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

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.

29 pages, 4 figures. Final version, to appear in International Journal of Mathematics

References in corpus (2)

• Schémas de Hilbert invariants et théorie classique des invariants
• Moduli spaces of (G,h)-constellations