  We employ the inductive structure of determinantal varieties to calculate the
mixed Hodge module structure of local cohomology modules with determinantal
support. We show that the weight of a simple composition factor is uniquely
determined by its support and cohomological degree. As a consequence, we obtain
the equivariant structure of the Hodge filtration on each local cohomology
module. Finally, as an application, we provide a formula for the generation
level of the Hodge filtration on these modules.
