I am currently reading a book on symplectic topology. I may have skipped some pages so find it confusing about the Duistermaat-Heckman theorem. In the book it states that
Assume Hamiltonian function $H:M \rightarrow \mathbb{R}$ is a Morse function which generates a circle action, then $$ \int_{M} e^{-\hbar H} \frac{\omega^{n}}{n !}=\sum_{p} \frac{e^{-\hbar H(p)}}{\hbar^{n} e(p)} $$ for $\hbar \in \mathbb{C}$. The right sum runs over all critical points of $H$, $e(p) \in \mathbb{Z}$ is the product of weights at $p$.
I didn't find the definition of weight near few pages. I wonder anyone can give me a reference to understand this phrase. Any comment is appreciated.
The formula you wrote assumes that the action has isolated fixed points, so that the right hand side is a finite sum. Let $p$ be a fixed point of the action, then the circle $S^{1}$ acts on $T_{p}(M)$ by taking the differential of the corresponding mappings. This gives $T_{p}(M)$ the structure of a $S^{1}$-representation. Minor technichal point: actually it is a complex reprentation since we can always pick an invariant almost complex structure compatible with the symplectic form (this is also proved in Mcduff-Salamon).
Anyway, the irreducible (complex) representations of $S^{1}$ are naturally identified with $\mathbb{Z}$, where the representation corresponding to $n \in \mathbb{Z}$ is the reprenation $z.w = z^{n}w$ for $z \in S^{1}=U(1)$ and $ w \in \mathbb{C}$. This $n$ is called the weight of the representation. In general we split a $2n$-dimensional representation into irreducibles and the weights are just the $n$-tuple consisting of the weights of each summand. The weight at $p$ is just the weight of the representation on $T_{p}(M)$