The Kazhdan-Lusztig polynomials encoded a good deal of topological information concerning Schubert varieties. For example, in
Kazhdan, David; Lusztig, George. Schubert varieties and Poincaré duality. Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), pp. 185-203, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980.
it is proved that if $W$ is a Weyl group, the coefficients of the Kazhdan-Lusztig polynomial $P_{u,v}(q)$ equal the dimension of the local intersection homology spaces of the Schubert variety $X_v$ at a point on the Schubert cell indexed by $u$.
You may want to see chapters 5 and 6 of the following book for more applications and references:
Björner, Anders; Brenti, Francesco. Combinatorics of Coxeter groups. Graduate Texts in Mathematics, 231. Springer, New York, 2005. xiv+363 pp. ISBN: 978-3540-442387; 3-540-44238-3
