  Let $k$ be an algebraically closed field of characteristic 0, $Y=k^{r}\times
{(k^{\times})}^{s}$ and let $G$ be an algebraic torus acting diagonally on the
ring of differential operators $\cD (Y)^G$. We give necessary and sufficient
conditions for $\cD (Y)^G$ to have enough simple finite dimensional
representations, in the sense that the intersection of the kernels of all the
simple finite dimensional representations is zero. As an application we show
that if $K\longrightarrow GL(V)$ is a representation of a reductive group $K$
and if zero is not a weight of a maximal torus of $K$ on $V$, then $\cD (V)^K$
has enough finite dimensional representations. We also construct examples of
FCR- algebras with any GK dimension $\geq 3$.
