The 2-sphere, endowed with the round Riemann metric with constant curvature 1, is a symplectic manifolds.
My question is: Is the group of symplectic automorphisms of $S^2$ with respect to this symplectic structure connected?
My motivation to this question is as follows. If the answer to this question is yes, then any symplectic automorphism of $S^2$ can be connected by a path with the identity map. Then the study of the automorphism can be reduced to the studty of a family of smooth functions. And then we are possibly allowed to enter the area of complex analysis since smooth funtions on $S^2$ can be approximated by holomoprhic functions on some complex manifold containg $S^2$ as a totally real submanifold.
The above question can be asked for compact Hermitian symmetric spaces, or more generally for coadjoint orbits of semi-simple Lie groups. So could you recommand any references about the topological structure of symplectic automorphisms of these spaces?
Symplectic geometry is not my major. Thanks a lot for your help!