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I think, that these

These two symplectic manifolds are canonically symplectomorphic. The zeros of

Notice first, that the moment map $\mu$ form vanishes on the sub-bundle $T_h^* M$ of $T^* M$ of 1-forms vanishing on the fibers of the fibration $M\to X$. There Let us call this sub-bundle by $T_h ^* M$ (h- for horizontal).

To construct the symplectomorphism notice that there is a natural an obvious projection $\pi: T_h^* M \to T^* X$and the . 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 $T^* M$. I think, this proves Since the projection $\pi$ just produces the quotient of $T_h^*M$ by the action of $G$, now everything ?follows from definitions.
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I think, that these two symplectic manifolds are canonically symplectomorphic. You can identify the The zeros of the moment map $\mu$ with form the sub-bundle $T_h^* M$ of $T^* M$ of 1-forms vanishing on the fibers of the fibration $M\to X$. There is a natural projection $\pi: T_h^* M \to T^* X$ and 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 commutes with the action of $G$ and $G$ preserves the symplectic form on $T^* M$. I think, this proves everything?
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I think, that these two symplectic manifolds are canonically symplectomorphic. You can identify the zeros of the moment map $\mu$ with the sub-bundle $T_h^* M$ of $T^* M$ of 1-forms vanishing on the fibers of the fibration $M\to X$. There is a natural projection $\pi: T_h^* M \to T^* X$ and the restriction of the symplectic form of $T^* M$ to $ T_h^* M$ equals to the pullback of the symplectic form $T^* X$ under $\pi$. The projection commutes with the action of $G$ and $G$ preserves the symplectic form on $T^* M$. I think, this proves everything?