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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.

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    $\begingroup$ 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. $\endgroup$
    – reuns
    May 6, 2021 at 12:20
  • $\begingroup$ @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 :) $\endgroup$ May 6, 2021 at 15:25

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