Topology of intersections of Schubert cells and Hecke algebra

We consider intersections of Schubert cells script B signα · script B sign and σscript B signσ^- β · script B sign in the space of complete flags F = SL/script B sign, where script B sign denotes the Borel subgroup of upper triangular matrices, while α, β and σ belong to the Weyl group W (coinciding with the symmetric group). We obtain a special decomposition of F which subdivides all script B signα · script B sign ∩ σscript B signσ^- β · script B sign into strata of a simple form. It enables us to establish a new geometrical interpretation of the structure constants for the corresponding Hecke algebra and in particular of the so-called R-polynomials used in Kazhdan-Lusztig theory. Structure constants of the Hecke algebra appear to be the alternating sums of the Hodge numbers for the mixed Hodge structure in the cohomology with compact supports of the above intersections. We derive a new efficient combinatorial algorithm calculating the R-polynomials and structure constants in general.