# finite generation of $G$-equivariant holomorphic maps by polynomials?

Let $V$ and $W$ be two complex vector spaces with an action of a finite group $G$. The $G$-equivariant polynomial maps from $V$ to $W$ are finitely generated as a module over the ring of $G$-invariant polynomials on $V$. In other words, there exist $G$-equivariant polynomial maps $p_1,\dotsc,p_n$ so that any $G$-equivariant polynomial map may be written as $$q_1p_1+\dotsb +q_np_n$$ where the $q_i$ are $G$-invariant polynomials on $V$.

Is it true that the $G$-equivariant holomorphic maps $V\longrightarrow W$ are finitely generated as a module over the ring of $G$-invariant holomorphic functions on $V$, and that the generators may be taken as $G$-equivariant polynomial maps? In other words, may we also write any $G$-equivariant holomorphic map as $$f_1p_1+\dotsb +f_np_n$$ where the $f_i$ are now $G$-invariant holomorphic functions?

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I believe the answer is yes for $C^\infty$ maps and actions of compact (not necessarily finite) Lie groups. I think it is due to Poénaru and can be found in his book Singularités $C^\infty$ en présence de symétrie Lecture Notes in Mathematics, Vol. 510.
(edit to reply to Brett's comment): Poénaru's theorem is not holomorphic. However, I believe it should not be hard to mimic its proof to extract the holomorphic version. I should note that I am not much of an expert on this area of mathematics and I know it more or less as a collection of black boxes. My impression, however, is that in going from polynomial versions the results (which is classical invariant theory) to $C^\infty$ version the main difficulty is in dealing with smooth invariant functions that vanish to infinite order. Going from polynomials to power series is not hard. And holomorphic maps from $V$ to $W$ are power series, aren't they?
Note also that in your example there is a big difference between complex $\mathbb Z/3$ invariant polynomials on $\mathbb C$ and real invariant polynomials on $\mathbb C$: $\mathbb C [\mathbb C]^{\mathbb Z/3}$ is generated by $z^3$ while $\mathbb R[\mathbb C]^{\mathbb Z/3}$ is generated by $Re(z^3), Im (z^3)$ and $|z|^2$.
Thanks Eugene - I assume you mean that $C^\infty$ equivariant maps are finitely generated as a module over $C^\infty$ equivariant functions. On the other hand, I am pretty sure that in general the generators can't be complex polynomial maps. The counter example is as follows: $V$ and $W$ are $\mathbb C$ with an action of $\mathbb Z_3$ by multiplication by $e^{2\pi i/3}$ and $e^{4\pi i/3}$ respectively. $z\mapsto \bar z$ is not equal to a sum of smooth functions times equivariant complex polynomial maps, because those are generated by the map $z\mapsto z^2$. –  Brett Parker Aug 3 '12 at 0:23