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.-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]. Also, we construct Harder-Narasimhan filtrations for (G, h)-constellations with respect to both stability notions (namely, the mu(theta)-HN and mu(D)-HN filtrations). Even though these filtrations do not coincide in general, we prove that they are strongly related: the mu(theta)-HN filtration is a subfiltration of the mu(D)-HN filtration, and the polygons of the mu D-HN filtrations converge to the polygon of the mu(theta)-HN filtration when D grows.