Topology of intersections of Schubert cells and Hecke algebra

We consider intersections of Schubert cells BB and B ?1 B in the space of complete ags F = SL=B, where B denotes the Borel subgroup of upper triangular matrices, while , and belong to the Weyl group W (coinciding with the symmetric group). We obtain a special decomposition of F which subdivides all B B \ B ?1 B 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 eecient combinatorial algorithm calculating the R-polynomials and structure constants in general. x1. Preliminaries Intersections of pairs and, more generally, of k-tuples of Schubert cells each belonging to its own Schubert cell decompositions of a ag space appeared in many articles, see e.g. BB], KL1],, KL2], De1-2], GS]. Topological properties of such intersections are of particular importance in representation theory. Intersections of some special interesting arrangements of Schubert cells are directly related to the problem of representability of matroids, see GS]. Most likely, for somewhat general class of arrangements of Schubert cells their intersections and complements to such intersections do not have nice topological properties. Even the problem of nonemp-tyness of such intersections in the complex ag varieties is hard. However, in the important case of pairs of Schubert cells topology of their intersections appears to be more accessible. Namely, one can obtain a special decomposition of such intersections (generalizing the standard Schubert cell decomposition) into products of algebraic tori and linear subspaces of diierent dimensions. These strata in their turn 1991 Mathematics Subject Classiication. Primary 14M15, Secondary 20F55.