A symplectic principal bundle is a principal bundle $(X,B, G)$ with projection map $q:X\to B$ such that $X$ and $B$ are symplectic manifolds and the right action of $G$ preserves the symplectic structure of $X$.

1)Does this imply that $G$ admit a symplectic structure?

2)Under what conditions the following map preserves the standard Poisson Lie structures ?

$$ T:C^{\infty} (X) \to C^{\infty} (B)\\ T(f)(b)= \int_{q^{-1}(b)} fd\mu $$

Here $\mu$ is the natural normalized Haar measure of the fibers of $X$.

Remark: Considering the natural principal bundle structure $S^3 \to S^2$, the above construction provides a Lie bracket structure on $C^{\infty}(S^3)$ with $[f,g]=[T(f),T(g)]\circ q$.(But does it satisfy Leibniz rule?)

With a modification of the same construction one can give an alternative Lie bracket structure on $\chi^{\infty}(S^3)$.