  We introduce deformations of Kazhdan-Lusztig elements and specialised
nonsymmetric 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.
