Moduli spaces of (G,h)-constellations

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