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.
Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Math.\ Notes, vol
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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.