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You are asking several questions here, so it may be useful to separate out what is going on first in the setting of affine reflection groups. This is independent of the application to algebraic grou …

While the question and answers have explained different aspects of the story for complex groups and homogeneous spaces, it might be worthwhile to place the question in a broader context. The complex …

For algebraic groups or Lie groups, the subject of Levi decompositions tends to be surrounded by some mystery in the literature (and in an older question raised here). While I postpone further my in …

To supplement what Francois Ziegler says, I'd point out that the structure of semisimple complex Lie groups has been developed piecemeal over a century or so. The basic results on nilpotent elements …

Like Jay, I don't see any reasonable way to address all parts of your wide-ranging question. You are looking at the intersection of numerous lines of research, motivated in different ways for differ …

This determination of component groups goes back to Elashvili and Alexeevskii, but has been improved somewhat in a 1998 IMRN paper by Eric Sommers and a later joint paper by him and George McNinch her …

EDIT: The question and some of the responses are out of focus, so it's worth clarifying a few of the issues here. First, the intended notion of "regular element" is unclear. In Kostant's early pa …

I'm a bit skeptical as to how simple and low-tech a proof can be, since a fair amount of Lie group theory has to be in hand. Short of going into the full details of Iwasawa decomposition, you might fi …

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 …

It's helpful to point out the original source, in one of Kostant's influential early papers: "The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group", Amer. J. …

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 …

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 …

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 …

The character formula should be viewed here as a purely formal statement about weight multiplicities in the irreducible representation, so the analytic-looking exponential notation for compact Lie gro …