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 [9]. 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. (C) 2017 Published by Elsevier Inc.