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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 …

For a single group of relatively small order, computer methods are available, as pointed out by John Wiltshire-Gordon. For a general theoretical approach, there is much less to go on in terms of me …

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 …

There is unfortunately no "formula" for tensor products in prime characteristic. Instead you can derive a list of composition factors $L(\lambda)$ (with multiplicity) by recursion. When $p=2$ there …

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 …

In their seminal 1979 paper Representations of Coxeter groups and Hecke algebras (Invent. Math. 53, doi:10.1007/BF01390031), Kazhdan and Lusztig studied an arbitrary Coxeter group $(W,S)$ and the corr …

Like some of his other important ideas, Bernstein's presentation has mostly been disseminated through the papers of other people. Probably the most influential is the 1989 JAMS paper by Lusztig, fre …

The questions here have certainly been explored (though not definitively) in many recent papers or preprints on arXiv. Look for example at the arXiv paper by Stephen Doty Link, as well as many othe …

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 …

The answers to your several closely related questions are not yet known, though many parts of the story have emerged. In particular, there is no "formula" for the number of simple $U_\chi(\mathfrak{g …

First I'd comment that there are quite a few questions on MO related to this one, but apparently not quite identical. (It's hard to search the site efficiently.) In any case I won't attempt a detail …

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 …

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. …