  We compute the GL-equivariant description of the local cohomology modules
with support in the ideal of maximal minors of a generic matrix, as well as of
those with support in the ideal of 2n x 2n Pfaffians of a (2n+1)x(2n+1) generic
skew-symmetric matrix. As an application, we characterize the Cohen-Macaulay
modules of covariants for the action of the special linear group SL(G) on G^m.
The main tool we develop is a method for computing certain Ext modules based on
the geometric technique for computing syzygies and on Matlis duality.
