A footnote to a paper of Deodhar

Let $X\subseteq G\text{slash}B$ be a Schubert variety in a flag manifold and let $\pi :\tilde{X}\to X$ be a Bott-Samelson resolution of $X$. In this paper we prove an effective version of the decomposition theorem for the derived pushforward $R{\pi }_{\ast }{\mathbb{Q}}_{\tilde{X}}$. As a by-product, we obtain recursive procedure to extract Kazhdan-Lusztig polynomials from the polynomials introduced by V. Deodhar in \cite{Deo}, 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.