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I have never really concentrated on Langlands, which explains my poor level of understanding of it. But I have read quite a few introductory papers related to Langlands, and to the circle of ideas surrounding it. I finally feel like I know what it is I don't know, and I can finally articulate what I need references for.

The following phenomena/processes are mysterious to me:

  1. My understanding is pretty vague when it comes to direct integrals. I would like a reference that would explain their importance in understanding representations.
  2. While I have done some reading about reductive groups, and linear algebraic groups more generally, I am still very vague about the intuition behind parabolic subgroups, and why they should be related to cusp forms. As I understand it, the definition of cuspidal representation is that there exists a central representation $\omega$ such that the given representation is an irreducible unitary subrepresentation of $\{f|f\in L^2(Z(\mathbb{A})\mathbb{G}(K)\backslash\mathbb{G}(\mathbb{A}),\omega), \forall K$-parabolic subgroup $\mathbb{P}$ of $\mathbb{G}$ the integral $\int_{N(K)\backslash N(\mathbb{A})}f(gn)dn$ is equal to $0\}$. I don't at all understand how this relates to cusp forms, and how I should be thinking of parabolic subgroups. I would love a reference that gives intuition about this. At the moment "parabolic subgroup" is a completely technical definition, as far as I'm concerned.
  3. I understand that Eisenstein series can somehow be thought of in terms of scattering theory. (Some theory in physics about how lines scatter in manifolds?) I don't understand any of this, and I would love a reference that would put Eisenstein series in their proper context, in a nice, readable manner. All I know about Eisenstein series is their definition in the classical modular forms case, and that somehow they are related to the continuous spectrum of $L^2(Z(\mathbb{A})\mathbb{G}(K)\backslash\mathbb{G}(\mathbb{A}),\omega)$, but I really don't understand the process or the philosophy...
  4. I don't understand what is meant by "cusp form" in general. I certainly know what a cusp form is in the classical modular forms case, and I know the definition for (though, not the intuition of) cuspidal representations for general reductive groups, but I don't know what is meant by "cusp form" for a general reductive group... I'm given to understand that they're supposed to somehow be related to embeddings into projective space, just like sections of very ample bundles? I would love a reference that explain some of this context.
  5. I have read about spherical Hecke algebras, and the Satake isomorphism. I know that L-functions of automorphic representations are given by the Satake parameters, which are given by the Satake isomorphism. This definition seems extremely unnatural and contorted to me! Do you have a reference that explains why it is natural to construct L-functions in this way?
  6. Anything that has to do with the word "growth" is unfamiliar to me: "rapid decay", "slow growth". I know that these notions are somehow important, but I managed to never understand their importance... What is a reference that explains the intuition behind caring about growth, and how it relates to the bigger picture?
  7. All of the above was on the automorphic side. This request for a reference is on the motivic side, or, at least, about the Langlands correspondence itself. The Local Langlands conjecture (theorem!) for $GL_n$ says that there is a bijection between the set (class?) of smooth, irreducible, complex representations of $GL_n(F)$ and equivalence classes of $n$-dimensional semi-simple Weil-Deligne representations. The definition of "semi-simple Weil-Deligne representations" is quite technical! I don't understand where this definition is coming from, and why that is the reasonable thing to consider. Is there a reference that explains the context in which this definition arose, and why it is a reasonable thing to consider?
  8. As I said in question 7, the Local Langlands for $GL_n$ gives a bijection between the set of smooth, irreducible, complex representations of $GL_n(F)$ and equivalence classes of $n$-dimensional semi-simple Weil-Deligne representations. What would irreducible semi-simple Weil-Deligne representations corrrespond to? Cuspidal representations? (I think I heard that once.) Why? Where can I read about the intuition of this?

I asked for references for a lot of things. I'm sorry if that is inappropriate. I'm new to the site, so I don't know the proper etiquette yet. I feel like things are finally falling into place for me about Langlands, but that I'm still extremely rough around the edges. I thought that this would be a great opportunity for me to get some leads about what I should read next.

I should mention that I have a huge preference for texts that give intuition. I would also prefer expository papers to books, just because it's easier for me to go through. But if you think that the best explanation is in a book, then please write that.

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2 Answers 2

Callum, if my experience is any guide, the main thing you need to get over all this material is time and work and more time and more work. There is no royal reference or small set of references which will give you an easy access to all this.

I can however suggest some references that were helpful, besides Corvallis which contains a lot of material but is not an easy read and doesn't provide much intuition.

For 1, you can read the little book of Mackey, The theory of Unitary Group Representations (Chicago Lectures in Mathematics).

For the others in general, it is important to understand well the case of $GL_2$ before dealing with the general case, especially if you have some familiarity with classical modular forms and want to use it as a basis for your future learning. For this there are references which are simpler than Corvallis: I suggest Gelbart, autonorphic forms for Gl_2, and the book by Knapp.

For 2, it seems that you're not familiar enough with the structure theory of reductive groups, and you should perhaps spend some time learning it for its own sake. Also for 2 and 4, see this question: What is the intuition behind the definition of cuspidal representations?

For 3, it's true that there is a gap between the theory of Eisenstein forms as explain in standard courses in classical modular forms (where Eisensteins series form a discrete set of objects which are holomorphic function on the upper-half plane) and the automorphic point of view on the Eisenstein series (where they form continuous family of non-holomorphic function). Perhaps the treatment of Miyake can make the gap shorter.

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I think you'll find answers to most of your questions in the Corvallis proceedings. More specifically,

A. Borel, W. Casselman, eds. Automorphic Forms, Representations and L-functions, Proc. Sympos. Pure Math. 33 Amer. Math. Soc., 1979.

It sounds like you have already read some survey type articles. These articles all have extensive bibliographies, so provide plenty to read if you want more detail.

A. Knapp also has several nice surveys, which in particular discuss how to go from the classical setting to the automorphic setting, along the way explaining why parabolic subgroups are relevant when defining the cuspidal condition. One in particular also has a nice discussion of Weil groups and statements of the main conjectures. I can't seem to find it right now, but if you would like something specific send me a message.

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Hi Mike. Thanks, I'll take a look. I don't know how to send private messages on this website, but I'll be very grateful if you could find specific references for me! –  Callum Fitzpatrick Apr 27 at 21:42
    
@Callum: it is not possible to send private messages on StackExchange sites. This was a deliberate design choice; the point is that MO is not for private conversation but for the public asking and answering of specific and focused questions. –  Qiaochu Yuan Apr 28 at 0:32
    
@Callum: You can find the surveys mentioned above in the Proceedings book edited by Knapp and Bailey. –  Mike B Apr 30 at 19:56

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