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It's probably too soon to expect a good historical overview, but for example Steve Kleiman has already written a scholarly article (The development of intersection homology theory) emphasizing the ori …

EDIT: I misunderstood at first what your basic question is but now understand it better. One cautionary case comes from older work of Richard Block here, which includes the rank 1 simple Lie algeb …

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 …

I'm not quite sure what you are looking for, but Green's work (though combinatorial and influential) was only one of the inputs for the Deligne-Lusztig paper of 1976. It might help for example to …

The problem with your highlighted formulation is that it's wrong as stated, unless for example you require that an "irreducible" representation be finite dimensional or have an integral highest weight …

Probably the earliest reference is the 1956-58 Chevalley seminar, available online in typed format, which has been republished in 2005 as a typeset book edited by P. Cartier: see Chapter 21. (No speci …

This area of the subject is somewhat frustrating, since there is a lot of literature but not many satisfactory results involving the entire cohomology ring. For what it's worth, I'll point you to a …

The basic set-up here makes sense in the theory of abstract root systems if one brings (integral) weights into the picture, but it may be more natural to think about the classical characteristic 0 the …

Your conclusion about direct sums is false. It's helpful here to have some examples in mind, such as a typical projective/injective module for the Lie algebra of $G:=\mathrm{SL}_2$, lifted to $G$. Th …

The case of an affine Weyl group is apparently the only one which has been looked at closely. But it may be hard to answer your specific question. As far as I know, there are two relevant papers, …

The main problem I see with this line of questioning is the lack of a clear definition of "inner automorphism group of a finite Lie algebra". Many Lie algebras don't occur as Lie algebras of linear …

I'd be extremely surprised if such tables or database existed, mainly because the number of possible reduced decompositions for a Weyl group element tenda to grow very large as the rank increases. …

For the finite groups GL$_n(\mathbb{F}_q)$ there is an early paper by Lusztig well worth checking out here. This predates his broader work on finite groups of Lie type with Deligne (1976), where they …

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 …

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 …