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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 …

The Weyl character is expressed as a quotient, which is elegant but does not easily tell you how to express the character as a nonnegative $\mathbb{Z}$-linear combination of $W$-orbits labelled by dom …

This may not be precisely what you ask for, but my colleague Paul Gunnells wrote a nice article for the May 2006 AMS Notices with some color pictures involving Kazhdan-Lusztig cells in small rank Coxe …

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 …

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 …

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 …

My understanding is that Soergel's approach applies just to finite Weyl groups and not directly to other finite Coxeter groups (or more generally), since what he can actually prove depends on some of …

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 …

[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 …

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 …

To replace my somewhat fuzzy comment, maybe I can formulate a skeptical semi-answer. At any rate your question probably doesn't have a clearcut answer unless you impose strong enough restrictive cond …

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 …

[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 …

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 …

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 …