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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 …

Though I can't answer the question directly, it may be helpful to clarify some of the issues here. First, it's usually best to focus on the case when $W$ (or the root system of the Lie algebra) is …

The significance of $\rho$ (and the associated dot-action of the Weyl group or other Coxeter group) in representation theory is discussed from many angles in the earlier linked question. When deal …

Maybe I can provide a belated kind of answer to my own question, which I came across when looking for something else in the older literature. Vinay Deodhar published a paper in 1990 here (just before …

[EDIT] Maybe it's useful after all this time to give a more complete and uniform answer to both of the questions asked, by referring to Theorem 3.4(i) in Springer's 1974 paper on regular elements of f …

It's not clear exactly what your first sentence is asking for (there are a variety of surveys and books). For instance, are you only interested in finite Coxeter groups? The embeddings of the non …

Here are some cautionary remarks, plus references. You ask: Is there a more comprehensive list of such polynomials? The answer seems to be no. Lists get long very quickly, and as I commented ea …

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 …

[EDIT] Concerning your specific question, my earlier answer was too offhand. After trying this with pencil and paper, I'm very doubtful that the multiplicity of the zero weight in the finite dimensi …

The question is natural but looks difficult to approach strictly within the combinatorial definitions. Maybe it's helpful here to suggest a broader geometric framework, which applies more generally …

The formulation is somewhat out of focus, starting with the notation $a_n(x)$ for characteristic polynomial (what is $n$?). The roots indicated do occur in Coxeter's formulation, but not as the eig …

The question may be somewhat open-ended, but perhaps I can focus some aspects of it. First, I'm not sure which papers by Littelmann you've looked at, but probably the most definitive treatment of his …

As Koen S points out, the longest element of an irreducible Weyl group is treated in an earlier question (in fact, it comes up in several questions). The question asked here presupposes a standard li …

It's worth keeping in mind here that the 1979 paper by Kazhdan and Lusztig dealt quite generally with Iwahori-Hecke algebras of arbitrary Coxeter groups, not just the finite Weyl groups (or even the W …

The question is stated a bit loosely, but the basic literature goes back about four decades to work of Brieskorn and Deligne. Since I'm not an expert on these matters I can only refer to the basic …