  We provide an explicit parameterization (a.k.a. "reconstruction") of the
permutation-invariant big $\mathcal{J}$-function of partial flag varieties,
treated as a non-abelian GIT quotient of a linear space, by applying
pseudo-finite-difference operators on the big $\mathcal{J}$-function of the
associated abelian quotient. The technique employed in this paper which is
based on the toric fixed point localization and recursion relations is in
principle similar to that in the grassmannian case [20], but the main
complication comes in the form of possibly non-isolated 1-dimensional toric
orbits in the abelian quotient. A substantial part of the paper is dedicated to
developing the tools for dealing with non-isolated orbits, including a
vanishing result for their contributions to the recursion coefficients.
Theories twisted by Eulerian modifications and level structures are also
considered, and explicit formulas of values of their $\mathcal{J}$-functions
are computed.
