TITLE: What information is contained in the Kazhdan-Lusztig polynomials?
QUESTION [21 upvotes]: The Kazhdan-Lusztig polynomials contain all kinds of representation theoretic (and other kinds of) informations. 
For example the character of a simple module over a Lie algebra with Weyl group $W$ can be read of from the KL-polynomials in the following way:
$$ch(L_w)=\sum_y (-1)^{l(w)-l(y)}P_{y,w}(1) ch(M_y)$$
here $L_w$ resp. $M_w$ denote the simple resp. Verma module of highest weight $-w(\rho) -\rho$.
What are other examples of important information encoded in KL-Polynomials?

REPLY [3 votes]: I'm not sure whether this would be considered an answer but there is a "concrete" interpretation of the coefficients given in http://arxiv.org/abs/1212.0791 .