Let $G$ be a complex simple reductive group. Then the set of isomorphy classes $Irr G$ is isomorphic to the set of dominant weights $\Lambda_+$ in the weight lattice of the maximal torus of a Borel subgroup of $G$. Further, $\Lambda_+$ is isomorphic to $\mathbb{N}_0^\ell$ for some integer $\ell$. Thus we can write $\rho = n_1\lambda_1 + ... + n_{\ell}\lambda_{\ell} = \underline{n}$ for any irreducible representation $\rho\colon G \to End(V_{\rho})$.

Now let $R$ be a ring with $G$-action, $M$ an $R$-$G$-module with isotypic decomposition $M \cong \bigoplus_{\underline{n} \in \mathbb{N}_0^{\ell}} M_{\underline{n}}\otimes_{\mathbb{C}} V_{\underline{n}}$. Then one defines the Hilbert function of $M$ as $h(\underline{n}) := rk (M_{\underline{n}})$. Further, consider the function $P(M,\underline{z}) := \sum_{\underline{n} \in D} h(\underline{n})z_1^{n_1}...z_{\ell}^{n_\ell}$ for some finite subset $D \subset \mathbb{N}_0^{\ell}$.

What is the behaviour of $P(M,\underline{z})$ if $D$ is sufficiently large? Is it a rational function?

If $M'$ is a submodule of $M$, how are $P(M',\underline{z})$ and $P(M,\underline{z})$ or $h'$ and $h$ related? Does $\frac{h'(\underline{n})}{h(\underline{n})}$ converge for $\underline{n} \not\in D$ as $D$ becomes large? Which assumptions on $R$ and $M$ are necessary? What is the limit?

I am especially interested in the case $G = Sl_2$ (i.e. $\ell = 1$) but I would also like to know the answer in the general case.