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?



These two symplectic manifolds are canonically symplectomorphic. Notice first, that the map $\mu$ vanishes on the subbundle of $T^* M$ of 1forms vanishing on the fibers of the fibration $M\to X$. Let us call this subbundle by $T_h ^* M$ (h for horizontal). To construct the symplectomorphism notice that there is an obvious projection $\pi: T_h^* M \to T^* X$. The restriction of the symplectic form of $T^* M$ to $ T_h^* M$ equals to the pullback of the symplectic form of $T^* X$ under $\pi$. The projection $\pi$ commutes with the action of $G$ and $G$ preserves the symplectic form on $T^* M$. Since the projection $\pi$ just produces the quotient of $T_h^*M$ by the action of $G$, now everything follows from definitions. 


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 covectors 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 nonAbelian $G$ ($H$ still trivial), it is easier to explain things in Poisson terms. 


In addition to Dmitri's answer: when you're doing reduction at nonzero momentum some interesting things happen: the symplectic reduced space $J^{1}(\mu)/G_\mu$ then becomes a fiber bundle over $T^\ast X$ with typical fiber the coadjoint orbit $\mathcal{O}_\mu$. In the case $\mu = 0$, this reduces to the case discussed previously. This realization is not canonical, though, and depends on the choice of a connection in $M \to M/G$. The idea of the proof is to relate $J^{1}(\mu)$ with $J^{1}(0)$ using the connection oneform, and then to use the isomorphism mentioned previously. Afterwards, the curvature shows up in the reduced symplectic form on $J^{1}(\mu)/G_\mu$, which is then the sum of the canonical form on $T^\ast X$ and a twoform constructed from the curvature. In the case where $M = G$, you can choose the MaurerCartan form as your connection, and then the isomorphism is $J^{1}(\mu)/G_\mu = \mathcal{O}_\mu$ with the KKS symplectic form, which is in some sense the "curvature" of the MaurerCartan "connection". 


No. Take $M=G$, with action by left translations. Then $T^*G//G$ gives the coadjoint orbits with the Kirilov symplectic str (eg for G=U(n) the complex flag manifolds with the standard Kahler str), while $X$ is just a point. 

