  We generalize Koll\'ar's package (including torsion freeness, injectivity
theorem, vanishing theorem and decomposition theorem) to polystable locally
abelian parabolic Higgs bundles twisted by a multiplier ideal sheaf associated
with an $\mathbb{R}$-divisor. This gives a uniform treatment for various kinds
of Koll\'ar's package in different topics in complex geometry. As applications,
the weakly positivity (in the sense of Viehweg) and the generic vanishing
property for higher direct image sheaves are deduced.
