# Relation between invariant meromorphic and invariant holomorphic functions

Given an affine space $$\mathbb{C}^n$$ (more generally a Stein space), and an action of a complex Lie group $$G$$ on it.

1. Is there a relation between (sheaves of) invariant holomorphic and invariant meromorphic functions? (Obviously $$\operatorname{Frac}(\mathcal{O}^G)\subset \mathcal{M}^G$$)
2. 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.

• What is your group $G$ and the action? With weaker restriction we can embed $\Bbb{Z}[i]$ into $(\Bbb{C}^*)^2$ with $\Bbb{Z}[i]$ acting on $\Bbb{C}$ by translation, there will be no non-constant invariant entire function but $\wp_i(z)$ is meromorphic invariant non-rational. May 6, 2021 at 12:20
• @reuns Well, my group is connected (an image under an exponential map of a linear subspace in $\mathbb{C}^n$). I have edited my question accordingly. Thank you :) May 6, 2021 at 15:25