Let $G$ be an algebraic group over an algebraically closed field of characteristic zero and let $W$ be a finite dimensional representation of $G$. Let $k[W]$ be the algebra of polynomial functions on $W$. The Hochster Roberts theorem asserts that the ring of invariants $k[W]^G$ is Cohen-Macaulay.
Let $U$ be some other finite dimensional $G$-representation. A rather natural question, in view of the HR-theorem, is whether $(U\otimes k[W])^G$ is a Cohen-Macaulay $k[W]^G$-module. This problem was first posed by Stanley in the case that $G$ is a torus, in connection with his work on linear diophantine equations.
It is easy to see however, that the answer is negative, even in the case $G={\bf G}_m$. On the other hand, work of Stanley and the author shows that the answer is positive in the case that $U$ is in some sense small with respect to $W$.
To get a precise answer one has to compute the local cohomology of $(U\otimes k[W]^G)$. In the case that $G$ is a torus this had been accomplished by Stanley. We have been able to do a similar computation in the case that $G$ is general and $W$ is "not too small". Our methods are based upon the computation of the cohomology of $k[W]$ with support in the unstable locus of $W$.