Suppose we have a reductive group $H$ and a representation $V$. Let $G$ be a group containing $H$ as a closed subgroup and let $W=G\times_H V$. The rings of differential operators $D(V)$ and $D(W)$ have natural actions of the groups $H$ and $G$ respectively.
Is there a natural map between the invariants rings: $$D(V)^H \text{ and } D(W)^G?$$
I think in general there are many maps once can construct from $D(V)$ to $D(W)$, but intuitively it seems that there must be a unique one between the ring of invariants. I tried computing the associated graded versions of the algebras, with respect to the degree filtration, but that wasn't of any help.