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

Abstract

Given an infinite reductive algebraic group [Formula: see text], we consider [Formula: see text]-equivariant coherent sheaves with prescribed multiplicities, called [Formula: see text]-constellations, for which two stability notions arise. The first one is analogous to the [Formula: see text]-stability defined for quiver representations by King [Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser.[Formula: see text]2) 45(180) (1994) 515–530] and for [Formula: see text]-constellations by Craw and Ishii [Flops of [Formula: see text]-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 [Formula: see text]-constellations, and depends on some finite subset [Formula: see text] of the isomorphy classes of irreducible representations of [Formula: see text]. We show that these two stability notions do not coincide, answering negatively a question raised in [Becker and Terpereau, Moduli spaces of [Formula: see text]-constellations, Transform. Groups 20(2) (2015) 335–366]. Also, we construct Harder–Narasimhan filtrations for [Formula: see text]-constellations with respect to both stability notions (namely, the [Formula: see text]-HN and [Formula: see text]-HN filtrations). Even though these filtrations do not coincide in general, we prove that they are strongly related: the [Formula: see text]-HN filtration is a subfiltration of the [Formula: see text]-HN filtration, and the polygons of the [Formula: see text]-HN filtrations converge to the polygon of the [Formula: see text]-HN filtration when [Formula: see text] grows.