This paper is concerned with local cohomology sheaves on generalized flag varieties supported in closed Schubert varieties, which carry natural structures as (mixed Hodge) $\mathcal{D}$-modules. We employ Kazhdan–Lusztig theory and Saito’s theory of mixed Hodge modules to describe a general strategy to calculate the simple composition factors, Hodge filtration, and weight filtration on these modules. Our main tool is the Grothendieck–Cousin complex, introduced by Kempf, which allows us to relate the local cohomology modules in question to parabolic Verma modules over the corresponding Lie algebra. We show that this complex underlies a complex of mixed Hodge modules, and is thus endowed with Hodge and weight filtrations. We execute this strategy to calculate the composition factors and weight filtration for Schubert varieties in the Grassmannian; in particular, showing that the weight filtration is controlled by the admissible augmented Dyck patterns of Raicu and Weyman. As an application, upon restriction to the opposite big cell, we recover the composition factors and weight filtration on local cohomology with support in generic determinantal varieties.

We endow certain GKZ-hypergeometric systems with a natural structure of a mixed Hodge module, which is compatible with the mixed Hodge module structure on the Gauß–Manin system of an associated family of Laurent polynomials. As an application we show that the underlying perverse sheaf of a GKZ-system with rational parameter has quasi-unipotent local monodromy.