Moduli spaces of (G,h)-constellations

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.