Some geometric consequences of the symmetric reduction

In this article, we derived some consequences to the symmetrization process developed in {Deng23}. This consists a geometric derivation of part of the properties which uniquely determines the Kazhdan-Lusztig polynomials of type Aₙ as well as an interpretations of the last property by the decomposition theorem of {BBD}. Finally, the relation of geometry of graded nilpotent classes to Parabolic Kazhdan-Lusztig polynomials is discussed for the purpose of later applications to Zelevinsky's derivatives.