Conjectures about certain parabolic Kazhdan--Lusztig polynomials

Irreducibility results for parabolic induction of representations of the general linear group over a local non-archimedean field can be formulated in terms of Kazhdan–Lusztig polynomials of type $A$. Spurred by these results and some computer calculations, we conjecture that certain alternating sums of Kazhdan–Lusztig polynomials known as parabolic Kazhdan–Lusztig polynomials satisfy properties analogous to those of the ordinary ones.