In a moduli space, usually, we impose a notion of stability for the objects and, when constructing the moduli space by using Geometric Invariant Theory, another notion of GIT stability appears, showing during the construction of the moduli that both notions do coincide at the stable and semistable level. For an object which is unstable (this is, contradicting the stability condition) there exists a canonical unique filtration, called the Harder-Narasimhan filtration. Onthe other habd, GIT stability is checked by 1-parameter subgroups, by the classical Hilbert-Mumford criterion, and it turns out that there exists a unique 1-parameter subgroup giving some notion of maximal unstability in the GIT sense. We show how to prove that this special 1-parameter subgroup can be converted into a filtration of the object and coincides with the Harder-Narasimhan filtration, hence both notions of maximal unstability are the same, for the moduli problem of classifying coherent sheaves on a smooth complex projective variety (cf. [GSZ]). A similar treatment can be used to prove the analogous correspondence for holomorphic pairs, Higgs sheaves, rank 2 tensors and finite dimensional quiver representations.

[GSZ] T. Gómez, I. Sols, A. Zamora, A GIT characterization of the Harder-Narasimhan filtration, arXiv:1112.1886v2, (Preprint 2012)