TITLE: What are immediate applications of the classification of connected reductive groups?
QUESTION [18 upvotes]: 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.

REPLY [2 votes]: 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'.