# Lie groups, finite groups of Lie type, and analogous geometric constructs

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...)

• @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. Nov 5, 2021 at 0:47
• @Alex Nelson: They build representations of finite groups of Lie type using algebraic geometry ( $l$-adic cohomology). Also, the exposition is fairly streamlined. Nov 5, 2021 at 1:32