Non-negativity properties of RR-polynomials

An explicit formula expressing the coefficients of the R̃-polynomials as a linear combinations of those of the corresponding RR-polynomials is given. Several systems of inequalities satisfied by the coefficients of the RR-polynomials are shown and applied to obtain a lower bound for the coefficient of q2q2 in Kazhdan–Lusztig polynomials. Finally we present a basis of the space of polynomials of a given degree with respect to which both Kazhdan–Lusztig and RR-polynomials have some nice (conjectural) non-negativity properties.