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I'm currently studying automorphic representations on Bump's book "Automorphic forms and representations" and on Gelbart's "Automorphic forms on Adele groups". And I have some problems in understanding the general philosophy behind automorphic representations. I apologize if I'm too foggy even for a "soft question".

If I'm not mistaken, in order to get a global Langlands correspondence, one wants to study admissible representations for $\mathrm{GL}_2(\mathbb{A}_\mathbb{Q})$. Question 1: Is it correct? Are there other reasons?

Then it seems to me that, once we have the notion of admissible representations, someone noticed that classical objects, such as modular forms of various kinds, could be reinterpreted as admissible representations for $\mathrm{GL}_2(\mathbb{A}_\mathbb{Q})$. Therefore, inspired by this interpretation, one is lead to define a space of automorphic forms on $\mathrm{GL}_2(\mathbb{A}_\mathbb{Q})$, where those automorphic representations sit. Question 2: Is it correct?

Since the importance given to automorphic representations, I was lead to think that they have some special "pure representation theoretic" meaning in the world of admissible representation. But I couldn't find anything about it. Question 3: Do they have such a special role? Or they are special just because they arise from other areas of Mathematics and/or they are easy to study?

Thank you for your interest!

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First, one should be aware that the automorphic forms on $GL_2$, and the Langlands-program questions about $GL_2$, are special cases of a larger pattern of issues for more general reductive groups in place of $GL_2$.

The automorphic representations are very special among all admissible representations of $GL_2(\mathbb A)$. One proves that, whether or not it is automorphic, an irreducible admissible $\pi$ of $GL_2(\mathbb A)$ is a kind of tensor product of irreducible admissibles $\pi_v$ of the "local" groups $GL_2(\mathbb Q_v)$. Indeed, this factorization depends on knowing or assuming admissibility of a useful class of irreducibles of the local groups. By now, we know that irreducible unitaries are admissible, for example.

Admissible representations of a topological group $G$ with compact subgroup $K$, meaning that the multiplicities of $K$-isotypes are finite, etc., are more tractable than representations not known to have this property, because they require less analytical finesse. For example, in the Lie case, it turns out that $\mathfrak g,K$-module structures suffice for many purposes.

Thus, the irreducibles that are admissible are the most tractable, from general representation-theoretic viewpoints. And, again, for reductive Lie or p-adic groups, irreducible unitaries are admissible.

Yes, in the 1950s and early 1960s people realized that representation theory worked well enough that it was beneficial to think in those terms to study automorphic forms and L-functions. Further, this viewpoint unified treatment of holomorphic elliptic modular forms and waveforms, for example, though these things looked quite different in more primitive terms. And, in light of the tensor product factorization, it is very useful that, while an individual automorphic function itself does not factor over primes (except in the extreme case of abelian groups), the (irreducible) automorphic representations do factor.

Automorphic representations are essentially/simply the representations generated by automorphic forms/functions under right translation, modulo technicalities. Thus, they arise naturally. It would not be at all accurate to say they're "simpler" than other things, since, for example, studying representations of the local groups is considerably easier, that is, "local questions are easier than global" (typically).

In fact, it is a difficult question to understand which collections of local repns arise in the automorphic context. Such collections are very special in many ways.

One kind of special-ness is that proofs for analytic continuation of L-functions (attached locally to local data) use the automorphic-ness in a very serious way. (In fact, "converse theorems" show that if L-functions behave sufficiently like those attached to automorphic forms, they are attached to automorphic forms...)

There is obviously much more to say... but, yes, often the process of parsing the technicalities leaves discussion of motivation behind...

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  • $\begingroup$ Thanks! This is the kind of answer I was looking for! In specific I find extremely enlightening the observation about "factorization of automorphic forms" vs "factorization of automorphic representation", I completely oversaw it while studying! $\endgroup$ Jun 25, 2013 at 16:32
  • $\begingroup$ You're welcome! :) $\endgroup$ Jun 25, 2013 at 16:44
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Question 1 seems to have a misunderstanding.

The global Langlands correspondence studies automorphic representations of the adele group. These are the ones that have to do with Galois reps., motives, and other aspects of global number theory.

The factorization of admissible adelic reps. that Paul Garrett discusses in his answer shows that the classification of admissible adelic reps. reduces to a local question. It is not inherently global in nature.

It is the automorphic reps. that are of a global nature (because they depend on the way the global $GL_2(\mathbb Q)$ embeds into the adelic group).

For more background on the Langlands program, you could look at some the links here.

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