| Let G be a reductive affine algebraic group over an algebraically closed field and V and W finite dimensional G-modules. These data induce a G-action on IP(V)xIP(W) and, for any pair (m,n) of positive integers, a linearization of this action in the very ample line bundle O(m,n). When the ratio n/m becomes sufficiently large, the GIT notion of (semi)stability with respect to the linearization in O(m,n) becomes independent of (m,n) and has a nice description. This elementary fact plays an important role in understanding semistability for decorated sheaves, e.g., quiver sheaves. We will firstly review these facts and give, as an application, a proof of the recent Hilbert-Mumford criterion of Gulbrandsen, Halle and Hulek in relative GIT when the base is of finite type. Secondly, we will explain a related result on the instability flag and review how this is applied to moduli problems. |