Given a reductive group $G$ and an affine $G$-scheme $X$, constellations are $G$-equivariant sheaves over $X$ such that their module of global sections has finite multiplicities. Prescribing these multiplicities by a function $h$, and imposing a stability condition $\theta$ there is a moduli space for $\theta$-stable constellations constructed by Becker and Terpereau, using Geometric Invariant Theory. This construction will depend on a finite subset $D$ of the set of irreducible representations of $G$. By reformulating the stability condition $\theta$ and the GIT stability condition, in terms of a slope condition (say $\mu_{\theta}$ and $\mu_D$) we are able to construct Harder-Narasimhan filtrations from both points of view, and prove a precise relation between the two filtrations. Finally, we show that the associated polygons to the $\mu_D$-filtrations converge to the one associated to the $\mu_{\theta}$-filtrations when $D$ grows.

These results are joint work with Ronan Terpereau.