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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 …

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 …

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 to the question is "yes", allowing for a generous interpretation of "direct way". This will follow from the recently posted work of Ben Elias and Geordie Williamson on non-negativity of c …

Here you are working over $\mathbb{C}$ (or perhaps any other splitting field of characteristic 0 for $G$). So the representation you are starting with is just the direct sum over all characters $\ch …

The early work of Borel showed in effect how to interpret the cohomology algebra of a flag variety as the coinvariant algebra associated to the Weyl group, which affords the regular representation of …

The answer to both questions is positive (since mathematicians tend to leave no stone unturned). See for example: Geck, Meinolf; Lambropoulou, Sofia. Markov traces and knot invariants related to Iw …

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 …

There are many relevant papers, but the most convenient book to consult is: MR1778802 (2002k:20017) 20C15 (20C08 20F55), Geck, Meinolf (F-LYON-GD); Pfeiffer,G¨otz (IRL-GLWY) Characters of finite Coxet …

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 …