Let $G$ be a linear algebraic group (not necessarily reductive) and let $X$ be an affine variety with a regular $G$-action (everything defined over the field of complex numbers $\mathbb{C}$). Denote by $\mathrm{Vec}^G(X)$ the vector space of algebraic vector fields that are $G$-invariant, i.e. those algebraic sections $\xi \colon X \to TX$ of the tangent bundle $T X \to X$ such that $\mathrm{d}_x \varphi_g(\xi(x)) = \xi(gx)$ for all $x \in X$ where $\varphi_g \colon X \to X$ denotes the automorphism given by multiplication with the group element $g \in G$. Denote by $\mathcal{O}(X)^G$ the ring of $G$-invariant regular functions on $X$. The vector space $\mathrm{Vec}^G(X)$ has a natural structure of an $\mathcal{O}(X)^G$-module, given by $$ (f \cdot\xi)(x) = f(x) \xi(x) \quad \textrm{for all $x \in X$} $$ where $f \in \mathcal{O}(X)^G$ and $\xi \in \mathrm{Vec}^G(X)$. Here comes my question:

Is $\mathrm{Vec}^G(X)$ a finitely generated $\mathcal{O}(X)^G$-module? If this is not true in general, does it hold if $\mathcal{O}(X)^G$ is a finitely generated $\mathbb{C}$-algebra?

Any proof or counter-example or any reference to a text book would be perfect.


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