  We compute the local cohomology modules H_Y^(X,O_X) in the case when X is the
complex vector space of n x n symmetric, respectively skew-symmetric matrices,
and Y is the closure of the GL-orbit consisting of matrices of any fixed rank,
for the natural action of the general linear group GL on X. We describe the
D-module composition factors of the local cohomology modules, and compute their
multiplicities explicitly in terms of generalized binomial coefficients. One
consequence of our work is a formula for the cohomological dimension of ideals
of even minors of a generic symmetric matrix: in the case of odd minors, this
was obtained by Barile in the 90s. Another consequence of our work is that we
obtain a description of the decomposition into irreducible GL-representations
of the local cohomology modules (the analogous problem in the case when X is
the vector space of m x n matrices was treated in earlier work of the authors).
