finite generation of $G$-equivariant holomorphic maps by polynomials? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T20:54:00Z http://mathoverflow.net/feeds/question/103659 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103659/finite-generation-of-g-equivariant-holomorphic-maps-by-polynomials finite generation of $G$-equivariant holomorphic maps by polynomials? Brett Parker 2012-08-01T05:30:39Z 2012-08-03T14:30:26Z <p>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 $V$.</p> <p>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?</p> http://mathoverflow.net/questions/103659/finite-generation-of-g-equivariant-holomorphic-maps-by-polynomials/103805#103805 Answer by Eugene Lerman for finite generation of $G$-equivariant holomorphic maps by polynomials? Eugene Lerman 2012-08-02T18:09:10Z 2012-08-03T14:30:26Z <p>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 <em>Singularités $C^\infty$ en présence de symétrie</em> Lecture Notes in Mathematics, Vol. 510. </p> <p>See also Lemma 6.6.1 in <em>Dynamics and symmetry</em> by Michael Field. (ICP Advanced Texts in Mathematics, 3. Imperial College Press, London, 2007. xiv+478 pp. ISBN: 978-1-86094-828-2)</p> <p>(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?</p> <p>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 $|z|^2$.</p>