Let (X,L) be a polarized variety and G be a subtorus of its group of automorphisms. For any linearization of the G action, we consider the GIT quotient Y=X//G. In this talk, we relate the notions of slope stability for sheaves on X, on Y, and the properties of the representation of G in Aut(X,L) in the equivariant context of toric varieties.

We build a family of fully faithfull functors from the category of equivariant reflexive sheaves on Y to the category of equivariant reflexive sheaves on X, and show that these functors preserve slope stability if and only if the pair (X,G) satisfies a combinatorial criterion.

As an application, we relate moduli spaces of equivariant stable sheaves on any toric orbifold to moduli spaces of equivariant stable sheaves on weighted projective spaces. We also relate stable sheaves on toric varieties and on projective bundles over them.

This is a joint work with Andrew Clarke.