TITLE: cotangent bundle symplectic reduction and fibre bundles
QUESTION [15 upvotes]: Suppose a compact Lie group $G$ acts on a manifold $M$ with only one orbit type $G/H$ ($H$ denotes the stabiliser group). Then the manifold $M$ becomes a fibre bundle over the quotient manifold $X:=M/G$ with typical fibre $G/H$ and structure group $G$.
On the one hand one could look at the cotangent bundle $T^* X$ of the quotient (which carries a natural symplectic structure).
On the other hand consider the lifted action of $G$ on the cotangent bundle $T^* M$ with moment map $\mu: T^* M\to \mathfrak{g}^* .$ The symplectic quotient $T^* M//G:=\mu^{-1}(0)/G$ inherits the structure of a symplectic manifold.  Here comes the question: Are $T^* X$ and $T^* M//G$ (canonically) symplectomorphic?

REPLY [4 votes]: This is more a comment towards Gourishankar than an answer to the original question. 
It was part of my thesis, (UCB, about 1986), so, apologies,  I chime in. For simplicity,
I take the case $G$ Abelian, and $H = $ trivial.
 To map $J^{-1} (\mu)$ equivariantly to $J^{-1}(0)$ subtract $\mu \cdot A$ where $A$ is any $G$-connection for $\pi: X \to X/G$.  $J^{-1} (0)/G = T^* (X/G)$
canonically, independent of connection.    The map `momentum shift map
of subtracing  $\mu \cdot A$ from co-vectors  is not symplectic,
relative to the standard structure, but it    becomes symplectic if you subtract
$\mu \pi^* F_A$, where  $F_A = curv(A)$, from the standard structure. So the reduced space at  $\mu$ is $T^*(X/G)$ with the standard structure minus the ``magnetic term''
$\mu F_A$.  
For non-Abelian $G$ ($H$ still trivial),  it is easier to explain things in Poisson terms.
$T^* X/G$ is a Poisson manifold whose symplectic leaves
are the reduced spaces in question. The momentum
shift trick turns it  into  $T^* (X/G) \oplus Ad^* (X)$  where $Ad^* (X) \to X/G$
is the   co-adjoint bundle associated to $X \to X/G$ -- its fibers are the 
dual Lie algebras for $G$. This direct sum bundle admits coordinates $s_i, p_i, \xi_a$ where $s_i, p_i$ are canonical coordinates on $T^*(X/G)$ induced by  coordinates $s_i$ on $X/G$
and where $\xi_a$ are fiber-linear coordinates on the co-adjoint bundle induced
by a choice of local section of $\pi$. Then the main tricky part of the bracket is that
the bracket of $p_i$ with $p_j$ is  $\Sigma \xi_a F^a _{ij}$,  $F$ being the curvature of the connection
relative to the choice of local section.  The symplectic leaves = reduced spaces
are of the form $T^*(X/G) \oplus $(co-adjoint orbit bundle).