On the Computation of the Cohomological Invariants of Bott–Samelson Resolutions of Schubert Varieties

Let X ⊆ G / B be a Schubert variety in a flag manifold and let π : X ~ → X be a Bott–Samelson resolution of X . In this paper, we prove an effective version of the decomposition theorem for the derived pushforward R π ∗ Q X ~ . As a by-product, we obtain recursive procedure to extract Kazhdan–Lusztig polynomials from the polynomials introduced by Deodhar [ 7 ], which does not require prior knowledge of a minimal set. We also observe that any family of equivariant resolutions of Schubert varieties allows to define a new basis in the Hecke algebra and we show a way to compute the transition matrix, from the Kazhdan–Lusztig basis to the new one.