Many characteristic classes of singular varieties are defined using resolution of singularities. I will show that the stable basis for a generalized flag variety may be constructed in this way. The geometry of Bott-Samelson resolution of singularities induces a Hecke-type algebra action. It allows for an effective computation of stable basis. This formula was previously known in the special case of canonical slope. The talk is based on a joint work with A. Weber.