  We study the structure of local cohomology with support in Pfaffian varieties
as a module over the Weyl algebra D_X of differential operators on the space X
of n x n complex skew-symmetric matrices. The simple composition factors of
these modules are known by the work of Raicu-Weyman in 2016, and when n is odd,
the general theory implies that the local cohomology modules are semi-simple.
When n is even, we show that the local cohomology is a direct sum of
indecomposable modules coming from the pole order filtration of the Pfaffian
hypersurface. We then determine the Lyubeznik numbers for Pfaffian rings by
computing local cohomology with support in the homogeneous maximal ideal of the
indecomposable summands referred to above.
