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

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

On Fields of rationality for automorphic representations

This paper proves two results on the field of rationality \Q(\pi) for an automorphic representation $\pi$, which is the subfield of \C fixed under the subgroup of \Aut(\C) stabilizing the isomorphism class of the finite part of $\pi$. For general linear groups and classical groups, our first main result is the finiteness of the set of discrete automorphic representations $\pi$ such that $\pi$ is unramified away from a fixed finite set of places, $\pi_\infty$ has a fixed infinitesimal character, and [\Q(\pi):\Q] is bounded. The second main result is that for classical groups, [\Q(\pi):\Q] grows to infinity in a family of automorphic representations in level aspect whose infinite components are discrete series in a fixed $L$-packet under mild conditions