Let S be the polynomial ring on the space of non-square generic matrices or
the space of odd-sized skew-symmetric matrices, and let I be the
determinantal ideal of maximal minors or Pf the ideal of
sub-maximal Pfaffians, respectively. Using desingularizations and
representation theory of the general linear group we expand upon work of
Raicu--Weyman--Witt to determine the S-module structures of
ExtSjβ(S/It,S) and
ExtSjβ(S/Pft,S), from which we get the
degrees of generators of these Ext modules. As a consequence,
via graded local duality we answer a question of Wenliang Zhang on the socle
degrees of local cohomology modules of the form Hmjβ(S/It).Comment: Final version. Comments welcome