There is an old conjecture by Stanley which gives a numerical
criterion for a module of covariants to be Cohen-Macaulay.
To prove this conjecture one has to be able to compute, or
at least estimate the local cohomology with support in the nullcone,
a question that has regained interest recently. For this reason I will
discuss my old work on this in which I give a precise (but complicated)
formula for the local cohomology with support in the nullcone as a
G-equivariant D-module, under some reasonably generic conditions.