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After years of putting it off, I finally sat down, read, and understood the classification of connected reductive groups via root data.

That's a non-trivial theory! I'm hoping that now that I am done I can reap some benefits. What are some immediate applications/corollaries of this theory? Now that I understand this classification, what can I do with it? At the moment, I don't really feel like I can do anything I couldn't do before...

Number theoretic and algebro-geometric applications are great! But I'm open to anything.

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    $\begingroup$ Worth noting is that much of the machinery used to classify reductive groups is also extremely useful in classifying/studying representations of these groups. $\endgroup$
    – Sam Hopkins
    Commented Sep 1, 2020 at 23:49
  • $\begingroup$ Since this seems not to be attracting much of a big-list, maybe it should be retagged gr.group theory? $\endgroup$
    – Mark Wildon
    Commented Sep 4, 2020 at 19:07
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    $\begingroup$ There are at least two different ways to interpret this question. One is, what are some applications of the machinery (e.g., root systems) that was developed to classify reductive groups? A second is, what corollaries are there of the classification theorem itself? The latter includes theorems proved by checking each group case by case. As for the former, I'm tempted to say that it'd be easier to list major results that don't make use of the root system. I'm not an expert but it feels like every paper in this area begins, "Let $\mathfrak g$ be a Lie algebra and $\Phi$ its root system..." $\endgroup$
    – Timothy Chow
    Commented Sep 5, 2020 at 22:51

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An immediate application is the existence of the Langlands complex dual group. If $G$ is a connected reductive group over a field $F$ and $\overline{F}/F$ is an algebraic closure, then $G^\vee$ is the connected reductive group over $\mathbb{C}$ whose root datum is dual to that of $G_{\overline{F}}$, i.e. is obtained by interchanging roots with coroots and simple roots with simple coroots.

In the case when $F$ is a local or global field, Langlands's discovery of $G^\vee$ and the related $L$-group ${}^LG$ in 1966 lead him to, and these groups figure prominently in: the local and global Langlands correspondences, the general definition of local and automorphic $L$-functions, and the principle of functoriality.

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Come that far, one can begin to exploit the subgroups, turn one's attention to symmetric spaces, of which there are the compact and noncompact ones, the riemannian and the hermitian ones. And one can begin to exploit the arithmetic subgroups and so delve into locally symmetric varieties with various geometric and arithmetic properties. Come that far, whole universes open up: examples, counterexamples, theorems in Differential Geometry, and the whole garden of Number Theory and Arithmetic Geometry with Shimura varieties, compactificaions of locally symmetric varieties of all sorts relating to moduli spaces, Langlands conjectures...

Needless to say, quite useless to give references - such a list would be endless and beyond. So it's up to you to get lost your own way.

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Given a prime $p$ and a connected reductive algebraic group $G$ over $\mathbb{F}_p^{\mathrm{alg}}$ with a Frobenius map $F$, the fixed points $G^F$ are a finite group 'of Lie type'. The finite groups of Lie type are the main case in the Classification Theorem of Finite Simple Groups. They can all be obtained in a uniform way by this construction, except for the Suzuki and Ree groups. (Roughly speaking, these are also obtained using the data from root systems and Dynkin diagrams, but they require further automorphisms that do not descend from the algebraic group.) Various structural properties of finite groups of Lie type follow easily from their analogues in the algebraic group, for example that they have $BN$ pairs.

I'm surprised this wasn't already an answer. It seems close enough to an 'application' to me to be worth mentioning. I admit it is far from an 'immediate application'.

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  • $\begingroup$ Is this really an application of the classification? In the following sense: does the proof of CSFG actually use the fact that we have an exhaustive list of reductive groups, or does it only use the constructions of the reductive groups without the fact that the classification is exhaustive? $\endgroup$
    – Qiaochu Yuan
    Commented Sep 4, 2020 at 22:39
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    $\begingroup$ The latter, according to my understanding. (And I don't think this will be changed by any of the programmes to revise the proof of CFSG.) But I would defend my answer on the grounds that many useful structural properties of the finite groups do follow fairly immediately from the analogues in the algebraic group. E.g. existence of tori (and conjugation, although split/non-split tori in the finite case complicate things), $BN$ pairs, and the (related) Bruhat decomposition. $\endgroup$
    – Mark Wildon
    Commented Sep 5, 2020 at 11:25
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check work of David Vogan, and a book of Trappa Vogan ...

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    $\begingroup$ "Work of David Vogan" is only marginally more specific than "the papers of George Lusztig". It would probably be more helpful to mention some good entrées. $\endgroup$
    – LSpice
    Commented Sep 3, 2020 at 0:43
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    $\begingroup$ By "a book of Trappa Vogan" do you mean Representations of Reductive Groups: In Honor of the 60th Birthday of David A. Vogan, Jr., edited by Monica Nevins and Peter E. Trapa? $\endgroup$
    – Timothy Chow
    Commented Sep 3, 2020 at 21:28
  • $\begingroup$ Sorry i confused you, i mean the book The Langlands classification and irreducible characters (D. Barbasch, J. Adams and D. Vogan), Birkhauser (1992), Boston-Basel-Berlin. $\endgroup$
    – jorge vargas
    Commented Sep 4, 2020 at 12:16

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