What are immediate applications of the classification of connected reductive groups? 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.
 A: 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.
A: 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.
A: 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'.
A: check work of David Vogan, and a book of Trappa Vogan ...
