Background: Let $K/F$ be a degree $r$ extension of number fields. It is conjectured that an automorphic representation of GL$_n$ associated to $K$ induces an automorphic representation of GL$_{rn}$ associated to $F$.
The book by Arthur and Clozel (1989) claims to prove automorphic induction for arbitary $n$ and arbitrary cyclic extensions $K/F$. Work by Lapid and Rogawski in the late 1990s showed that the proof is incorrect and that the result of AC is valid only for cyclic extensions of prime degree (i.e. the same class of fields considered earlier by Langlands in the case of GL$_2$). In the context of automorphic motives this restriction is much too strong because in general the fields that appear are not cyclic, not to mention cyclic of prime degree. This leads to the
Question:
What results are known for automorphic induction of GL$_n$ automorphic forms for various choices of $n$ and various types of extensions $K$?
In the context of automorphic motives even results for the smallest groups GL$_1$ and GL$_2$ would already be of interest.
Best Answer
I'm quite interested in this myself. I'll try to answer, in hope that someone else can complete it if I'm missing something important.
The only two major cases of automorphic induction known still are:
Local fields (Henniart-Herb)
Cyclic Galois extension of prime degree (Arthur-Clozel)
For a recent source that only mentions these two examples, see Colin Bushnell's paper for the 2014 book "Automorphic Forms and Galois Representations, Volumen 1".
Some other known instances of automorphic induction include:
Non-normal cubic extension (Jacquet-Piatetski-Shapiro-Shalika)
Non-normal extensions with solvable Galois closure for certain Hecke characters (Harris)
A case of non-normal quintic extension with non-solvable closure (Kim)
Feel free to comment or edit if you have knowledge of any other relevant result.
References:
Hervé Jacquet, Ilja Iosifovitch Piatetski-Shapiro, Joseph Shalika, Automorphic forms on $GL(3)$, I and II (1979)
James Arthur, Laurent Clozel: Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula (1989)
Guy Henniart, Rebecca Herb, Automorphic induction for $GL(n)$ (over local nonarchimedean fields) (1995)
Michael Harris, The local Langlands conjecture for $GL(n)$ over a p-adic field, $n < p$ (1996)
Henery Jim, An example of non-normal quintic automorphic induction and modularity of symmetric powers of cusp forms of icosahedral type (2003)