I was reading a book on symplectic topology and get confused of the following sentence.
The action of $S^{1}$ on $T_{p}M$ is conjugate to an n-fold product of circle actions on the complex plane by $$z \mapsto e^{-2\pi itk_{j}}z$$ for $t \in S^{1}$ and we define the weight $e(p)=k_{1}...k_{n}$
For your reference, this appreared in the Duistermaat-Heckman formula i.e. the localization formula. I wonder what it means and how it works out.
Thanks for any comment.
Two topological group actions $\phi : G\times X \to X$ and $\psi : G\times X \to X$ are conjugate if there is a homeomorphism $f : X \to X$ such that $f(\phi(g, x)) = \psi(g, f(x))$, or equivalently, $\phi(g, x) = f^{-1}(\psi(g, f(x)))$.
There are induced group homomorphisms $\phi' : G \to \operatorname{Homeo}(X)$ and $\psi' : G \to \operatorname{Homeo}(X)$ given by $g \mapsto \phi(g, \cdot)$ and $g \mapsto \psi(g, \cdot)$ respectively. The equation $f(\phi(g, x)) = \psi(g, f(x))$ is equivalent to the commutativity of the following diagram:
$$\require{AMScd} \begin{CD} G @>{\psi'}>> \operatorname{Homeo}(X)\\ @V{\phi'}VV @VV{\_\,\circ f}V \\ \operatorname{Homeo}(X) @>{f\circ\,\_}>> \operatorname{Homeo}(X) \end{CD}$$
Note that $f\circ\_$ and $\_\circ f$ are not group homomorphisms, so this is only a diagram of sets. On the other hand, $f^{-1}\circ\_\circ f : \operatorname{Homeo}(X) \to \operatorname{Homeo}(X)$ is a group homomorphism. The equation $\phi(g, x) = f^{-1}(\psi(g, f(x)))$ is equivalent to the commutativity of the following diagram of groups:
$$\require{AMScd} \begin{CD} G @>{\psi'}>> \operatorname{Homeo}(X)\\ @V{\phi'}VV @VV{f^{-1}\circ\,\_\,\circ f}V \\ \operatorname{Homeo}(X) @>{\operatorname{id}}>> \operatorname{Homeo}(X) \end{CD}$$
(I would have omitted the identity arrow and drawn this diagram as a triangle, but I don't know how to do that in MathJax.)