Isomorphisms of Hecke modules and parabolic Kazhdan–Lusztig polynomials

We define and study some Hecke modules which generalize the ones defined by Deodhar in [10] and we find isomorphisms between them. As consequences, we obtain equalities between parabolic Kazhdan–Lusztig polynomials of different Coxeter groups. In particular, we show that any parabolic Kazhdan–Lusztig polynomial equals a parabolic polynomial of a maximal quotient, and we find equalities between the parabolic polynomials of quasi-minuscule quotients and those of certain maximal quotients of the corresponding affine Weyl groups.