Just a disclaimer, I am not a mathematician so I am sorry for being less than formal.
Let us say we have a Hilbert series $\mathcal{H}(K[V]^G)$. Now I know that by eliminating some parameters in a given theory I get the freedom of a subgroup $H < G$. Now I may compute another Hilbert series $\mathcal{H}(K[V]^H)$. I have noticed that by computing $\mathcal{H}(K[V]^G) / \mathcal{H}(K[V]^H)$ I get what I am interpreting as the Hilbert series corresponding to the invariants of $G$ but not of $H$.
As an example, let us have complex invariants under $G$ and now I find that the $\mathbb{R}$ version of this Hilbert series corresponds to invariants under $H$. What is the significance of the quotient of these two Hilbert series?