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

Let X subset of G/B be a Schubert variety in a flag manifold and let 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 . 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.