Quiver representations, parabolic bundles, and the Deligne-Simpson problem

The "very good" property for smooth complex equidimensional algebraic stacks was introduced by Beilinson and Drinfeld in their paper "The Quantization of Hitchin's Integrable System and Hecke Eigensheaves".  They proved that for a semisimple complex group G, the moduli stack of G-bundles over a smooth complex projective curve X is "very good" as long as X has genus g > 1.  We will define the "very good" property in the context of a group action on an algebraic variety, prove it for a moduli space of parabolic bundles arising from quiver representations, and examine the implications this has for the spaces of solutions to the Deligne-Simpson problem.