Under the free-action assumption on the stable locus, components of the torus fixed-point locus on a GIT quotient are GIT quotients of linear subspaces by Levi subgroups.
Weist, Localization in quiver moduli spaces, Represent.\ Theory 17 (2013), 382--425
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Torus Actions on Quotients of Affine Spaces
Under the free-action assumption on the stable locus, components of the torus fixed-point locus on a GIT quotient are GIT quotients of linear subspaces by Levi subgroups.