Positivity of some intersections in K0(G/B)

Let G be a reductive group. Each Schubert subvariety Sw of the flag variety G/B determines a class [Sw] of its Grothendieck group K0(G/B). Indeed the set of all these classes, which are parametrized by the elements w of the Weyl group W, is a well-known Z-basis of K0(G/B). Therefore, the multiplication by an element L in K0(G/B) is identified with a square matrix s(L)=(sxy(L)) with entries indexeded by the pairs (x,y) of elements of W. Equivalently, its rows are the coordinates of the intersection product L.[Sy]. The main result states that the entries of the matrix s(L) are all ≥0, whenever L is the class of an effective line bundle. Indeed we show that Sxy(L) occurs as multiplicities of B-modules in some filtrations, and therefore the proof relies on the existence result of such filtrations, which was proved in [12].