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

TL;DR

This paper explores two different stability notions for $(G,h)$-constellations, constructs their Harder-Narasimhan filtrations, and shows how these filtrations relate and converge as parameters vary.

Contribution

It demonstrates that the two stability notions do not coincide and establishes the relationship and convergence of their Harder-Narasimhan filtrations.

Findings

The two stability notions are not equivalent.

The $ ext{HN}$ filtrations are related: one is a subfiltration of the other.

The polygons of the $ ext{HN}$ filtrations converge as the parameter set grows.

Abstract

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 $\theta$-stability defined for quiver representations by King and for $G$-constellations by Craw and Ishii, 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 [BT15]. 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.