# The status of automorphic induction

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.

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:

Here are a few recent results which haven't been mentioned so far (in chronological order).

• Rajan, On the image and fibres of solvable base change (2002)

This relies on the result of Lapid and Rogawski for $\mathrm{GL}(2)$. The paper works with the generalization for $\mathrm{GL}(n)$ as a hypothesis. Remark 1 there says that, "Granting this, our theorem extends to $\mathrm{GL}_n$, and we present the proof in the general case assuming Statement B of Lapid-Rogawski."

• Henniart, Induction automorphe pour $\mathrm{GL}(n,\mathbb{C})$ (2009)

The result is the same as that obtained in the non-archimedean case.

• Henniart, Lemaire, Formules de caracteres pour l'induction automorphe (2010)

If $E/F$ is a degree $d$ cyclic extension of non-archimedean local fields then Henniart and Herb proved that if $\tau$ is a tempered representation of $\mathrm{GL}(m,E)$ then there is a tempered representation $\pi$ of $\mathrm{GL}(md,F)$ such that $$\mathrm{tr}\,\tau(f)=c\cdot\mathrm{tr}\,\pi(f^G)\circ A,$$ whenever $f$ and $f^G$ have matching orbital integrals. The constant $c$ depends on $\tau$ and the operator $A$ which intertwines $\pi$ and $\omega\pi$, where $\omega$ is a character of $F^\times$ determining the extension $E/F$. Henniart and Lemaire show that when one uses the canonical normalization of $A$ via Whittaker models then $c$ does not depend on $\tau$.

• Hiraga and Ichino, On the Kottwitz-Shelstad normalization of transfer factors for automorphic induction for $\mathrm{GL}_n$ (2012)

They show that this constant $c$ is in fact 1 when using normalized transfer factors.

• Henniart, Induction automorphe globale pour les corps de nombres (2012)

When $E/F$ is a cyclic extension of number fields of degree $d$ and $\tau$ is an induced from cuspidal representation of $\mathrm{GL}(m,\mathbb{A}_E)$, from the abstract,

"We prove that the representation $\pi$ automorphically induced from $\tau$ exists, and we study the fibres and the image of automorphic induction. For that we use and extend the results of Arthur and Clozel on base change, which corresponds to restricting Galois representations from $F$ to $E$, and we clarify the relations between the two processes. Moreover we prove that global automorphic induction is compatible, at finite places, with the local automorphic induction defined by R. Herb and the author."