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

2012-08-01T05:30:39Z

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?

Answer by Eugene Lerman for finite generation of $G$-equivariant holomorphic maps by polynomials?

2012-08-02T18:09:10Z

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.

See also Lemma 6.6.1 in Dynamics and symmetry by Michael Field. (ICP Advanced Texts in Mathematics, 3. Imperial College Press, London, 2007. xiv+478 pp. ISBN: 978-1-86094-828-2)

(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 $|z|^2$.

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

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

Answer by Eugene Lerman for finite generation of $G$-equivariant holomorphic maps by polynomials?