The talk will focus on two invariants of Schubert varieties which are polynomials defined on pairs of permutations in the symmetric group. The first invariant is the celebrated Kazhdan-Lusztig polynomials defined using Hecke algebras. The second invariant is the h-polynomials of the local rings of Schubert varieties. We introduced a combinatorial concept called"drift configuration" which characterizes the second invariant for covexillary Schubert varieties, and we use this characterization to give a relation between the above two invariants. This is joint work with Alex Yong.