Let $k$ be any base field and $A$ an affine infinite dimensional $k$-algebra.
Let $\mathcal{F}= \{ A_i \}_{i \geq 0}$ be a finite dimensional filtration for $A$: that is, $k \subset A_0$ and each $A_i$ is finite dimensional. Let $M$ be a finitely generated left $A$-module. A filtration $\Omega = \{ M_i \}_{i \geq 0}$ on $M$ is called finite dimensional if each $M_i$ is finite dimensional.
I say that $M$ has a Hilbert polynomial if there exists $p_M \in \mathbb{Q}[x]$ such that, for $n>>0$, $\operatorname{dim}_k \, M_n = p_M(n)$, the degree of $p_M$ independent of the filtrations $\mathcal{F}, \Omega$ and the leading coeficient independent of $\Omega$. In this case $GK \, M$ is a positive integer, equal to the degree of $p_M$.
The most general classes of algebras with Hilbert polynomial I am aware of are somewhat commutative algebras and filtered semi-commutative algebras whose graded associated algebra is generated by homogeneous elements of degree 1. (cf. McConnell, Robinson, Noncommutative Noetherian Rings, Chapter 8; McConnell Quantum groups, filtered rings and Gel’fand-Kirillov dimension)
Question 1 What are other general classes of algebras that have Hilbert polynomials?
Now, I say that $A$ is an algebra with multiplicity if the Gelfand-Kirillov dimension is exact for $A$, that is, given a short exact sequence of finitely generated modules
$0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$, $GK(M)=max \{ GK(M'), GK(M') \}$, and for f.g. modules the Gelfand-Kirillov dimension is a nonnegative integer. A multiplicity function is a map from $e()$ from f.g. modules $M$ to the rationals such that $e(M) \in \mathbb{Q}^+$, and moreover, given an exact sequence as above,
a) If $GK \, M' < GK \, M= GK \, M''$ then $e(M)=e(M'')$
b) If $GK \, M'=GK \, M > GK \, M''$ then $e(M)=e(M')$
c) If $GK \, M'=GK \, M= GK \, M''$, then $e(M)=e(M')+e(M'')$
The most general classes of algebras which I am aware of that are algebras with multiplicity (see M. Lorenz, Gelfand-Kirillov dimension and Poincaré series, Chapter 3; or V. Bavula Identification of the Hilbert function and Poincare series, and the dimension of modules over filtered rings) are those for which there is a $k \in \mathbb{Z}^+$ and $k$ polynomials $q_0, \ldots, q_{k-1} \in \mathbb{Q}[x]$, all with the same degree $d$ with $\operatorname{dim} M_i= q_j(i), \, i>>0$ and $i \equiv j (mod \, k)$. In this case, $d=GK(M)$.
As an example of algebras in this situation we have filtered semi-commutative algebras whose graded associative algebra is not generated by homogeneous elements of degree 1.
Question 2 Is there a even more generic condition on an algebra such that it is what I called an algebra with multiplicity? Maybe something related to its Poincaré series?
