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Lie groups can leverage quite a bit of useful methods from differential geometry, and vice-versa.

I've just started seriously reading Robert Wilson's Finite Simple Groups and I'm curious about whether we can do something analogous for finite groups of Lie type?

For finite groups of Lie type, is there some analogous interplay with (I guess) algebraic geometry over finite fields? Something which would make the following diagram commute (or something similar)?

$$\require{AMScd} \begin{CD} \begin{pmatrix}\mbox{Differential}\\ \mbox{Geometry}\end{pmatrix} @>{\text{generalize}}>> \begin{pmatrix}\mbox{Algebraic}\\ \mbox{Geometry}\end{pmatrix} @.\\ @VVV @VVV @.\\ \mbox{Lie Groups} @<{\mathbb{F}\in\{\mathbb{R},\mathbb{C}\}}<<\begin{pmatrix}\mbox{Linear}\\ \mbox{Algebraic Group}\end{pmatrix} @>{\mathbb{F}\ \text{finite}}>> \begin{pmatrix}\mbox{Finite Group}\\ \mbox{of Lie Type}\end{pmatrix} \end{CD} $$

Well, does this even work? That is to say, are the linear algebraic groups over finite fields even finite groups of Lie type? If so, are the vector fields on the algebraic variety the corresponding Lie algebra? What about Haar measures? Is the maximal torus some generalized "torus"?

Is this geometric aspect of finite groups of Lie type discussed in detail somewhere?

(The closest question like this I could find is Most general definition of Borel and parabolic Lie algebras? wherein an answer recommends Tavel and Yu's Lie Algebras and Algebraic Groups, which I'm looking into further, but I'm wondering if there's more to this than one book...)

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Try

Carter, Finite groups of Lie type

Digne & Michel, Representations of finite groups of Lie type

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    $\begingroup$ +1 Carter's book is stupendous, and I second this recommendation. $\endgroup$ Nov 4, 2021 at 3:21
  • $\begingroup$ OK, I found a copy of Carter's book (comparatively cheap, but it's shipping from Germany, so I should expect it arrive by December). Unrelated question: it feels like even the wreath product resembles a "discrete fiber bundle", is this coincidence? Or is there some deeper algebraic geometric reason? $\endgroup$ Nov 4, 2021 at 14:33
  • $\begingroup$ @Alex Nelson: It is a valuable book. The other question: in Lie group theory, every subgroup (normal or not) gives a principal $H$ bundle $G\to G/H$. So extensions and bundles should be related. Write down your ideas, later on they do clarify. $\endgroup$
    – orangeskid
    Nov 5, 2021 at 0:47
  • $\begingroup$ @orangeskid Thanks, I'll almost certainly have more questions in the future. Could I ask why you recommend Digne & Michel? $\endgroup$ Nov 5, 2021 at 1:21
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    $\begingroup$ @Alex Nelson: They build representations of finite groups of Lie type using algebraic geometry ( $l$-adic cohomology). Also, the exposition is fairly streamlined. $\endgroup$
    – orangeskid
    Nov 5, 2021 at 1:32

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