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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 $\theta$-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.
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Dates and versions

hal-01620368 , version 1 (20-10-2017)
hal-01620368 , version 2 (22-12-2017)



Ronan Terpereau, Alfonso Zamora. Stability conditions and related filtrations for $(G,h)$-constellations. International Journal of Mathematics, 2017, 28, n°14 (1750098), ⟨10.1142/S0129167X17500987⟩. ⟨hal-01620368v2⟩


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