TITLE: Special Cases of Duistermaat-Heckman Formula
QUESTION [9 upvotes]: The Duistermaat Heckman localization formula states how integrals over symplectic spaces with Hamiltonian $U(1)$ group actions.
$$ \int_M \frac{\omega^n}{n!} e^{-\mu} = \sum_{x_i \text{ fixed}} \frac{e^{-\mu(x_i)}}{e(x_i)} $$
Here $M$ is a symplectic manifold and here are a few invariants:

$\omega$ is the symplectic form
$\mu$ is the moment map of the $U(1)$ rotation action
$x_i$ are the fixed points of the group action
$e(x_i)$ is the product of the weights of the rotation at fixed point $x_i$


In a few cases, we can produce formulas an engineer might recognize:

$M = \mathbb{R}^2 \simeq \mathbb{C}$ and $\omega = dx \wedge dy$  
$U(1)$ the the rotation $z = x_1 + i x_2 \mapsto e^{i\theta}z$, the moment map is $\mu(z) = |z|^2 = x_1^2 + x_2^2$
Certainly $\mathbb{R}^{2n}$ is analogous.

The fixed point of the rotation is $x_i = 0$.  This leads to the formula for the Gaussian integral:
$$ \int (dx \wedge dy) \,e^{-(x^2 + y^2)} = \frac{e^{-\mu(0)}}{e(0)} = \frac{1}{2\pi}$$
I'm not actually sure that $e(0) = 2\pi$ but I'm guessing.  There's not too many cases were all the ingredients are known.  It is possible to write a manifold (or orbifold or variety) with moment map $\mu$.  
My understanding is that we can rarely write $\omega$ down explicitly.  For the sphere $S^2 = \{ x^2 + y^2 + z^2 = 1\}$ there's a symplectic form $\omega = \frac{dx \wedge dy}{z}$ which is invariant under rotating aroun the $z$ axis.
Are there other cases where the DH formula specializes in a particularly nice way? I'm guessing that typically we bypass explicitly computing $\omega$.

REPLY [6 votes]: Nice examples are worked out in Audin (2004, §VI.3.d), Arvanitoyeorgos (1999)(pdf), McDuff-Salamon (1998, §5.6). It’s not true that $\omega$ is rarely explicit: e.g. on all coadjoint orbits (including $\smash{S^2}$) it is, and DH gives a formula of Harish-Chandra for their Fourier transforms; see Berline-Vergne (1983), Vergne (1983), or Guillemin-Sternberg (1984, ends of §§33 and 34).