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The case of an affine Weyl group is apparently the only one which has been looked at closely. But it may be hard to answer your specific question. As far as I know, there are two relevant papers, …

Already in the case of finite symmetric groups, one can find any polynomial with non-negative integral coefficients and constant term 1 as KL polynomial for some pair of group elements. See the pa …

The answer is yes, for fairly elementary reasons, though it's not easy to give a reference. The point is partly that the polynomials are undefined for two elements of the Weyl group not related by t …

Contrary to Verma's initial impression, the internal structure of a "Verma module" (Dixmier's terminology) tends to be extremely complicated. However, this complexity only shows up in ranks 3 or hig …

Concerning the basic question (a), my first reaction is to be skeptical. Though as you say miracles sometimes do occur in this subject. I don't have a counterexample at my fingertips. Beyond the …

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 …

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 …

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 …