A twisted duality for parabolic Kazhdan–Lusztig R-polynomials

We prove a duality result for the parabolic Kazhdan-Lusztig R-polynomials of a finite Coxeter system. This duality is similar to, but different from, the one obtained in [J. Douglass, Comm. Algebra, 18 (1990), 371-387.]. As a consequence of our duality we obtain an identity between the parabolic Kazhdan-Lusztig and inverse Kazhdan-Lusztig polynomials of a finite Coxeter system. We also obtain applications to certain modules defined by Deodhar and derive a result that gives evidence in favor of Marietti's combinatorial invariance conjecture for parabolic Kazhdan-Lusztig polynomials.