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

Given an infinite reductive algebraic group G, we consider G-equivariant coherent sheaves with prescribed multiplicities, called (G,h)-constellations, for which two stability notions arise. The first one is analogous to the 𝜃-stability defined for quiver representations by King [Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser.(2) 45(180) (1994) 515–530] and for G-constellations by Craw and Ishii [Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient, Duke Math. J. 124(2) (2004) 259–307], but depending on infinitely many parameters. The second one comes from Geometric Invariant Theory in the construction of a moduli space for (G,h)-constellations, and depends on some finite subset D of the isomorphy classes of irreducible representations of G. We show that these two stability notions do not coincide, answering negatively a question raised in [Becker and Terpereau, Moduli spaces of (G,h)-constellations, Transform. Groups 20(2) (2015) 335–366...