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Given an infinite reductive algebraic group $G$, we consider $G$-equivariantcoherent sheaves with prescribed multiplicities, called $(G,h)$-constellations,for which two stability notions arise. The first one is analogous to the$\theta$-stability defined for quiver representations by King and for$G$-constellations by Craw and Ishii, but depending on infinitely manyparameters. The second one comes from Geometric Invariant Theory in theconstruction of a moduli space for $(G,h)$-constellations, and depends on somefinite subset $D$ of the isomorphy classes of irreducible representations of$G$. We show that these two stability notions do not coincide, answeringnegatively a question raised in [BT15]. Also, we construct Harder-Narasimhanfiltrations for $(G,h)$-constellations with respect to both stability notions(namely, the $\mu_{\theta}$-HN and $\mu_D$-HN filtrations). Even though thesefiltrations do not coincide in general, we prove that they are stronglyrelated: the $\mu_{\theta}$-HN filtration is a subfiltration of the $\mu_D$-HNfiltration, and the polygons of the $\mu_D$-HN filtrations converge to thepolygon of the $\mu_{\theta}$-HN filtration when $D$ grows.

Stability conditions and related filtrations for $(G,h)$-constellations