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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 …

Start with the Dynkin diagram of an irreducible root system, typically associated with a simple Lie algebra over $\mathbb{C}$ or a simple algebraic group. Most of the simply-laced ADE diagrams admi …

This concerns one of those "well known" facts, referred to in a recent preprint I've been looking at. In principle it's elementary, but I can't pin down an explicit textbook reference for it. Star …

I'm not sure how best to answer the question formulated here, but I can comment further on references. As Dietrich says, there is a large literature. Ever since the foundational work by Jacobson an …

There is quite a bit of literature by now, in the classical characteristic 0 setting of finite dimensional Lie algebras. Looking up some of the papers listed below on arXiv (usually under math.RT) a …

In characteristic 0 or good prime characteristic, there are standard ways to relate the unipotent variety $\mathcal{U}$ of a simple algebraic group $G$ and the nilpotent variety $\mathcal{N}$ of its L …

There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl group …

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 …

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 …

By now there is a fairly long paper trail dealing with this kind of question, which is usually a byproduct of the study of representation theory over rings of $p$-adic integers, etc. I'm not aware of …

Rather than overload the question with side remarks, I'll add some background here in community-wiki format to indicate what can be gotten from Springer's relatively elementary treatment of regular el …

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 …

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 …