establishes a canonical bijection between Bernstein components of inner forms of $GL_n$ and $SL_n$ over non-archimedean fields and their respective extended quotients, respecting key naturality properties.

Contribution

It introduces a new canonical parametrization of Bernstein components for inner forms of $GL_n$ and $SL_n$ using extended quotients, compatible with key representation-theoretic structures.

Findings

Bernstein components of inner forms of $GL_n$ correspond to extended quotients.

Bernstein components of inner forms of $SL_n$ correspond to twisted extended quotients.

Bijections are compatible with tempered dual, parabolic induction, central characters, and Langlands correspondence.

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 $GL_n(F)$ 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 $SL_n(F)$ 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.