Unitary Dual of GL_n at archimedean places and global Jacquet-Langlands correspondence

TL;DR

This paper extends the local results of the global Jacquet-Langlands correspondence to include archimedean places by analyzing the unitary dual of general linear groups over real, complex, and quaternionic fields, removing previous assumptions.

Contribution

It generalizes the local results of the Jacquet-Langlands correspondence to all archimedean places, providing new insights into the unitary dual of GL_n over various fields.

Findings

Extended local results to archimedean places.

Collected new results on the unitary dual of GL_n over $bR$, $bC$, and $bH$.

Removed the split assumption at archimedean places in the correspondence.

Abstract

In [7], results about the global Jacquet-Langlands correspondence, (weak and strong) multiplicity-one theorems and the classification of automorphic representations for inner forms of the general linear group over a number field are established, under the condition that the local inner forms are split at archimedean places. In this paper, we extend the main local results of [7] to archimedean places so that this assumption can be removed. Along the way, we collect several results about the unitary dual of general linear groups over $\bbR$, $\bbC$ or $\bbH$ of independent interest.