Does anybody knows a reference for the following statement?


Let $S^1$ acts on $\mathbb{C}^n$ in the usual (diagonal) way and $f:\mathbb{C}^n\to\mathbb{R}$ a smooth $S^1$-invariant function defined in a neighborhood of $0$ with a non-degenerate critical point at 0.
Then, there exists a local $S^1$-equivariant diffeomorphism $\Phi:U\to V\subset \mathbb{C}^n$ such that 


$$f(\Phi(z\_1,\ldots,z\_n))=|z\_1|^2+\cdots+|z\_p|^2-|z\_{p+1}|^2-\cdots-|z\_n|^2$$
It looks like Moser's proof of Morse lemma extends easily to this version, but I would be happy to know a place where it is actually done. Thanks for any help.



A reference for an equivariant Morse Lemma
Check Wasserman, Arthur G. Equivariant differential topology. Topology 8 1969 127--150. MR0250324 (40 #3563). In particular see Lemma 4.1 for an equivariant Morse lemma.


