We introduce deformations of Kazhdan-Lusztig elements and specialised non-symmetric Macdonald polynomials, both of which form a distinguished basis of the polynomial representation of a maximal parabolic subalgebra of the Hecke algebra. We give explicit integral formula for these polynomials, and explicitly describe the transition matrices between classes of polynomials. We further develop a combinatorial interpretation of homogeneous evaluations using an expansion in terms of Schubert polynomials in the deformation parameters. © 2011.
De Gier, J., Lascoux, A., & Sorrell, M. (2012). Deformed Kazhdan-Lusztig elements and Macdonald polynomials. Journal of Combinatorial Theory. Series A, 119(1), 183–211. https://doi.org/10.1016/j.jcta.2011.08.002