My question is concerned with filtered rings. It is a classical result that if $R$ is a finitely generated commutative ring graded by a semigroup $S$ then $S$ is also finitely generated.
The reverse is of course not true. However there are cases when one can still say something for the reverse. For instance, I think the following holds true:
Let $R$ be a commutative ring filtered by a finitely generated totally ordered semigroup $S$ such that the associated graded ring has $1$ - dimensional graded components. Then $R$ is finitely generated.
It could be that the property of having $1$ - dimensional graded components could be relaxed somewhat, but for my purposes I'll work with this property and call such rings "$S$ - spherical".
An example of a $S$ - spherical ring is $k[x_1, \ldots, x_n]^{S_n}$ where $S = \mathbb{Z}_{\geq 0} \times \mathbb{Z}_{\geq 0}^n$ and $S_n$ is the symmetric group. Here the filtration is done first by total degree then by the highest multidegree in lexicographic order.
My question is kind of vague but here it goes: what do we know about spherical rings in general? I'm not really interested in Homological/Commutative Algebra type of results but rather about results about the underlying semigroup. Here of interest to me is, the convex geometry of the supporting cone $C(S)$; the lattice geometry of $S$ and $C(S)$ as well as connections between lattice geometric properties of $S$ and properties of $R$.
The motivation for this question is the following statement.
$\textbf{Conjecture. }$. Let $\mathfrak{g}$ be a reductive (complex) Lie Algebra of rank $n$ and let $\mathfrak{U}(\mathfrak{g})$ be its universal enveloping algebra. Let $Z(\mathfrak{g})$ be the center of $\mathfrak{U}(\mathfrak{g})$. Then there exists a semigroup $S \subset \mathbb{Z}_{\geq 0}\times\mathbb{Z}^{n}_{\geq 0}$ which can be explicitly computed such that $Z(\mathfrak{g})$ is $S$ - spherical.
The above example is an instance for this conjecture for $\mathfrak{g} = gl(n, \mathbb{C})$. Of particular interest to me is the case $\mathfrak{g} = sl(n, \mathbb{C})$ for which the semigroup $S$ is quite hard to compute. "Reasonable" non-finitely generated extensions of $S$ exists, but I do not know what kind of ring theoretical information I can extract from them, hence the vaguely formulated question.
