Moduli spaces of (G,h)-constellations

TL;DR

This paper constructs a moduli space of heta-stable (G,h)-constellations for an infinite reductive group acting on an affine scheme, generalizing previous invariant Hilbert schemes and potentially resolving quotient singularities.

Contribution

It introduces a new moduli space for (G,h)-constellations that extends existing frameworks to infinite reductive groups and proposes a morphism to the quotient, aiming at singularity resolution.

Findings

Constructed the moduli space M_{ heta}(X) for (G,h)-constellations.

Established a morphism from M_{ heta}(X) to the GIT quotient X//G.

Proposed this moduli space as a candidate for resolving singularities of the quotient.

Abstract

Given an infinite reductive group G acting on an affine scheme X over C and a Hilbert function h: Irr G \to N_0, we construct the moduli space M_{\theta}(X) of \theta-stable (G,h)-constellations on X, which is a generalization of the invariant Hilbert scheme after Alexeev and Brion and an analogue of the moduli space of \theta-stable G-constellations for finite groups introduced by Craw and Ishii. Our construction of a morphism M_{\theta}(X) \to X//G makes this moduli space a candidate for a resolution of singularities of the quotient X//G.