Equivariant volumes for linearized actions

Suppose given a linearized action on a polarized complex projective manifold (M,L), and assume that the stable locus is non-empty. We study the leading asymptotics of the dimension of the equivariant summands appearing in the space of global sections of high powers of L. We use Kirwan resolutions and an elementary algebro-geometric argument to reduce the problem to the case of non-singular actions (that is, actions for which the stable and semistable loci coincide and are non-empty).