A well-known and fundamental property of the Macdonald polynomials P_λ (x;q, t) is their invariance under the transformation sending (q, t) to (q^-1, t^-1). Recently, Concha and Lapointe showed that this property extends in an interesting, nontrivial way to an identity for partially symmetric Macdonald polynomials. Their identity played a key role in the work of Bechtloff Weising and Orr linking partially symmetric Macdonald polynomials to parabolic flag Hilbert schemes. In this paper, we refine the Concha-Lapointe identity to a sub-family of Alexandersson's permuted basement Macdonald polynomials and give a combinatorial proof of the refined identity. We show also that the Concha-Lapointe identity is equivalent to the assertion that (normalized) partially symmetric Macdonald polynomials are fixed under the Kazhdan-Lusztig involution.
Orr et al. (Thu,) studied this question.