  For positive integers m >= n >= p, we compute the GL_m x GL_n-equivariant
description of the local cohomology modules of the polynomial ring S of
functions on the space of m x n matrices, with support in the ideal of p x p
minors. Our techniques allow us to explicitly compute all the modules
Ext_S(S/I_x,S), for x a partition and I_x the ideal generated by the
irreducible sub-representation of S indexed by x. In particular we determine
the regularity of the ideals I_x, and we deduce that the only ones admitting a
linear free resolution are the powers of the ideal of maximal minors of the
generic matrix, as well as the products between such powers and the maximal
ideal of S.
