I would recommend a look at chapter 8 of the **third** edition of *Geometric invariant theory* by Mumford, Forgarty and Kirwan.  It describes a connection between GIT and Hamiltonian group actions in symplectic geometry.  

(edit) You may also like [Moment maps and geometric invariant theory][1] by Chris Woodward.


  [1]: http://arxiv.org/abs/0912.1132