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InformationsWhatinformationis contained in KazhdanLusztigtheKazhdan-Lusztig polynomials?

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 $-w(\rho) -\rho$.