Given an affine space $\mathbb{C}^n$ (more generally a Stein space), and an action of a complex Lie group $G$ on it.
- Is there a relation between (sheaves of) invariant holomorphic and invariant meromorphic functions? (Obviously $\operatorname{Frac}(\mathcal{O}^G)\subset \mathcal{M}^G$)
- Provided all holomorphic invariants are polynomials is it true that all meromorphic invariants are rational functions?
Maybe it will help that in my case the group $G$ is a closed subgroup of the torus $(\mathbb{C}^\times)^n$ hence abelian and with a very nice action.
EDIT: the case I am interested in is of the group $G$ being an image under $\operatorname{exp}:\mathbb{C}^n\to (\mathbb{C}^\times)^n$ of a linear subspace in $\mathbb{C}^n.$
I know the answer for the case when $G$ is a subtorus so the interesting part of the question is about images of non-rational subsapces.