TITLE: How important are automorphic representations among admissible ones?
QUESTION [9 upvotes]: 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!

REPLY [2 votes]: 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.