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.
Brion, Introduction to actions of algebraic groups, Les cours du CIRM 1 (2010), no
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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.