Deformed Kazhdan-Lusztig elements and Macdonald polynomials

We introduce deformations of Kazhdan-Lusztig elements and degenerate nonsymmetric Macdonald polynomials, both of which form a distinguished basis of the polynomial representation of the 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.