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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 by Chris Woodward.
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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.