Equivariant stable sheaves and toric GIT

For $(X,\,L)$ a polarized toric variety and $G\subset \mathrm {Aut}(X,\,L)$ torus, denote by $Y$ the GIT quotient $X/\!\!/G$ . We define family of fully faithful functors from category torus equivariant reflexive sheaves on to $X$ show, under genericity assumption $G$ , that slope stability is preserved these if only pair $((X,\,L),\,G)$ satisfies combinatorial criterion. As an application, when orbifold dimension $n$ we relate stable certain $(n-1)$ -dimensional weighted projective spaces