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I have finished learning about linear algebraic groups (minus their representation theory) and the associated algebraic structures (root data, root systems, etc.), and will next attempt to summarize for myself the main components related to their representation theory.

It's quite confusing for the uninitiated!

  1. I want the beginning of the story to be "the easy case", by which I mean the case for which classification of irreducible representations is done via the Theorem of the Heighest Weight. Sources that I've glanced at discuss two types of cases: the semisimple Lie algebra case (which I choose not to care about), and the compact real Lie group case. I somehow care about neither one... I want to discuss (split) reductive groups over a general field. Over the reals, the reductive groups correspond to the real compact Lie groups... Is it correct to say that the Theorem of Heighest Weight applies in general to split reductive groups over a general field? And that this is the "easy case"? Would it apply to reductive or semisimple groups?

  2. I'm somewhat confused in general about at what point it is necessary to restrict to unitary representations. This is my understanding: for finite groups and for compact groups all group representations can be given an inner product in such a manner as to make them unitary, and this is essentially the proof that the category of representations in these cases are semisimple. So I guess the point is that for general reductive groups, even though their category of representations is semisimple, not all representations can be made unitary... Or am I confused, and somehow being reductive should be seen as a generalization of being compact?

  3. On the one hand, it appears that the classification of irreducible (unitary?) representations of reductive groups is classified using the Theorem of Highest Weight and is therefore "the easy case". But I guess the point is that once you look at $G(K)$ for some ring $K$ then this stops being the easy case? For example: $K=\mathbb{R}$, or the adeles, or $\mathbb{C}$. So let's start with an easy question: is the representation theory of $G(\mathbb{C})$ the same as the representation theory of $G$?

  4. Can you put into context for me the following phrases: cuspidal representations - is that a term that only applies to the representation theory of the adelic points of $G$? What about tempered representations? Smooth representations? Admissible representations? Are they only for $G(\mathbb{R})$? Are there several unrelated notions of admissible/smooth representations? I see them arise with very different definitions in different context, and I'm not sure if I need to think of them as specific examples of one phenomenon. What are these good for, and why is it not covered by the Theorem of the Highest Weight? Is it hopeless to classify unitary representations that are not smooth/admissible?

  5. The Langlands classification "is a description of the irreducible representations of a reductive Lie group G". Why was that not already covered by the Theorem of the Highest Weight? Is that point that here we are dealing with a reductive Lie group as opposed to a reductive linear algebraic group? Or is that point that we're looking at $G(\mathbb{R})$? It's very hard for me to draw the line between what is easy and what is difficult...

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    $\begingroup$ It sounds like you care only about algebraic representations. In that case, you don't have to worry about anything in 4 or 5 - those are for non-algebraic representations. So if you are looking at representations of $G(\mathbb{R})$, this is happening in a different category (Lie group representations) and so you get many more (infinite-dimensional) representations. Unitarity is another concept that isn't very applicable in the algebraic setting. $\endgroup$ – dhy Oct 20 '20 at 0:18
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    $\begingroup$ For split reductive groups over an arbitrary field, see Chapter 22 of Milne's book on Algebraic Groups (CUP 2017). $\endgroup$ – anon Oct 20 '20 at 0:27
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    $\begingroup$ Anthony Knapp has written some roadmaps for learning about the Langlands Program. $\endgroup$ – Kimball Oct 20 '20 at 2:19
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    $\begingroup$ At first, I thought the title was asking for the appropriate choice of morphisms for a category. :-) $\endgroup$ – LSpice Dec 7 '20 at 17:21
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Probably someone will step in with a more detailed answer soon... but here are some comments.

I think the line you are looking for between easy and hard could be the following:

  1. The algebraic representations of a split reductive algebraic group $G$, and

  2. The representations of some associated Lie groups $G(\mathbb R)$ or $G(\mathbb C)$ (or indeed p-adic groups $G(\mathbb Q_p)$, or adelic groups $G(\mathbb A)$, ...).

In case 1), the algebraic representations (of a split reductive algebraic group over a field, say) are determined by highest weight theory. In this case all irreducible representations are finite dimensional. The classification of representations of compact Lie groups is the same (so, for example, the algebraic representations of $SL_2(\mathbb R)$ and $SL_2(\mathbb C)$ are the same as the Lie group representations of $SU(2)$). The book of Fulton and Harris covers this topic in some detail.

In case 2), say $G$ is defined and split over $\mathbb R$, then we have an associated Lie group $G(\mathbb R)$. This is a non-compact Lie group, and it will typically have infinite dimensional irreducible representations. This theory is much more intricate. For example, one must think about what kind of topologies you want to consider on the underlying vector space of the representation.

Amongst such representations, we have the class of admissible representations. A key point about admissible representations is that they they are determined by their Harish-Chandra $(\mathfrak g, K)$-module, which is a purely algebraic gadget. The Langlands classification for real reductive groups is about admissible representations (one version of which reduces the classification to so-called tempered representations).

Amongst admissible representations, unitary representations (those that can be represented by unitary operators on a Hilbert space) are of particular importance and interest. The classification of such is more subtle and less well-understood.

There are a bunch of textbooks and lecture notes (e.g. Knapp, Trapa). One approach is to focus on the case of $SL_2(\mathbb R)$. David Ben-Zvi taught a class in this direction at UT Austin some years ago - you can find notes here: https://web.ma.utexas.edu/users/benzvi/GRASP/lectures/benzvi/mylectures.html

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  • $\begingroup$ This is a great answer, but aren’t (at least holomorphic) representations of $G(\mathbb C)$ also on the easy side? I’d expect it only really to be when you move to rational points over non-algebraically closed fields that you see the real difficulty. $\endgroup$ – LSpice Dec 7 '20 at 17:25
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    $\begingroup$ Thanks for the comment! In category 2, had in mind (possibly infinite dimensional, admissible) representations of $G(\mathbb C)$ as a real Lie group. These should correspond to $(\mathfrak g,\times \mathfrak g,G)$-modules, i.e. Harish-Chandra bimodules. I guess the complex case is easier than the general (e.g. all maximal tori are conjugate etc.), but I wanted to distinguish it from the study of algebraic representations of the algebraic group $G/\mathbb C$ (which is governed by highest weight theory) as that seemed to be a point of confusion for the OP. $\endgroup$ – Sam Gunningham Dec 7 '20 at 18:28
  • $\begingroup$ You're quite right—representations of $G(\mathbb C)$ as a real group (i.e., smooth, not necessarily holomorphic) are more interesting. $\endgroup$ – LSpice Dec 7 '20 at 19:26

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