We compute the local cohomology modules $$\mathcal {H}_Y^{\bullet }(X,\mathcal {O}_X)$$ in the case when $$X$$ is the complex vector space of $$n\times n$$ symmetric (respectively, skew-symmetric matrices) and $$Y$$ is the closure of the $$\hbox {GL}$$-orbit consisting of matrices of any fixed rank, for the natural action of the general linear group $$\hbox {GL}$$ on $$X$$. We describe the $$\mathcal {D}$$-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 1990s. Another consequence of our work is that we obtain a description of the decomposition into irreducible $$\hbox {GL}$$-representations of the local cohomology modules (the analogous problem in the case when $$X$$ is the vector space of $$m\times n$$ matrices was treated in earlier work of the authors).

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