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As Derek Holt comments, cohomology has complications even for fimite general linear groups. Probably you are using the term "reductive" too casually and should replace it by "simple" or perhaps "semis …

First of all, I'd inquire what role the characteristic of the field plays here. It's true that the finite twisted groups rely on characteristics 2, 3 especiallu, but infinite twisted groups include …

Chapter VI of my old Springer Lecture Notes in Mathematics 789 Arithmetic Groups (1980) is in English and gives a version of H. Matsumoto's and C. Moore's arguments for the subgroups of $\operatorname …

There are probably multiple ways to see that $G(\mathbb{R})$ is non-compact when $G$ is a Chevalley group (in either the narrow sense of Chevalley or the broader sense of Steinberg's lectures). One …

In the original sense, Chevalley groups are generated by copies of the additive group of the field and are in fact simple as abstract groups if the field is not too small. (This was the motive for …

Concerning the first question in the header (and some of your preparatory remarks), it's useful to keep in mind the Planche VII for $E_8$ at the end of Chapters 4-6 of Bourbaki's treatise Groupes et a …

I agree that the question needs a better formulation. In any case, an approach by Demetris Deriziotis might be useful because it's based on a different kind of analysis: see here (freely availab …

This is just a comment but in community wiki format. Most studies of semisimple (or reductive) algebraic groups and finite groups of Lie type emphasize counting the number of classes of various el …

Keep in mind that it's tricky to assemble a character table for all finite groups of Lie type in a family, and in particular the thesis work by Srinivasan at Manchester (supervised by J.A. Green) excl …

[EDIT] Since my various edits got quite long, maybe I can answer the question more directly and refer to previous versions for elaboration. A key elementary result can be found in Feit's 1982 monogra …

Say $(W,S)$ is a finite Coxeter group, such as a Weyl group (which satisfies an additional crystallographic condition). Assume also that $W$ is irreducible. Then it has a longest element $w_o$ r …

You are asking many questions here, about which much is written down in the rather large literature of the subject. Maybe I can point to some of the answers. In the 1955 paper of J.A. Green on fini …

Let me add a few comments in community-wiki format. There doesn't seem to be a convenient reference, apart from the one in Digne-Michel which Jay Taylor cites. But even here, the authors don't give …

I'll add an overlong comment to provide more perspective, in community-wiki format. The question is probably a bit misguided, even taken as a purely pedagogical one (the older mathematics involved h …

Steinberg's lecture at the 1966 ICM in Moscow here surveyed his work on regular elements of semisimple algebraic groups, while also formulating a number of then-open questions as "problems" (with posi …