  We give an explicit recipe for determining iterated local cohomology groups
with support in ideals of minors of a generic matrix in characteristic zero,
expressing them as direct sums of indecomposable D-modules. For non-square
matrices these indecomposables are simple, but this is no longer true for
square matrices where the relevant indecomposables arise from the pole order
filtration associated with the determinant hypersurface. Specializing our
results to a single iteration, we determine the Lyubeznik numbers for all
generic determinantal rings, thus answering a question of Hochster.
